mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Spectrum.Prime.Topology

Goal state before:
pp: 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
h : b.coeff 0 = 1
⊢ b.natTrailingDegree = 0

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝ : Group G
P Q : Sylow p G
h : ↑P = ↑Q
⊢ P = Q

Tactic:
by

Goal state after:
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
a : α
b : ↥(zpowers a)
⊢ ↑b ^ orderOf a = 1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
P : Subgroup G
hP : IsPGroup p ↥P
H : Subgroup G
inst✝ : H.Normal
h_cop : (Nat.card ↥H).Coprime H.index
⊢ P ≤ H ∨ Disjoint H P

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝¹ : Group G
inst✝ : Fact (Nat.Prime p)
P : Subgroup G
hP1 : IsPGroup p ↥P
hP2 : ¬p ∣ P.index
⊢ ∀ {Q : Subgroup G}, IsPGroup p ↥Q → P ≤ Q → Q = P

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝¹ : Group G
inst✝ : Fact (Nat.Prime p)
n : ℕ
hGpn : Nat.card G = p ^ n
hn : 0 < n
⊢ ∃ k > 0, Nat.card ↥(center G) = p ^ k

Tactic:
by

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

Goal state before:
pp: 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
⊢ p.natDegree ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: p✝ : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Finite G
p : ℕ
inst✝ : Fact (Nat.Prime p)
H : Subgroup G
card_eq : Nat.card ↥H = p ^ (Nat.card G).factorization p
⊢ ¬p ∣ H.index

Tactic:
by

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
K : Type u_2
inst✝ : Group K
ϕ : K →* G
N : Subgroup G
hϕ : IsPGroup p ↥ϕ.ker
h : ↑P ≤ ϕ.range
Q : Subgroup K
hQ : IsPGroup p ↥Q
hle : comap ϕ ↑P ≤ Q
⊢ Q = comap ϕ ↑P

Tactic:
by

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
K : Type u_2
inst✝ : Group K
ϕ : K →* G
N : Subgroup G
h : ↑P ≤ N
⊢ ↑P ≤ N.subtype.range

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝ : Group G
N : Subgroup G
P Q : Sylow p G
hP : ↑P ≤ N
hQ : ↑Q ≤ N
h : P.subtype hP = Q.subtype hQ
⊢ P = Q

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝ : Group G
P : Subgroup G
hP : IsPGroup p ↥P
c : Set (Subgroup G)
hc1 : c ⊆ {Q | IsPGroup p ↥Q}
hc2 : IsChain (fun x1 x2 ↦ x1 ≤ x2) c
Q : Subgroup G
hQ : Q ∈ c
x✝ : ↥{ carrier := ⋃ R, ↑↑R, mul_mem' := ⋯, one_mem' := ⋯, inv_mem' := ⋯ }
g : G
S : ↑c
hg : g ∈ (fun R ↦ ↑↑R) S
⊢ ∃ k, ⟨g, ⋯⟩ ^ p ^ k = 1

Tactic:
by

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

Goal state before:
pp: 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
⊢ (X ^ 0).natDegree ≤ 1 + c.natDegree

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝¹ : Group G
inst✝ : Fact (Nat.Prime p)
hG : Nat.card G = p ^ 2
⊢ IsCyclic (G ⧸ center G)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: n : ℕ
G : Type u_2
inst✝¹ : AddCommGroup G
inst✝ : Module (ZMod n) G
x✝ : G
⊢ Multiplicative.ofAdd x✝ ^ n ^ 1 = 1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.PGroup

Goal state before:
pp: n : ℕ
G : Type u_2
inst✝¹ : AddCommGroup G
inst✝ : Module (ZMod n) G
⊢ IsPGroup n (Multiplicative G)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.ZMod

Goal state before:
pp: n : ℕ
M : Type u_1
M₁ : Type u_2
inst✝¹ : NeZero n
inst✝ : AddCommMonoid M
h : ∀ (x : M), n • x = 0
c : ℕ
x : M
this : (c % n + c / n * n) • x = c • x
⊢ (c % n) • x = c • x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.ZMod

Goal state before:
pp: n : ℕ
M : Type u_1
M₁ : Type u_2
inst✝¹ : NeZero n
inst✝ : AddCommMonoid M
h : ∀ (x : M), n • x = 0
c : ℕ
x : M
⊢ (c % n) • x = c • x

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.ZMod

Goal state before:
pp: n : ℕ
M : Type u_1
M₁ : Type u_2
inst✝¹ : NeZero n
inst✝ : AddCommMonoid M
h : ∀ (x : M), n • x = 0
⊢ Module (ZMod n) M

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.ZMod

Goal state before:
pp: n : ℕ
M : Type u_1
M₁ : Type u_2
G : Type u_3
inst✝ : AddCommGroup G
H : AddSubgroup G
hH : ∀ (x : G), n • x ∈ H
⊢ ∀ (x : G ⧸ H), n • x = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.ZMod

Goal state before:
pp: n : ℕ
M : Type u_1
M₁ : Type u_2
F : Type u_3
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup M₁
inst✝³ : FunLike F M M₁
inst✝² : AddMonoidHomClass F M M₁
inst✝¹ : Module (ZMod n) M
inst✝ : Module (ZMod n) M₁
f : F
c : ZMod n
x : M
⊢ f (c • x) = c • f x

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝ : Group G
ι : Type u_2
s : Set ι
H : ι → Subgroup G
h : ∀ i ∈ s, IsPGroup p ↥(H i)
hn : ∀ i ∈ s, (H i).Normal
⊢ IsPGroup p ↥(⨆ i ∈ s, H i)

Tactic:
by

Goal state after:
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 ∈ d.properDivisors, #{a | orderOf a = m} = ∑ m ∈ d.properDivisors, φ m

Tactic:
by

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

Goal state before:
pp: 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
⊢ 1 ≤ b.natDegree + 1

Tactic:
by

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 ↥(sSup Hs)

Tactic:
by

Goal state after:
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 • x ∈ K

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.ZMod

Goal state before:
pp: n : ℕ
M : Type u_1
M₁ : Type u_2
F : Type u_3
S : Type u_4
inst✝⁷ : AddCommGroup M
inst✝⁶ : AddCommGroup M₁
inst✝⁵ : FunLike F M M₁
inst✝⁴ : AddMonoidHomClass F M M₁
inst✝³ : Module (ZMod n) M
inst✝² : Module (ZMod n) M₁
inst✝¹ : SetLike S M
inst✝ : AddSubgroupClass S M
x : M
K : S
f₁ f₂ : M →+ M₁
⊢ toZModLinearMap n (f₁ + f₂) = toZModLinearMap n f₁ + toZModLinearMap n f₂

Tactic:
by

Goal state after:
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
⊢ ↑P = ↑Q

Tactic:
by

Goal state after:
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
⊢ ∑ m ∈ d.divisors, #{a | orderOf a = m} = ∑ m ∈ d.divisors, φ m

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Module.ZMod

Goal state before:
pp: p : ℕ
G : Type u_5
inst✝¹ : AddCommGroup G
hp : Nat.Prime p
inst✝ : Module (ZMod p) G
H : Submodule (ZMod p) G
k : ℕ
hk : k ≤ Nat.card ↥H
h'k : k ≠ 0
⊢ ∃ H', Nat.card ↥H' ≤ k ∧ k < p * Nat.card ↥H' ∧ H' ≤ H

Tactic:
by

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
⊢ Finite (Sylow p G)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝ : Group G
g : G
P : Sylow p G
h : G
x✝ : h ∈ g • P
a : G
b : a ∈ ↑↑P
c : ((MulDistribMulAction.toMonoidEnd (MulAut G) G) (MulAut.conj g)) a = h
⊢ g⁻¹ * ((MulDistribMulAction.toMonoidEnd (MulAut G) G) (MulAut.conj g)) a * g ∈ ↑P

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝ : Group G
g : G
P : Sylow p G
⊢ g • P = P ↔ g ∈ normalizer ↑P

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝ : Group G
g : G
P : Sylow p G
h : (↑P).Normal
⊢ g • P = P

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Rank

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : Subsingleton G
⊢ rank G = 0

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Rank

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : FG G
⊢ rank G = 0 ↔ Subsingleton G

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Rank

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : Nontrivial G
inst✝ : FG G
⊢ 0 < rank G

Tactic:
by

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

Goal state before:
pp: 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
⊢ (C r).natDegree ≤ b.natDegree

Tactic:
by

Goal state after:
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
hpos : 0 < #{a | orderOf a = d}
⊢ #{a | orderOf a = d} = φ d

Tactic:
by

Goal state after:
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
⊢ rank H ≤ rank G

Tactic:
by

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

Goal state before:
pp: 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
⊢ X.natDegree ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝ : Group G
H : Subgroup G
P : Sylow p G
⊢ P ∈ fixedPoints (↥H) (Sylow p G) ↔ H ≤ normalizer ↑P

Tactic:
by

Goal state after:
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
⊢ #{a | orderOf a = d} = φ d

Tactic:
have hc0 : 0 < c := Fintype.card_pos_iff.2 ⟨1⟩

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
⊢ #{a | orderOf a = d} = φ d
Module: Mathlib.GroupTheory.Rank

Goal state before:
pp: G : Type u_1
inst✝² : Group G
H K : Subgroup G
inst✝¹ : Group.FG ↥H
inst✝ : Group.FG ↥K
h : H = K
⊢ rank ↥H = rank ↥K

Tactic:
by

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 ↑t = ⊤

Tactic:
by

Goal state after:
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
⊢ Q ∈ fixedPoints (↥P) (Sylow p G) ↔ P ≤ ↑Q

Tactic:
by

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

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
h : IsClosedMap (comap (mapRingHom f))
⊢ f.IsIntegral

Tactic:
by

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

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Algebra.IsIntegral R S
inst✝ : FaithfulSMul R S
p : Ideal R
hp : p.IsPrime
hinj : Function.Injective ⇑(algebraMap R S)
⊢ Ideal.comap (algebraMap R S) ⊥ ≤ p

Tactic:
by

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

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Algebra.IsIntegral R S
inst✝ : FaithfulSMul R S
⊢ Function.Surjective (comap (algebraMap R S))

Tactic:
by

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
⊢ t.card ≤ s.card

Tactic:
by

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

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
hf : f.IsIntegral
hinj : Function.Injective ⇑f
⊢ Function.Surjective (comap f)

Tactic:
by

Goal state after:
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
⊢ ↑I.minimalPrimes ≃o (↑(irreducibleComponents ↑(zeroLocus ↑I)))ᵒᵈ

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Rank

Goal state before:
pp: G : Type u_1
inst✝ : Group G
s : Finset G
⊢ rank ↥(closure ↑s) ≤ s.card

Tactic:
by

Goal state after:
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} = ∅
⊢ c = ∑ m ∈ c.divisors, #{a | orderOf a = m}

Tactic:
by

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

Goal state before:
pp: R : Type u
S : Type v
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
⊢ ↑(minimalPrimes R) ≃o (↑(irreducibleComponents (PrimeSpectrum R)))ᵒᵈ

Tactic:
by

Goal state after:
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 = Q

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Rank

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

Tactic:
by

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

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

Tactic:
by

Goal state after:
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
⊢ ∃ g, g • P = Q

Tactic:
by

Goal state after:
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 : ℕ
hm₁ : m ∈ c.divisors
hm₂ : m ∉ c.divisors.erase d
⊢ m = d

Tactic:
by

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

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
⊢ zeroLocus ∘ SetLike.coe '' minimalPrimes R = irreducibleComponents (PrimeSpectrum R)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Rank

Goal state before:
pp: G : Type u_1
inst✝ : Group G
g : G
⊢ Nat.card ↥(centralizer {g}) = Nat.card ↥(MulAction.stabilizer (ConjAct G) g)

Tactic:
by

Goal state after:
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, #{a | orderOf a = m} = ∑ m ∈ c.divisors.erase d, #{a | orderOf a = m}

Tactic:
by

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

Goal state before:
pp: R : Type u
inst✝ : CommSemiring R
s : Set (PrimeSpectrum R)
⊢ vanishingIdeal s ∈ minimalPrimes R ↔ closure s ∈ irreducibleComponents (PrimeSpectrum R)

Tactic:
by

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

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

Tactic:
by

Goal state after:
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 : ℕ
hm : m ∈ c.divisors.erase d
⊢ m ∣ c

Tactic:
by

Goal state after:
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
⊢ IsLocalHom f ↔ PrimeSpectrum.comap f (closedPoint S) = closedPoint R

Tactic:
by

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

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

Tactic:
by

Goal state after:
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) x = closedPoint T

Tactic:
by

Goal state after:
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
⊢ ∃ g, g • P = Q

Tactic:
by

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

Goal state before:
pp: R : Type u
inst✝¹ : CommSemiring R
inst✝ : IsLocalRing R
⊢ IsClosed {closedPoint R}

Tactic:
by

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

Goal state before:
pp: R : Type u
inst✝ : CommRing R
e : R
he : IsIdempotentElem e
⊢ e * (1 - e) = 0

Tactic:
by

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

Goal state before:
pp: R : Type u
inst✝ : CommRing R
e : R
he : IsIdempotentElem e
⊢ e + (1 - e) = 1

Tactic:
by

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
this : fixedPoints (↥↑P) (Sylow p G) = {P}
⊢ Nat.card ↑(fixedPoints (↥↑P) (Sylow p G)) = 1

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: R : Type u
inst✝ : CommRing R
s : Set (PrimeSpectrum R)
⊢ IsClopen s ↔ ∃ e, IsIdempotentElem e ∧ s = ↑(basicOpen e)

Tactic:
by

Goal state after:
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.erase d, φ m

Tactic:
by

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
this✝ : fixedPoints (↥↑P) (Sylow p G) = {P}
this : Nat.card ↑(fixedPoints (↥↑P) (Sylow p G)) = 1
⊢ Nat.card ↑(fixedPoints (↥↑P) (Sylow p G)) ≡ 1 [MOD p]

Tactic:
by

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)
⊢ Nat.card (Sylow p G) ≡ 1 [MOD p]

Tactic:
by

Goal state after:
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
⊢ #{a | orderOf a = d} = φ d

Tactic:
by

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 ≃* ↥↑Q

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: p : ℕ
G : Type u_1
inst✝ : Group G
P : Sylow p G
⊢ stabilizer G P = normalizer ↑P

Tactic:
by

Goal state after:
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 < #{a | orderOf a = Nat.card α}

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝² : Group α
inst✝¹ : IsCyclic α
inst✝ : Fintype α
d : ℕ
hd : d ∣ Fintype.card α
⊢ #{a | orderOf a = d} = φ d

Tactic:
by

Goal state after: