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

Goal state before:
pp: G : Type u_2
inst✝¹ : Infinite G
inst✝ : AddGroup G
g : G
hg : zmultiples g = ⊤
k : ℤ
⊢ (intEquivOfZMultiplesEqTop g hg).symm (k • g) = k

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Torsion

Goal state before:
pp: G : Type u_1
H : Type u_2
R : Type u_3
M : Type u_4
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
r : R
m : M
x✝ : m ∈ (AddCommGroup.torsion M).carrier
n : ℕ
hn : n > 0
hn' : Function.IsPeriodicPt (fun x ↦ m + x) n 0
⊢ Function.IsPeriodicPt (fun x ↦ r • m + x) n 0

Tactic:
by

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

Goal state before:
pp: G : Type u_2
inst✝¹ : Group G
inst✝ : Finite G
h : IsCyclic G
⊢ Nat.card (MulAut G) = (Nat.card G).totient

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Torsion

Goal state before:
pp: G : Type u_1
H : Type u_2
R : Type u_3
M : Type u_4
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
S : Submodule R M :=
  let __src := AddCommGroup.torsion M;
  { toAddSubmonoid := __src.toAddSubmonoid, smul_mem' := ⋯ }
⊢ AddSubgroup.map (↑(AddEquiv.refl M)) (AddCommGroup.torsion M) = S.toAddSubgroup

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Torsion

Goal state before:
pp: M : Type u_3
inst✝¹ : CommMonoid M
inst✝ : HasDistribNeg M
⊢ (-1) ^ 2 = 1

Tactic:
by

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

Goal state before:
pp: G : Type u_2
inst✝¹ : CommGroup G
hG : IsCyclic G
inst✝ : Finite G
d : ℕ
⊢ Nat.card ↥(powMonoidHom d).range = Nat.card G / (Nat.card G).gcd d

Tactic:
by

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

Goal state before:
pp: G : Type u_2
inst✝² : CommGroup G
inst✝¹ : IsCyclic G
inst✝ : Finite G
d : ℕ
⊢ (powMonoidHom d).ker.index = Nat.card G / (Nat.card G).gcd d

Tactic:
by

Goal state after:
Module: Mathlib.Order.Atoms.Finite

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : LE α
inst✝¹ : BoundedOrder α
inst✝ : IsSimpleOrder α
⊢ Finite α

Tactic:
by

Goal state after:
Module: Mathlib.Order.Atoms.Finite

Goal state before:
pp: α : Type u_1
inst✝³ : LE α
inst✝² : BoundedOrder α
inst✝¹ : IsSimpleOrder α
inst✝ : DecidableEq α
⊢ Finset.univ = {⊤, ⊥}

Tactic:
by

Goal state after:
Module: Mathlib.Order.Atoms.Finite

Goal state before:
pp: α : Type u_1
β : Type u_2
a : Bool
⊢ a = ⊥ ∨ a = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Order.Atoms.Finite

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : Preorder α
inst✝ : LocallyFiniteOrder α
a b : α
hab : a < b
⊢ b ∈ LocallyFiniteOrder.finsetIoc a b

Tactic:
by

Goal state after:
Module: Mathlib.Order.Atoms.Finite

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : Preorder α
inst✝ : LocallyFiniteOrder α
a b : α
hab : a < b
⊢ ∃ x, a ⋖ x ∧ x ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Order.Atoms.Finite

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : Preorder α
inst✝ : LocallyFiniteOrder α
⊢ IsStronglyCoatomic α

Tactic:
by

Goal state after:
Module: Mathlib.Order.Atoms.Finite

Goal state before:
pp: α : Type u_1
inst✝¹ : PartialOrder α
a : α
inst✝ : IsStronglyAtomic α
ha : (Set.Ici a).Infinite
hfin : {x | a ⋖ x}.Finite
h : ∀ (b : α), a ⋖ b → (Set.Ici b).Finite
⊢ ∀ i ∈ {x | a ⋖ x}, (Set.Ici i).Finite

Tactic:
by

Goal state after:
Module: Mathlib.Order.Atoms.Finite

Goal state before:
pp: α : Type u_1
inst✝¹ : PartialOrder α
a : α
inst✝ : IsStronglyAtomic α
ha : (Set.Ici a).Infinite
hfin : {x | a ⋖ x}.Finite
⊢ ∃ b, a ⋖ b ∧ (Set.Ici b).Infinite

Tactic:
by

Goal state after:
Module: Mathlib.Order.Atoms.Finite

Goal state before:
pp: α : Type u_1
inst✝¹ : PartialOrder α
a : α
inst✝ : IsStronglyCoatomic α
ha : (Set.Iic a).Infinite
hfin : {x | x ⋖ a}.Finite
⊢ ∃ b, b ⋖ a ∧ (Set.Iic b).Infinite

Tactic:
by

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

Goal state before:
pp: G : Type u_2
inst✝² : CommGroup G
inst✝¹ : IsCyclic G
inst✝ : Finite G
d : ℕ
⊢ Nat.card ↥(powMonoidHom d).ker = (Nat.card G).gcd d

Tactic:
by

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

Goal state before:
pp: G : Type u_2
inst✝² : CommGroup G
inst✝¹ : IsCyclic G
inst✝ : Finite G
d : ℕ
⊢ (powMonoidHom d).range.index = (Nat.card G).gcd d

Tactic:
by

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

Goal state before:
pp: α : Type u_1
G : Type u_2
G' : Type u_3
a : α
inst✝¹ : Group G
inst✝ : Group G'
g : G
hg : ∀ (x : G), x ∈ zpowers g
g' : G'
hg' : orderOf g' ∣ orderOf g
x y : G
⊢ g' ^ Classical.choose ⋯ = g' ^ Classical.choose ⋯ * g' ^ Classical.choose ⋯

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u_1
inst✝³ : Group G
p : ℕ
inst✝² : Fact (Nat.Prime p)
N : Subgroup G
inst✝¹ : N.Normal
inst✝ : Finite (Sylow p ↥N)
P : Sylow p ↥N
⊢ normalizer ↑(map N.subtype ↑P) ⊔ N = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u_1
inst✝³ : Group G
p : ℕ
inst✝² : Fact (Nat.Prime p)
N : Subgroup G
inst✝¹ : N.Normal
inst✝ : Finite (Sylow p ↥N)
P : Sylow p G
hP : ↑P ≤ N
⊢ normalizer ↑P ⊔ N = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝ : Group G
s : Subgroup G
t : Set (G ⧸ s)
⊢ Nat.card ↑(mk ⁻¹' t) = Nat.card ↥s * Nat.card ↑t

Tactic:
by

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

Goal state before:
pp: G : Type u_2
G' : Type u_3
inst✝¹ : Group G
inst✝ : Group G'
g : G
hg : ∀ (x : G), x ∈ zpowers g
g' : G'
hg' : orderOf g' ∣ orderOf g
⊢ (monoidHomOfForallMemZpowers hg hg') g = g'

Tactic:
by

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

Goal state before:
pp: G : Type u_2
G' : Type u_3
inst✝¹ : Group G
inst✝ : Group G'
g : G
hg : ∀ (x : G), x ∈ zpowers g
f₁ f₂ : G →* G'
⊢ f₁ = f₂ ↔ f₁ g = f₂ g

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝¹ : Group G
H : Subgroup G
inst✝ : Finite ↑↑H
x : G
hx : ∀ (n : G), n ∈ H ↔ x * n * x⁻¹ ∈ H
y✝ : G ⧸ H
y : G
hy : Quotient.mk'' y ∈ orbit ↥H ↑x
b : G
hb₁ : b ∈ H
hb₂✝ : (fun m ↦ m • ↑x) ⟨b, hb₁⟩ = Quotient.mk'' y
hb₂ : (b * x)⁻¹ * y ∈ H
⊢ b⁻¹ * (x * (y⁻¹ * x)⁻¹ * x⁻¹) ∈ H

