P. 196 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 19501-19600   *Has distinct variable group(s)

Type	Label	Description
Statement
Syntax	cpgp 19501	Extend class notation to include the class of all p-groups.
class pGrp
Syntax	cslw 19502	Extend class notation to include the class of all Sylow p-subgroups of a group.
class pSyl
Definition	df-od 19503*	Define the order of an element in a group. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by Stefan O'Rear, 4-Sep-2015.) (Revised by AV, 5-Oct-2020.)
⊢ od = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ ⦋{𝑛 ∈ ℕ ∣ (𝑛(.g‘𝑔)𝑥) = (0g‘𝑔)} / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
Definition	df-gex 19504*	Define the exponent of a group. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by Stefan O'Rear, 4-Sep-2015.) (Revised by AV, 26-Sep-2020.)
⊢ gEx = (𝑔 ∈ V ↦ ⦋{𝑛 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑛(.g‘𝑔)𝑥) = (0g‘𝑔)} / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
Definition	df-pgp 19505*	Define the set of p-groups, which are groups such that every element has a power of 𝑝 as its order. (Contributed by Mario Carneiro, 15-Jan-2015.) (Revised by AV, 5-Oct-2020.)
⊢ pGrp = {⟨𝑝, 𝑔⟩ ∣ ((𝑝 ∈ ℙ ∧ 𝑔 ∈ Grp) ∧ ∀𝑥 ∈ (Base‘𝑔)∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝↑𝑛))}
Definition	df-slw 19506*	Define the set of Sylow p-subgroups of a group 𝑔. A Sylow p-subgroup is a p-group that is not a subgroup of any other p-groups in 𝑔. (Contributed by Mario Carneiro, 16-Jan-2015.)
⊢ pSyl = (𝑝 ∈ ℙ, 𝑔 ∈ Grp ↦ {ℎ ∈ (SubGrp‘𝑔) ∣ ∀𝑘 ∈ (SubGrp‘𝑔)((ℎ ⊆ 𝑘 ∧ 𝑝 pGrp (𝑔 ↾s 𝑘)) ↔ ℎ = 𝑘)})
Theorem	odfval 19507*	Value of the order function. For a shorter proof using ax-rep 5212, see odfvalALT 19508. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by AV, 5-Oct-2020.) Remove dependency on ax-rep 5212. (Revised by Rohan Ridenour, 17-Aug-2023.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    ⇒   ⊢ 𝑂 = (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
Theorem	odfvalALT 19508*	Shorter proof of odfval 19507 using ax-rep 5212. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by AV, 5-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    ⇒   ⊢ 𝑂 = (𝑥 ∈ 𝑋 ↦ ⦋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
Theorem	odval 19509*	Second substitution for the group order definition. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by Stefan O'Rear, 5-Sep-2015.) (Revised by AV, 5-Oct-2020.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝐼 = {𝑦 ∈ ℕ ∣ (𝑦 · 𝐴) = 0 }    ⇒   ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
Theorem	odlem1 19510*	The group element order is either zero or a nonzero multiplier that annihilates the element. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.) (Revised by AV, 5-Oct-2020.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝐼 = {𝑦 ∈ ℕ ∣ (𝑦 · 𝐴) = 0 }    ⇒   ⊢ (𝐴 ∈ 𝑋 → (((𝑂‘𝐴) = 0 ∧ 𝐼 = ∅) ∨ (𝑂‘𝐴) ∈ 𝐼))
Theorem	odcl 19511	The order of a group element is always a nonnegative integer. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) ∈ ℕ0)
Theorem	odf 19512	Functionality of the group element order. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 5-Oct-2020.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    ⇒   ⊢ 𝑂:𝑋⟶ℕ0
Theorem	odid 19513	Any element to the power of its order is the identity. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑋 → ((𝑂‘𝐴) · 𝐴) = 0 )
Theorem	odlem2 19514	Any positive annihilator of a group element is an upper bound on the (positive) order of the element. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 5-Oct-2020.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ ∧ (𝑁 · 𝐴) = 0 ) → (𝑂‘𝐴) ∈ (1...𝑁))
Theorem	odmodnn0 19515	Reduce the argument of a group multiple by modding out the order of the element. (Contributed by Mario Carneiro, 23-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) ∧ (𝑂‘𝐴) ∈ ℕ) → ((𝑁 mod (𝑂‘𝐴)) · 𝐴) = (𝑁 · 𝐴))
Theorem	mndodconglem 19516	Lemma for mndodcong 19517. (Contributed by Mario Carneiro, 23-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑀 < (𝑂‘𝐴))    &   ⊢ (𝜑 → 𝑁 < (𝑂‘𝐴))    &   ⊢ (𝜑 → (𝑀 · 𝐴) = (𝑁 · 𝐴))    ⇒   ⊢ ((𝜑 ∧ 𝑀 ≤ 𝑁) → 𝑀 = 𝑁)
Theorem	mndodcong 19517	If two multipliers are congruent relative to the base point's order, the corresponding multiples are the same. (Contributed by Mario Carneiro, 23-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋) ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ (𝑂‘𝐴) ∈ ℕ) → ((𝑂‘𝐴) ∥ (𝑀 − 𝑁) ↔ (𝑀 · 𝐴) = (𝑁 · 𝐴)))
Theorem	mndodcongi 19518	If two multipliers are congruent relative to the base point's order, the corresponding multiples are the same. For monoids, the reverse implication is false for elements with infinite order. For example, the powers of 2 mod 10 are 1,2,4,8,6,2,4,8,6,... so that the identity 1 never repeats, which is infinite order by our definition, yet other numbers like 6 appear many times in the sequence. (Contributed by Mario Carneiro, 23-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ (𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)) → ((𝑂‘𝐴) ∥ (𝑀 − 𝑁) → (𝑀 · 𝐴) = (𝑁 · 𝐴)))
Theorem	oddvdsnn0 19519	The only multiples of 𝐴 that are equal to the identity are the multiples of the order of 𝐴. (Contributed by Mario Carneiro, 23-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → ((𝑂‘𝐴) ∥ 𝑁 ↔ (𝑁 · 𝐴) = 0 ))
Theorem	odnncl 19520	If a nonzero multiple of an element is zero, the element has positive order. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑁 ≠ 0 ∧ (𝑁 · 𝐴) = 0 )) → (𝑂‘𝐴) ∈ ℕ)
Theorem	odmod 19521	Reduce the argument of a group multiple by modding out the order of the element. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 6-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) ∈ ℕ) → ((𝑁 mod (𝑂‘𝐴)) · 𝐴) = (𝑁 · 𝐴))
Theorem	oddvds 19522	The only multiples of 𝐴 that are equal to the identity are the multiples of the order of 𝐴. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → ((𝑂‘𝐴) ∥ 𝑁 ↔ (𝑁 · 𝐴) = 0 ))
Theorem	oddvdsi 19523	Any group element is annihilated by any multiple of its order. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) ∥ 𝑁) → (𝑁 · 𝐴) = 0 )
Theorem	odcong 19524	If two multipliers are congruent relative to the base point's order, the corresponding multiples are the same. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑂‘𝐴) ∥ (𝑀 − 𝑁) ↔ (𝑀 · 𝐴) = (𝑁 · 𝐴)))
Theorem	odeq 19525*	The oddvds 19522 property uniquely defines the group order. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → (𝑁 = (𝑂‘𝐴) ↔ ∀𝑦 ∈ ℕ0 (𝑁 ∥ 𝑦 ↔ (𝑦 · 𝐴) = 0 )))
Theorem	odval2 19526*	A non-conditional definition of the group order. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) = (℩𝑥 ∈ ℕ0 ∀𝑦 ∈ ℕ0 (𝑥 ∥ 𝑦 ↔ (𝑦 · 𝐴) = 0 )))
Theorem	odcld 19527	The order of a group element is always a nonnegative integer, deduction form of odcl 19511. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℕ0)
Theorem	odm1inv 19528	The (order-1)th multiple of an element is its inverse. (Contributed by SN, 31-Jan-2025.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (((𝑂‘𝐴) − 1) · 𝐴) = (𝐼‘𝐴))
Theorem	odmulgid 19529	A relationship between the order of a multiple and the order of the basepoint. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ 𝐾 ∈ ℤ) → ((𝑂‘(𝑁 · 𝐴)) ∥ 𝐾 ↔ (𝑂‘𝐴) ∥ (𝐾 · 𝑁)))
Theorem	odmulg2 19530	The order of a multiple divides the order of the base point. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → (𝑂‘(𝑁 · 𝐴)) ∥ (𝑂‘𝐴))
Theorem	odmulg 19531	Relationship between the order of an element and that of a multiple. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) → (𝑂‘𝐴) = ((𝑁 gcd (𝑂‘𝐴)) · (𝑂‘(𝑁 · 𝐴))))
Theorem	odmulgeq 19532	A multiple of a point of finite order only has the same order if the multiplier is relatively prime. (Contributed by Stefan O'Rear, 12-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑂‘𝐴) ∈ ℕ) → ((𝑂‘(𝑁 · 𝐴)) = (𝑂‘𝐴) ↔ (𝑁 gcd (𝑂‘𝐴)) = 1))
Theorem	odbezout 19533*	If 𝑁 is coprime to the order of 𝐴, there is a modular inverse 𝑥 to cancel multiplication by 𝑁. (Contributed by Mario Carneiro, 27-Apr-2016.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ (((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑁 ∈ ℤ) ∧ (𝑁 gcd (𝑂‘𝐴)) = 1) → ∃𝑥 ∈ ℤ (𝑥 · (𝑁 · 𝐴)) = 𝐴)
Theorem	od1 19534	The order of the group identity is one. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)
⊢ 𝑂 = (od‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → (𝑂‘ 0 ) = 1)
Theorem	odeq1 19535	The group identity is the unique element of a group with order one. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)
⊢ 𝑂 = (od‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑋 = (Base‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝑂‘𝐴) = 1 ↔ 𝐴 = 0 ))
Theorem	odinv 19536	The order of the inverse of a group element. (Contributed by Mario Carneiro, 20-Oct-2015.)
⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    &   ⊢ 𝑋 = (Base‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑂‘(𝐼‘𝐴)) = (𝑂‘𝐴))
Theorem	odf1 19537*	The multiples of an element with infinite order form an infinite cyclic subgroup of 𝐺. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝑂‘𝐴) = 0 ↔ 𝐹:ℤ–1-1→𝑋))
Theorem	odinf 19538*	The multiples of an element with infinite order form an infinite cyclic subgroup of 𝐺. (Contributed by Mario Carneiro, 14-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → ¬ ran 𝐹 ∈ Fin)
Theorem	dfod2 19539*	An alternative definition of the order of a group element is as the cardinality of the cyclic subgroup generated by the element. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐹 = (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴))    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) = if(ran 𝐹 ∈ Fin, (♯‘ran 𝐹), 0))
Theorem	odcl2 19540	The order of an element of a finite group is finite. (Contributed by Mario Carneiro, 14-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℕ)
Theorem	oddvds2 19541	The order of an element of a finite group divides the order (cardinality) of the group. Corollary of Lagrange's theorem for the order of a subgroup. (Contributed by Mario Carneiro, 14-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) ∥ (♯‘𝑋))
Theorem	finodsubmsubg 19542*	A submonoid whose elements have finite order is a subgroup. (Contributed by SN, 31-Jan-2025.)
⊢ 𝑂 = (od‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺))    &   ⊢ (𝜑 → ∀𝑎 ∈ 𝑆 (𝑂‘𝑎) ∈ ℕ)    ⇒   ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))
Theorem	0subgALT 19543	A shorter proof of 0subg 19127 using df-od 19503. (Contributed by SN, 31-Jan-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺))
Theorem	submod 19544	The order of an element is the same in a submonoid. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by AV, 5-Oct-2020.)
