mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.GroupTheory.SpecificGroups.Cyclic

Goal state before:
pp: α : Type u_1
inst✝² : Group α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hn : ∀ (n : ℕ), 0 < n → #{a | a ^ n = 1} ≤ n
d : ℕ
IH : ∀ m < d, m ∣ Fintype.card α → 0 < #{a | orderOf a = m} → #{a | orderOf a = m} = φ m
hd : d ∣ Fintype.card α
hpos : 0 < #{a | orderOf a = d}
hd0 : d ≠ 0
a : α
ha' : a ∈ {a | orderOf a = d}
ha : orderOf a = d
m : ℕ
hm : m ∣ d ∧ m < d
⊢ m = m

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝ : Group G
Hs : Set (Subgroup G)
h : ∀ H ∈ Hs, IsPGroup p ↥H
hn : ∀ H ∈ Hs, H.Normal
⊢ IsPGroup p ↥(⨆ a ∈ Hs, a)

Tactic:
]

Goal state after:
p : ℕ
G : Type u_1
inst✝ : Group G
Hs : Set (Subgroup G)
h : ∀ H ∈ Hs, IsPGroup p ↥H
hn : ∀ H ∈ Hs, H.Normal
⊢ IsPGroup p ↥(⨆ a ∈ Hs, a)
Module: Mathlib.Algebra.Module.ZMod

Goal state before:
pp: n : ℕ
M : Type u_1
S : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module (ZMod n) M
inst✝¹ : SetLike S M
inst✝ : AddSubgroupClass S M
x : M
K : S
hx : x ∈ K
c : ZMod n
⊢ c.cast • x ∈ K

Tactic:
]

Goal state after:
n : ℕ
M : Type u_1
S : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module (ZMod n) M
inst✝¹ : SetLike S M
inst✝ : AddSubgroupClass S M
x : M
K : S
hx : x ∈ K
c : ZMod n
⊢ c.cast • x ∈ K
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝² : Group G
H : Type u_2
inst✝¹ : Group H
f : H →* G
hf : IsPGroup p ↥f.ker
inst✝ : Finite (Sylow p G)
h_exists : ∀ (P : Sylow p H), ∃ Q, comap f ↑Q = ↑P := fun P ↦ exists_comap_eq_of_ker_isPGroup P hf
g : Sylow p H → Sylow p G := fun P ↦ Classical.choose ⋯
hg : ∀ (P : Sylow p H), comap f ↑(g P) = ↑P
P Q : Sylow p H
h : g P = g Q
⊢ comap f ↑(g Q) = ↑Q

Tactic:
]

Goal state after:
p : ℕ
G : Type u_1
inst✝² : Group G
H : Type u_2
inst✝¹ : Group H
f : H →* G
hf : IsPGroup p ↥f.ker
inst✝ : Finite (Sylow p G)
h_exists : ∀ (P : Sylow p H), ∃ Q, comap f ↑Q = ↑P := fun P ↦ exists_comap_eq_of_ker_isPGroup P hf
g : Sylow p H → Sylow p G := fun P ↦ Classical.choose ⋯
hg : ∀ (P : Sylow p H), comap f ↑(g P) = ↑P
P Q : Sylow p H
h : g P = g Q
⊢ comap f ↑(g Q) = ↑Q
Module: Mathlib.GroupTheory.SpecificGroups.Cyclic

Goal state before:
pp: α : Type u_1
inst✝² : Group α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hn : ∀ (n : ℕ), 0 < n → #{a | a ^ n = 1} ≤ n
d : ℕ
IH : ∀ m < d, m ∣ Fintype.card α → 0 < #{a | orderOf a = m} → #{a | orderOf a = m} = φ m
hd : d ∣ Fintype.card α
hpos : 0 < #{a | orderOf a = d}
hd0 : d ≠ 0
a : α
ha' : a ∈ {a | orderOf a = d}
ha : orderOf a = d
h1 : ∑ m ∈ d.properDivisors, #{a | orderOf a = m} = ∑ m ∈ d.properDivisors, φ m
⊢ orderOf a = orderOf a

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝¹ : Group G
P : Sylow p G
inst✝ : (↑P).FiniteIndex
⊢ IsPGroup p ↥(↑P).normalCore

Tactic:
]

Goal state after:
p : ℕ
G : Type u_1
inst✝¹ : Group G
P : Sylow p G
inst✝ : (↑P).FiniteIndex
⊢ IsPGroup p ↥(↑P).normalCore
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝ : Group G
g : G
P : Sylow p G
⊢ (∀ (x : G), x ∈ P ↔ x ∈ g • P) ↔ ∀ (h : G), h ∈ ↑P ↔ g⁻¹ * h * g ∈ ↑P

Tactic:
]

Goal state after:
p : ℕ
G : Type u_1
inst✝ : Group G
g : G
P : Sylow p G
⊢ (∀ (x : G), x ∈ P ↔ x ∈ g • P) ↔ ∀ (h : G), h ∈ ↑P ↔ g⁻¹ * h * g ∈ ↑P
Module: Mathlib.GroupTheory.Rank

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : Subsingleton G
⊢ rank G ≤ ∅.card

pp: G : Type u_1
inst✝¹ : Group G
inst✝ : Subsingleton G
⊢ Type ?u.15

Tactic:
]

Goal state after:
G : Type u_1
inst✝¹ : Group G
inst✝ : Subsingleton G
⊢ rank G ≤ ∅.card

G : Type u_1
inst✝¹ : Group G
inst✝ : Subsingleton G
⊢ Type ?u.15
Module: Mathlib.GroupTheory.Rank

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : FG G
h : rank G = 0
s : Finset G
hs : s = ∅
hs' : Subgroup.closure ↑s = ⊤
⊢ Subsingleton G

Tactic:
]

Goal state after:
G : Type u_1
inst✝¹ : Group G
inst✝ : FG G
h : rank G = 0
s : Finset G
hs : s = ∅
hs' : Subgroup.closure ↑s = ⊤
⊢ Subsingleton G
Module: Mathlib.GroupTheory.Rank

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : Nontrivial G
inst✝ : FG G
⊢ Nontrivial G

Tactic:
]

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : Nontrivial G
inst✝ : FG G
⊢ Nontrivial G
Module: Mathlib.GroupTheory.Rank

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝³ : Group G
inst✝² : Group H
inst✝¹ : FG G
inst✝ : FG H
f : G →* H
hf : Surjective ⇑f
S : Finset G
hS1 : S.card = rank G
hS2 : Subgroup.closure ↑S = ⊤
⊢ ⊤ = ⊤

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.Rank

Goal state before:
pp: G : Type u_1
inst✝ : Group G
s : Finset G
t : Finset ↥(closure ↑s) := s.preimage Subtype.val ⋯
⊢ closure (Subtype.val ⁻¹' ↑s) = ⊤

Tactic:
]

Goal state after:
G : Type u_1
inst✝ : Group G
s : Finset G
t : Finset ↥(closure ↑s) := s.preimage Subtype.val ⋯
⊢ closure (Subtype.val ⁻¹' ↑s) = ⊤
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: case refine_2
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
h : IsClosedMap (comap (mapRingHom f))
algInst✝ : Algebra R S := f.toAlgebra
a✝¹ : Nontrivial R
a✝ : Nontrivial S
r : S
p : S[X] := C r * X - 1
this✝ : 1 ∈ Ideal.span {X} ⊔ Ideal.comap (mapRingHom f) (Ideal.span {p})
a b : R[X]
hb : b ∈ Ideal.comap (mapRingHom f) (Ideal.span {p})
e : a * X + b = 1
c : S[X]
hc : map (algebraMap R S) b = p * c
this : p.natDegree ≤ 1
⊢ eval r (reflect b.natDegree (C r) * reflect 1 X - reflect (b.natDegree + 1) 1) * eval r (reflect c.natDegree c) = 0

Tactic:
]

Goal state after:
case refine_2
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
h : IsClosedMap (comap (mapRingHom f))
algInst✝ : Algebra R S := f.toAlgebra
a✝¹ : Nontrivial R
a✝ : Nontrivial S
r : S
p : S[X] := C r * X - 1
this✝ : 1 ∈ Ideal.span {X} ⊔ Ideal.comap (mapRingHom f) (Ideal.span {p})
a b : R[X]
hb : b ∈ Ideal.comap (mapRingHom f) (Ideal.span {p})
e : a * X + b = 1
c : S[X]
hc : map (algebraMap R S) b = p * c
this : p.natDegree ≤ 1
⊢ eval r (reflect b.natDegree (C r) * reflect 1 X - reflect (b.natDegree + 1) 1) * eval r (reflect c.natDegree c) = 0
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝ : Group G
P : Subgroup G
hP : IsPGroup p ↥P
Q : Sylow p G
⊢ P ≤ ↑Q ↔ P ≤ ↑Q

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.Rank

Goal state before:
pp: G : Type u_1
inst✝ : Group G
s : Finset G
t : Finset ↥(closure ↑s) := s.preimage Subtype.val ⋯
ht : closure ↑t = ⊤
H : Set.InjOn Subtype.val ↑t
⊢ {x ∈ s | x ∈ Set.range Subtype.val}.card ≤ s.card

Tactic:
]

Goal state after:
G : Type u_1
inst✝ : Group G
s : Finset G
t : Finset ↥(closure ↑s) := s.preimage Subtype.val ⋯
ht : closure ↑t = ⊤
H : Set.InjOn Subtype.val ↑t
⊢ {x ∈ s | x ∈ Set.range Subtype.val}.card ≤ s.card
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
this : (fixedPoints ↥↑Q ↑(orbit G P)).Nonempty
R : ↑(orbit G P)
hR : R ∈ fixedPoints ↥↑Q ↑(orbit G P)
⊢ ∃ g, g • P = ↑R

Tactic:
]

Goal state after:
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
this : (fixedPoints ↥↑Q ↑(orbit G P)).Nonempty
R : ↑(orbit G P)
hR : R ∈ fixedPoints ↥↑Q ↑(orbit G P)
⊢ ∃ g, g • P = ↑R
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u
S : Type v
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
I : Ideal R
e : ↑{p | p.IsPrime ∧ I ≤ p} ≃o ↑(zeroLocus ↑I) :=
  { toFun := fun x ↦ ⟨{ asIdeal := ↑x, isPrime := ⋯ }, ⋯⟩, invFun := fun x ↦ ⟨(↑x).asIdeal, ⋯⟩, left_inv := ⋯,
    right_inv := ⋯, map_rel_iff' := ⋯ }
⊢ ↑I.minimalPrimes ≃o (↑{s | Maximal (fun x ↦ IsClosed x ∧ IsIrreducible x) s})ᵒᵈ

Tactic:
]

Goal state after:
R : Type u
S : Type v
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
I : Ideal R
e : ↑{p | p.IsPrime ∧ I ≤ p} ≃o ↑(zeroLocus ↑I) :=
  { toFun := fun x ↦ ⟨{ asIdeal := ↑x, isPrime := ⋯ }, ⋯⟩, invFun := fun x ↦ ⟨(↑x).asIdeal, ⋯⟩, left_inv := ⋯,
    right_inv := ⋯, map_rel_iff' := ⋯ }
⊢ ↑I.minimalPrimes ≃o (↑{s | Maximal (fun x ↦ IsClosed x ∧ IsIrreducible x) s})ᵒᵈ
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u
S : Type v
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
e : ↑{p | p.IsPrime ∧ ⊥ ≤ p} ≃o PrimeSpectrum R :=
  { toFun := fun x ↦ { asIdeal := ↑x, isPrime := ⋯ }, invFun := fun x ↦ ⟨x.asIdeal, ⋯⟩, left_inv := ⋯, right_inv := ⋯,
    map_rel_iff' := ⋯ }
⊢ ↑(minimalPrimes R) ≃o (↑{s | Maximal (fun x ↦ IsClosed x ∧ IsIrreducible x) s})ᵒᵈ

Tactic:
]

Goal state after:
R : Type u
S : Type v
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
e : ↑{p | p.IsPrime ∧ ⊥ ≤ p} ≃o PrimeSpectrum R :=
  { toFun := fun x ↦ { asIdeal := ↑x, isPrime := ⋯ }, invFun := fun x ↦ ⟨x.asIdeal, ⋯⟩, left_inv := ⋯, right_inv := ⋯,
    map_rel_iff' := ⋯ }
⊢ ↑(minimalPrimes R) ≃o (↑{s | Maximal (fun x ↦ IsClosed x ∧ IsIrreducible x) s})ᵒᵈ
Module: Mathlib.GroupTheory.Rank

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
s : Set G
inst✝ : Finite ↑s
this : Fintype ↑s
⊢ rank ↥(closure ↑s.toFinset) ≤ s.toFinset.card

Tactic:
]

Goal state after:
G : Type u_1
inst✝¹ : Group G
s : Set G
inst✝ : Finite ↑s
this : Fintype ↑s
⊢ rank ↥(closure ↑s.toFinset) ≤ s.toFinset.card
Module: Mathlib.GroupTheory.SpecificGroups.Cyclic

Goal state before:
pp: α : Type u_1
inst✝² : Group α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hn : ∀ (n : ℕ), 0 < n → #{a | a ^ n = 1} ≤ n
d : ℕ
hd : d ∣ Fintype.card α
c : ℕ := Fintype.card α
hc0 : 0 < c
h0 : {a | orderOf a = d} = ∅
⊢ ∑ m ∈ c.divisors.erase d, #{a | orderOf a = m} = ∑ m ∈ c.divisors, #{a | orderOf a = m}

Tactic:
]

Goal state after:
α : Type u_1
inst✝² : Group α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hn : ∀ (n : ℕ), 0 < n → #{a | a ^ n = 1} ≤ n
d : ℕ
hd : d ∣ Fintype.card α
c : ℕ := Fintype.card α
hc0 : 0 < c
h0 : {a | orderOf a = d} = ∅
⊢ ∑ m ∈ c.divisors.erase d, #{a | orderOf a = m} = ∑ m ∈ c.divisors, #{a | orderOf a = m}
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
⊢ {x | Minimal (fun x ↦ ∃ x₀, (IsClosed x₀ ∧ IsIrreducible x₀) ∧ vanishingIdeal x₀ = x) x} =
    {x | Minimal (Membership.mem (vanishingIdeal '' {s | IsClosed s ∧ IsIrreducible s})) x}

Tactic:
]

Goal state after:
R : Type u
inst✝ : CommSemiring R
⊢ {x | Minimal (fun x ↦ ∃ x₀, (IsClosed x₀ ∧ IsIrreducible x₀) ∧ vanishingIdeal x₀ = x) x} =
    {x | Minimal (Membership.mem (vanishingIdeal '' {s | IsClosed s ∧ IsIrreducible s})) x}
Module: Mathlib.GroupTheory.Rank

Goal state before:
pp: G : Type u_1
inst✝ : Group G
g : G
⊢ Nat.card ↥(comap ConjAct.toConjAct.toMonoidHom (MulAction.stabilizer (ConjAct G) g)) =
    Nat.card ↥(MulAction.stabilizer (ConjAct G) g)

Tactic:
]

Goal state after:
G : Type u_1
inst✝ : Group G
g : G
⊢ Nat.card ↥(comap ConjAct.toConjAct.toMonoidHom (MulAction.stabilizer (ConjAct G) g)) =
    Nat.card ↥(MulAction.stabilizer (ConjAct G) g)
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
⊢ irreducibleComponents (PrimeSpectrum R) = irreducibleComponents (PrimeSpectrum R)

pp: R : Type u
inst✝ : CommSemiring R
⊢ Set.EqOn ((zeroLocus ∘ SetLike.coe) ∘ vanishingIdeal) id (irreducibleComponents (PrimeSpectrum R))

Tactic:
]

Goal state after:
R : Type u
inst✝ : CommSemiring R
⊢ Set.EqOn ((zeroLocus ∘ SetLike.coe) ∘ vanishingIdeal) id (irreducibleComponents (PrimeSpectrum R))
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: case mp
R : Type u
inst✝ : CommSemiring R
s : Set (PrimeSpectrum R)
⊢ vanishingIdeal s ∈ minimalPrimes R → zeroLocus ↑(vanishingIdeal s) ∈ zeroLocus ∘ SetLike.coe '' minimalPrimes R

Tactic:
]

Goal state after:
case mp
R : Type u
inst✝ : CommSemiring R
s : Set (PrimeSpectrum R)
⊢ vanishingIdeal s ∈ minimalPrimes R → zeroLocus ↑(vanishingIdeal s) ∈ zeroLocus ∘ SetLike.coe '' minimalPrimes R
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: case mpr
R : Type u
inst✝ : CommSemiring R
s : Set (PrimeSpectrum R)
⊢ closure s ∈ irreducibleComponents (PrimeSpectrum R) →
    vanishingIdeal (closure s) ∈ vanishingIdeal '' irreducibleComponents (PrimeSpectrum R)

Tactic:
]

Goal state after:
case mpr
R : Type u
inst✝ : CommSemiring R
s : Set (PrimeSpectrum R)
⊢ closure s ∈ irreducibleComponents (PrimeSpectrum R) →
    vanishingIdeal (closure s) ∈ vanishingIdeal '' irreducibleComponents (PrimeSpectrum R)
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
I : Ideal R
⊢ zeroLocus ↑I ∈ irreducibleComponents (PrimeSpectrum R) ↔ vanishingIdeal (zeroLocus ↑I) ∈ minimalPrimes R

Tactic:
]

Goal state after:
R : Type u
inst✝ : CommSemiring R
I : Ideal R
⊢ zeroLocus ↑I ∈ irreducibleComponents (PrimeSpectrum R) ↔ vanishingIdeal (zeroLocus ↑I) ∈ minimalPrimes R
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
h : p ∣ Nat.card ↑(orbit G P)
⊢ Nat.card ↑(fixedPoints ↥↑P ↑(orbit G P)) = Nat.card ↑{⟨P, ⋯⟩}

Tactic:
]

Goal state after:
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
h : p ∣ Nat.card ↑(orbit G P)
⊢ Nat.card ↑(fixedPoints ↥↑P ↑(orbit G P)) = Nat.card ↑{⟨P, ⋯⟩}
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u
inst✝³ : CommSemiring R
inst✝² : IsLocalRing R
S : Type v
inst✝¹ : CommSemiring S
inst✝ : IsLocalRing S
f : R →+* S
this : IsLocalHom f ↔ Ideal.comap f (maximalIdeal S) = maximalIdeal R
⊢ Ideal.comap f (maximalIdeal S) = maximalIdeal R ↔
    (PrimeSpectrum.comap f (closedPoint S)).asIdeal = (closedPoint R).asIdeal

Tactic:
]

Goal state after:
R : Type u
inst✝³ : CommSemiring R
inst✝² : IsLocalRing R
S : Type v
inst✝¹ : CommSemiring S
inst✝ : IsLocalRing S
f : R →+* S
this : IsLocalHom f ↔ Ideal.comap f (maximalIdeal S) = maximalIdeal R
⊢ Ideal.comap f (maximalIdeal S) = maximalIdeal R ↔
    (PrimeSpectrum.comap f (closedPoint S)).asIdeal = (closedPoint R).asIdeal
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: case mp
R : Type u
inst✝¹ : CommSemiring R
inst✝ : IsLocalRing R
U : TopologicalSpace.Opens (PrimeSpectrum R)
⊢ closedPoint R ∈ U → ⊤ ≤ U

Tactic:
]

Goal state after:
case mp
R : Type u
inst✝¹ : CommSemiring R
inst✝ : IsLocalRing R
U : TopologicalSpace.Opens (PrimeSpectrum R)
⊢ closedPoint R ∈ U → ⊤ ≤ U
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
h : p ∣ Nat.card ↑(orbit G P)
⊢ ⟨P, ⋯⟩ ∈ fixedPoints ↥↑P ↑(orbit G P) ∧ ∀ x ∈ fixedPoints ↥↑P ↑(orbit G P), x = ⟨P, ⋯⟩

Tactic:
]

Goal state after:
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
h : p ∣ Nat.card ↑(orbit G P)
⊢ ⟨P, ⋯⟩ ∈ fixedPoints ↥↑P ↑(orbit G P) ∧ ∀ x ∈ fixedPoints ↥↑P ↑(orbit G P), x = ⟨P, ⋯⟩
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: T : Type u
inst✝¹ : CommRing T
inst✝ : IsLocalRing T
x : PrimeSpectrum (ResidueField T)
⊢ PrimeSpectrum.comap (residue T) ⊥ = closedPoint T

Tactic:
]

Goal state after:
T : Type u
inst✝¹ : CommRing T
inst✝ : IsLocalRing T
x : PrimeSpectrum (ResidueField T)
⊢ PrimeSpectrum.comap (residue T) ⊥ = closedPoint T
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u
inst✝¹ : CommSemiring R
inst✝ : IsLocalRing R
⊢ { asIdeal := maximalIdeal R, isPrime := ⋯ }.asIdeal.IsMaximal

Tactic:
]

Goal state after:
R : Type u
inst✝¹ : CommSemiring R
inst✝ : IsLocalRing R
⊢ { asIdeal := maximalIdeal R, isPrime := ⋯ }.asIdeal.IsMaximal
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u
inst✝ : CommRing R
e : R
he : IsIdempotentElem e
⊢ zeroLocus {e} = zeroLocus {e}

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u
inst✝ : CommRing R
e : R
he : IsIdempotentElem e
⊢ IsClosed (zeroLocus {1 - e})

Tactic:
]

Goal state after:
R : Type u
inst✝ : CommRing R
e : R
he : IsIdempotentElem e
⊢ IsClosed (zeroLocus {1 - e})
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
this✝ : fixedPoints (↥↑P) (Sylow p G) = {P}
this : Nat.card ↑(fixedPoints (↥↑P) (Sylow p G)) = 1
⊢ 1 ≡ 1 [MOD p]

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
⊢ ↥↑P ≃* ↥↑(Classical.choose ⋯ • P)

Tactic:
]

Goal state after:
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
⊢ ↥↑P ≃* ↥↑(Classical.choose ⋯ • P)
Module: Mathlib.GroupTheory.SpecificGroups.Cyclic

Goal state before:
pp: α : Type u_1
inst✝² : Group α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hn : ∀ (n : ℕ), 0 < n → #{a | a ^ n = 1} ≤ n
⊢ 0 < Fintype.card α

Tactic:
]

Goal state after:
α : Type u_1
inst✝² : Group α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hn : ∀ (n : ℕ), 0 < n → #{a | a ^ n = 1} ≤ n
⊢ 0 < Fintype.card α
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
hy : g⁻¹ * x * g ∈ centralizer ↑P
⊢ x ∈ centralizer ↑↑P

Tactic:
]

Goal state after:
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
hy : g⁻¹ * x * g ∈ centralizer ↑P
⊢ x ∈ centralizer ↑↑P
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
hy : g⁻¹ * x * g ∈ centralizer ↑P
h1 : ↑P ≤ centralizer ↑(zpowers x)
⊢ x ∈ centralizer ↑↑(g • P)

Tactic:
]

Goal state after:
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
hy : g⁻¹ * x * g ∈ centralizer ↑P
h1 : ↑P ≤ centralizer ↑(zpowers x)
⊢ x ∈ centralizer ↑↑(g • P)
Module: Mathlib.GroupTheory.SpecificGroups.Cyclic

Goal state before:
pp: G : Type u_2
G' : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : IsCyclic G'
f : G →* G'
hf : f.ker ≤ center G
a b : G
x : G'
y : G
hxy : f y = x
hx : ∀ (a : ↥f.range), a ∈ zpowers ⟨x, ⋯⟩
m : ℤ
hm✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) m = ⟨f a, ⋯⟩
n : ℤ
hn✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) n = ⟨f b, ⋯⟩
hm : x ^ m = f a
hn : x ^ n = f b
⊢ (x ^ m)⁻¹ * f a = 1

Tactic:
hm,

Goal state after:
G : Type u_2
G' : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : IsCyclic G'
f : G →* G'
hf : f.ker ≤ center G
a b : G
x : G'
y : G
hxy : f y = x
hx : ∀ (a : ↥f.range), a ∈ zpowers ⟨x, ⋯⟩
m : ℤ
hm✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) m = ⟨f a, ⋯⟩
n : ℤ
hn✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) n = ⟨f b, ⋯⟩
hm : x ^ m = f a
hn : x ^ n = f b
⊢ (f a)⁻¹ * f a = 1
Module: Mathlib.GroupTheory.SpecificGroups.Cyclic

Goal state before:
pp: G : Type u_2
G' : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : IsCyclic G'
f : G →* G'
hf : f.ker ≤ center G
a b : G
x : G'
y : G
hxy : f y = x
hx : ∀ (a : ↥f.range), a ∈ zpowers ⟨x, ⋯⟩
m : ℤ
hm✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) m = ⟨f a, ⋯⟩
n : ℤ
hn✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) n = ⟨f b, ⋯⟩
hm : x ^ m = f a
hn : x ^ n = f b
⊢ 1 = 1

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.SpecificGroups.Cyclic

Goal state before:
pp: G : Type u_2
G' : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : IsCyclic G'
f : G →* G'
hf : f.ker ≤ center G
a b : G
x : G'
y : G
hxy : f y = x
hx : ∀ (a : ↥f.range), a ∈ zpowers ⟨x, ⋯⟩
m : ℤ
hm✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) m = ⟨f a, ⋯⟩
n : ℤ
hn✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) n = ⟨f b, ⋯⟩
hm : x ^ m = f a
hn : x ^ n = f b
ha : y ^ (-m) * a ∈ center G
⊢ 1 = 1

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
h1 : ↑P ≤ centralizer ↑(zpowers x)
z : G
hy : g * z * g⁻¹ * x = x * (g * z * g⁻¹)
hz : z ∈ ↑↑P
⊢ ((MulDistribMulAction.toMonoidEnd (MulAut G) G) (MulAut.conj g)) z * x =
    x * ((MulDistribMulAction.toMonoidEnd (MulAut G) G) (MulAut.conj g)) z

Tactic:
]

Goal state after:
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
h1 : ↑P ≤ centralizer ↑(zpowers x)
z : G
hy : g * z * g⁻¹ * x = x * (g * z * g⁻¹)
hz : z ∈ ↑↑P
⊢ ((MulDistribMulAction.toMonoidEnd (MulAut G) G) (MulAut.conj g)) z * x =
    x * ((MulDistribMulAction.toMonoidEnd (MulAut G) G) (MulAut.conj g)) z
Module: Mathlib.GroupTheory.SpecificGroups.Cyclic

Goal state before:
pp: G : Type u_2
G' : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : IsCyclic G'
f : G →* G'
hf : f.ker ≤ center G
a b : G
x : G'
y : G
hxy : f y = x
hx : ∀ (a : ↥f.range), a ∈ zpowers ⟨x, ⋯⟩
m : ℤ
hm✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) m = ⟨f a, ⋯⟩
n : ℤ
hn✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) n = ⟨f b, ⋯⟩
hm : x ^ m = f a
hn : x ^ n = f b
ha : y ^ (-m) * a ∈ center G
hb : y ^ (-n) * b ∈ center G
⊢ y ^ m * (y ^ (-m) * a * y ^ n) * (y ^ (-n) * b) = y ^ m * (y ^ (-m) * a * y ^ n) * (y ^ (-n) * b)

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
hy : g⁻¹ * x * g ∈ centralizer ↑P
h1 : ↑P ≤ centralizer ↑(zpowers x)
h2 : ↑(g • P) ≤ centralizer ↑(zpowers x)
h : ↥(centralizer ↑(zpowers x))
hh : (h • g • P).subtype ⋯ = P.subtype h1
⊢ ∃ n ∈ normalizer ↑P, g⁻¹ * x * g = n⁻¹ * x * n

Tactic:
Subgroup.smul_def,

Goal state after:
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
x g : G
hx : x ∈ centralizer ↑P
hy : g⁻¹ * x * g ∈ centralizer ↑P
h1 : ↑P ≤ centralizer ↑(zpowers x)
h2 : ↑(g • P) ≤ centralizer ↑(zpowers x)
h : ↥(centralizer ↑(zpowers x))
hh : (↑h • g • P).subtype ⋯ = P.subtype h1
⊢ ∃ n ∈ normalizer ↑P, g⁻¹ * x * g = n⁻¹ * x * n
Module: Mathlib.GroupTheory.SpecificGroups.Cyclic

Goal state before:
pp: G : Type u_2
G' : Type u_3
inst✝² : Group G
inst✝¹ : Group G'
inst✝ : IsCyclic G'
f : G →* G'
hf : f.ker ≤ center G
a b : G
x : G'
y : G
hxy : f y = x
hx : ∀ (a : ↥f.range), a ∈ zpowers ⟨x, ⋯⟩
m : ℤ
hm✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) m = ⟨f a, ⋯⟩
n : ℤ
hn✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) n = ⟨f b, ⋯⟩
hm : x ^ m = f a
hn : x ^ n = f b
ha : y ^ (-m) * a ∈ center G
hb : y ^ (-n) * b ∈ center G
⊢ y ^ m * y ^ n * y ^ (-m) * (y ^ (-n) * b * a) = y ^ m * y ^ n * y ^ (-m) * (y ^ (-n) * b * a)

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.SpecificGroups.Cyclic

Goal state before:
pp: α : Type u_1
inst✝¹ : CommGroup α
inst✝ : IsSimpleGroup α
g : α
hg : ∀ (x : α), x ∈ Subgroup.zpowers g
hα : orderOf g ≠ 1
⊢ Nat.Prime (orderOf g)

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : CommGroup α
inst✝ : IsSimpleGroup α
g : α
hg : ∀ (x : α), x ∈ Subgroup.zpowers g
hα : orderOf g ≠ 1
⊢ Nat.Prime (orderOf g)
Module: Mathlib.GroupTheory.Torsion

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝ : Monoid G
tG : IsMulTorsion G
g : G
⊢ 1 = 1

pp: G : Type u_1
H : Type u_2
inst✝ : Monoid G
tG : IsMulTorsion G
g : G
⊢ 1 ≤ orderOf g

Tactic:
]

Goal state after:
G : Type u_1
H : Type u_2
inst✝ : Monoid G
tG : IsMulTorsion G
g : G
⊢ 1 ≤ orderOf g