url
stringclasses 148
values | commit
stringclasses 148
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
100
| 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
|
|---|---|---|---|---|---|---|---|---|
null
|
null
|
Mathlib/MeasureTheory/Measure/AEMeasurable.lean
|
aemeasurable_Ioi_of_forall_Ioc
|
[317, 1]
|
[331, 36]
|
rw [Ioi_eq_iUnion, aemeasurable_iUnion_iff]
|
case intro
ι : Type ?u.4550285
α : Type u_2
β✝ : Type ?u.4550291
γ : Type ?u.4550294
δ : Type ?u.4550297
R : Type ?u.4550300
m0 : MeasurableSpace α
inst✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : MeasureTheory.Measure α
β : Type u_1
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : IsCountablyGenerated atTop
x : α
g : α → β
g_meas : ∀ (t : α), t > x → AEMeasurable g
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
Ioi_eq_iUnion : Ioi x = ⋃ (n : ℕ), Ioc x (u n)
⊢ AEMeasurable g
|
case intro
ι : Type ?u.4550285
α : Type u_2
β✝ : Type ?u.4550291
γ : Type ?u.4550294
δ : Type ?u.4550297
R : Type ?u.4550300
m0 : MeasurableSpace α
inst✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : MeasureTheory.Measure α
β : Type u_1
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : IsCountablyGenerated atTop
x : α
g : α → β
g_meas : ∀ (t : α), t > x → AEMeasurable g
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
Ioi_eq_iUnion : Ioi x = ⋃ (n : ℕ), Ioc x (u n)
⊢ ∀ (i : ℕ), AEMeasurable g
|
null
|
null
|
Mathlib/MeasureTheory/Measure/AEMeasurable.lean
|
aemeasurable_Ioi_of_forall_Ioc
|
[317, 1]
|
[331, 36]
|
intro n
|
case intro
ι : Type ?u.4550285
α : Type u_2
β✝ : Type ?u.4550291
γ : Type ?u.4550294
δ : Type ?u.4550297
R : Type ?u.4550300
m0 : MeasurableSpace α
inst✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : MeasureTheory.Measure α
β : Type u_1
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : IsCountablyGenerated atTop
x : α
g : α → β
g_meas : ∀ (t : α), t > x → AEMeasurable g
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
Ioi_eq_iUnion : Ioi x = ⋃ (n : ℕ), Ioc x (u n)
⊢ ∀ (i : ℕ), AEMeasurable g
|
case intro
ι : Type ?u.4550285
α : Type u_2
β✝ : Type ?u.4550291
γ : Type ?u.4550294
δ : Type ?u.4550297
R : Type ?u.4550300
m0 : MeasurableSpace α
inst✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : MeasureTheory.Measure α
β : Type u_1
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : IsCountablyGenerated atTop
x : α
g : α → β
g_meas : ∀ (t : α), t > x → AEMeasurable g
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
Ioi_eq_iUnion : Ioi x = ⋃ (n : ℕ), Ioc x (u n)
n : ℕ
⊢ AEMeasurable g
|
null
|
null
|
Mathlib/MeasureTheory/Measure/AEMeasurable.lean
|
aemeasurable_Ioi_of_forall_Ioc
|
[317, 1]
|
[331, 36]
|
cases' lt_or_le x (u n) with h h
|
case intro
ι : Type ?u.4550285
α : Type u_2
β✝ : Type ?u.4550291
γ : Type ?u.4550294
δ : Type ?u.4550297
R : Type ?u.4550300
m0 : MeasurableSpace α
inst✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : MeasureTheory.Measure α
β : Type u_1
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : IsCountablyGenerated atTop
x : α
g : α → β
g_meas : ∀ (t : α), t > x → AEMeasurable g
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
Ioi_eq_iUnion : Ioi x = ⋃ (n : ℕ), Ioc x (u n)
n : ℕ
⊢ AEMeasurable g
|
case intro.inl
ι : Type ?u.4550285
α : Type u_2
β✝ : Type ?u.4550291
γ : Type ?u.4550294
δ : Type ?u.4550297
R : Type ?u.4550300
m0 : MeasurableSpace α
inst✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : MeasureTheory.Measure α
β : Type u_1
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : IsCountablyGenerated atTop
x : α
g : α → β
g_meas : ∀ (t : α), t > x → AEMeasurable g
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
Ioi_eq_iUnion : Ioi x = ⋃ (n : ℕ), Ioc x (u n)
n : ℕ
h : x < u n
⊢ AEMeasurable g
case intro.inr
ι : Type ?u.4550285
α : Type u_2
β✝ : Type ?u.4550291
γ : Type ?u.4550294
δ : Type ?u.4550297
R : Type ?u.4550300
m0 : MeasurableSpace α
inst✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : MeasureTheory.Measure α
β : Type u_1
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : IsCountablyGenerated atTop
x : α
g : α → β
g_meas : ∀ (t : α), t > x → AEMeasurable g
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
Ioi_eq_iUnion : Ioi x = ⋃ (n : ℕ), Ioc x (u n)
n : ℕ
h : u n ≤ x
⊢ AEMeasurable g
|
null
|
null
|
Mathlib/MeasureTheory/Measure/AEMeasurable.lean
|
aemeasurable_Ioi_of_forall_Ioc
|
[317, 1]
|
[331, 36]
|
rw [iUnion_Ioc_eq_Ioi_self_iff.mpr _]
|
ι : Type ?u.4550285
α : Type u_2
β✝ : Type ?u.4550291
γ : Type ?u.4550294
δ : Type ?u.4550297
R : Type ?u.4550300
m0 : MeasurableSpace α
inst✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : MeasureTheory.Measure α
β : Type u_1
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : IsCountablyGenerated atTop
x : α
g : α → β
g_meas : ∀ (t : α), t > x → AEMeasurable g
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
⊢ Ioi x = ⋃ (n : ℕ), Ioc x (u n)
|
ι : Type ?u.4550285
α : Type u_2
β✝ : Type ?u.4550291
γ : Type ?u.4550294
δ : Type ?u.4550297
R : Type ?u.4550300
m0 : MeasurableSpace α
inst✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : MeasureTheory.Measure α
β : Type u_1
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : IsCountablyGenerated atTop
x : α
g : α → β
g_meas : ∀ (t : α), t > x → AEMeasurable g
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
⊢ ∀ (x_1 : α), x < x_1 → ∃ i, x_1 ≤ u i
|
null
|
null
|
Mathlib/MeasureTheory/Measure/AEMeasurable.lean
|
aemeasurable_Ioi_of_forall_Ioc
|
[317, 1]
|
[331, 36]
|
exact fun y _ => (hu_tendsto.eventually (eventually_ge_atTop y)).exists
|
ι : Type ?u.4550285
α : Type u_2
β✝ : Type ?u.4550291
γ : Type ?u.4550294
δ : Type ?u.4550297
R : Type ?u.4550300
m0 : MeasurableSpace α
inst✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : MeasureTheory.Measure α
β : Type u_1
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : IsCountablyGenerated atTop
x : α
g : α → β
g_meas : ∀ (t : α), t > x → AEMeasurable g
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
⊢ ∀ (x_1 : α), x < x_1 → ∃ i, x_1 ≤ u i
|
no goals
|
null
|
null
|
Mathlib/MeasureTheory/Measure/AEMeasurable.lean
|
aemeasurable_Ioi_of_forall_Ioc
|
[317, 1]
|
[331, 36]
|
exact g_meas (u n) h
|
case intro.inl
ι : Type ?u.4550285
α : Type u_2
β✝ : Type ?u.4550291
γ : Type ?u.4550294
δ : Type ?u.4550297
R : Type ?u.4550300
m0 : MeasurableSpace α
inst✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : MeasureTheory.Measure α
β : Type u_1
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : IsCountablyGenerated atTop
x : α
g : α → β
g_meas : ∀ (t : α), t > x → AEMeasurable g
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
Ioi_eq_iUnion : Ioi x = ⋃ (n : ℕ), Ioc x (u n)
n : ℕ
h : x < u n
⊢ AEMeasurable g
|
no goals
|
null
|
null
|
Mathlib/MeasureTheory/Measure/AEMeasurable.lean
|
aemeasurable_Ioi_of_forall_Ioc
|
[317, 1]
|
[331, 36]
|
rw [Ioc_eq_empty (not_lt.mpr h), Measure.restrict_empty]
|
case intro.inr
ι : Type ?u.4550285
α : Type u_2
β✝ : Type ?u.4550291
γ : Type ?u.4550294
δ : Type ?u.4550297
R : Type ?u.4550300
m0 : MeasurableSpace α
inst✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : MeasureTheory.Measure α
β : Type u_1
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : IsCountablyGenerated atTop
x : α
g : α → β
g_meas : ∀ (t : α), t > x → AEMeasurable g
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
Ioi_eq_iUnion : Ioi x = ⋃ (n : ℕ), Ioc x (u n)
n : ℕ
h : u n ≤ x
⊢ AEMeasurable g
|
case intro.inr
ι : Type ?u.4550285
α : Type u_2
β✝ : Type ?u.4550291
γ : Type ?u.4550294
δ : Type ?u.4550297
R : Type ?u.4550300
m0 : MeasurableSpace α
inst✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : MeasureTheory.Measure α
β : Type u_1
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : IsCountablyGenerated atTop
x : α
g : α → β
g_meas : ∀ (t : α), t > x → AEMeasurable g
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
Ioi_eq_iUnion : Ioi x = ⋃ (n : ℕ), Ioc x (u n)
n : ℕ
h : u n ≤ x
⊢ AEMeasurable g
|
null
|
null
|
Mathlib/MeasureTheory/Measure/AEMeasurable.lean
|
aemeasurable_Ioi_of_forall_Ioc
|
[317, 1]
|
[331, 36]
|
exact aemeasurable_zero_measure
|
case intro.inr
ι : Type ?u.4550285
α : Type u_2
β✝ : Type ?u.4550291
γ : Type ?u.4550294
δ : Type ?u.4550297
R : Type ?u.4550300
m0 : MeasurableSpace α
inst✝⁴ : MeasurableSpace β✝
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
f g✝ : α → β✝
μ ν : MeasureTheory.Measure α
β : Type u_1
mβ : MeasurableSpace β
inst✝¹ : LinearOrder α
inst✝ : IsCountablyGenerated atTop
x : α
g : α → β
g_meas : ∀ (t : α), t > x → AEMeasurable g
this : Nonempty α
u : ℕ → α
hu_tendsto : Tendsto u atTop atTop
Ioi_eq_iUnion : Ioi x = ⋃ (n : ℕ), Ioc x (u n)
n : ℕ
h : u n ≤ x
⊢ AEMeasurable g
|
no goals
|
null
|
null
|
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.extendWith_single_zero
|
[1647, 1]
|
[1654, 83]
|
ext (_ | j)
|
ι : Type u
γ : Type w
β : ι → Type v
β₁ : ι → Type v₁
β₂ : ι → Type v₂
dec : DecidableEq ι
κ : Type ?u.661354
α : Option ι → Type v
inst✝¹ : DecidableEq ι
inst✝ : (i : Option ι) → Zero (α i)
i : ι
x : α (some i)
⊢ extendWith 0 (single i x) = single (some i) x
|
case h.none
ι : Type u
γ : Type w
β : ι → Type v
β₁ : ι → Type v₁
β₂ : ι → Type v₂
dec : DecidableEq ι
κ : Type ?u.661354
α : Option ι → Type v
inst✝¹ : DecidableEq ι
inst✝ : (i : Option ι) → Zero (α i)
i : ι
x : α (some i)
⊢ ↑(extendWith 0 (single i x)) none = ↑(single (some i) x) none
case h.some
ι : Type u
γ : Type w
β : ι → Type v
β₁ : ι → Type v₁
β₂ : ι → Type v₂
dec : DecidableEq ι
κ : Type ?u.661354
α : Option ι → Type v
inst✝¹ : DecidableEq ι
inst✝ : (i : Option ι) → Zero (α i)
i : ι
x : α (some i)
j : ι
⊢ ↑(extendWith 0 (single i x)) (some j) = ↑(single (some i) x) (some j)
|
null
|
null
|
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.extendWith_single_zero
|
[1647, 1]
|
[1654, 83]
|
rw [extendWith_none, single_eq_of_ne (Option.some_ne_none _)]
|
case h.none
ι : Type u
γ : Type w
β : ι → Type v
β₁ : ι → Type v₁
β₂ : ι → Type v₂
dec : DecidableEq ι
κ : Type ?u.661354
α : Option ι → Type v
inst✝¹ : DecidableEq ι
inst✝ : (i : Option ι) → Zero (α i)
i : ι
x : α (some i)
⊢ ↑(extendWith 0 (single i x)) none = ↑(single (some i) x) none
|
no goals
|
null
|
null
|
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.extendWith_single_zero
|
[1647, 1]
|
[1654, 83]
|
rw [extendWith_some]
|
case h.some
ι : Type u
γ : Type w
β : ι → Type v
β₁ : ι → Type v₁
β₂ : ι → Type v₂
dec : DecidableEq ι
κ : Type ?u.661354
α : Option ι → Type v
inst✝¹ : DecidableEq ι
inst✝ : (i : Option ι) → Zero (α i)
i : ι
x : α (some i)
j : ι
⊢ ↑(extendWith 0 (single i x)) (some j) = ↑(single (some i) x) (some j)
|
case h.some
ι : Type u
γ : Type w
β : ι → Type v
β₁ : ι → Type v₁
β₂ : ι → Type v₂
dec : DecidableEq ι
κ : Type ?u.661354
α : Option ι → Type v
inst✝¹ : DecidableEq ι
inst✝ : (i : Option ι) → Zero (α i)
i : ι
x : α (some i)
j : ι
⊢ ↑(single i x) j = ↑(single (some i) x) (some j)
|
null
|
null
|
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.extendWith_single_zero
|
[1647, 1]
|
[1654, 83]
|
obtain rfl | hij := Decidable.eq_or_ne i j
|
case h.some
ι : Type u
γ : Type w
β : ι → Type v
β₁ : ι → Type v₁
β₂ : ι → Type v₂
dec : DecidableEq ι
κ : Type ?u.661354
α : Option ι → Type v
inst✝¹ : DecidableEq ι
inst✝ : (i : Option ι) → Zero (α i)
i : ι
x : α (some i)
j : ι
⊢ ↑(single i x) j = ↑(single (some i) x) (some j)
|
case h.some.inl
ι : Type u
γ : Type w
β : ι → Type v
β₁ : ι → Type v₁
β₂ : ι → Type v₂
dec : DecidableEq ι
κ : Type ?u.661354
α : Option ι → Type v
inst✝¹ : DecidableEq ι
inst✝ : (i : Option ι) → Zero (α i)
i : ι
x : α (some i)
⊢ ↑(single i x) i = ↑(single (some i) x) (some i)
case h.some.inr
ι : Type u
γ : Type w
β : ι → Type v
β₁ : ι → Type v₁
β₂ : ι → Type v₂
dec : DecidableEq ι
κ : Type ?u.661354
α : Option ι → Type v
inst✝¹ : DecidableEq ι
inst✝ : (i : Option ι) → Zero (α i)
i : ι
x : α (some i)
j : ι
hij : i ≠ j
⊢ ↑(single i x) j = ↑(single (some i) x) (some j)
|
null
|
null
|
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.extendWith_single_zero
|
[1647, 1]
|
[1654, 83]
|
rw [single_eq_same, single_eq_same]
|
case h.some.inl
ι : Type u
γ : Type w
β : ι → Type v
β₁ : ι → Type v₁
β₂ : ι → Type v₂
dec : DecidableEq ι
κ : Type ?u.661354
α : Option ι → Type v
inst✝¹ : DecidableEq ι
inst✝ : (i : Option ι) → Zero (α i)
i : ι
x : α (some i)
⊢ ↑(single i x) i = ↑(single (some i) x) (some i)
|
no goals
|
null
|
null
|
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.extendWith_single_zero
|
[1647, 1]
|
[1654, 83]
|
rw [single_eq_of_ne hij, single_eq_of_ne ((Option.some_injective _).ne hij)]
|
case h.some.inr
ι : Type u
γ : Type w
β : ι → Type v
β₁ : ι → Type v₁
β₂ : ι → Type v₂
dec : DecidableEq ι
κ : Type ?u.661354
α : Option ι → Type v
inst✝¹ : DecidableEq ι
inst✝ : (i : Option ι) → Zero (α i)
i : ι
x : α (some i)
j : ι
hij : i ≠ j
⊢ ↑(single i x) j = ↑(single (some i) x) (some j)
|
no goals
|
null
|
null
|
Mathlib/Data/Rat/Defs.lean
|
Rat.coe_int_div_eq_divInt
|
[503, 1]
|
[505, 54]
|
repeat' rw [coe_int_eq_divInt]
|
a b c : ℚ
n d : ℤ
⊢ ↑n / ↑d = n /. d
|
a b c : ℚ
n d : ℤ
⊢ n /. 1 / (d /. 1) = n /. d
|
null
|
null
|
Mathlib/Data/Rat/Defs.lean
|
Rat.coe_int_div_eq_divInt
|
[503, 1]
|
[505, 54]
|
exact divInt_div_divInt_cancel_left one_ne_zero n d
|
a b c : ℚ
n d : ℤ
⊢ n /. 1 / (d /. 1) = n /. d
|
no goals
|
null
|
null
|
Mathlib/Data/Rat/Defs.lean
|
Rat.coe_int_div_eq_divInt
|
[503, 1]
|
[505, 54]
|
rw [coe_int_eq_divInt]
|
a b c : ℚ
n d : ℤ
⊢ n /. 1 / (d /. 1) = n /. d
|
a b c : ℚ
n d : ℤ
⊢ n /. 1 / (d /. 1) = n /. d
|
null
|
null
|
Mathlib/Data/Matrix/Basic.lean
|
Matrix.smul_eq_diagonal_mul
|
[1048, 1]
|
[1051, 7]
|
ext
|
l : Type ?u.279030
m : Type u_1
n : Type u_2
o : Type ?u.279039
m' : o → Type ?u.279044
n' : o → Type ?u.279049
R : Type ?u.279052
S : Type ?u.279055
α : Type v
β : Type w
γ : Type ?u.279062
inst✝² : NonUnitalNonAssocSemiring α
inst✝¹ : Fintype m
inst✝ : DecidableEq m
M : Matrix m n α
a : α
⊢ a • M = (diagonal fun x => a) ⬝ M
|
case a.h
l : Type ?u.279030
m : Type u_1
n : Type u_2
o : Type ?u.279039
m' : o → Type ?u.279044
n' : o → Type ?u.279049
R : Type ?u.279052
S : Type ?u.279055
α : Type v
β : Type w
γ : Type ?u.279062
inst✝² : NonUnitalNonAssocSemiring α
inst✝¹ : Fintype m
inst✝ : DecidableEq m
M : Matrix m n α
a : α
i✝ : m
x✝ : n
⊢ (a • M) i✝ x✝ = ((diagonal fun x => a) ⬝ M) i✝ x✝
|
null
|
null
|
Mathlib/Data/Matrix/Basic.lean
|
Matrix.smul_eq_diagonal_mul
|
[1048, 1]
|
[1051, 7]
|
simp
|
case a.h
l : Type ?u.279030
m : Type u_1
n : Type u_2
o : Type ?u.279039
m' : o → Type ?u.279044
n' : o → Type ?u.279049
R : Type ?u.279052
S : Type ?u.279055
α : Type v
β : Type w
γ : Type ?u.279062
inst✝² : NonUnitalNonAssocSemiring α
inst✝¹ : Fintype m
inst✝ : DecidableEq m
M : Matrix m n α
a : α
i✝ : m
x✝ : n
⊢ (a • M) i✝ x✝ = ((diagonal fun x => a) ⬝ M) i✝ x✝
|
no goals
|
null
|
null
|
Mathlib/MeasureTheory/Integral/CircleIntegral.lean
|
cauchyPowerSeries_apply
|
[533, 1]
|
[538, 27]
|
simp only [cauchyPowerSeries, ContinuousMultilinearMap.mkPiField_apply, Fin.prod_const,
div_eq_mul_inv, mul_pow, mul_smul, circleIntegral.integral_smul]
|
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
R : ℝ
n : ℕ
w : ℂ
⊢ (↑(cauchyPowerSeries f c R n) fun x => w) = (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (w / (z - c)) ^ n • (z - c)⁻¹ • f z
|
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
R : ℝ
n : ℕ
w : ℂ
⊢ (w ^ n • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z) =
(2 * ↑π * I)⁻¹ • w ^ n • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z
|
null
|
null
|
Mathlib/MeasureTheory/Integral/CircleIntegral.lean
|
cauchyPowerSeries_apply
|
[533, 1]
|
[538, 27]
|
rw [← smul_comm (w ^ n)]
|
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
R : ℝ
n : ℕ
w : ℂ
⊢ (w ^ n • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z) =
(2 * ↑π * I)⁻¹ • w ^ n • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z
|
no goals
|
null
|
null
|
Mathlib/Data/Nat/Log.lean
|
Nat.clog_mono_right
|
[334, 1]
|
[339, 37]
|
cases' le_or_lt b 1 with hb hb
|
b n m : ℕ
h : n ≤ m
⊢ clog b n ≤ clog b m
|
case inl
b n m : ℕ
h : n ≤ m
hb : b ≤ 1
⊢ clog b n ≤ clog b m
case inr
b n m : ℕ
h : n ≤ m
hb : 1 < b
⊢ clog b n ≤ clog b m
|
null
|
null
|
Mathlib/Data/Nat/Log.lean
|
Nat.clog_mono_right
|
[334, 1]
|
[339, 37]
|
rw [clog_of_left_le_one hb]
|
case inl
b n m : ℕ
h : n ≤ m
hb : b ≤ 1
⊢ clog b n ≤ clog b m
|
case inl
b n m : ℕ
h : n ≤ m
hb : b ≤ 1
⊢ 0 ≤ clog b m
|
null
|
null
|
Mathlib/Data/Nat/Log.lean
|
Nat.clog_mono_right
|
[334, 1]
|
[339, 37]
|
exact zero_le _
|
case inl
b n m : ℕ
h : n ≤ m
hb : b ≤ 1
⊢ 0 ≤ clog b m
|
no goals
|
null
|
null
|
Mathlib/Data/Nat/Log.lean
|
Nat.clog_mono_right
|
[334, 1]
|
[339, 37]
|
rw [← le_pow_iff_clog_le hb]
|
case inr
b n m : ℕ
h : n ≤ m
hb : 1 < b
⊢ clog b n ≤ clog b m
|
case inr
b n m : ℕ
h : n ≤ m
hb : 1 < b
⊢ n ≤ b ^ clog b m
|
null
|
null
|
Mathlib/Data/Nat/Log.lean
|
Nat.clog_mono_right
|
[334, 1]
|
[339, 37]
|
exact h.trans (le_pow_clog hb _)
|
case inr
b n m : ℕ
h : n ≤ m
hb : 1 < b
⊢ n ≤ b ^ clog b m
|
no goals
|
null
|
null
|
Mathlib/ModelTheory/Syntax.lean
|
FirstOrder.Language.BoundedFormula.not_all_isAtomic
|
[687, 1]
|
[688, 12]
|
cases con
|
L : Language
L' : Language
M : Type w
N : Type ?u.109819
P : Type ?u.109822
inst✝² : Structure L M
inst✝¹ : Structure L N
inst✝ : Structure L P
α : Type u'
β : Type v'
γ : Type ?u.109850
n l : ℕ
φ✝ ψ : BoundedFormula L α l
θ : BoundedFormula L α (Nat.succ l)
v : α → M
xs : Fin l → M
φ : BoundedFormula L α (n + 1)
con : IsAtomic (all φ)
⊢ False
|
no goals
|
null
|
null
|
Std/Data/Int/Lemmas.lean
|
Int.neg_add_lt_left_of_lt_add
|
[1091, 11]
|
[1093, 36]
|
rw [Int.add_comm]
|
a b c : Int
h : a < b + c
⊢ -b + a < c
|
a b c : Int
h : a < b + c
⊢ a + -b < c
|
null
|
null
|
Std/Data/Int/Lemmas.lean
|
Int.neg_add_lt_left_of_lt_add
|
[1091, 11]
|
[1093, 36]
|
exact Int.sub_left_lt_of_lt_add h
|
a b c : Int
h : a < b + c
⊢ a + -b < c
|
no goals
|
null
|
null
|
Std/Data/Option/Lemmas.lean
|
Option.isSome_iff_exists
|
[55, 1]
|
[55, 87]
|
cases x <;> simp [isSome]
|
α✝ : Type u_1
x : Option α✝
⊢ isSome x = true ↔ ∃ a, x = some a
|
no goals
|
null
|
null
|
Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean
|
AlgebraicTopology.DoldKan.hσ'_naturality
|
[161, 1]
|
[171, 8]
|
have h : n + 1 = m := hnm
|
C : Type u_2
inst✝¹ : Category C
inst✝ : Preadditive C
X✝ : SimplicialObject C
q n m : ℕ
hnm : ComplexShape.Rel c m n
X Y : SimplicialObject C
f : X ⟶ Y
⊢ f.app [n].op ≫ hσ' q n m hnm = hσ' q n m hnm ≫ f.app [m].op
|
C : Type u_2
inst✝¹ : Category C
inst✝ : Preadditive C
X✝ : SimplicialObject C
q n m : ℕ
hnm : ComplexShape.Rel c m n
X Y : SimplicialObject C
f : X ⟶ Y
h : n + 1 = m
⊢ f.app [n].op ≫ hσ' q n m hnm = hσ' q n m hnm ≫ f.app [m].op
|
null
|
null
|
Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean
|
AlgebraicTopology.DoldKan.hσ'_naturality
|
[161, 1]
|
[171, 8]
|
subst h
|
C : Type u_2
inst✝¹ : Category C
inst✝ : Preadditive C
X✝ : SimplicialObject C
q n m : ℕ
hnm : ComplexShape.Rel c m n
X Y : SimplicialObject C
f : X ⟶ Y
h : n + 1 = m
⊢ f.app [n].op ≫ hσ' q n m hnm = hσ' q n m hnm ≫ f.app [m].op
|
C : Type u_2
inst✝¹ : Category C
inst✝ : Preadditive C
X✝ : SimplicialObject C
q n : ℕ
X Y : SimplicialObject C
f : X ⟶ Y
hnm : ComplexShape.Rel c (n + 1) n
⊢ f.app [n].op ≫ hσ' q n (n + 1) hnm = hσ' q n (n + 1) hnm ≫ f.app [n + 1].op
|
null
|
null
|
Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean
|
AlgebraicTopology.DoldKan.hσ'_naturality
|
[161, 1]
|
[171, 8]
|
simp only [hσ', eqToHom_refl, comp_id]
|
C : Type u_2
inst✝¹ : Category C
inst✝ : Preadditive C
X✝ : SimplicialObject C
q n : ℕ
X Y : SimplicialObject C
f : X ⟶ Y
hnm : ComplexShape.Rel c (n + 1) n
⊢ f.app [n].op ≫ hσ' q n (n + 1) hnm = hσ' q n (n + 1) hnm ≫ f.app [n + 1].op
|
C : Type u_2
inst✝¹ : Category C
inst✝ : Preadditive C
X✝ : SimplicialObject C
q n : ℕ
X Y : SimplicialObject C
f : X ⟶ Y
hnm : ComplexShape.Rel c (n + 1) n
⊢ f.app [n].op ≫ hσ q n = hσ q n ≫ f.app [n + 1].op
|
null
|
null
|
Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean
|
AlgebraicTopology.DoldKan.hσ'_naturality
|
[161, 1]
|
[171, 8]
|
unfold hσ
|
C : Type u_2
inst✝¹ : Category C
inst✝ : Preadditive C
X✝ : SimplicialObject C
q n : ℕ
X Y : SimplicialObject C
f : X ⟶ Y
hnm : ComplexShape.Rel c (n + 1) n
⊢ f.app [n].op ≫ hσ q n = hσ q n ≫ f.app [n + 1].op
|
C : Type u_2
inst✝¹ : Category C
inst✝ : Preadditive C
X✝ : SimplicialObject C
q n : ℕ
X Y : SimplicialObject C
f : X ⟶ Y
hnm : ComplexShape.Rel c (n + 1) n
⊢ (f.app [n].op ≫ if n < q then 0 else (-1) ^ (n - q) • σ Y { val := n - q, isLt := (_ : n - q < Nat.succ n) }) =
(if n < q then 0 else (-1) ^ (n - q) • σ X { val := n - q, isLt := (_ : n - q < Nat.succ n) }) ≫ f.app [n + 1].op
|
null
|
null
|
Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean
|
AlgebraicTopology.DoldKan.hσ'_naturality
|
[161, 1]
|
[171, 8]
|
split_ifs
|
C : Type u_2
inst✝¹ : Category C
inst✝ : Preadditive C
X✝ : SimplicialObject C
q n : ℕ
X Y : SimplicialObject C
f : X ⟶ Y
hnm : ComplexShape.Rel c (n + 1) n
⊢ (f.app [n].op ≫ if n < q then 0 else (-1) ^ (n - q) • σ Y { val := n - q, isLt := (_ : n - q < Nat.succ n) }) =
(if n < q then 0 else (-1) ^ (n - q) • σ X { val := n - q, isLt := (_ : n - q < Nat.succ n) }) ≫ f.app [n + 1].op
|
case inl
C : Type u_2
inst✝¹ : Category C
inst✝ : Preadditive C
X✝ : SimplicialObject C
q n : ℕ
X Y : SimplicialObject C
f : X ⟶ Y
hnm : ComplexShape.Rel c (n + 1) n
h✝ : n < q
⊢ f.app [n].op ≫ 0 = 0 ≫ f.app [n + 1].op
case inr
C : Type u_2
inst✝¹ : Category C
inst✝ : Preadditive C
X✝ : SimplicialObject C
q n : ℕ
X Y : SimplicialObject C
f : X ⟶ Y
hnm : ComplexShape.Rel c (n + 1) n
h✝ : ¬n < q
⊢ f.app [n].op ≫ ((-1) ^ (n - q) • σ Y { val := n - q, isLt := (_ : n - q < Nat.succ n) }) =
((-1) ^ (n - q) • σ X { val := n - q, isLt := (_ : n - q < Nat.succ n) }) ≫ f.app [n + 1].op
|
null
|
null
|
Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean
|
AlgebraicTopology.DoldKan.hσ'_naturality
|
[161, 1]
|
[171, 8]
|
rw [zero_comp, comp_zero]
|
case inl
C : Type u_2
inst✝¹ : Category C
inst✝ : Preadditive C
X✝ : SimplicialObject C
q n : ℕ
X Y : SimplicialObject C
f : X ⟶ Y
hnm : ComplexShape.Rel c (n + 1) n
h✝ : n < q
⊢ f.app [n].op ≫ 0 = 0 ≫ f.app [n + 1].op
|
no goals
|
null
|
null
|
Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean
|
AlgebraicTopology.DoldKan.hσ'_naturality
|
[161, 1]
|
[171, 8]
|
simp only [zsmul_comp, comp_zsmul]
|
case inr
C : Type u_2
inst✝¹ : Category C
inst✝ : Preadditive C
X✝ : SimplicialObject C
q n : ℕ
X Y : SimplicialObject C
f : X ⟶ Y
hnm : ComplexShape.Rel c (n + 1) n
h✝ : ¬n < q
⊢ f.app [n].op ≫ ((-1) ^ (n - q) • σ Y { val := n - q, isLt := (_ : n - q < Nat.succ n) }) =
((-1) ^ (n - q) • σ X { val := n - q, isLt := (_ : n - q < Nat.succ n) }) ≫ f.app [n + 1].op
|
case inr
C : Type u_2
inst✝¹ : Category C
inst✝ : Preadditive C
X✝ : SimplicialObject C
q n : ℕ
X Y : SimplicialObject C
f : X ⟶ Y
hnm : ComplexShape.Rel c (n + 1) n
h✝ : ¬n < q
⊢ (-1) ^ (n - q) • f.app [n].op ≫ σ Y { val := n - q, isLt := (_ : n - q < Nat.succ n) } =
(-1) ^ (n - q) • σ X { val := n - q, isLt := (_ : n - q < Nat.succ n) } ≫ f.app [n + 1].op
|
null
|
null
|
Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean
|
AlgebraicTopology.DoldKan.hσ'_naturality
|
[161, 1]
|
[171, 8]
|
erw [f.naturality]
|
case inr
C : Type u_2
inst✝¹ : Category C
inst✝ : Preadditive C
X✝ : SimplicialObject C
q n : ℕ
X Y : SimplicialObject C
f : X ⟶ Y
hnm : ComplexShape.Rel c (n + 1) n
h✝ : ¬n < q
⊢ (-1) ^ (n - q) • f.app [n].op ≫ σ Y { val := n - q, isLt := (_ : n - q < Nat.succ n) } =
(-1) ^ (n - q) • σ X { val := n - q, isLt := (_ : n - q < Nat.succ n) } ≫ f.app [n + 1].op
|
case inr
C : Type u_2
inst✝¹ : Category C
inst✝ : Preadditive C
X✝ : SimplicialObject C
q n : ℕ
X Y : SimplicialObject C
f : X ⟶ Y
hnm : ComplexShape.Rel c (n + 1) n
h✝ : ¬n < q
⊢ (-1) ^ (n - q) • f.app [n].op ≫ σ Y { val := n - q, isLt := (_ : n - q < Nat.succ n) } =
(-1) ^ (n - q) • f.app [n].op ≫ Y.map (SimplexCategory.σ { val := n - q, isLt := (_ : n - q < Nat.succ n) }).op
|
null
|
null
|
Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean
|
AlgebraicTopology.DoldKan.hσ'_naturality
|
[161, 1]
|
[171, 8]
|
rfl
|
case inr
C : Type u_2
inst✝¹ : Category C
inst✝ : Preadditive C
X✝ : SimplicialObject C
q n : ℕ
X Y : SimplicialObject C
f : X ⟶ Y
hnm : ComplexShape.Rel c (n + 1) n
h✝ : ¬n < q
⊢ (-1) ^ (n - q) • f.app [n].op ≫ σ Y { val := n - q, isLt := (_ : n - q < Nat.succ n) } =
(-1) ^ (n - q) • f.app [n].op ≫ Y.map (SimplexCategory.σ { val := n - q, isLt := (_ : n - q < Nat.succ n) }).op
|
no goals
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.isCycleOn_support_cycleOf
|
[1183, 1]
|
[1193, 36]
|
refine fun _ ↦ ⟨fun h ↦ mem_support_cycleOf_iff.2 ?_, fun h ↦ mem_support_cycleOf_iff.2 ?_⟩
|
ι : Type ?u.2583510
α : Type u_1
β : Type ?u.2583516
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f✝ g : Perm α
x✝ y : α
f : Perm α
x : α
⊢ ∀ (a : α), ↑f a ∈ ↑(support (cycleOf f x)) ↔ a ∈ ↑(support (cycleOf f x))
|
case refine_1
ι : Type ?u.2583510
α : Type u_1
β : Type ?u.2583516
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f✝ g : Perm α
x✝¹ y : α
f : Perm α
x x✝ : α
h : ↑f x✝ ∈ ↑(support (cycleOf f x))
⊢ SameCycle f x x✝ ∧ x ∈ support f
case refine_2
ι : Type ?u.2583510
α : Type u_1
β : Type ?u.2583516
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f✝ g : Perm α
x✝¹ y : α
f : Perm α
x x✝ : α
h : x✝ ∈ ↑(support (cycleOf f x))
⊢ SameCycle f x (↑f x✝) ∧ x ∈ support f
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.isCycleOn_support_cycleOf
|
[1183, 1]
|
[1193, 36]
|
exact ⟨sameCycle_apply_right.1 (mem_support_cycleOf_iff.1 h).1,
(mem_support_cycleOf_iff.1 h).2⟩
|
case refine_1
ι : Type ?u.2583510
α : Type u_1
β : Type ?u.2583516
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f✝ g : Perm α
x✝¹ y : α
f : Perm α
x x✝ : α
h : ↑f x✝ ∈ ↑(support (cycleOf f x))
⊢ SameCycle f x x✝ ∧ x ∈ support f
|
no goals
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.isCycleOn_support_cycleOf
|
[1183, 1]
|
[1193, 36]
|
exact ⟨sameCycle_apply_right.2 (mem_support_cycleOf_iff.1 h).1,
(mem_support_cycleOf_iff.1 h).2⟩
|
case refine_2
ι : Type ?u.2583510
α : Type u_1
β : Type ?u.2583516
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f✝ g : Perm α
x✝¹ y : α
f : Perm α
x x✝ : α
h : x✝ ∈ ↑(support (cycleOf f x))
⊢ SameCycle f x (↑f x✝) ∧ x ∈ support f
|
no goals
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.isCycleOn_support_cycleOf
|
[1183, 1]
|
[1193, 36]
|
rw [mem_coe, mem_support_cycleOf_iff] at ha hb
|
ι : Type ?u.2583510
α : Type u_1
β : Type ?u.2583516
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f✝ g : Perm α
x✝ y : α
f : Perm α
x a : α
ha : a ∈ ↑(support (cycleOf f x))
b : α
hb : b ∈ ↑(support (cycleOf f x))
⊢ SameCycle f a b
|
ι : Type ?u.2583510
α : Type u_1
β : Type ?u.2583516
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f✝ g : Perm α
x✝ y : α
f : Perm α
x a : α
ha : SameCycle f x a ∧ x ∈ support f
b : α
hb : SameCycle f x b ∧ x ∈ support f
⊢ SameCycle f a b
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.isCycleOn_support_cycleOf
|
[1183, 1]
|
[1193, 36]
|
exact ha.1.symm.trans hb.1
|
ι : Type ?u.2583510
α : Type u_1
β : Type ?u.2583516
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f✝ g : Perm α
x✝ y : α
f : Perm α
x a : α
ha : SameCycle f x a ∧ x ∈ support f
b : α
hb : SameCycle f x b ∧ x ∈ support f
⊢ SameCycle f a b
|
no goals
|
null
|
null
|
Mathlib/Data/Finset/Lattice.lean
|
Finset.iInf_singleton
|
[1935, 1]
|
[1935, 92]
|
simp
|
F : Type ?u.430806
α : Type u_2
β : Type u_1
γ : Type ?u.430815
ι : Type ?u.430818
κ : Type ?u.430821
inst✝ : CompleteLattice β
a : α
s : α → β
⊢ (⨅ (x : α) (_ : x ∈ {a}), s x) = s a
|
no goals
|
null
|
null
|
src/lean/Init/SimpLemmas.lean
|
ite_congr
|
[56, 1]
|
[60, 63]
|
cases Decidable.em c with
| inl h => rw [if_pos h]; subst b; rw [if_pos h]; exact h₂ h
| inr h => rw [if_neg h]; subst b; rw [if_neg h]; exact h₃ h
|
α : Sort u_1
b c : Prop
x y u v : α
s : Decidable b
inst✝ : Decidable c
h₁ : b = c
h₂ : c → x = u
h₃ : ¬c → y = v
⊢ ite b x y = ite c u v
|
no goals
|
null
|
null
|
src/lean/Init/SimpLemmas.lean
|
ite_congr
|
[56, 1]
|
[60, 63]
|
rw [if_pos h]
|
case inl
α : Sort u_1
b c : Prop
x y u v : α
s : Decidable b
inst✝ : Decidable c
h₁ : b = c
h₂ : c → x = u
h₃ : ¬c → y = v
h : c
⊢ ite b x y = ite c u v
|
case inl
α : Sort u_1
b c : Prop
x y u v : α
s : Decidable b
inst✝ : Decidable c
h₁ : b = c
h₂ : c → x = u
h₃ : ¬c → y = v
h : c
⊢ ite b x y = u
|
null
|
null
|
src/lean/Init/SimpLemmas.lean
|
ite_congr
|
[56, 1]
|
[60, 63]
|
subst b
|
case inl
α : Sort u_1
b c : Prop
x y u v : α
s : Decidable b
inst✝ : Decidable c
h₁ : b = c
h₂ : c → x = u
h₃ : ¬c → y = v
h : c
⊢ ite b x y = u
|
case inl
α : Sort u_1
c : Prop
x y u v : α
inst✝ : Decidable c
h₂ : c → x = u
h₃ : ¬c → y = v
h : c
s : Decidable c
⊢ ite c x y = u
|
null
|
null
|
src/lean/Init/SimpLemmas.lean
|
ite_congr
|
[56, 1]
|
[60, 63]
|
rw [if_pos h]
|
case inl
α : Sort u_1
c : Prop
x y u v : α
inst✝ : Decidable c
h₂ : c → x = u
h₃ : ¬c → y = v
h : c
s : Decidable c
⊢ ite c x y = u
|
case inl
α : Sort u_1
c : Prop
x y u v : α
inst✝ : Decidable c
h₂ : c → x = u
h₃ : ¬c → y = v
h : c
s : Decidable c
⊢ x = u
|
null
|
null
|
src/lean/Init/SimpLemmas.lean
|
ite_congr
|
[56, 1]
|
[60, 63]
|
exact h₂ h
|
case inl
α : Sort u_1
c : Prop
x y u v : α
inst✝ : Decidable c
h₂ : c → x = u
h₃ : ¬c → y = v
h : c
s : Decidable c
⊢ x = u
|
no goals
|
null
|
null
|
src/lean/Init/SimpLemmas.lean
|
ite_congr
|
[56, 1]
|
[60, 63]
|
rw [if_neg h]
|
case inr
α : Sort u_1
b c : Prop
x y u v : α
s : Decidable b
inst✝ : Decidable c
h₁ : b = c
h₂ : c → x = u
h₃ : ¬c → y = v
h : ¬c
⊢ ite b x y = ite c u v
|
case inr
α : Sort u_1
b c : Prop
x y u v : α
s : Decidable b
inst✝ : Decidable c
h₁ : b = c
h₂ : c → x = u
h₃ : ¬c → y = v
h : ¬c
⊢ ite b x y = v
|
null
|
null
|
src/lean/Init/SimpLemmas.lean
|
ite_congr
|
[56, 1]
|
[60, 63]
|
subst b
|
case inr
α : Sort u_1
b c : Prop
x y u v : α
s : Decidable b
inst✝ : Decidable c
h₁ : b = c
h₂ : c → x = u
h₃ : ¬c → y = v
h : ¬c
⊢ ite b x y = v
|
case inr
α : Sort u_1
c : Prop
x y u v : α
inst✝ : Decidable c
h₂ : c → x = u
h₃ : ¬c → y = v
h : ¬c
s : Decidable c
⊢ ite c x y = v
|
null
|
null
|
src/lean/Init/SimpLemmas.lean
|
ite_congr
|
[56, 1]
|
[60, 63]
|
rw [if_neg h]
|
case inr
α : Sort u_1
c : Prop
x y u v : α
inst✝ : Decidable c
h₂ : c → x = u
h₃ : ¬c → y = v
h : ¬c
s : Decidable c