⊢ 𝐻 = (𝐺 ↾s 𝑌)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝑃 = (od‘𝐻)    ⇒   ⊢ ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴 ∈ 𝑌) → (𝑂‘𝐴) = (𝑃‘𝐴))
Theorem	subgod 19545	The order of an element is the same in a subgroup. (Contributed by Mario Carneiro, 14-Jan-2015.) (Proof shortened by Stefan O'Rear, 12-Sep-2015.)
⊢ 𝐻 = (𝐺 ↾s 𝑌)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝑃 = (od‘𝐻)    ⇒   ⊢ ((𝑌 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑌) → (𝑂‘𝐴) = (𝑃‘𝐴))
Theorem	odsubdvds 19546	The order of an element of a subgroup divides the order of the subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
⊢ 𝑂 = (od‘𝐺)    ⇒   ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ∈ Fin ∧ 𝐴 ∈ 𝑆) → (𝑂‘𝐴) ∥ (♯‘𝑆))
Theorem	odf1o1 19547*	An element with zero order has infinitely many multiples. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → (𝑥 ∈ ℤ ↦ (𝑥 · 𝐴)):ℤ–1-1-onto→(𝐾‘{𝐴}))
Theorem	odf1o2 19548*	An element with nonzero order has as many multiples as its order. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑥 ∈ (0..^(𝑂‘𝐴)) ↦ (𝑥 · 𝐴)):(0..^(𝑂‘𝐴))–1-1-onto→(𝐾‘{𝐴}))
Theorem	odhash 19549	An element of zero order generates an infinite subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) = 0) → (♯‘(𝐾‘{𝐴})) = +∞)
Theorem	odhash2 19550	If an element has nonzero order, it generates a subgroup with size equal to the order. (Contributed by Stefan O'Rear, 12-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) ∈ ℕ) → (♯‘(𝐾‘{𝐴})) = (𝑂‘𝐴))
Theorem	odhash3 19551	An element which generates a finite subgroup has order the size of that subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝐾‘{𝐴}) ∈ Fin) → (𝑂‘𝐴) = (♯‘(𝐾‘{𝐴})))
Theorem	odngen 19552*	A cyclic subgroup of size (𝑂‘𝐴) has (ϕ‘(𝑂‘𝐴)) generators. (Contributed by Stefan O'Rear, 12-Sep-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    &   ⊢ 𝐾 = (mrCls‘(SubGrp‘𝐺))    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑂‘𝐴) ∈ ℕ) → (♯‘{𝑥 ∈ (𝐾‘{𝐴}) ∣ (𝑂‘𝑥) = (𝑂‘𝐴)}) = (ϕ‘(𝑂‘𝐴)))
Theorem	gexval 19553*	Value of the exponent of a group. (Contributed by Mario Carneiro, 23-Apr-2016.) (Revised by AV, 26-Sep-2020.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝐸 = (gEx‘𝐺)    &   ⊢ 𝐼 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 }    ⇒   ⊢ (𝐺 ∈ 𝑉 → 𝐸 = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
Theorem	gexlem1 19554*	The group element order is either zero or a nonzero multiplier that annihilates the element. (Contributed by Mario Carneiro, 23-Apr-2016.) (Proof shortened by AV, 26-Sep-2020.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝐸 = (gEx‘𝐺)    &   ⊢ 𝐼 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 }    ⇒   ⊢ (𝐺 ∈ 𝑉 → ((𝐸 = 0 ∧ 𝐼 = ∅) ∨ 𝐸 ∈ 𝐼))
Theorem	gexcl 19555	The exponent of a group is a nonnegative integer. (Contributed by Mario Carneiro, 23-Apr-2016.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐸 = (gEx‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑉 → 𝐸 ∈ ℕ0)
Theorem	gexid 19556	Any element to the power of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐸 = (gEx‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (𝐴 ∈ 𝑋 → (𝐸 · 𝐴) = 0 )
Theorem	gexlem2 19557*	Any positive annihilator of all the group elements is an upper bound on the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.) (Proof shortened by AV, 26-Sep-2020.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐸 = (gEx‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑁 ∈ ℕ ∧ ∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 ) → 𝐸 ∈ (1...𝑁))
Theorem	gexdvdsi 19558	Any group element is annihilated by any multiple of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐸 = (gEx‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐸 ∥ 𝑁) → (𝑁 · 𝐴) = 0 )
Theorem	gexdvds 19559*	The only 𝑁 that annihilate all the elements of the group are the multiples of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐸 = (gEx‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → (𝐸 ∥ 𝑁 ↔ ∀𝑥 ∈ 𝑋 (𝑁 · 𝑥) = 0 ))
Theorem	gexdvds2 19560*	An integer divides the group exponent iff it divides all the group orders. In other words, the group exponent is the LCM of the orders of all the elements. (Contributed by Mario Carneiro, 24-Apr-2016.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐸 = (gEx‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ) → (𝐸 ∥ 𝑁 ↔ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∥ 𝑁))
Theorem	gexod 19561	Any group element is annihilated by any multiple of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐸 = (gEx‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) ∥ 𝐸)
Theorem	gexcl3 19562*	If the order of every group element is bounded by 𝑁, the group has finite exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐸 = (gEx‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∈ (1...𝑁)) → 𝐸 ∈ ℕ)
Theorem	gexnnod 19563	Every group element has finite order if the exponent is finite. (Contributed by Mario Carneiro, 24-Apr-2016.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐸 = (gEx‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝐸 ∈ ℕ ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℕ)
Theorem	gexcl2 19564	The exponent of a finite group is finite. (Contributed by Mario Carneiro, 24-Apr-2016.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐸 = (gEx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → 𝐸 ∈ ℕ)
Theorem	gexdvds3 19565	The exponent of a finite group divides the order (cardinality) of the group. Corollary of Lagrange's theorem for the order of a subgroup. (Contributed by Mario Carneiro, 24-Apr-2016.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐸 = (gEx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → 𝐸 ∥ (♯‘𝑋))
Theorem	gex1 19566	A group or monoid has exponent 1 iff it is trivial. (Contributed by Mario Carneiro, 24-Apr-2016.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐸 = (gEx‘𝐺)    ⇒   ⊢ (𝐺 ∈ Mnd → (𝐸 = 1 ↔ 𝑋 ≈ 1o))
Theorem	ispgp 19567*	A group is a 𝑃-group if every element has some power of 𝑃 as its order. (Contributed by Mario Carneiro, 15-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    ⇒   ⊢ (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 ∃𝑛 ∈ ℕ0 (𝑂‘𝑥) = (𝑃↑𝑛)))
Theorem	pgpprm 19568	Reverse closure for the first argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.)
⊢ (𝑃 pGrp 𝐺 → 𝑃 ∈ ℙ)
Theorem	pgpgrp 19569	Reverse closure for the second argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.)
⊢ (𝑃 pGrp 𝐺 → 𝐺 ∈ Grp)
Theorem	pgpfi1 19570	A finite group with order a power of a prime 𝑃 is a 𝑃-group. (Contributed by Mario Carneiro, 16-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → ((♯‘𝑋) = (𝑃↑𝑁) → 𝑃 pGrp 𝐺))
Theorem	pgp0 19571	The identity subgroup is a 𝑃-group for every prime 𝑃. (Contributed by Mario Carneiro, 19-Jan-2015.)
⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ ℙ) → 𝑃 pGrp (𝐺 ↾s { 0 }))
Theorem	subgpgp 19572	A subgroup of a p-group is a p-group. (Contributed by Mario Carneiro, 27-Apr-2016.)
⊢ ((𝑃 pGrp 𝐺 ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑃 pGrp (𝐺 ↾s 𝑆))
Theorem	sylow1lem1 19573*	Lemma for sylow1 19578. The p-adic valuation of the size of 𝑆 is equal to the number of excess powers of 𝑃 in (♯‘𝑋) / (𝑃↑𝑁). (Contributed by Mario Carneiro, 15-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → (𝑃↑𝑁) ∥ (♯‘𝑋))    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)}    ⇒   ⊢ (𝜑 → ((♯‘𝑆) ∈ ℕ ∧ (𝑃 pCnt (♯‘𝑆)) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
Theorem	sylow1lem2 19574*	Lemma for sylow1 19578. The function ⊕ is a group action on 𝑆. (Contributed by Mario Carneiro, 15-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → (𝑃↑𝑁) ∥ (♯‘𝑋))    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)}    &   ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑆 ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))    ⇒   ⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct 𝑆))
Theorem	sylow1lem3 19575*	Lemma for sylow1 19578. One of the orbits of the group action has p-adic valuation less than the prime count of the set 𝑆. (Contributed by Mario Carneiro, 15-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → (𝑃↑𝑁) ∥ (♯‘𝑋))    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)}    &   ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑆 ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))    &   ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑆 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}    ⇒   ⊢ (𝜑 → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (♯‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
Theorem	sylow1lem4 19576*	Lemma for sylow1 19578. The stabilizer subgroup of any element of 𝑆 is at most 𝑃↑𝑁 in size. (Contributed by Mario Carneiro, 15-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → (𝑃↑𝑁) ∥ (♯‘𝑋))    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)}    &   ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑆 ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))    &   ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑆 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐵) = 𝐵}    ⇒   ⊢ (𝜑 → (♯‘𝐻) ≤ (𝑃↑𝑁))
Theorem	sylow1lem5 19577*	Lemma for sylow1 19578. Using Lagrange's theorem and the orbit-stabilizer theorem, show that there is a subgroup with size exactly 𝑃↑𝑁. (Contributed by Mario Carneiro, 16-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → (𝑃↑𝑁) ∥ (♯‘𝑋))    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)}    &   ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑆 ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))    &   ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑆 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐵) = 𝐵}    &   ⊢ (𝜑 → (𝑃 pCnt (♯‘[𝐵] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))    ⇒   ⊢ (𝜑 → ∃ℎ ∈ (SubGrp‘𝐺)(♯‘ℎ) = (𝑃↑𝑁))
Theorem	sylow1 19578*	Sylow's first theorem. If 𝑃↑𝑁 is a prime power that divides the cardinality of 𝐺, then 𝐺 has a supgroup with size 𝑃↑𝑁. This is part of Metamath 100 proof #72. (Contributed by Mario Carneiro, 16-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → (𝑃↑𝑁) ∥ (♯‘𝑋))    ⇒   ⊢ (𝜑 → ∃𝑔 ∈ (SubGrp‘𝐺)(♯‘𝑔) = (𝑃↑𝑁))
Theorem	odcau 19579*	Cauchy's theorem for the order of an element in a group. A finite group whose order divides a prime 𝑃 contains an element of order 𝑃. (Contributed by Mario Carneiro, 16-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑂 = (od‘𝐺)    ⇒   ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) ∧ 𝑃 ∥ (♯‘𝑋)) → ∃𝑔 ∈ 𝑋 (𝑂‘𝑔) = 𝑃)
Theorem	pgpfi 19580*	The converse to pgpfi1 19570. A finite group is a 𝑃-group iff it has size some power of 𝑃. (Contributed by Mario Carneiro, 16-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ ∃𝑛 ∈ ℕ0 (♯‘𝑋) = (𝑃↑𝑛))))
Theorem	pgpfi2 19581	Alternate version of pgpfi 19580. (Contributed by Mario Carneiro, 27-Apr-2016.)
⊢ 𝑋 = (Base‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin) → (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ (♯‘𝑋) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))))
Theorem	pgphash 19582	The order of a p-group. (Contributed by Mario Carneiro, 27-Apr-2016.)
⊢ 𝑋 = (Base‘𝐺)    ⇒   ⊢ ((𝑃 pGrp 𝐺 ∧ 𝑋 ∈ Fin) → (♯‘𝑋) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
Theorem	isslw 19583*	The property of being a Sylow subgroup. A Sylow 𝑃-subgroup is a 𝑃-group which has no proper supersets that are also 𝑃-groups. (Contributed by Mario Carneiro, 16-Jan-2015.)
⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ ∀𝑘 ∈ (SubGrp‘𝐺)((𝐻 ⊆ 𝑘 ∧ 𝑃 pGrp (𝐺 ↾s 𝑘)) ↔ 𝐻 = 𝑘)))
Theorem	slwprm 19584	Reverse closure for the first argument of a Sylow 𝑃-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.) (Revised by Mario Carneiro, 2-May-2015.)
⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 ∈ ℙ)
Theorem	slwsubg 19585	A Sylow 𝑃-subgroup is a subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝐻 ∈ (SubGrp‘𝐺))
Theorem	slwispgp 19586	Defining property of a Sylow 𝑃-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
⊢ 𝑆 = (𝐺 ↾s 𝐾)    ⇒   ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ((𝐻 ⊆ 𝐾 ∧ 𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾))
Theorem	slwpss 19587	A proper superset of a Sylow subgroup is not a 𝑃-group. (Contributed by Mario Carneiro, 16-Jan-2015.)
⊢ 𝑆 = (𝐺 ↾s 𝐾)    ⇒   ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻 ⊊ 𝐾) → ¬ 𝑃 pGrp 𝑆)
Theorem	slwpgp 19588	A Sylow 𝑃-subgroup is a 𝑃-group. (Contributed by Mario Carneiro, 16-Jan-2015.)
⊢ 𝑆 = (𝐺 ↾s 𝐻)    ⇒   ⊢ (𝐻 ∈ (𝑃 pSyl 𝐺) → 𝑃 pGrp 𝑆)
Theorem	pgpssslw 19589*	Every 𝑃-subgroup is contained in a Sylow 𝑃-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑆 = (𝐺 ↾s 𝐻)    &   ⊢ 𝐹 = (𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾s 𝑦) ∧ 𝐻 ⊆ 𝑦)} ↦ (♯‘𝑥))    ⇒   ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∃𝑘 ∈ (𝑃 pSyl 𝐺)𝐻 ⊆ 𝑘)
Theorem	slwn0 19590	Every finite group contains a Sylow 𝑃-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) → (𝑃 pSyl 𝐺) ≠ ∅)
Theorem	subgslw 19591	A Sylow subgroup that is contained in a larger subgroup is also Sylow with respect to the subgroup. (The converse need not be true.) (Contributed by Mario Carneiro, 19-Jan-2015.)
⊢ 𝐻 = (𝐺 ↾s 𝑆)    ⇒   ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐾 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ⊆ 𝑆) → 𝐾 ∈ (𝑃 pSyl 𝐻))
Theorem	sylow2alem1 19592*	Lemma for sylow2a 19594. An equivalence class of fixed points is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct 𝑌))    &   ⊢ (𝜑 → 𝑃 pGrp 𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑌 ∈ Fin)    &   ⊢ 𝑍 = {𝑢 ∈ 𝑌 ∣ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢}    &   ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → [𝐴] ∼ = {𝐴})
Theorem	sylow2alem2 19593*	Lemma for sylow2a 19594. All the orbits which are not for fixed points have size ∣ 𝐺 ∣ / ∣ 𝐺𝑥 ∣ (where 𝐺𝑥 is the stabilizer subgroup) and thus are powers of 𝑃. And since they are all nontrivial (because any orbit which is a singleton is a fixed point), they all divide 𝑃, and so does the sum of all of them. (Contributed by Mario Carneiro, 17-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct 𝑌))    &   ⊢ (𝜑 → 𝑃 pGrp 𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑌 ∈ Fin)    &   ⊢ 𝑍 = {𝑢 ∈ 𝑌 ∣ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢}    &   ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}    ⇒   ⊢ (𝜑 → 𝑃 ∥ Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧))
Theorem	sylow2a 19594*	A named lemma of Sylow's second and third theorems. If 𝐺 is a finite 𝑃-group that acts on the finite set 𝑌, then the set 𝑍 of all points of 𝑌 fixed by every element of 𝐺 has cardinality equivalent to the cardinality of 𝑌, mod 𝑃. (Contributed by Mario Carneiro, 17-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct 𝑌))    &   ⊢ (𝜑 → 𝑃 pGrp 𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑌 ∈ Fin)    &   ⊢ 𝑍 = {𝑢 ∈ 𝑌 ∣ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢}    &   ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}    ⇒   ⊢ (𝜑 → 𝑃 ∥ ((♯‘𝑌) − (♯‘𝑍)))
Theorem	sylow2blem1 19595*	Lemma for sylow2b 19598. Evaluate the group action on a left coset. (Contributed by Mario Carneiro, 17-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺))    &   ⊢ + = (+g‘𝐺)    &   ⊢ ∼ = (𝐺 ~QG 𝐾)    &   ⊢ · = (𝑥 ∈ 𝐻, 𝑦 ∈ (𝑋 / ∼ ) ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))    ⇒   ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐻 ∧ 𝐶 ∈ 𝑋) → (𝐵 · [𝐶] ∼ ) = [(𝐵 + 𝐶)] ∼ )
Theorem	sylow2blem2 19596*	Lemma for sylow2b 19598. Left multiplication in a subgroup 𝐻 is a group action on the set of all left cosets of 𝐾. (Contributed by Mario Carneiro, 17-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺))    &   ⊢ + = (+g‘𝐺)    &   ⊢ ∼ = (𝐺 ~QG 𝐾)    &   ⊢ · = (𝑥 ∈ 𝐻, 𝑦 ∈ (𝑋 / ∼ ) ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))    ⇒   ⊢ (𝜑 → · ∈ ((𝐺 ↾s 𝐻) GrpAct (𝑋 / ∼ )))
Theorem	sylow2blem3 19597*	Sylow's second theorem. Putting together the results of sylow2a 19594 and the orbit-stabilizer theorem to show that 𝑃 does not divide the set of all fixed points under the group action, we get that there is a fixed point of the group action, so that there is some 𝑔 ∈ 𝑋 with ℎ𝑔𝐾 = 𝑔𝐾 for all ℎ ∈ 𝐻. This implies that invg(𝑔)ℎ𝑔 ∈ 𝐾, so ℎ is in the conjugated subgroup 𝑔𝐾invg(𝑔). (Contributed by Mario Carneiro, 18-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺))    &   ⊢ + = (+g‘𝐺)    &   ⊢ ∼ = (𝐺 ~QG 𝐾)    &   ⊢ · = (𝑥 ∈ 𝐻, 𝑦 ∈ (𝑋 / ∼ ) ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))    &   ⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾s 𝐻))    &   ⊢ (𝜑 → (♯‘𝐾) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
Theorem	sylow2b 19598*	Sylow's second theorem. Any 𝑃-group 𝐻 is a subgroup of a conjugated 𝑃-group 𝐾 of order 𝑃↑𝑛 ∥ (♯‘𝑋) with 𝑛 maximal. This is usually stated under the assumption that 𝐾 is a Sylow subgroup, but we use a slightly different definition, whose equivalence to this one requires this theorem. This is part of Metamath 100 proof #72. (Contributed by Mario Carneiro, 18-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺))    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾s 𝐻))    &   ⊢ (𝜑 → (♯‘𝐾) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
Theorem	slwhash 19599	A sylow subgroup has cardinality equal to the maximum power of 𝑃 dividing the group. (Contributed by Mario Carneiro, 18-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐻 ∈ (𝑃 pSyl 𝐺))    ⇒   ⊢ (𝜑 → (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
Theorem	fislw 19600	The sylow subgroups of a finite group are exactly the groups which have cardinality equal to the maximum power of 𝑃 dividing the group. (Contributed by Mario Carneiro, 16-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ) → (𝐻 ∈ (𝑃 pSyl 𝐺) ↔ (𝐻 ∈ (SubGrp‘𝐺) ∧ (♯‘𝐻) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))))