Tactic:
by

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

Goal state before:
pp: α : Type u_1
G : Type u_2
G' : Type u_3
a : α
inst✝¹ : Group G
inst✝ : Group G'
g : G
hg : ∀ (x : G), x ∈ zpowers g
g' : G'
hg' : ∀ (x : G'), x ∈ zpowers g'
h : orderOf g = orderOf g'
⊢ (monoidHomOfForallMemZpowers hg' ⋯).comp (monoidHomOfForallMemZpowers hg ⋯) = MonoidHom.id G

Tactic:
by

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

Goal state before:
pp: α : Type u_1
G : Type u_2
G' : Type u_3
a : α
inst✝¹ : Group G
inst✝ : Group G'
g : G
hg : ∀ (x : G), x ∈ zpowers g
g' : G'
hg' : ∀ (x : G'), x ∈ zpowers g'
h : orderOf g = orderOf g'
⊢ (monoidHomOfForallMemZpowers hg ⋯).comp (monoidHomOfForallMemZpowers hg' ⋯) = MonoidHom.id G'

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝¹ : Group G
H : Subgroup G
inst✝ : Finite ↑↑H
⊢ ∀ (x y : ↥(normalizer ↑H)), (leftRel (comap (normalizer ↑H).subtype H)) x y ↔ (leftRel H) ↑x ↑y

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝¹ : Group G
inst✝ : Finite G
p n : ℕ
hp : Fact (Nat.Prime p)
H : Subgroup G
hH : Nat.card ↥H = p ^ n
⊢ Nat.card (↥(normalizer ↑H) ⧸ comap (normalizer ↑H).subtype H) ≡ Nat.card (G ⧸ H) [MOD p]

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝¹ : Group G
inst✝ : Finite G
p n : ℕ
hp : Fact (Nat.Prime p)
H : Subgroup G
hH : Nat.card ↥H = p ^ n
⊢ Nat.card ↥(normalizer ↑H) ≡ Nat.card G [MOD p ^ (n + 1)]

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝² : Group G
inst✝¹ : Finite G
p n : ℕ
inst✝ : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
⊢ Nat.card (G ⧸ H) * Nat.card ↥H = s * p * Nat.card ↥H

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompactlyGenerated.Intervals

Goal state before:
pp: α : Type u_2
inst✝ : CompleteLattice α
a : α
b : ↑(Iic a)
h : ∀ (ι : Type u_2) (s : ι → α), ↑b ≤ iSup s → ∃ t, ↑b ≤ ⨆ a ∈ t, s a
ι : Type u_2
s : ι → ↑(Iic a)
hb : ↑b ≤ iSup (Subtype.val ∘ s)
t : Finset ι
ht : ↑b ≤ ⨆ a_1 ∈ t, (Subtype.val ∘ s) a_1
⊢ ↑b ≤ ↑(⨆ a_1 ∈ t, s a_1)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝² : Group G
inst✝¹ : Finite G
p n : ℕ
inst✝ : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
hcard : Nat.card (G ⧸ H) = s * p
hm : s * p % p = Nat.card (↥(normalizer ↑H) ⧸ comap (normalizer ↑H).subtype H) % p
⊢ Nat.card (↥(normalizer ↑H) ⧸ comap (normalizer ↑H).subtype H) % p = 0

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompactlyGenerated.Intervals

Goal state before:
pp: α : Type u_2
inst✝ : CompleteLattice α
a : α
b : ↑(Iic a)
h : IsCompactElement ↑b
⊢ IsCompactElement b

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝¹ : Group G
inst✝ : Finite G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
⊢ Nat.card (G ⧸ H) * Nat.card ↥H = s * p * Nat.card ↥H

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝¹ : Group G
inst✝ : Finite G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
hcard : Nat.card (G ⧸ H) = s * p
hm : s * p % p = Nat.card (↥(normalizer ↑H) ⧸ H.subgroupOf (normalizer ↑H)) % p
⊢ Nat.card (↥(normalizer ↑H) ⧸ H.subgroupOf (normalizer ↑H)) % p = 0

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝¹ : Group G
inst✝ : Finite G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
hcard : Nat.card (G ⧸ H) = s * p
hm : s * p % p = Nat.card (↥(normalizer ↑H) ⧸ H.subgroupOf (normalizer ↑H)) % p
hm' : p ∣ Nat.card (↥(normalizer ↑H) ⧸ H.subgroupOf (normalizer ↑H))
x : ↥(normalizer ↑H) ⧸ H.subgroupOf (normalizer ↑H)
hx : orderOf x = p
hequiv : ↥H ≃ ↥(H.subgroupOf (normalizer ↑H))
this : Nat.card ↑(Subtype.val '' ?m.374) = p ^ (n + 1)
⊢ Nat.card ↥(Subgroup.map (normalizer ↑H).subtype (comap (mk' (H.subgroupOf (normalizer ↑H))) (zpowers x))) =
    p ^ (n + 1)

Tactic:
by

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

Goal state before:
pp: M : Type u_4
N : Type u_5
inst✝³ : CommGroup M
inst✝² : Group N
f : M →* N
h : (Nat.card ↥f.ker).Coprime (Nat.card ↥f.range)
inst✝¹ : IsCyclic ↥f.ker
inst✝ : IsCyclic ↥f.range
⊢ IsCyclic M

Tactic:
by

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

Goal state before:
pp: M : Type u_4
N : Type u_5
inst✝² : CommGroup M
inst✝¹ : Group N
f : M →* N
h : (Nat.card ↥f.ker).Coprime (Nat.card N)
inst✝ : IsCyclic ↥f.ker
hN : IsCyclic N
hf : Function.Surjective ⇑f
⊢ IsCyclic M

Tactic:
by

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

Goal state before:
pp: M : Type u_4
N : Type u_5
inst✝³ : Group M
inst✝² : Group N
cyc : IsCyclic (M × N)
inst✝¹ : Finite M
inst✝ : Finite N
⊢ (Nat.card M).Coprime (Nat.card N)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝¹ : Group G
inst✝ : Finite G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
hcard : Nat.card (G ⧸ H) = s * p
hm : s * p % p = Nat.card (↥(normalizer ↑H) ⧸ H.subgroupOf (normalizer ↑H)) % p
hm' : p ∣ Nat.card (↥(normalizer ↑H) ⧸ H.subgroupOf (normalizer ↑H))
x : ↥(normalizer ↑H) ⧸ H.subgroupOf (normalizer ↑H)
hx : orderOf x = p
hequiv : ↥H ≃ ↥(H.subgroupOf (normalizer ↑H))
⊢ Nat.card ↑↑(comap (mk' (H.subgroupOf (normalizer ↑H))) (zpowers x)) = p ^ n * p

Tactic:
← hH,

Goal state after:
G : Type u
inst✝¹ : Group G
inst✝ : Finite G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
hcard : Nat.card (G ⧸ H) = s * p
hm : s * p % p = Nat.card (↥(normalizer ↑H) ⧸ H.subgroupOf (normalizer ↑H)) % p
hm' : p ∣ Nat.card (↥(normalizer ↑H) ⧸ H.subgroupOf (normalizer ↑H))
x : ↥(normalizer ↑H) ⧸ H.subgroupOf (normalizer ↑H)
hx : orderOf x = p
hequiv : ↥H ≃ ↥(H.subgroupOf (normalizer ↑H))
⊢ Nat.card ↑↑(comap (mk' (H.subgroupOf (normalizer ↑H))) (zpowers x)) = Nat.card ↥H * p
Module: Mathlib.GroupTheory.SpecificGroups.Cyclic

Goal state before:
pp: M : Type u_4
N : Type u_5
inst✝³ : AddGroup M
inst✝² : AddGroup N
inst✝¹ : Infinite M
inst✝ : Nontrivial N
hMN : IsAddCyclic (ZMod 0 × ZMod (Nat.card N))
h✝ : Finite N
ZN : Type := ZMod (Nat.card N)
this : IsAddCyclic (ZMod (Nat.card N) × ZN)
⊢ Nat.card N = 1

Tactic:
by

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

Goal state before:
pp: M : Type u_4
N : Type u_5
inst✝³ : AddGroup M
inst✝² : AddGroup N
inst✝¹ : Infinite M
inst✝ : Nontrivial N
hMN : IsAddCyclic (M × N)
⊢ False

Tactic:
by

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

Goal state before:
pp: M : Type u_4
N : Type u_5
inst✝³ : Group M
inst✝² : Group N
inst✝¹ : Infinite M
inst✝ : Nontrivial N
⊢ ¬IsCyclic (M × N)

Tactic:
by

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

Goal state before:
pp: M : Type u_4
N : Type u_5
inst✝¹ : Group M
inst✝ : Group N
x✝ : IsCyclic M ∧ IsCyclic N ∧ (Nat.card M).Coprime (Nat.card N)
hM : IsCyclic M
hN : IsCyclic N
h : (Nat.card M).Coprime (Nat.card N)
f : M × N →* N := MonoidHom.snd M N
⊢ ↥f.ker ≃* M

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompactlyGenerated.Intervals

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝¹ : CompleteLattice α
inst✝ : IsCompactlyGenerated α
a : α
⊢ IsCompactlyGenerated ↑(Iic a)

Tactic:
by

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

Goal state before:
pp: M : Type u_4
N : Type u_5
inst✝¹ : Group M
inst✝ : Group N
⊢ IsCyclic (M × N) ↔ IsCyclic M ∧ IsCyclic N ∧ (Nat.card M).Coprime (Nat.card N)

Tactic:
by

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

Goal state before:
pp: α : Type u_1
G✝ : Type u_2
G' : Type u_3
a : α
G : Type u_4
inst✝¹ : Group G
inst✝ : IsCyclic G
⊢ IsCyclic (WithZero G)ˣ

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝¹ : Group G
inst✝ : Finite G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
hcard : Nat.card (G ⧸ H) = s * p
hm : s * p % p = Nat.card (↥(normalizer ↑H) ⧸ H.subgroupOf (normalizer ↑H)) % p
hm' : p ∣ Nat.card (↥(normalizer ↑H) ⧸ H.subgroupOf (normalizer ↑H))
x : ↥(normalizer ↑H) ⧸ H.subgroupOf (normalizer ↑H)
hx : orderOf x = p
hequiv : ↥H ≃ ↥(H.subgroupOf (normalizer ↑H))
⊢ Nat.card ↥(Subgroup.map (normalizer ↑H).subtype (comap (mk' (H.subgroupOf (normalizer ↑H))) (zpowers x))) =
    p ^ (n + 1)

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompactlyGenerated.Intervals

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝² : CompleteLattice α
inst✝¹ : IsModularLattice α
inst✝ : IsCompactlyGenerated α
s : Set ι
f : ι → α
h : ∀ i ∈ s, ComplementedLattice ↑(Iic (f i))
h' : ⨆ i ∈ s, f i = ⊤
i : ι
hi : i ∈ s
⊢ ∃ t, f i = sSup t ∧ ∀ a ∈ t, IsAtom a

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompactlyGenerated.Intervals

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝² : CompleteLattice α
inst✝¹ : IsModularLattice α
inst✝ : IsCompactlyGenerated α
s : Set ι
f : ι → α
h : ∀ i ∈ s, ComplementedLattice ↑(Iic (f i))
h' : ⨆ i ∈ s, f i = ⊤
t : (i : ι) → i ∈ s → Set α
ht : ∀ (i : ι) (a : i ∈ s), f i = sSup (t i a)
ht' : ∀ (i : ι) (a : i ∈ s), ∀ a_1 ∈ t i a, IsAtom a_1
u : Set α := ⋃ i, ⋃ (hi : i ∈ s), t i hi
⊢ u ⊆ {a | IsAtom a}

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompactlyGenerated.Intervals

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝² : CompleteLattice α
inst✝¹ : IsModularLattice α
inst✝ : IsCompactlyGenerated α
s : Set ι
f : ι → α
h : ∀ i ∈ s, ComplementedLattice ↑(Iic (f i))
h' : ⨆ i ∈ s, f i = ⊤
t : (i : ι) → i ∈ s → Set α
ht : ∀ (i : ι) (a : i ∈ s), f i = sSup (t i a)
ht' : ∀ (i : ι) (a : i ∈ s), ∀ a_1 ∈ t i a, IsAtom a_1
u : Set α := ⋃ i, ⋃ (hi : i ∈ s), t i hi
hu₁ : u ⊆ {a | IsAtom a}
⊢ sSup u = ⨆ i ∈ s, f i

Tactic:
by

Goal state after:
Module: Mathlib.Order.CompactlyGenerated.Intervals

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝² : CompleteLattice α
inst✝¹ : IsModularLattice α
inst✝ : IsCompactlyGenerated α
s : Set ι
f : ι → α
h : ∀ i ∈ s, ComplementedLattice ↑(Iic (f i))
h' : ⨆ i ∈ s, f i = ⊤
⊢ ComplementedLattice α

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝¹ : Group G
inst✝ : Finite G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
hcard : Nat.card (G ⧸ H) = s * p
hm : s * p % p = Nat.card (↥(normalizer ↑H) ⧸ H.subgroupOf (normalizer ↑H)) % p
hm' : p ∣ Nat.card (↥(normalizer ↑H) ⧸ H.subgroupOf (normalizer ↑H))
x : ↥(normalizer ↑H) ⧸ H.subgroupOf (normalizer ↑H)
hx : orderOf x = p
hequiv : ↥H ≃ ↥(H.subgroupOf (normalizer ↑H))
⊢ H ≤ Subgroup.map (normalizer ↑H).subtype (comap (mk' (H.subgroupOf (normalizer ↑H))) (zpowers x))

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝¹ : Group G
inst✝ : Finite G
p : ℕ
_hp✝ : Fact (Nat.Prime p)
n m : ℕ
hdvd : p ^ m ∣ Nat.card G
H : Subgroup G
hH : Nat.card ↥H = p ^ n
hnm✝ : n ≤ m
hnm : n < m
h0m : 0 < m
hnm1 : n ≤ m - 1
K : Subgroup G
hK : Nat.card ↥K = p ^ (m - 1) ∧ H ≤ K
⊢ p ^ (m - 1 + 1) ∣ Nat.card G

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝¹ : Group G
inst✝ : Finite G
p : ℕ
_hp✝ : Fact (Nat.Prime p)
n m : ℕ
hdvd : p ^ m ∣ Nat.card G
H : Subgroup G
hH : Nat.card ↥H = p ^ n
hnm✝ : n ≤ m
hnm : n < m
h0m : 0 < m
hnm1 : n ≤ m - 1
K : Subgroup G
hK : Nat.card ↥K = p ^ (m - 1) ∧ H ≤ K
hdvd' : p ^ (m - 1 + 1) ∣ Nat.card G
K' : Subgroup G
hK' : Nat.card ↥K' = p ^ (m - 1 + 1) ∧ K ≤ K'
⊢ Nat.card ↥K' = p ^ m

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝¹ : Group G
inst✝ : Finite G
p : ℕ
_hp✝ : Fact (Nat.Prime p)
n m : ℕ
hdvd : p ^ m ∣ Nat.card G
H : Subgroup G
hH : Nat.card ↥H = p ^ n
hnm✝ : n ≤ m
hnm : n = m
⊢ Nat.card ↥H = p ^ m ∧ H ≤ H

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝² : Group G
inst✝¹ : Finite G
p n : ℕ
inst✝ : Fact (Nat.Prime p)
hdvd : p ^ n ∣ Nat.card G
⊢ Nat.card ↥⊥ = p ^ 0

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
hn : p ^ n ≤ Nat.card G
this : Fact (Nat.Prime p)
⊢ Nat.card G ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
hn : p ^ n ≤ Nat.card G
⊢ ∃ H, Nat.card ↥H = p ^ n

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
H : Subgroup G
hn : p ^ n ≤ Nat.card ↥H
⊢ ∃ H' ≤ H, Nat.card ↥H' = p ^ n

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝ : Group G
k p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
H : Subgroup G
hk : k ≤ Nat.card ↥H
hk₀ : k ≠ 0
⊢ ∃ H' ≤ H, Nat.card ↥H' ≤ k ∧ k < p * Nat.card ↥H'

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝¹ : Group G
inst✝ : Finite G
p n : ℕ
hp : Fact (Nat.Prime p)
P : Sylow p G
hdvd : p ^ n ∣ Nat.card G
⊢ p ^ n ∣ Nat.card ↥↑P

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝² : Group G
inst✝¹ : Finite G
p : ℕ
inst✝ : Fact (Nat.Prime p)
P : Sylow p G
hdvd : p ∣ Nat.card G
⊢ p ∣ Nat.card ↥↑P

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝¹ : Group G
inst✝ : Finite G
p : ℕ
hp : Fact (Nat.Prime p)
P : Sylow p G
hdvd : p ∣ Nat.card G
⊢ ↑P ≠ ⊥

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝¹ : Group G
inst✝ : Finite G
p : ℕ
hp : Fact (Nat.Prime p)
P : Sylow p G
⊢ Nat.card ↥↑P = p ^ (Nat.card G).factorization p

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝¹ : Group G
inst✝ : Fintype G
p k : ℕ
hp : Nat.Prime p
h : p ^ k ∣ Nat.card G
this : Fact (Nat.Prime p)
H : Subgroup G
hH : Nat.card ↥H = p ^ k
g : G
hg : g ∈ (↑H \ {1}).toFinset
⊢ g ∈ H ∧ g ≠ 1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝¹ : Group G
inst✝ : Fintype G
p k : ℕ
hp : Nat.Prime p
h : p ^ k ∣ Nat.card G
this : Fact (Nat.Prime p)
H : Subgroup G
hH : Nat.card ↥H = p ^ k
g : G
hg✝ : g ∈ (↑H \ {1}).toFinset
hg : g ∈ H
hg1 : g ≠ 1
⊢ ⟨g, hg⟩ ≠ 1

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝¹ : Group G
inst✝ : Fintype G
p k : ℕ
hp : Nat.Prime p
h : p ^ k ∣ Nat.card G
this : Fact (Nat.Prime p)
H : Subgroup G
hH : Nat.card ↥H = p ^ k
g : G
hg : g ∈ (↑H \ {1}).toFinset
⊢ p ∣ orderOf g

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝¹ : Group G
inst✝ : Fintype G
⊢ Nat.card G ∣ ∏ g, orderOf g

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
h : (↑P).Normal
⊢ Unique (Sylow p G)

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝¹ : Group G
p : ℕ
inst✝ : Subsingleton (Sylow p G)
P : Sylow p G
⊢ (↑P).Characteristic

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
h : (↑P).Normal
⊢ (↑P).Characteristic

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
hn : (normalizer ↑P).Normal
⊢ (↑P).Normal

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝² : Group G
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
⊢ normalizer ↑(normalizer ↑P) = normalizer ↑P

Tactic:
by

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: case inr
G : Type u
inst✝³ : Group G
inst✝² : Finite G
hnc : ∀ (H : Subgroup G), IsCoatom H → H.Normal
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
K : Subgroup G
hK : IsCoatom K
hNK : normalizer ↑P ≤ K
⊢ normalizer ↑↑P = ⊤

Tactic:
have := hnc _ hK
        have hPK : P ≤ K := le_trans le_normalizer hNK
        refine (hK.1 ?_).elim
        rw [← sup_of_le_right hNK, P.normalizer_sup_eq_top' hPK]

Goal state after:
Module: Mathlib.GroupTheory.Sylow

Goal state before:
pp: G : Type u
inst✝³ : Group G
inst✝² : Finite G
hnc : ∀ (H : Subgroup G), IsCoatom H → H.Normal
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P : Sylow p G
⊢ normalizer ↑↑P = ⊤

Tactic:
by

Goal state after: