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

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝ : Group G
q : ℕ
hpq : p ∣ q
hp : ∀ (g : G), ∃ k, orderOf g ∣ p ^ k
⊢ ∀ (g : G), ∃ k, orderOf g ∣ q ^ k

Tactic:
]

Goal state after:
p : ℕ
G : Type u_1
inst✝ : Group G
q : ℕ
hpq : p ∣ q
hp : ∀ (g : G), ∃ k, orderOf g ∣ p ^ k
⊢ ∀ (g : G), ∃ k, orderOf g ∣ q ^ k
Module: Mathlib.GroupTheory.Exponent

Goal state before:
pp: G : Type u
inst✝ : CommMonoid G
hG : ExponentExists G
he : exponent G ≠ 0
hne : (Set.range orderOf).Nonempty
hfin : (Set.range orderOf).Finite
t : G
ht : orderOf t = sSup (Set.range orderOf)
p : ℕ
hp : Nat.Prime p
k : ℕ := (orderOf t).factorization p
hpe : k < (exponent G).factorization p
hk : k = (orderOf t).factorization p
g : G
hg : orderOf g = p ^ (exponent G).factorization p
hpk : p ^ k ∣ orderOf t
hpk' : orderOf (t ^ p ^ k) = orderOf t / p ^ k
a : ℕ
ha : (exponent G).factorization p = k + a + 1
⊢ p.Coprime (orderOf (t ^ p ^ k))

Tactic:
]

Goal state after:
G : Type u
inst✝ : CommMonoid G
hG : ExponentExists G
he : exponent G ≠ 0
hne : (Set.range orderOf).Nonempty
hfin : (Set.range orderOf).Finite
t : G
ht : orderOf t = sSup (Set.range orderOf)
p : ℕ
hp : Nat.Prime p
k : ℕ := (orderOf t).factorization p
hpe : k < (exponent G).factorization p
hk : k = (orderOf t).factorization p
g : G
hg : orderOf g = p ^ (exponent G).factorization p
hpk : p ^ k ∣ orderOf t
hpk' : orderOf (t ^ p ^ k) = orderOf t / p ^ k
a : ℕ
ha : (exponent G).factorization p = k + a + 1
⊢ p.Coprime (orderOf (t ^ p ^ k))
Module: Mathlib.GroupTheory.Exponent

Goal state before:
pp: G : Type u
inst✝ : CommMonoid G
hG : ExponentExists G
he : exponent G ≠ 0
hne : (Set.range orderOf).Nonempty
hfin : (Set.range orderOf).Finite
t : G
ht : orderOf t = sSup (Set.range orderOf)
p : ℕ
hp : Nat.Prime p
k : ℕ := (orderOf t).factorization p
hpe : k < (exponent G).factorization p
hk : k = (orderOf t).factorization p
g : G
hg : orderOf g = p ^ (exponent G).factorization p
hpk : p ^ k ∣ orderOf t
hpk' : orderOf (t ^ p ^ k) = orderOf t / p ^ k
a : ℕ
ha : (exponent G).factorization p = k + a + 1
⊢ ¬p ^ 1 ∣ orderOf (t ^ p ^ k)

Tactic:
]

Goal state after:
G : Type u
inst✝ : CommMonoid G
hG : ExponentExists G
he : exponent G ≠ 0
hne : (Set.range orderOf).Nonempty
hfin : (Set.range orderOf).Finite
t : G
ht : orderOf t = sSup (Set.range orderOf)
p : ℕ
hp : Nat.Prime p
k : ℕ := (orderOf t).factorization p
hpe : k < (exponent G).factorization p
hk : k = (orderOf t).factorization p
g : G
hg : orderOf g = p ^ (exponent G).factorization p
hpk : p ^ k ∣ orderOf t
hpk' : orderOf (t ^ p ^ k) = orderOf t / p ^ k
a : ℕ
ha : (exponent G).factorization p = k + a + 1
⊢ ¬p ^ 1 ∣ orderOf (t ^ p ^ k)
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝¹ : Group G
inst✝ : Finite G
k : G → ℕ
hk : ∀ (g : G), orderOf g ∣ p ^ k g
this : Fintype G
g : G
left✝ : g ∈ Finset.univ
hg : ∀ x' ∈ Finset.univ, k x' ≤ k g
g' : G
⊢ orderOf g' ∣ p ^ k g'

Tactic:
]

Goal state after:
p : ℕ
G : Type u_1
inst✝¹ : Group G
inst✝ : Finite G
k : G → ℕ
hk : ∀ (g : G), orderOf g ∣ p ^ k g
this : Fintype G
g : G
left✝ : g ∈ Finset.univ
hg : ∀ x' ∈ Finset.univ, k x' ≤ k g
g' : G
⊢ orderOf g' ∣ p ^ k g'
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h✝ : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus ↑I = sᶜ
hJ' : zeroLocus ↑J = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
⊢ (fun x ↦ x ^ n) '' ↑I ∪ (fun x ↦ x ^ n) '' ↑J ⊆ ↑(I ^ n ⊔ J ^ n)

pp: R : Type u
inst✝ : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h✝ : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus ↑I = sᶜ
hJ' : zeroLocus ↑J = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
⊢ Ideal.span (↑I ∪ ↑J) = ⊤

Tactic:
Set.union_subset_iff

Goal state after:
R : Type u
inst✝ : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h✝ : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus ↑I = sᶜ
hJ' : zeroLocus ↑J = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
⊢ (fun x ↦ x ^ n) '' ↑I ⊆ ↑(I ^ n ⊔ J ^ n) ∧ (fun x ↦ x ^ n) '' ↑J ⊆ ↑(I ^ n ⊔ J ^ n)

R : Type u
inst✝ : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h✝ : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus ↑I = sᶜ
hJ' : zeroLocus ↑J = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
⊢ Ideal.span (↑I ∪ ↑J) = ⊤
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h✝ : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus ↑I = sᶜ
hJ' : zeroLocus ↑J = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
⊢ (fun x ↦ x ^ n) '' ↑I ⊆ ↑(I ^ n ⊔ J ^ n) ∧ (fun x ↦ x ^ n) '' ↑J ⊆ ↑(I ^ n ⊔ J ^ n)

pp: R : Type u
inst✝ : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h✝ : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus ↑I = sᶜ
hJ' : zeroLocus ↑J = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
⊢ Ideal.span (↑I ∪ ↑J) = ⊤

Tactic:
]

Goal state after:
R : Type u
inst✝ : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h✝ : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus ↑I = sᶜ
hJ' : zeroLocus ↑J = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
⊢ (fun x ↦ x ^ n) '' ↑I ⊆ ↑(I ^ n ⊔ J ^ n) ∧ (fun x ↦ x ^ n) '' ↑J ⊆ ↑(I ^ n ⊔ J ^ n)

R : Type u
inst✝ : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h✝ : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus ↑I = sᶜ
hJ' : zeroLocus ↑J = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
⊢ Ideal.span (↑I ∪ ↑J) = ⊤
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: case refine_1.inr
p : ℕ
G : Type u_1
inst✝¹ : Group G
inst✝ : Finite G
h : p ≠ 0
x✝ : ∃ n, Nat.card G ∣ p ^ n
n : ℕ
hn : Nat.card G ∣ p ^ n
hn0 : n ≠ 0
⊢ (p ^ n).primeFactors ⊆ (p ^ n).primeFactors

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.Exponent

Goal state before:
pp: G : Type u
inst✝ : CommMonoid G
hG : ExponentExists G
he : exponent G ≠ 0
hne : (Set.range orderOf).Nonempty
hfin : (Set.range orderOf).Finite
t : G
ht : orderOf t = sSup (Set.range orderOf)
p : ℕ
hp : Nat.Prime p
k : ℕ := (orderOf t).factorization p
hpe : k < (exponent G).factorization p
hk : k = (orderOf t).factorization p
g : G
hg : orderOf g = p ^ (exponent G).factorization p
hpk : p ^ k ∣ orderOf t
hpk' : orderOf (t ^ p ^ k) = orderOf t / p ^ k
a : ℕ
ha : (exponent G).factorization p = k + a + 1
⊢ 1 = ((orderOf t).factorization - (p ^ k).factorization) p + 1

Tactic:
]

Goal state after:
G : Type u
inst✝ : CommMonoid G
hG : ExponentExists G
he : exponent G ≠ 0
hne : (Set.range orderOf).Nonempty
hfin : (Set.range orderOf).Finite
t : G
ht : orderOf t = sSup (Set.range orderOf)
p : ℕ
hp : Nat.Prime p
k : ℕ := (orderOf t).factorization p
hpe : k < (exponent G).factorization p
hk : k = (orderOf t).factorization p
g : G
hg : orderOf g = p ^ (exponent G).factorization p
hpk : p ^ k ∣ orderOf t
hpk' : orderOf (t ^ p ^ k) = orderOf t / p ^ k
a : ℕ
ha : (exponent G).factorization p = k + a + 1
⊢ 1 = ((orderOf t).factorization - (p ^ k).factorization) p + 1
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: case refine_2
p : ℕ
G : Type u_1
inst✝¹ : Group G
inst✝ : Finite G
h : p ≠ 0
hG : (Nat.card G).primeFactors ⊆ p.primeFactors
⊢ p ^ Nat.card G ∣ p ^ Nat.card G

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝¹ : Group G
hG : IsPGroup p G
H : Type u_2
inst✝ : Group H
ϕ : G →* H
hϕ : Function.Surjective ⇑ϕ
h : H
g : G
hg : ϕ g = h
k : ℕ
hk : g ^ p ^ k = 1
⊢ 1 = 1

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝ : Group G
hG : IsPGroup p G
n : ℕ
hn : p.Coprime n
g : G
⊢ (hG.powEquiv hn).symm g = g ^ (Nat.card ↥(Subgroup.zpowers g)).gcdB n

Tactic:
]

Goal state after:
p : ℕ
G : Type u_1
inst✝ : Group G
hG : IsPGroup p G
n : ℕ
hn : p.Coprime n
g : G
⊢ (hG.powEquiv hn).symm g = g ^ (Nat.card ↥(Subgroup.zpowers g)).gcdB n
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: case inl
p : ℕ
G : Type u_1
inst✝ : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
h✝ : Finite G
n : ℕ
hn : Nat.card G = p ^ n
⊢ p ^ n = 1 ∨ p ∣ p ^ n

Tactic:
]

Goal state after:
case inl
p : ℕ
G : Type u_1
inst✝ : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
h✝ : Finite G
n : ℕ
hn : Nat.card G = p ^ n
⊢ p ^ n = 1 ∨ p ∣ p ^ n
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝ : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
h✝ : Finite G
n : ℕ
hn : Nat.card G = p ^ (n + 1)
⊢ p * p ^ n = p * p ^ n

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: case inr
p : ℕ
G : Type u_1
inst✝ : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
h✝ : Infinite G
⊢ 0 = 1 ∨ p ∣ 0

Tactic:
]

Goal state after:
case inr
p : ℕ
G : Type u_1
inst✝ : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
h✝ : Infinite G
⊢ 0 = 1 ∨ p ∣ 0
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝¹ : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
inst✝ : Finite G
hGnt : Nontrivial G
k : ℕ
hk : Nat.card G = 1
hk0 : k = 0
⊢ False

Tactic:
]

Goal state after:
p : ℕ
G : Type u_1
inst✝¹ : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
inst✝ : Finite G
hGnt : Nontrivial G
k : ℕ
hk : Nat.card G = 1
hk0 : k = 0
⊢ False
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝² : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
α : Type u_2
inst✝¹ : MulAction G α
a : α
inst✝ : Finite ↑(orbit G a)
ϕ : ↑(orbit G a) ≃ G ⧸ stabilizer G a := orbitEquivQuotientStabilizer G a
this✝ : Finite (G ⧸ stabilizer G a)
this : (stabilizer G a).FiniteIndex
⊢ ∃ n, Nat.card (G ⧸ stabilizer G a) = p ^ n

Tactic:
]

Goal state after:
p : ℕ
G : Type u_1
inst✝² : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
α : Type u_2
inst✝¹ : MulAction G α
a : α
inst✝ : Finite ↑(orbit G a)
ϕ : ↑(orbit G a) ≃ G ⧸ stabilizer G a := orbitEquivQuotientStabilizer G a
this✝ : Finite (G ⧸ stabilizer G a)
this : (stabilizer G a).FiniteIndex
⊢ ∃ n, Nat.card (G ⧸ stabilizer G a) = p ^ n
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝² : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
this✝ : Fintype α
this : Fintype ↑(fixedPoints G α)
⊢ card α ≡ card ↑(fixedPoints G α) [MOD p]

Tactic:
]

Goal state after:
p : ℕ
G : Type u_1
inst✝² : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
this✝ : Fintype α
this : Fintype ↑(fixedPoints G α)
⊢ card α ≡ card ↑(fixedPoints G α) [MOD p]
Module: Mathlib.GroupTheory.Exponent

Goal state before:
pp: G : Type u
inst✝ : CommMonoid G
hG : ExponentExists G
he : exponent G ≠ 0
hne : (Set.range orderOf).Nonempty
hfin : (Set.range orderOf).Finite
t : G
ht : orderOf t = sSup (Set.range orderOf)
p : ℕ
hp : Nat.Prime p
k : ℕ := (orderOf t).factorization p
hpe : k < (exponent G).factorization p
hk : k = (orderOf t).factorization p
g : G
hg : orderOf g = p ^ (exponent G).factorization p
hpk : p ^ k ∣ orderOf t
hpk' : orderOf (t ^ p ^ k) = orderOf t / p ^ k
a : ℕ
ha : (exponent G).factorization p = k + a + 1
this : 1 = (orderOf (t ^ p ^ k)).factorization p + 1
⊢ ¬p ^ ((orderOf (t ^ p ^ k)).factorization p + 1) ∣ orderOf (t ^ p ^ k)

Tactic:
]

Goal state after:
G : Type u
inst✝ : CommMonoid G
hG : ExponentExists G
he : exponent G ≠ 0
hne : (Set.range orderOf).Nonempty
hfin : (Set.range orderOf).Finite
t : G
ht : orderOf t = sSup (Set.range orderOf)
p : ℕ
hp : Nat.Prime p
k : ℕ := (orderOf t).factorization p
hpe : k < (exponent G).factorization p
hk : k = (orderOf t).factorization p
g : G
hg : orderOf g = p ^ (exponent G).factorization p
hpk : p ^ k ∣ orderOf t
hpk' : orderOf (t ^ p ^ k) = orderOf t / p ^ k
a : ℕ
ha : (exponent G).factorization p = k + a + 1
this : 1 = (orderOf (t ^ p ^ k)).factorization p + 1
⊢ ¬p ^ ((orderOf (t ^ p ^ k)).factorization p + 1) ∣ orderOf (t ^ p ^ k)
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝² : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
this✝ : Fintype α
this : Fintype ↑(fixedPoints G α)
⊢ ∑ x, ↑(card { x_1 // Quotient.mk'' x_1 = x }) = ∑ x, ↑1

Tactic:
]

Goal state after:
p : ℕ
G : Type u_1
inst✝² : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
this✝ : Fintype α
this : Fintype ↑(fixedPoints G α)
⊢ ∑ x, ↑(card { x_1 // Quotient.mk'' x_1 = x }) = ∑ x, ↑1
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝² : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
this✝ : Fintype α
this : Fintype ↑(fixedPoints G α)
key : ∀ (x : α), card { y // Quotient.mk'' y = Quotient.mk'' x } = card ↑(orbit G x)
a : ↑(fixedPoints G α)
ha : a ∈ Finset.univ
x✝ : ↑1 ≠ 0
⊢ ↑1 = ↑1

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝² : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
this✝ : Fintype α
this : Fintype ↑(fixedPoints G α)
key : ∀ (x : α), card { y // Quotient.mk'' y = Quotient.mk'' x } = card ↑(orbit G x)
b✝ : Quotient (orbitRel G α)
b : α
x✝ : Quotient.mk'' b ∈ Finset.univ
hb : ↑(card { x // Quotient.mk'' x = Quotient.mk'' b }) ≠ 0
k : ℕ
hk : card ↑(orbit G b) = p ^ k
⊢ ∃ a, ∃ (_ : a ∈ Finset.univ) (_ : ↑1 ≠ 0), Quotient.mk'' ↑a = Quotient.mk'' b

Tactic:
]

Goal state after:
p : ℕ
G : Type u_1
inst✝² : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
this✝ : Fintype α
this : Fintype ↑(fixedPoints G α)
key : ∀ (x : α), card { y // Quotient.mk'' y = Quotient.mk'' x } = card ↑(orbit G x)
b✝ : Quotient (orbitRel G α)
b : α
x✝ : Quotient.mk'' b ∈ Finset.univ
hb : ↑(card { x // Quotient.mk'' x = Quotient.mk'' b }) ≠ 0
k : ℕ
hk : card ↑(orbit G b) = p ^ k
⊢ ∃ a, ∃ (_ : a ∈ Finset.univ) (_ : ↑1 ≠ 0), Quotient.mk'' ↑a = Quotient.mk'' b
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝² : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
this✝¹ : Fintype α
this✝ : Fintype ↑(fixedPoints G α)
key : ∀ (x : α), card { y // Quotient.mk'' y = Quotient.mk'' x } = card ↑(orbit G x)
b✝ : Quotient (orbitRel G α)
b : α
x✝ : Quotient.mk'' b ∈ Finset.univ
hb : ↑(card { x // Quotient.mk'' x = Quotient.mk'' b }) ≠ 0
k : ℕ
hk : card ↑(orbit G b) = p ^ k
this : k = 0
⊢ 1 = 1

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝¹ : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
α : Type u_3
inst✝ : MulAction G α
hpα : ¬p ∣ Nat.card α
this : Finite α
⊢ Nonempty ↑(fixedPoints G α)

Tactic:
← Finite.card_pos_iff,

Goal state after:
p : ℕ
G : Type u_1
inst✝¹ : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
α : Type u_3
inst✝ : MulAction G α
hpα : ¬p ∣ Nat.card α
this : Finite α
⊢ 0 < Nat.card ↑(fixedPoints G α)
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝¹ : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
α : Type u_3
inst✝ : MulAction G α
hpα : ¬p ∣ Nat.card α
this : Finite α
⊢ Nat.card ↑(fixedPoints G α) ≠ 0

Tactic:
]

Goal state after:
p : ℕ
G : Type u_1
inst✝¹ : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
α : Type u_3
inst✝ : MulAction G α
hpα : ¬p ∣ Nat.card α
this : Finite α
⊢ Nat.card ↑(fixedPoints G α) ≠ 0
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝¹ : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
α : Type u_3
inst✝ : MulAction G α
this : Finite α
hpα : Nat.card ↑(fixedPoints G α) = 0
⊢ Nat.card α ≡ Nat.card ↑(fixedPoints G α) [MOD p]

Tactic:
]

Goal state after:
p : ℕ
G : Type u_1
inst✝¹ : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
α : Type u_3
inst✝ : MulAction G α
this : Finite α
hpα : Nat.card ↑(fixedPoints G α) = 0
⊢ Nat.card α ≡ Nat.card ↑(fixedPoints G α) [MOD p]
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝² : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
hpα : p ∣ Nat.card α
a : α
ha : a ∈ fixedPoints G α
hpf : p ∣ Nat.card ↑(fixedPoints G α)
hα : Nontrivial ↑(fixedPoints G α)
⊢ ∃ b ∈ fixedPoints G α, a ≠ b

Tactic:
]

Goal state after:
p : ℕ
G : Type u_1
inst✝² : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
α : Type u_2
inst✝¹ : MulAction G α
inst✝ : Finite α
hpα : p ∣ Nat.card α
a : α
ha : a ∈ fixedPoints G α
hpf : p ∣ Nat.card ↑(fixedPoints G α)
hα : Nontrivial ↑(fixedPoints G α)
⊢ ∃ b ∈ fixedPoints G α, a ≠ b
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝² : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
inst✝¹ : Nontrivial G
inst✝ : Finite G
this : p ∣ Nat.card G → ∀ {a : G}, a ∈ ↑(Subgroup.center G) → ∃ b ∈ ↑(Subgroup.center G), a ≠ b
⊢ Nontrivial ↥(Subgroup.center G)

Tactic:
]

Goal state after:
p : ℕ
G : Type u_1
inst✝² : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
inst✝¹ : Nontrivial G
inst✝ : Finite G
this : p ∣ Nat.card G → ∀ {a : G}, a ∈ ↑(Subgroup.center G) → ∃ b ∈ ↑(Subgroup.center G), a ≠ b
⊢ Nontrivial ↥(Subgroup.center G)
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝² : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
inst✝¹ : Nontrivial G
inst✝ : Finite G
this : p ∣ Nat.card G → ∀ {a : G}, a ∈ ↑(Subgroup.center G) → ∃ b ∈ ↑(Subgroup.center G), a ≠ b
dvd : p ∣ Nat.card G
⊢ Nontrivial ↥(Subgroup.center G)

Tactic:
obtain ⟨g, hg⟩ := this dvd (Subgroup.center G).one_mem

Goal state after:
p : ℕ
G : Type u_1
inst✝² : Group G
hG : IsPGroup p G
hp : Fact (Nat.Prime p)
inst✝¹ : Nontrivial G
inst✝ : Finite G
this : p ∣ Nat.card G → ∀ {a : G}, a ∈ ↑(Subgroup.center G) → ∃ b ∈ ↑(Subgroup.center G), a ≠ b
dvd : p ∣ Nat.card G
g : G
hg : g ∈ ↑(Subgroup.center G) ∧ 1 ≠ g
⊢ Nontrivial ↥(Subgroup.center G)
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h✝ : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus ↑I = sᶜ
hJ' : zeroLocus ↑J = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
⊢ ∅ = ∅

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
hH : IsPGroup p ↥H
K : Type u_2
inst✝ : Group K
ϕ : G →* K
⊢ IsPGroup p ↥(ϕ.comp H.subtype).range

Tactic:
]

Goal state after:
p : ℕ
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
hH : IsPGroup p ↥H
K : Type u_2
inst✝ : Group K
ϕ : G →* K
⊢ IsPGroup p ↥(ϕ.comp H.subtype).range
Module: Mathlib.GroupTheory.Exponent

Goal state before:
pp: case h
G : Type u
inst✝ : CommMonoid G
hG : ExponentExists G
he : exponent G ≠ 0
hne : (Set.range orderOf).Nonempty
hfin : (Set.range orderOf).Finite
t : G
ht : orderOf t = sSup (Set.range orderOf)
p : ℕ
hp : Nat.Prime p
k : ℕ := (orderOf t).factorization p
hpe : k < (exponent G).factorization p
hk : k = (orderOf t).factorization p
g : G
hg : orderOf g = p ^ (exponent G).factorization p
hpk : p ^ k ∣ orderOf t
hpk' : orderOf (t ^ p ^ k) = orderOf t / p ^ k
a : ℕ
ha : (exponent G).factorization p = k + a + 1
hcoprime : (orderOf (t ^ p ^ k)).Coprime (orderOf g)
⊢ 1 < p ^ (a + 1)

pp: case h
G : Type u
inst✝ : CommMonoid G
hG : ExponentExists G
he : exponent G ≠ 0
hne : (Set.range orderOf).Nonempty
hfin : (Set.range orderOf).Finite
t : G
ht : orderOf t = sSup (Set.range orderOf)
p : ℕ
hp : Nat.Prime p
k : ℕ := (orderOf t).factorization p
hpe : k < (exponent G).factorization p
hk : k = (orderOf t).factorization p
g : G
hg : orderOf g = p ^ (exponent G).factorization p
hpk : p ^ k ∣ orderOf t
hpk' : orderOf (t ^ p ^ k) = orderOf t / p ^ k
a : ℕ
ha : (exponent G).factorization p = k + a + 1
hcoprime : (orderOf (t ^ p ^ k)).Coprime (orderOf g)
⊢ p ^ (orderOf t).factorization p ∣ orderOf t

Tactic:
]

Goal state after:
case h
G : Type u
inst✝ : CommMonoid G
hG : ExponentExists G
he : exponent G ≠ 0
hne : (Set.range orderOf).Nonempty
hfin : (Set.range orderOf).Finite
t : G
ht : orderOf t = sSup (Set.range orderOf)
p : ℕ
hp : Nat.Prime p
k : ℕ := (orderOf t).factorization p
hpe : k < (exponent G).factorization p
hk : k = (orderOf t).factorization p
g : G
hg : orderOf g = p ^ (exponent G).factorization p
hpk : p ^ k ∣ orderOf t
hpk' : orderOf (t ^ p ^ k) = orderOf t / p ^ k
a : ℕ
ha : (exponent G).factorization p = k + a + 1
hcoprime : (orderOf (t ^ p ^ k)).Coprime (orderOf g)
⊢ 1 < p ^ (a + 1)

case h
G : Type u
inst✝ : CommMonoid G
hG : ExponentExists G
he : exponent G ≠ 0
hne : (Set.range orderOf).Nonempty
hfin : (Set.range orderOf).Finite
t : G
ht : orderOf t = sSup (Set.range orderOf)
p : ℕ
hp : Nat.Prime p
k : ℕ := (orderOf t).factorization p
hpe : k < (exponent G).factorization p
hk : k = (orderOf t).factorization p
g : G
hg : orderOf g = p ^ (exponent G).factorization p
hpk : p ^ k ∣ orderOf t
hpk' : orderOf (t ^ p ^ k) = orderOf t / p ^ k
a : ℕ
ha : (exponent G).factorization p = k + a + 1
hcoprime : (orderOf (t ^ p ^ k)).Coprime (orderOf g)
⊢ p ^ (orderOf t).factorization p ∣ orderOf t
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: case inr
R : Type u
inst✝ : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h✝ : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus ↑I = sᶜ
hJ' : zeroLocus ↑J = s
this✝ : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
this : ∃ y ∈ I ^ n, ∃ z ∈ J ^ n, y + z = 1
⊢ ∃ e f, e * f = 0 ∧ e + f = 1 ∧ s = ↑(basicOpen e) ∧ sᶜ = ↑(basicOpen f)

Tactic:
]

Goal state after:
case inr
R : Type u
inst✝ : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h✝ : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus ↑I = sᶜ
hJ' : zeroLocus ↑J = s
this✝ : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
this : ∃ y ∈ I ^ n, ∃ z ∈ J ^ n, y + z = 1
⊢ ∃ e f, e * f = 0 ∧ e + f = 1 ∧ s = ↑(basicOpen e) ∧ sᶜ = ↑(basicOpen f)
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
hH : IsPGroup p ↥H
K : Type u_2
inst✝ : Group K
ϕ : K →* G
hϕ : IsPGroup p ↥ϕ.ker
g : ↥(Subgroup.comap ϕ H)
j : ℕ
hj : ϕ (↑g ^ p ^ j) = ↑1
⊢ ∃ k, g ^ p ^ k = 1

Tactic:
]

Goal state after:
p : ℕ
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
hH : IsPGroup p ↥H
K : Type u_2
inst✝ : Group K
ϕ : K →* G
hϕ : IsPGroup p ↥ϕ.ker
g : ↥(Subgroup.comap ϕ H)
j : ℕ
hj : ϕ (↑g ^ p ^ j) = ↑1
⊢ ∃ k, g ^ p ^ k = 1
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: case refine_1
R : Type u
inst✝ : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h✝ : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus ↑I = sᶜ
hJ' : zeroLocus ↑J = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
x : R
hx : x ∈ I ^ n
y : R
hy : y ∈ J ^ n
add : x + y = 1
mul : x * y = 0
⊢ zeroLocus ↑J ⊆ zeroLocus {y}

Tactic:
]

Goal state after:
case refine_1
R : Type u
inst✝ : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h✝ : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus ↑I = sᶜ
hJ' : zeroLocus ↑J = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
x : R
hx : x ∈ I ^ n
y : R
hy : y ∈ J ^ n
add : x + y = 1
mul : x * y = 0
⊢ zeroLocus ↑J ⊆ zeroLocus {y}
Module: Mathlib.GroupTheory.Exponent

Goal state before:
pp: G : Type u
inst✝ : CommMonoid G
h : ∀ (g : G), 0 < orderOf g
⊢ exponent G = sSup (Set.range fun g ↦ orderOf g)

Tactic:
]

Goal state after:
G : Type u
inst✝ : CommMonoid G
h : ∀ (g : G), 0 < orderOf g
⊢ exponent G = sSup (Set.range fun g ↦ orderOf g)