⊢ ite c x y = v
|
case inr
α : Sort u_1
c : Prop
x y u v : α
inst✝ : Decidable c
h₂ : c → x = u
h₃ : ¬c → y = v
h : ¬c
s : Decidable c
⊢ y = v
|
null
|
null
|
src/lean/Init/SimpLemmas.lean
|
ite_congr
|
[56, 1]
|
[60, 63]
|
exact h₃ h
|
case inr
α : Sort u_1
c : Prop
x y u v : α
inst✝ : Decidable c
h₂ : c → x = u
h₃ : ¬c → y = v
h : ¬c
s : Decidable c
⊢ y = v
|
no goals
|
null
|
null
|
Mathlib/Algebra/Order/Field/Basic.lean
|
add_halves
|
[497, 1]
|
[498, 70]
|
rw [div_add_div_same, ← two_mul, mul_div_cancel_left a two_ne_zero]
|
ι : Type ?u.89441
α : Type u_1
β : Type ?u.89447
inst✝ : LinearOrderedSemifield α
a✝ b c d e : α
m n : ℤ
a : α
⊢ a / 2 + a / 2 = a
|
no goals
|
null
|
null
|
Mathlib/Data/Finset/Basic.lean
|
Finset.subset_insert_iff_of_not_mem
|
[2002, 1]
|
[2003, 48]
|
rw [subset_insert_iff, erase_eq_of_not_mem h]
|
α : Type u_1
β : Type ?u.218847
γ : Type ?u.218850
inst✝ : DecidableEq α
s t u v : Finset α
a b : α
h : ¬a ∈ s
⊢ s ⊆ insert a t ↔ s ⊆ t
|
no goals
|
null
|
null
|
Mathlib/Probability/ConditionalProbability.lean
|
ProbabilityTheory.cond_pos_of_inter_ne_zero
|
[120, 1]
|
[124, 53]
|
rw [cond_apply _ hms]
|
Ω : Type u_1
m : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
s t : Set Ω
inst✝ : IsFiniteMeasure μ
hms : MeasurableSet s
hci : ↑↑μ (s ∩ t) ≠ 0
⊢ 0 < ↑↑(μ[|s]) t
|
Ω : Type u_1
m : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
s t : Set Ω
inst✝ : IsFiniteMeasure μ
hms : MeasurableSet s
hci : ↑↑μ (s ∩ t) ≠ 0
⊢ 0 < (↑↑μ s)⁻¹ * ↑↑μ (s ∩ t)
|
null
|
null
|
Mathlib/Probability/ConditionalProbability.lean
|
ProbabilityTheory.cond_pos_of_inter_ne_zero
|
[120, 1]
|
[124, 53]
|
refine' ENNReal.mul_pos _ hci
|
Ω : Type u_1
m : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
s t : Set Ω
inst✝ : IsFiniteMeasure μ
hms : MeasurableSet s
hci : ↑↑μ (s ∩ t) ≠ 0
⊢ 0 < (↑↑μ s)⁻¹ * ↑↑μ (s ∩ t)
|
Ω : Type u_1
m : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
s t : Set Ω
inst✝ : IsFiniteMeasure μ
hms : MeasurableSet s
hci : ↑↑μ (s ∩ t) ≠ 0
⊢ (↑↑μ s)⁻¹ ≠ 0
|
null
|
null
|
Mathlib/Probability/ConditionalProbability.lean
|
ProbabilityTheory.cond_pos_of_inter_ne_zero
|
[120, 1]
|
[124, 53]
|
exact ENNReal.inv_ne_zero.mpr (measure_ne_top _ _)
|
Ω : Type u_1
m : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
s t : Set Ω
inst✝ : IsFiniteMeasure μ
hms : MeasurableSet s
hci : ↑↑μ (s ∩ t) ≠ 0
⊢ (↑↑μ s)⁻¹ ≠ 0
|
no goals
|
null
|
null
|
Mathlib/Data/Set/Intervals/SurjOn.lean
|
surjOn_Ici_of_monotone_surjective
|
[79, 1]
|
[84, 52]
|
rw [← Ioi_union_left, ← Ioi_union_left]
|
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : PartialOrder β
f : α → β
h_mono : Monotone f
h_surj : Surjective f
a : α
⊢ SurjOn f (Ici a) (Ici (f a))
|
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : PartialOrder β
f : α → β
h_mono : Monotone f
h_surj : Surjective f
a : α
⊢ SurjOn f (Ioi a ∪ {a}) (Ioi (f a) ∪ {f a})
|
null
|
null
|
Mathlib/Data/Set/Intervals/SurjOn.lean
|
surjOn_Ici_of_monotone_surjective
|
[79, 1]
|
[84, 52]
|
exact
(surjOn_Ioi_of_monotone_surjective h_mono h_surj a).union_union
(@image_singleton _ _ f a ▸ surjOn_image _ _)
|
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : PartialOrder β
f : α → β
h_mono : Monotone f
h_surj : Surjective f
a : α
⊢ SurjOn f (Ioi a ∪ {a}) (Ioi (f a) ∪ {f a})
|
no goals
|
null
|
null
|
Mathlib/Algebra/Order/Ring/Defs.lean
|
mul_le_mul_of_nonpos_right
|
[355, 1]
|
[356, 92]
|
simpa only [mul_neg, neg_le_neg_iff] using mul_le_mul_of_nonneg_right h (neg_nonneg.2 hc)
|
α : Type u
β : Type ?u.34618
inst✝ : OrderedRing α
a b c d : α
h : b ≤ a
hc : c ≤ 0
⊢ a * c ≤ b * c
|
no goals
|
null
|
null
|
Mathlib/Data/Fin/Basic.lean
|
Fin.castAdd_lt
|
[1141, 1]
|
[1142, 7]
|
simp
|
n✝ m✝ m n : ℕ
i : Fin m
⊢ ↑(↑(castAdd n) i) < m
|
no goals
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
cases nonempty_fintype α
|
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ π ρ : Perm α
hc1 : IsConj σ π
hc2 : IsConj τ ρ
hd1 : Disjoint σ τ
hd2 : Disjoint π ρ
⊢ IsConj (σ * τ) (π * ρ)
|
case intro
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ π ρ : Perm α
hc1 : IsConj σ π
hc2 : IsConj τ ρ
hd1 : Disjoint σ τ
hd2 : Disjoint π ρ
val✝ : Fintype α
⊢ IsConj (σ * τ) (π * ρ)
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
obtain ⟨f, rfl⟩ := isConj_iff.1 hc1
|
case intro
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ π ρ : Perm α
hc1 : IsConj σ π
hc2 : IsConj τ ρ
hd1 : Disjoint σ τ
hd2 : Disjoint π ρ
val✝ : Fintype α
⊢ IsConj (σ * τ) (π * ρ)
|
case intro.intro
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ ρ : Perm α
hc2 : IsConj τ ρ
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) ρ
⊢ IsConj (σ * τ) (f * σ * f⁻¹ * ρ)
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
obtain ⟨g, rfl⟩ := isConj_iff.1 hc2
|
case intro.intro
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ ρ : Perm α
hc2 : IsConj τ ρ
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) ρ
⊢ IsConj (σ * τ) (f * σ * f⁻¹ * ρ)
|
case intro.intro.intro
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
⊢ IsConj (σ * τ) (f * σ * f⁻¹ * (g * τ * g⁻¹))
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
have hd1' := coe_inj.2 hd1.support_mul
|
case intro.intro.intro
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
⊢ IsConj (σ * τ) (f * σ * f⁻¹ * (g * τ * g⁻¹))
|
case intro.intro.intro
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ ∪ support τ)
⊢ IsConj (σ * τ) (f * σ * f⁻¹ * (g * τ * g⁻¹))
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
have hd2' := coe_inj.2 hd2.support_mul
|
case intro.intro.intro
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ ∪ support τ)
⊢ IsConj (σ * τ) (f * σ * f⁻¹ * (g * τ * g⁻¹))
|
case intro.intro.intro
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ ∪ support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹) ∪ support (g * τ * g⁻¹))
⊢ IsConj (σ * τ) (f * σ * f⁻¹ * (g * τ * g⁻¹))
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
rw [coe_union] at *
|
case intro.intro.intro
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ ∪ support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹) ∪ support (g * τ * g⁻¹))
⊢ IsConj (σ * τ) (f * σ * f⁻¹ * (g * τ * g⁻¹))
|
case intro.intro.intro
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
⊢ IsConj (σ * τ) (f * σ * f⁻¹ * (g * τ * g⁻¹))
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
have hd1'' := disjoint_coe.2 (disjoint_iff_disjoint_support.1 hd1)
|
case intro.intro.intro
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
⊢ IsConj (σ * τ) (f * σ * f⁻¹ * (g * τ * g⁻¹))
|
case intro.intro.intro
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
⊢ IsConj (σ * τ) (f * σ * f⁻¹ * (g * τ * g⁻¹))
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
have hd2'' := disjoint_coe.2 (disjoint_iff_disjoint_support.1 hd2)
|
case intro.intro.intro
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
⊢ IsConj (σ * τ) (f * σ * f⁻¹ * (g * τ * g⁻¹))
|
case intro.intro.intro
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
⊢ IsConj (σ * τ) (f * σ * f⁻¹ * (g * τ * g⁻¹))
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
refine' isConj_of_support_equiv _ _
|
case intro.intro.intro
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
⊢ IsConj (σ * τ) (f * σ * f⁻¹ * (g * τ * g⁻¹))
|
case intro.intro.intro.refine'_1
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
⊢ { x // x ∈ ↑(support (σ * τ)) } ≃ { x // x ∈ ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) }
case intro.intro.intro.refine'_2
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
⊢ ∀ (x : α) (hx : x ∈ ↑(support (σ * τ))),
↑(↑?intro.intro.intro.refine'_1 { val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) }) =
↑(f * σ * f⁻¹ * (g * τ * g⁻¹)) ↑(↑?intro.intro.intro.refine'_1 { val := x, property := hx })
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
simp only [Set.mem_image, toEmbedding_apply, exists_eq_right, support_conj, coe_map,
apply_eq_iff_eq]
|
case intro.intro.intro.refine'_1.refine'_2
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
a : α
⊢ a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹))
|
no goals
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
intro x hx
|
case intro.intro.intro.refine'_2
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
⊢ ∀ (x : α) (hx : x ∈ ↑(support (σ * τ))),
↑(↑(((Set.ofEq hd1').trans (Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))).trans
((Equiv.sumCongr (subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹))))
(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹))))).trans
((Set.ofEq hd2').trans (Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥))).symm))
{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) }) =
↑(f * σ * f⁻¹ * (g * τ * g⁻¹))
↑(↑(((Set.ofEq hd1').trans (Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))).trans
((Equiv.sumCongr (subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹))))
(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹))))).trans
((Set.ofEq hd2').trans
(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥))).symm))
{ val := x, property := hx })
|
case intro.intro.intro.refine'_2
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
⊢ ↑(↑(((Set.ofEq hd1').trans (Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))).trans
((Equiv.sumCongr (subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹))))
(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹))))).trans
((Set.ofEq hd2').trans (Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥))).symm))
{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) }) =
↑(f * σ * f⁻¹ * (g * τ * g⁻¹))
↑(↑(((Set.ofEq hd1').trans (Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))).trans
((Equiv.sumCongr (subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹))))
(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹))))).trans
((Set.ofEq hd2').trans (Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥))).symm))
{ val := x, property := hx })
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
simp only [trans_apply, symm_trans_apply, Equiv.Set.ofEq_apply, Equiv.Set.ofEq_symm_apply,
Equiv.sumCongr_apply]
|
case intro.intro.intro.refine'_2
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
⊢ ↑(↑(((Set.ofEq hd1').trans (Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))).trans
((Equiv.sumCongr (subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹))))
(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹))))).trans
((Set.ofEq hd2').trans (Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥))).symm))
{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) }) =
↑(f * σ * f⁻¹ * (g * τ * g⁻¹))
↑(↑(((Set.ofEq hd1').trans (Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))).trans
((Equiv.sumCongr (subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹))))
(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹))))).trans
((Set.ofEq hd2').trans (Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥))).symm))
{ val := x, property := hx })
|
case intro.intro.intro.refine'_2
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))
{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) }))) =
↑(f * σ * f⁻¹ * (g * τ * g⁻¹))
↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))
{ val := x,
property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) })))
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
rw [hd1', Set.mem_union] at hx
|
case intro.intro.intro.refine'_2
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))
{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) }))) =
↑(f * σ * f⁻¹ * (g * τ * g⁻¹))
↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))
{ val := x,
property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) })))
|
case intro.intro.intro.refine'_2
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx✝ : x ∈ ↑(support (σ * τ))
hx : x ∈ ↑(support σ) ∨ x ∈ ↑(support τ)
⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))
{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) }))) =
↑(f * σ * f⁻¹ * (g * τ * g⁻¹))
↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))
{ val := x,
property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx✝ } ∈ ↑(support σ) ∪ ↑(support τ)) })))
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
cases' hx with hxσ hxτ
|
case intro.intro.intro.refine'_2
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx✝ : x ∈ ↑(support (σ * τ))
hx : x ∈ ↑(support σ) ∨ x ∈ ↑(support τ)
⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))
{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) }))) =
↑(f * σ * f⁻¹ * (g * τ * g⁻¹))
↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))
{ val := x,
property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx✝ } ∈ ↑(support σ) ∪ ↑(support τ)) })))
|
case intro.intro.intro.refine'_2.inl
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxσ : x ∈ ↑(support σ)
⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))
{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) }))) =
↑(f * σ * f⁻¹ * (g * τ * g⁻¹))
↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))
{ val := x,
property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) })))
case intro.intro.intro.refine'_2.inr
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxτ : x ∈ ↑(support τ)
⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))
{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) }))) =
↑(f * σ * f⁻¹ * (g * τ * g⁻¹))
↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))
{ val := x,
property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) })))
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
rw [mem_coe, mem_support] at hxσ
|
case intro.intro.intro.refine'_2.inl
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxσ : x ∈ ↑(support σ)
⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))
{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) }))) =
↑(f * σ * f⁻¹ * (g * τ * g⁻¹))
↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))
{ val := x,
property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) })))
|
case intro.intro.intro.refine'_2.inl
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxσ : ↑σ x ≠ x
⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))
{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) }))) =
↑(f * σ * f⁻¹ * (g * τ * g⁻¹))
↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))
{ val := x,
property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) })))
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
rw [Set.union_apply_left hd1''.le_bot _, Set.union_apply_left hd1''.le_bot _]
|
case intro.intro.intro.refine'_2.inl
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxσ : ↑σ x ≠ x
⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))
{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) }))) =
↑(f * σ * f⁻¹ * (g * τ * g⁻¹))
↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))
{ val := x,
property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) })))
|
case intro.intro.intro.refine'_2.inl
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxσ : ↑σ x ≠ x
⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(Sum.inl
{
val :=
↑{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) },
property := ?m.3053395 }))) =
↑(f * σ * f⁻¹ * (g * τ * g⁻¹))
↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(Sum.inl
{
val :=
↑{ val := x,
property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) },
property := ?m.3053769 })))
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxσ : ↑σ x ≠ x
⊢ ↑{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) } ∈
↑(support σ)
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxσ : ↑σ x ≠ x
⊢ ↑{ val := x, property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) } ∈
↑(support σ)
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxσ : ↑σ x ≠ x
⊢ ↑{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) } ∈
↑(support σ)
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
simp only [subtypeEquiv_apply, Perm.coe_mul, Sum.map_inl, comp_apply,
Set.union_symm_apply_left, Subtype.coe_mk, apply_eq_iff_eq]
|
case intro.intro.intro.refine'_2.inl
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxσ : ↑σ x ≠ x
⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(Sum.inl
{
val :=
↑{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) },
property := ?m.3053395 }))) =
↑(f * σ * f⁻¹ * (g * τ * g⁻¹))
↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(Sum.inl
{
val :=
↑{ val := x,
property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) },
property := ?m.3053769 })))
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxσ : ↑σ x ≠ x
⊢ ↑{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) } ∈
↑(support σ)
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxσ : ↑σ x ≠ x
⊢ ↑{ val := x, property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) } ∈
↑(support σ)
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxσ : ↑σ x ≠ x
⊢ ↑{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) } ∈
↑(support σ)
|
case intro.intro.intro.refine'_2.inl
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxσ : ↑σ x ≠ x
⊢ ↑τ x = ↑f⁻¹ (↑g (↑τ (↑g⁻¹ (↑f x))))
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxσ : ↑σ x ≠ x
⊢ ↑{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) } ∈
↑(support σ)
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxσ : ↑σ x ≠ x
⊢ ↑{ val := x, property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) } ∈
↑(support σ)
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxσ : ↑σ x ≠ x
⊢ ↑{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) } ∈
↑(support σ)
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
have h := (hd2 (f x)).resolve_left ?_
|
case intro.intro.intro.refine'_2.inl
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxσ : ↑σ x ≠ x
⊢ ↑τ x = ↑f⁻¹ (↑g (↑τ (↑g⁻¹ (↑f x))))
|
case intro.intro.intro.refine'_2.inl.refine_2
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxσ : ↑σ x ≠ x
h : ↑(g * τ * g⁻¹) (↑f x) = ↑f x
⊢ ↑τ x = ↑f⁻¹ (↑g (↑τ (↑g⁻¹ (↑f x))))
case intro.intro.intro.refine'_2.inl.refine_1
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxσ : ↑σ x ≠ x
⊢ ¬↑(f * σ * f⁻¹) (↑f x) = ↑f x
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
rw [mul_apply, mul_apply] at h
|
case intro.intro.intro.refine'_2.inl.refine_2
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxσ : ↑σ x ≠ x
h : ↑(g * τ * g⁻¹) (↑f x) = ↑f x
⊢ ↑τ x = ↑f⁻¹ (↑g (↑τ (↑g⁻¹ (↑f x))))
|
case intro.intro.intro.refine'_2.inl.refine_2
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxσ : ↑σ x ≠ x
h : ↑g (↑τ (↑g⁻¹ (↑f x))) = ↑f x
⊢ ↑τ x = ↑f⁻¹ (↑g (↑τ (↑g⁻¹ (↑f x))))
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
rw [h, inv_apply_self, (hd1 x).resolve_left hxσ]
|
case intro.intro.intro.refine'_2.inl.refine_2
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxσ : ↑σ x ≠ x
h : ↑g (↑τ (↑g⁻¹ (↑f x))) = ↑f x
⊢ ↑τ x = ↑f⁻¹ (↑g (↑τ (↑g⁻¹ (↑f x))))
|
no goals
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
rwa [mul_apply, mul_apply, inv_apply_self, apply_eq_iff_eq]
|
case intro.intro.intro.refine'_2.inl.refine_1
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxσ : ↑σ x ≠ x
⊢ ¬↑(f * σ * f⁻¹) (↑f x) = ↑f x
|
no goals
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
rwa [Subtype.coe_mk, Perm.mul_apply, (hd1 x).resolve_left hxσ, mem_coe,
apply_mem_support, mem_support]
|
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxσ : ↑σ x ≠ x
⊢ ↑{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) } ∈
↑(support σ)
|
no goals
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
rwa [Subtype.coe_mk, mem_coe, mem_support]
|
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxσ : ↑σ x ≠ x
⊢ ↑{ val := x, property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) } ∈
↑(support σ)
|
no goals
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
rw [mem_coe, ← apply_mem_support, mem_support] at hxτ
|
case intro.intro.intro.refine'_2.inr
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxτ : x ∈ ↑(support τ)
⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))
{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) }))) =
↑(f * σ * f⁻¹ * (g * τ * g⁻¹))
↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))
{ val := x,
property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) })))
|
case intro.intro.intro.refine'_2.inr
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxτ : ↑τ (↑τ x) ≠ ↑τ x
⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))
{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) }))) =
↑(f * σ * f⁻¹ * (g * τ * g⁻¹))
↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))
{ val := x,
property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) })))
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
rw [Set.union_apply_right hd1''.le_bot _, Set.union_apply_right hd1''.le_bot _]
|
case intro.intro.intro.refine'_2.inr
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxτ : ↑τ (↑τ x) ≠ ↑τ x
⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))
{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) }))) =
↑(f * σ * f⁻¹ * (g * τ * g⁻¹))
↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(↑(Set.union (_ : ↑(support σ) ⊓ ↑(support τ) ≤ ⊥))
{ val := x,
property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) })))
|
case intro.intro.intro.refine'_2.inr
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxτ : ↑τ (↑τ x) ≠ ↑τ x
⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(Sum.inr
{
val :=
↑{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) },
property := ?m.3057896 }))) =
↑(f * σ * f⁻¹ * (g * τ * g⁻¹))
↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(Sum.inr
{
val :=
↑{ val := x,
property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) },
property := ?m.3058005 })))
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxτ : ↑τ (↑τ x) ≠ ↑τ x
⊢ ↑{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) } ∈
↑(support τ)
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxτ : ↑τ (↑τ x) ≠ ↑τ x
⊢ ↑{ val := x, property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) } ∈
↑(support τ)
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxτ : ↑τ (↑τ x) ≠ ↑τ x
⊢ ↑{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) } ∈
↑(support τ)
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
simp only [subtypeEquiv_apply, Perm.coe_mul, Sum.map_inr, comp_apply,
Set.union_symm_apply_right, Subtype.coe_mk, apply_eq_iff_eq]
|
case intro.intro.intro.refine'_2.inr
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxτ : ↑τ (↑τ x) ≠ ↑τ x
⊢ ↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(Sum.inr
{
val :=
↑{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) },
property := ?m.3057896 }))) =
↑(f * σ * f⁻¹ * (g * τ * g⁻¹))
↑(↑(Set.union (_ : ↑(support (f * σ * f⁻¹)) ⊓ ↑(support (g * τ * g⁻¹)) ≤ ⊥)).symm
(Sum.map (↑(subtypeEquiv f (_ : ∀ (a : α), a ∈ ↑(support σ) ↔ ↑f a ∈ ↑(support (f * σ * f⁻¹)))))
(↑(subtypeEquiv g (_ : ∀ (a : α), a ∈ ↑(support τ) ↔ ↑g a ∈ ↑(support (g * τ * g⁻¹)))))
(Sum.inr
{
val :=
↑{ val := x,
property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) },
property := ?m.3058005 })))
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxτ : ↑τ (↑τ x) ≠ ↑τ x
⊢ ↑{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) } ∈
↑(support τ)
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxτ : ↑τ (↑τ x) ≠ ↑τ x
⊢ ↑{ val := x, property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) } ∈
↑(support τ)
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxτ : ↑τ (↑τ x) ≠ ↑τ x
⊢ ↑{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) } ∈
↑(support τ)
|
case intro.intro.intro.refine'_2.inr
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxτ : ↑τ (↑τ x) ≠ ↑τ x
⊢ ↑g (↑σ (↑τ x)) = ↑f (↑σ (↑f⁻¹ (↑g (↑τ (↑g⁻¹ (↑g x))))))
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxτ : ↑τ (↑τ x) ≠ ↑τ x
⊢ ↑{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) } ∈
↑(support τ)
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxτ : ↑τ (↑τ x) ≠ ↑τ x
⊢ ↑{ val := x, property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) } ∈
↑(support τ)
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxτ : ↑τ (↑τ x) ≠ ↑τ x
⊢ ↑{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) } ∈
↑(support τ)
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
have h := (hd2 (g (τ x))).resolve_right ?_
|
case intro.intro.intro.refine'_2.inr
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxτ : ↑τ (↑τ x) ≠ ↑τ x
⊢ ↑g (↑σ (↑τ x)) = ↑f (↑σ (↑f⁻¹ (↑g (↑τ (↑g⁻¹ (↑g x))))))
|
case intro.intro.intro.refine'_2.inr.refine_2
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxτ : ↑τ (↑τ x) ≠ ↑τ x
h : ↑(f * σ * f⁻¹) (↑g (↑τ x)) = ↑g (↑τ x)
⊢ ↑g (↑σ (↑τ x)) = ↑f (↑σ (↑f⁻¹ (↑g (↑τ (↑g⁻¹ (↑g x))))))
case intro.intro.intro.refine'_2.inr.refine_1
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxτ : ↑τ (↑τ x) ≠ ↑τ x
⊢ ¬↑(g * τ * g⁻¹) (↑g (↑τ x)) = ↑g (↑τ x)
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
rw [mul_apply, mul_apply] at h
|
case intro.intro.intro.refine'_2.inr.refine_2
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxτ : ↑τ (↑τ x) ≠ ↑τ x
h : ↑(f * σ * f⁻¹) (↑g (↑τ x)) = ↑g (↑τ x)
⊢ ↑g (↑σ (↑τ x)) = ↑f (↑σ (↑f⁻¹ (↑g (↑τ (↑g⁻¹ (↑g x))))))
|
case intro.intro.intro.refine'_2.inr.refine_2
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxτ : ↑τ (↑τ x) ≠ ↑τ x
h : ↑f (↑σ (↑f⁻¹ (↑g (↑τ x)))) = ↑g (↑τ x)
⊢ ↑g (↑σ (↑τ x)) = ↑f (↑σ (↑f⁻¹ (↑g (↑τ (↑g⁻¹ (↑g x))))))
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
rw [inv_apply_self, h, (hd1 (τ x)).resolve_right hxτ]
|
case intro.intro.intro.refine'_2.inr.refine_2
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxτ : ↑τ (↑τ x) ≠ ↑τ x
h : ↑f (↑σ (↑f⁻¹ (↑g (↑τ x)))) = ↑g (↑τ x)
⊢ ↑g (↑σ (↑τ x)) = ↑f (↑σ (↑f⁻¹ (↑g (↑τ (↑g⁻¹ (↑g x))))))
|
no goals
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
rwa [mul_apply, mul_apply, inv_apply_self, apply_eq_iff_eq]
|
case intro.intro.intro.refine'_2.inr.refine_1
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxτ : ↑τ (↑τ x) ≠ ↑τ x
⊢ ¬↑(g * τ * g⁻¹) (↑g (↑τ x)) = ↑g (↑τ x)
|
no goals
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
rwa [Subtype.coe_mk, Perm.mul_apply, (hd1 (τ x)).resolve_right hxτ,
mem_coe, mem_support]
|
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxτ : ↑τ (↑τ x) ≠ ↑τ x
⊢ ↑{ val := ↑(σ * τ) x,
property :=
(_ :
↑(Equiv.refl α) ↑{ val := ↑(σ * τ) x, property := (_ : ↑(σ * τ) x ∈ support (σ * τ)) } ∈
↑(support σ) ∪ ↑(support τ)) } ∈
↑(support τ)
|
no goals
|
null
|
null
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.Disjoint.isConj_mul
|
[1771, 1]
|
[1815, 74]
|
rwa [Subtype.coe_mk, mem_coe, ← apply_mem_support, mem_support]
|
ι : Type ?u.3042712
α✝ : Type ?u.3042715
β : Type ?u.3042718
inst✝² : DecidableEq α✝
inst✝¹ : Fintype α✝
σ✝ τ✝ : Perm α✝
α : Type u_1
inst✝ : Finite α
σ τ : Perm α
hd1 : Disjoint σ τ
val✝ : Fintype α
f : Perm α
hc1 : IsConj σ (f * σ * f⁻¹)
g : Perm α
hc2 : IsConj τ (g * τ * g⁻¹)
hd2 : Disjoint (f * σ * f⁻¹) (g * τ * g⁻¹)
hd1' : ↑(support (σ * τ)) = ↑(support σ) ∪ ↑(support τ)
hd2' : ↑(support (f * σ * f⁻¹ * (g * τ * g⁻¹))) = ↑(support (f * σ * f⁻¹)) ∪ ↑(support (g * τ * g⁻¹))
hd1'' : _root_.Disjoint ↑(support σ) ↑(support τ)
hd2'' : _root_.Disjoint ↑(support (f * σ * f⁻¹)) ↑(support (g * τ * g⁻¹))
x : α
hx : x ∈ ↑(support (σ * τ))
hxτ : ↑τ (↑τ x) ≠ ↑τ x
⊢ ↑{ val := x, property := (_ : ↑(Equiv.refl α) ↑{ val := x, property := hx } ∈ ↑(support σ) ∪ ↑(support τ)) } ∈
↑(support τ)
|
no goals
|
null
|
null
|
Mathlib/MeasureTheory/Group/Pointwise.lean
|
MeasurableSet.const_smul₀
|
[42, 1]
|
[47, 85]
|
rcases eq_or_ne a 0 with (rfl | ha)
|
G₀ : Type u_1
α : Type u_2
inst✝⁶ : GroupWithZero G₀
inst✝⁵ : Zero α
inst✝⁴ : MulActionWithZero G₀ α
inst✝³ : MeasurableSpace G₀
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSMul G₀ α
inst✝ : MeasurableSingletonClass α
s : Set α
hs : MeasurableSet s
a : G₀
⊢ MeasurableSet (a • s)
|
case inl
G₀ : Type u_1
α : Type u_2
inst✝⁶ : GroupWithZero G₀
inst✝⁵ : Zero α
inst✝⁴ : MulActionWithZero G₀ α
inst✝³ : MeasurableSpace G₀
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSMul G₀ α
inst✝ : MeasurableSingletonClass α
s : Set α
hs : MeasurableSet s
⊢ MeasurableSet (0 • s)
case inr
G₀ : Type u_1
α : Type u_2
inst✝⁶ : GroupWithZero G₀
inst✝⁵ : Zero α
inst✝⁴ : MulActionWithZero G₀ α
inst✝³ : MeasurableSpace G₀
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSMul G₀ α
inst✝ : MeasurableSingletonClass α
s : Set α
hs : MeasurableSet s
a : G₀
ha : a ≠ 0
⊢ MeasurableSet (a • s)
|
null
|
null
|
Mathlib/MeasureTheory/Group/Pointwise.lean
|
MeasurableSet.const_smul₀
|
[42, 1]
|
[47, 85]
|
exacts [(subsingleton_zero_smul_set s).measurableSet, hs.const_smul_of_ne_zero ha]
|
case inl
G₀ : Type u_1
α : Type u_2
inst✝⁶ : GroupWithZero G₀
inst✝⁵ : Zero α
inst✝⁴ : MulActionWithZero G₀ α
inst✝³ : MeasurableSpace G₀
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSMul G₀ α
inst✝ : MeasurableSingletonClass α
s : Set α
hs : MeasurableSet s
⊢ MeasurableSet (0 • s)
case inr
G₀ : Type u_1
α : Type u_2
inst✝⁶ : GroupWithZero G₀
inst✝⁵ : Zero α
inst✝⁴ : MulActionWithZero G₀ α
inst✝³ : MeasurableSpace G₀
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSMul G₀ α
inst✝ : MeasurableSingletonClass α
s : Set α
hs : MeasurableSet s
a : G₀
ha : a ≠ 0
⊢ MeasurableSet (a • s)
|
no goals
|
null
|
null
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean
|
Real.range_arcsin
|
[46, 1]
|
[48, 13]
|
rw [arcsin, range_comp Subtype.val]
|
⊢ range arcsin = Icc (-(π / 2)) (π / 2)
|
⊢ Subtype.val '' range (IccExtend arcsin.proof_2 ↑(OrderIso.symm sinOrderIso)) = Icc (-(π / 2)) (π / 2)
|
null
|
null
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean
|
Real.range_arcsin
|
[46, 1]
|
[48, 13]
|
simp [Icc]
|
⊢ Subtype.val '' range (IccExtend arcsin.proof_2 ↑(OrderIso.symm sinOrderIso)) = Icc (-(π / 2)) (π / 2)
|
no goals
|
null
|
null
|
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.tendsto_Iio_atBot
|
[1600, 1]
|
[1602, 44]
|
rw [atBot_Iio_eq, tendsto_comap_iff]
|
ι : Type ?u.334826
ι' : Type ?u.334829
α : Type u_1
β : Type u_2
γ : Type ?u.334838
inst✝ : SemilatticeInf α
a : α
f : β → ↑(Iio a)
l : Filter β
⊢ Tendsto f l atBot ↔ Tendsto (fun x => ↑(f x)) l atBot
|
ι : Type ?u.334826
ι' : Type ?u.334829
α : Type u_1
β : Type u_2
γ : Type ?u.334838
inst✝ : SemilatticeInf α
a : α
f : β → ↑(Iio a)
l : Filter β
⊢ Tendsto (Subtype.val ∘ f) l atBot ↔ Tendsto (fun x => ↑(f x)) l atBot
|
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