url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
|
|---|---|---|---|---|---|---|---|---|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_inv_rev
|
[192, 1]
|
[193, 8]
|
sorry
|
A : Type ?u.3654
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ (a * b)⁻¹ = b⁻¹ * a⁻¹
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_inv_eq_iff_eq_mul
|
[195, 1]
|
[197, 8]
|
sorry
|
A : Type ?u.3816
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b c : G
⊢ a * b⁻¹ = c ↔ a = c * b
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_inv_eq_one
|
[199, 1]
|
[200, 8]
|
sorry
|
A : Type ?u.3960
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a * b⁻¹ = 1 ↔ a = b
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_mul_eq_one
|
[202, 1]
|
[203, 8]
|
sorry
|
A : Type ?u.4111
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a⁻¹ * b = 1 ↔ a = b
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture1.lean
|
Tutorial.Subgroup.inv_mem_iff
|
[264, 1]
|
[265, 8]
|
sorry
|
G : Type
inst✝ : Group G
H : Subgroup G
a : G
⊢ a⁻¹ ∈ H ↔ a ∈ H
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_mul_cancel_left
|
[133, 1]
|
[145, 11]
|
calc
a⁻¹ * (a * b) = (a⁻¹ * a) * b := by
rw [mul_assoc]
_ = 1 * b := by
simp
_ = b := by
simp
|
A : Type ?u.2656
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a⁻¹ * (a * b) = b
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_mul_cancel_left
|
[133, 1]
|
[145, 11]
|
rw [mul_assoc]
|
A : Type ?u.2656
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a⁻¹ * (a * b) = a⁻¹ * a * b
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_mul_cancel_left
|
[133, 1]
|
[145, 11]
|
simp
|
A : Type ?u.2656
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a⁻¹ * a * b = 1 * b
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_mul_cancel_left
|
[133, 1]
|
[145, 11]
|
simp
|
A : Type ?u.2656
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ 1 * b = b
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_left_cancel
|
[153, 1]
|
[165, 21]
|
intro h
|
A : Type ?u.4453
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x y : G
⊢ a * x = a * y → x = y
|
A : Type ?u.4453
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x y : G
h : a * x = a * y
⊢ x = y
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_left_cancel
|
[153, 1]
|
[165, 21]
|
calc
x = 1 * x := by simp
_ = (a⁻¹ * a) * x := by simp
_ = a⁻¹ * (a * x) := by rw [mul_assoc]
_ = a⁻¹ * (a * y) := by rw [h]
_ = (a⁻¹ * a) * y := by rw [← mul_assoc]
_ = 1 * y := by simp
_ = y := by simp
|
A : Type ?u.4453
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x y : G
h : a * x = a * y
⊢ x = y
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_left_cancel
|
[153, 1]
|
[165, 21]
|
simp
|
A : Type ?u.4453
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x y : G
h : a * x = a * y
⊢ x = 1 * x
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_left_cancel
|
[153, 1]
|
[165, 21]
|
simp
|
A : Type ?u.4453
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x y : G
h : a * x = a * y
⊢ 1 * x = a⁻¹ * a * x
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_left_cancel
|
[153, 1]
|
[165, 21]
|
rw [mul_assoc]
|
A : Type ?u.4453
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x y : G
h : a * x = a * y
⊢ a⁻¹ * a * x = a⁻¹ * (a * x)
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_left_cancel
|
[153, 1]
|
[165, 21]
|
rw [h]
|
A : Type ?u.4453
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x y : G
h : a * x = a * y
⊢ a⁻¹ * (a * x) = a⁻¹ * (a * y)
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_left_cancel
|
[153, 1]
|
[165, 21]
|
rw [← mul_assoc]
|
A : Type ?u.4453
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x y : G
h : a * x = a * y
⊢ a⁻¹ * (a * y) = a⁻¹ * a * y
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_left_cancel
|
[153, 1]
|
[165, 21]
|
simp
|
A : Type ?u.4453
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x y : G
h : a * x = a * y
⊢ a⁻¹ * a * y = 1 * y
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_left_cancel
|
[153, 1]
|
[165, 21]
|
simp
|
A : Type ?u.4453
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x y : G
h : a * x = a * y
⊢ 1 * y = y
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_one
|
[170, 1]
|
[180, 7]
|
apply mul_left_cancel (a := a⁻¹)
|
A : Type ?u.7038
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a : G
⊢ a * 1 = a
|
A : Type ?u.7038
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a : G
⊢ a⁻¹ * (a * 1) = a⁻¹ * a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_one
|
[170, 1]
|
[180, 7]
|
simp
|
A : Type ?u.7038
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a : G
⊢ a⁻¹ * (a * 1) = a⁻¹ * a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_inv_self
|
[185, 1]
|
[188, 7]
|
apply mul_left_cancel (a := a⁻¹)
|
A : Type ?u.9053
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a : G
⊢ a * a⁻¹ = 1
|
A : Type ?u.9053
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a : G
⊢ a⁻¹ * (a * a⁻¹) = a⁻¹ * 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_inv_self
|
[185, 1]
|
[188, 7]
|
simp
|
A : Type ?u.9053
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a : G
⊢ a⁻¹ * (a * a⁻¹) = a⁻¹ * 1
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_inv_cancel_left
|
[193, 1]
|
[196, 7]
|
rw [← mul_assoc]
|
A : Type ?u.10914
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a * (a⁻¹ * b) = b
|
A : Type ?u.10914
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a * a⁻¹ * b = b
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_inv_cancel_left
|
[193, 1]
|
[196, 7]
|
simp
|
A : Type ?u.10914
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a * a⁻¹ * b = b
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_inv_cancel_right
|
[200, 1]
|
[202, 19]
|
simp [mul_assoc]
|
A : Type ?u.12143
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a * b * b⁻¹ = a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_mul_cancel_right
|
[206, 1]
|
[208, 19]
|
simp [mul_assoc]
|
A : Type ?u.13426
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a * b⁻¹ * b = a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_right_cancel
|
[212, 1]
|
[222, 21]
|
intro h
|
A : Type ?u.14883
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x y : G
⊢ x * a = y * a → x = y
|
A : Type ?u.14883
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x y : G
h : x * a = y * a
⊢ x = y
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_right_cancel
|
[212, 1]
|
[222, 21]
|
calc
x = x * 1 := by simp
_ = x * (a * a⁻¹) := by simp
_ = (x * a) * a⁻¹ := by rw [← mul_assoc]
_ = (y * a) * a⁻¹ := by rw [h]
_ = y * (a * a⁻¹) := by rw [mul_assoc]
_ = y * 1 := by simp
_ = y := by simp
|
A : Type ?u.14883
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x y : G
h : x * a = y * a
⊢ x = y
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_right_cancel
|
[212, 1]
|
[222, 21]
|
simp
|
A : Type ?u.14883
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x y : G
h : x * a = y * a
⊢ x = x * 1
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_right_cancel
|
[212, 1]
|
[222, 21]
|
simp
|
A : Type ?u.14883
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x y : G
h : x * a = y * a
⊢ x * 1 = x * (a * a⁻¹)
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_right_cancel
|
[212, 1]
|
[222, 21]
|
rw [← mul_assoc]
|
A : Type ?u.14883
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x y : G
h : x * a = y * a
⊢ x * (a * a⁻¹) = x * a * a⁻¹
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_right_cancel
|
[212, 1]
|
[222, 21]
|
rw [h]
|
A : Type ?u.14883
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x y : G
h : x * a = y * a
⊢ x * a * a⁻¹ = y * a * a⁻¹
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_right_cancel
|
[212, 1]
|
[222, 21]
|
rw [mul_assoc]
|
A : Type ?u.14883
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x y : G
h : x * a = y * a
⊢ y * a * a⁻¹ = y * (a * a⁻¹)
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_right_cancel
|
[212, 1]
|
[222, 21]
|
simp
|
A : Type ?u.14883
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x y : G
h : x * a = y * a
⊢ y * (a * a⁻¹) = y * 1
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_right_cancel
|
[212, 1]
|
[222, 21]
|
simp
|
A : Type ?u.14883
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x y : G
h : x * a = y * a
⊢ y * 1 = y
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_eq_of_mul_eq_one_left
|
[226, 1]
|
[231, 7]
|
intro h
|
A : Type ?u.17097
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x : G
⊢ x * a = 1 → a⁻¹ = x
|
A : Type ?u.17097
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x : G
h : x * a = 1
⊢ a⁻¹ = x
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_eq_of_mul_eq_one_left
|
[226, 1]
|
[231, 7]
|
apply mul_right_cancel a
|
A : Type ?u.17097
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x : G
h : x * a = 1
⊢ a⁻¹ = x
|
A : Type ?u.17097
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x : G
h : x * a = 1
⊢ a⁻¹ * a = x * a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_eq_of_mul_eq_one_left
|
[226, 1]
|
[231, 7]
|
rw [h]
|
A : Type ?u.17097
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x : G
h : x * a = 1
⊢ a⁻¹ * a = x * a
|
A : Type ?u.17097
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x : G
h : x * a = 1
⊢ a⁻¹ * a = 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_eq_of_mul_eq_one_left
|
[226, 1]
|
[231, 7]
|
simp
|
A : Type ?u.17097
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x : G
h : x * a = 1
⊢ a⁻¹ * a = 1
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_one
|
[239, 1]
|
[242, 7]
|
apply inv_eq_of_mul_eq_one_left
|
A : Type ?u.18067
G : Type
inst✝¹ : Ring A
inst✝ : Group G
⊢ 1⁻¹ = 1
|
case a
A : Type ?u.18067
G : Type
inst✝¹ : Ring A
inst✝ : Group G
⊢ 1 * 1 = 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_one
|
[239, 1]
|
[242, 7]
|
simp
|
case a
A : Type ?u.18067
G : Type
inst✝¹ : Ring A
inst✝ : Group G
⊢ 1 * 1 = 1
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_inv
|
[246, 1]
|
[249, 7]
|
apply inv_eq_of_mul_eq_one_left
|
A : Type ?u.18884
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a : G
⊢ a⁻¹⁻¹ = a
|
case a
A : Type ?u.18884
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a : G
⊢ a * a⁻¹ = 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_inv
|
[246, 1]
|
[249, 7]
|
simp
|
case a
A : Type ?u.18884
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a : G
⊢ a * a⁻¹ = 1
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_inv_rev
|
[254, 1]
|
[258, 7]
|
apply inv_eq_of_mul_eq_one_left
|
A : Type ?u.19458
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ (a * b)⁻¹ = b⁻¹ * a⁻¹
|
case a
A : Type ?u.19458
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ b⁻¹ * a⁻¹ * (a * b) = 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_inv_rev
|
[254, 1]
|
[258, 7]
|
rw [← mul_assoc]
|
case a
A : Type ?u.19458
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ b⁻¹ * a⁻¹ * (a * b) = 1
|
case a
A : Type ?u.19458
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ b⁻¹ * a⁻¹ * a * b = 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_inv_rev
|
[254, 1]
|
[258, 7]
|
simp
|
case a
A : Type ?u.19458
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ b⁻¹ * a⁻¹ * a * b = 1
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_inv_eq_iff_eq_mul
|
[261, 1]
|
[270, 9]
|
constructor
|
A : Type ?u.20667
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b c : G
⊢ a * b⁻¹ = c ↔ a = c * b
|
case mp
A : Type ?u.20667
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b c : G
⊢ a * b⁻¹ = c → a = c * b
case mpr
A : Type ?u.20667
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b c : G
⊢ a = c * b → a * b⁻¹ = c
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_inv_eq_iff_eq_mul
|
[261, 1]
|
[270, 9]
|
intro h
|
case mp
A : Type ?u.20667
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b c : G
⊢ a * b⁻¹ = c → a = c * b
|
case mp
A : Type ?u.20667
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b c : G
h : a * b⁻¹ = c
⊢ a = c * b
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_inv_eq_iff_eq_mul
|
[261, 1]
|
[270, 9]
|
apply mul_right_cancel (a := b⁻¹)
|
case mp
A : Type ?u.20667
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b c : G
h : a * b⁻¹ = c
⊢ a = c * b
|
case mp
A : Type ?u.20667
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b c : G
h : a * b⁻¹ = c
⊢ a * b⁻¹ = c * b * b⁻¹
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_inv_eq_iff_eq_mul
|
[261, 1]
|
[270, 9]
|
simp [h]
|
case mp
A : Type ?u.20667
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b c : G
h : a * b⁻¹ = c
⊢ a * b⁻¹ = c * b * b⁻¹
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_inv_eq_iff_eq_mul
|
[261, 1]
|
[270, 9]
|
intro h
|
case mpr
A : Type ?u.20667
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b c : G
⊢ a = c * b → a * b⁻¹ = c
|
case mpr
A : Type ?u.20667
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b c : G
h : a = c * b
⊢ a * b⁻¹ = c
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_inv_eq_iff_eq_mul
|
[261, 1]
|
[270, 9]
|
rw [h]
|
case mpr
A : Type ?u.20667
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b c : G
h : a = c * b
⊢ a * b⁻¹ = c
|
case mpr
A : Type ?u.20667
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b c : G
h : a = c * b
⊢ c * b * b⁻¹ = c
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_inv_eq_iff_eq_mul
|
[261, 1]
|
[270, 9]
|
simp
|
case mpr
A : Type ?u.20667
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b c : G
h : a = c * b
⊢ c * b * b⁻¹ = c
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_inv_eq_one
|
[273, 1]
|
[276, 30]
|
suffices a * b⁻¹ = 1 ↔ a = 1 * b by simpa using this
|
A : Type ?u.21692
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a * b⁻¹ = 1 ↔ a = b
|
A : Type ?u.21692
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a * b⁻¹ = 1 ↔ a = 1 * b
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_inv_eq_one
|
[273, 1]
|
[276, 30]
|
exact mul_inv_eq_iff_eq_mul
|
A : Type ?u.21692
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a * b⁻¹ = 1 ↔ a = 1 * b
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_inv_eq_one
|
[273, 1]
|
[276, 30]
|
simpa using this
|
A : Type ?u.21692
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
this : a * b⁻¹ = 1 ↔ a = 1 * b
⊢ a * b⁻¹ = 1 ↔ a = b
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_mul_eq_one
|
[279, 1]
|
[289, 9]
|
constructor
|
A : Type ?u.23889
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a⁻¹ * b = 1 ↔ a = b
|
case mp
A : Type ?u.23889
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a⁻¹ * b = 1 → a = b
case mpr
A : Type ?u.23889
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a = b → a⁻¹ * b = 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_mul_eq_one
|
[279, 1]
|
[289, 9]
|
intro h
|
case mp
A : Type ?u.23889
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a⁻¹ * b = 1 → a = b
|
case mp
A : Type ?u.23889
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
h : a⁻¹ * b = 1
⊢ a = b
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_mul_eq_one
|
[279, 1]
|
[289, 9]
|
have h : a⁻¹ * b⁻¹⁻¹ = 1 := by simp [h]
|
case mp
A : Type ?u.23889
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
h : a⁻¹ * b = 1
⊢ a = b
|
case mp
A : Type ?u.23889
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
h✝ : a⁻¹ * b = 1
h : a⁻¹ * b⁻¹⁻¹ = 1
⊢ a = b
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_mul_eq_one
|
[279, 1]
|
[289, 9]
|
rw [mul_inv_eq_one] at h
|
case mp
A : Type ?u.23889
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
h✝ : a⁻¹ * b = 1
h : a⁻¹ * b⁻¹⁻¹ = 1
⊢ a = b
|
case mp
A : Type ?u.23889
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
h✝ : a⁻¹ * b = 1
h : a⁻¹ = b⁻¹
⊢ a = b
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_mul_eq_one
|
[279, 1]
|
[289, 9]
|
suffices a⁻¹⁻¹ = b⁻¹⁻¹ by simpa using this
|
case mp
A : Type ?u.23889
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
h✝ : a⁻¹ * b = 1
h : a⁻¹ = b⁻¹
⊢ a = b
|
case mp
A : Type ?u.23889
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
h✝ : a⁻¹ * b = 1
h : a⁻¹ = b⁻¹
⊢ a⁻¹⁻¹ = b⁻¹⁻¹
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_mul_eq_one
|
[279, 1]
|
[289, 9]
|
rw [h]
|
case mp
A : Type ?u.23889
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
h✝ : a⁻¹ * b = 1
h : a⁻¹ = b⁻¹
⊢ a⁻¹⁻¹ = b⁻¹⁻¹
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_mul_eq_one
|
[279, 1]
|
[289, 9]
|
simp [h]
|
A : Type ?u.23889
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
h : a⁻¹ * b = 1
⊢ a⁻¹ * b⁻¹⁻¹ = 1
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_mul_eq_one
|
[279, 1]
|
[289, 9]
|
simpa using this
|
A : Type ?u.23889
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
h✝ : a⁻¹ * b = 1
h : a⁻¹ = b⁻¹
this : a⁻¹⁻¹ = b⁻¹⁻¹
⊢ a = b
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_mul_eq_one
|
[279, 1]
|
[289, 9]
|
intro h
|
case mpr
A : Type ?u.23889
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a = b → a⁻¹ * b = 1
|
case mpr
A : Type ?u.23889
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
h : a = b
⊢ a⁻¹ * b = 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_mul_eq_one
|
[279, 1]
|
[289, 9]
|
rw [h]
|
case mpr
A : Type ?u.23889
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
h : a = b
⊢ a⁻¹ * b = 1
|
case mpr
A : Type ?u.23889
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
h : a = b
⊢ b⁻¹ * b = 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_mul_eq_one
|
[279, 1]
|
[289, 9]
|
simp
|
case mpr
A : Type ?u.23889
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
h : a = b
⊢ b⁻¹ * b = 1
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.Subgroup.inv_mem_iff
|
[351, 1]
|
[357, 18]
|
constructor
|
G : Type
inst✝ : Group G
H : Subgroup G
a : G
⊢ a⁻¹ ∈ H ↔ a ∈ H
|
case mp
G : Type
inst✝ : Group G
H : Subgroup G
a : G
⊢ a⁻¹ ∈ H → a ∈ H
case mpr
G : Type
inst✝ : Group G
H : Subgroup G
a : G
⊢ a ∈ H → a⁻¹ ∈ H
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.Subgroup.inv_mem_iff
|
[351, 1]
|
[357, 18]
|
intro h
|
case mp
G : Type
inst✝ : Group G
H : Subgroup G
a : G
⊢ a⁻¹ ∈ H → a ∈ H
|
case mp
G : Type
inst✝ : Group G
H : Subgroup G
a : G
h : a⁻¹ ∈ H
⊢ a ∈ H
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.Subgroup.inv_mem_iff
|
[351, 1]
|
[357, 18]
|
have h := inv_mem h
|
case mp
G : Type
inst✝ : Group G
H : Subgroup G
a : G
h : a⁻¹ ∈ H
⊢ a ∈ H
|
case mp
G : Type
inst✝ : Group G
H : Subgroup G
a : G
h✝ : a⁻¹ ∈ H
h : a⁻¹⁻¹ ∈ H
⊢ a ∈ H
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.Subgroup.inv_mem_iff
|
[351, 1]
|
[357, 18]
|
simpa using h
|
case mp
G : Type
inst✝ : Group G
H : Subgroup G
a : G
h✝ : a⁻¹ ∈ H
h : a⁻¹⁻¹ ∈ H
⊢ a ∈ H
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture1.lean
|
Tutorial.Subgroup.inv_mem_iff
|
[351, 1]
|
[357, 18]
|
exact inv_mem
|
case mpr
G : Type
inst✝ : Group G
H : Subgroup G
a : G
⊢ a ∈ H → a⁻¹ ∈ H
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture3.lean
|
Tutorial.inv_smul_smul
|
[53, 1]
|
[56, 7]
|
rw [smul_smul]
|
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
a : G
x : X
⊢ a⁻¹ • a • x = x
|
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
a : G
x : X
⊢ (a⁻¹ * a) • x = x
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture3.lean
|
Tutorial.inv_smul_smul
|
[53, 1]
|
[56, 7]
|
simp
|
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
a : G
x : X
⊢ (a⁻¹ * a) • x = x
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture3.lean
|
Tutorial.smul_inv_smul
|
[60, 1]
|
[63, 7]
|
rw [smul_smul]
|
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
a : G
x : X
⊢ a • a⁻¹ • x = x
|
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
a : G
x : X
⊢ (a * a⁻¹) • x = x
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture3.lean
|
Tutorial.smul_inv_smul
|
[60, 1]
|
[63, 7]
|
simp
|
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
a : G
x : X
⊢ (a * a⁻¹) • x = x
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture3.lean
|
Tutorial.GroupAction.injective
|
[67, 1]
|
[73, 35]
|
intro x y (h : a • x = a • y)
|
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
a : G
⊢ Function.Injective fun x => a • x
|
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
a : G
x y : X
h : a • x = a • y
⊢ x = y
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture3.lean
|
Tutorial.GroupAction.injective
|
[67, 1]
|
[73, 35]
|
calc x = a⁻¹ • a • x := by rw [inv_smul_smul]
_ = a⁻¹ • a • y := by rw [h]
_ = y := by rw [inv_smul_smul]
|
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
a : G
x y : X
h : a • x = a • y
⊢ x = y
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture3.lean
|
Tutorial.GroupAction.injective
|
[67, 1]
|
[73, 35]
|
rw [inv_smul_smul]
|
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
a : G
x y : X
h : a • x = a • y
⊢ x = a⁻¹ • a • x
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture3.lean
|
Tutorial.GroupAction.injective
|
[67, 1]
|
[73, 35]
|
rw [h]
|
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
a : G
x y : X
h : a • x = a • y
⊢ a⁻¹ • a • x = a⁻¹ • a • y
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture3.lean
|
Tutorial.GroupAction.injective
|
[67, 1]
|
[73, 35]
|
rw [inv_smul_smul]
|
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
a : G
x y : X
h : a • x = a • y
⊢ a⁻¹ • a • y = y
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture3.lean
|
Tutorial.GroupAction.surjective
|
[82, 1]
|
[86, 22]
|
intro y
|
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
a : G
⊢ Function.Surjective fun x => a • x
|
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
a : G
y : X
⊢ ∃ a_1, (fun x => a • x) a_1 = y
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture3.lean
|
Tutorial.GroupAction.surjective
|
[82, 1]
|
[86, 22]
|
use a⁻¹ • y
|
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
a : G
y : X
⊢ ∃ a_1, (fun x => a • x) a_1 = y
|
case h
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
a : G
y : X
⊢ (fun x => a • x) (a⁻¹ • y) = y
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture3.lean
|
Tutorial.GroupAction.surjective
|
[82, 1]
|
[86, 22]
|
exact smul_inv_smul
|
case h
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
a : G
y : X
⊢ (fun x => a • x) (a⁻¹ • y) = y
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture3.lean
|
Tutorial.orbit_eq_orbit_iff_mem_orbit
|
[291, 1]
|
[308, 22]
|
constructor
|
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
⊢ orbit G x = orbit G y ↔ y ∈ orbit G x
|
case mp
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
⊢ orbit G x = orbit G y → y ∈ orbit G x
case mpr
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
⊢ y ∈ orbit G x → orbit G x = orbit G y
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture3.lean
|
Tutorial.orbit_eq_orbit_iff_mem_orbit
|
[291, 1]
|
[308, 22]
|
intro h
|
case mp
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
⊢ orbit G x = orbit G y → y ∈ orbit G x
|
case mp
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
h : orbit G x = orbit G y
⊢ y ∈ orbit G x
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture3.lean
|
Tutorial.orbit_eq_orbit_iff_mem_orbit
|
[291, 1]
|
[308, 22]
|
rw [h]
|
case mp
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
h : orbit G x = orbit G y
⊢ y ∈ orbit G x
|
case mp
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
h : orbit G x = orbit G y
⊢ y ∈ orbit G y
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture3.lean
|
Tutorial.orbit_eq_orbit_iff_mem_orbit
|
[291, 1]
|
[308, 22]
|
exists 1
|
case mp
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
h : orbit G x = orbit G y
⊢ y ∈ orbit G y
|
case mp
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
h : orbit G x = orbit G y
⊢ 1 • y = y
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture3.lean
|
Tutorial.orbit_eq_orbit_iff_mem_orbit
|
[291, 1]
|
[308, 22]
|
simp
|
case mp
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
h : orbit G x = orbit G y
⊢ 1 • y = y
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture3.lean
|
Tutorial.orbit_eq_orbit_iff_mem_orbit
|
[291, 1]
|
[308, 22]
|
intro ⟨a, ha⟩
|
case mpr
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
⊢ y ∈ orbit G x → orbit G x = orbit G y
|
case mpr
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
a : G
ha : a • x = y
⊢ orbit G x = orbit G y
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture3.lean
|
Tutorial.orbit_eq_orbit_iff_mem_orbit
|
[291, 1]
|
[308, 22]
|
ext z
|
case mpr
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
a : G
ha : a • x = y
⊢ orbit G x = orbit G y
|
case mpr.h
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
a : G
ha : a • x = y
z : X
⊢ z ∈ orbit G x ↔ z ∈ orbit G y
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture3.lean
|
Tutorial.orbit_eq_orbit_iff_mem_orbit
|
[291, 1]
|
[308, 22]
|
constructor
|
case mpr.h
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
a : G
ha : a • x = y
z : X
⊢ z ∈ orbit G x ↔ z ∈ orbit G y
|
case mpr.h.mp
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
a : G
ha : a • x = y
z : X
⊢ z ∈ orbit G x → z ∈ orbit G y
case mpr.h.mpr
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
a : G
ha : a • x = y
z : X
⊢ z ∈ orbit G y → z ∈ orbit G x
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture3.lean
|
Tutorial.orbit_eq_orbit_iff_mem_orbit
|
[291, 1]
|
[308, 22]
|
intro ⟨b, hb⟩
|
case mpr.h.mp
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
a : G
ha : a • x = y
z : X
⊢ z ∈ orbit G x → z ∈ orbit G y
|
case mpr.h.mp
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
a : G
ha : a • x = y
z : X
b : G
hb : b • x = z
⊢ z ∈ orbit G y
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture3.lean
|
Tutorial.orbit_eq_orbit_iff_mem_orbit
|
[291, 1]
|
[308, 22]
|
rw [← hb, ← ha]
|
case mpr.h.mp
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
a : G
ha : a • x = y
z : X
b : G
hb : b • x = z
⊢ z ∈ orbit G y
|
case mpr.h.mp
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
a : G
ha : a • x = y
z : X
b : G
hb : b • x = z
⊢ b • x ∈ orbit G (a • x)
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture3.lean
|
Tutorial.orbit_eq_orbit_iff_mem_orbit
|
[291, 1]
|
[308, 22]
|
exists b * a⁻¹
|
case mpr.h.mp
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
a : G
ha : a • x = y
z : X
b : G
hb : b • x = z
⊢ b • x ∈ orbit G (a • x)
|
case mpr.h.mp
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
a : G
ha : a • x = y
z : X
b : G
hb : b • x = z
⊢ (b * a⁻¹) • a • x = b • x
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture3.lean
|
Tutorial.orbit_eq_orbit_iff_mem_orbit
|
[291, 1]
|
[308, 22]
|
simp [mul_smul]
|
case mpr.h.mp
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
a : G
ha : a • x = y
z : X
b : G
hb : b • x = z
⊢ (b * a⁻¹) • a • x = b • x
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture3.lean
|
Tutorial.orbit_eq_orbit_iff_mem_orbit
|
[291, 1]
|
[308, 22]
|
intro ⟨b, hb⟩
|
case mpr.h.mpr
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
a : G
ha : a • x = y
z : X
⊢ z ∈ orbit G y → z ∈ orbit G x
|
case mpr.h.mpr
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
a : G
ha : a • x = y
z : X
b : G
hb : b • y = z
⊢ z ∈ orbit G x
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture3.lean
|
Tutorial.orbit_eq_orbit_iff_mem_orbit
|
[291, 1]
|
[308, 22]
|
rw [← hb, ← ha]
|
case mpr.h.mpr
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
a : G
ha : a • x = y
z : X
b : G
hb : b • y = z
⊢ z ∈ orbit G x
|
case mpr.h.mpr
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
a : G
ha : a • x = y
z : X
b : G
hb : b • y = z
⊢ b • a • x ∈ orbit G x
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture3.lean
|
Tutorial.orbit_eq_orbit_iff_mem_orbit
|
[291, 1]
|
[308, 22]
|
exists b * a
|
case mpr.h.mpr
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
a : G
ha : a • x = y
z : X
b : G
hb : b • y = z
⊢ b • a • x ∈ orbit G x
|
case mpr.h.mpr
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
a : G
ha : a • x = y
z : X
b : G
hb : b • y = z
⊢ (b * a) • x = b • a • x
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Algebra/Lecture3.lean
|
Tutorial.orbit_eq_orbit_iff_mem_orbit
|
[291, 1]
|
[308, 22]
|
simp [mul_smul]
|
case mpr.h.mpr
G X : Type
inst✝¹ : Group G
inst✝ : GroupAction G X
x y : X
a : G
ha : a • x = y
z : X
b : G
hb : b • y = z
⊢ (b * a) • x = b • a • x
|
no goals
|
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