P. 184 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 18301-18400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	psgnunilem4 18301	Lemma for psgnuni 18303. An odd-length representation of the identity is impossible, as it could be repeatedly shortened to a length of 1, but a length 1 permutation must be a transposition. (Contributed by Stefan O'Rear, 25-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑊 ∈ Word 𝑇)    &   ⊢ (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))    ⇒   ⊢ (𝜑 → (-1↑(♯‘𝑊)) = 1)
Theorem	m1expaddsub 18302	Addition and subtraction of parities are the same. (Contributed by Stefan O'Rear, 27-Aug-2015.)
⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (-1↑(𝑋 − 𝑌)) = (-1↑(𝑋 + 𝑌)))
Theorem	psgnuni 18303	If the same permutation can be written in more than one way as a product of transpositions, the parity of those products must agree; otherwise the product of one with the inverse of the other would be an odd representation of the identity. (Contributed by Stefan O'Rear, 27-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑊 ∈ Word 𝑇)    &   ⊢ (𝜑 → 𝑋 ∈ Word 𝑇)    &   ⊢ (𝜑 → (𝐺 Σg 𝑊) = (𝐺 Σg 𝑋))    ⇒   ⊢ (𝜑 → (-1↑(♯‘𝑊)) = (-1↑(♯‘𝑋)))
Theorem	psgnfval 18304*	Function definition of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐹 = {𝑝 ∈ 𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin}    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ 𝑁 = (𝑥 ∈ 𝐹 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))
Theorem	psgnfn 18305*	Functionality and domain of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐹 = {𝑝 ∈ 𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin}    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ 𝑁 Fn 𝐹
Theorem	psgndmsubg 18306	The finitary permutations are a subgroup. (Contributed by Stefan O'Rear, 28-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ (𝐷 ∈ 𝑉 → dom 𝑁 ∈ (SubGrp‘𝐺))
Theorem	psgneldm 18307	Property of being a finitary permutation. (Contributed by Stefan O'Rear, 28-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝑃 ∈ dom 𝑁 ↔ (𝑃 ∈ 𝐵 ∧ dom (𝑃 ∖ I ) ∈ Fin))
Theorem	psgneldm2 18308*	The finitary permutations are the span of the transpositions. (Contributed by Stefan O'Rear, 28-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ (𝐷 ∈ 𝑉 → (𝑃 ∈ dom 𝑁 ↔ ∃𝑤 ∈ Word 𝑇𝑃 = (𝐺 Σg 𝑤)))
Theorem	psgneldm2i 18309	A sequence of transpositions describes a finitary permutation. (Contributed by Stefan O'Rear, 28-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → (𝐺 Σg 𝑊) ∈ dom 𝑁)
Theorem	psgneu 18310*	A finitary permutation has exactly one parity. (Contributed by Stefan O'Rear, 28-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ (𝑃 ∈ dom 𝑁 → ∃!𝑠∃𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))
Theorem	psgnval 18311*	Value of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ (𝑃 ∈ dom 𝑁 → (𝑁‘𝑃) = (℩𝑠∃𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))
Theorem	psgnvali 18312*	A finitary permutation has at least one representation for its parity. (Contributed by Stefan O'Rear, 28-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ (𝑃 ∈ dom 𝑁 → ∃𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ (𝑁‘𝑃) = (-1↑(♯‘𝑤))))
Theorem	psgnvalii 18313	Any representation of a permutation is length matching the permutation sign. (Contributed by Stefan O'Rear, 28-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ Word 𝑇) → (𝑁‘(𝐺 Σg 𝑊)) = (-1↑(♯‘𝑊)))
Theorem	psgnpmtr 18314	All transpositions are odd. (Contributed by Stefan O'Rear, 29-Aug-2015.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ (𝑃 ∈ 𝑇 → (𝑁‘𝑃) = -1)
Theorem	psgn0fv0 18315	The permutation sign function for an empty set at an empty set is 1. (Contributed by AV, 27-Feb-2019.)
⊢ ((pmSgn‘∅)‘∅) = 1
Theorem	sygbasnfpfi 18316	The class of non-fixed points of a permutation of a finite set is finite. (Contributed by AV, 13-Jan-2019.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ ((𝐷 ∈ Fin ∧ 𝑃 ∈ 𝐵) → dom (𝑃 ∖ I ) ∈ Fin)
Theorem	psgnfvalfi 18317*	Function definition of the permutation sign function for permutations of finite sets. (Contributed by AV, 13-Jan-2019.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ (𝐷 ∈ Fin → 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑠∃𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))))
Theorem	psgnvalfi 18318*	Value of the permutation sign function for permutations of finite sets. (Contributed by AV, 13-Jan-2019.)
⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ ((𝐷 ∈ Fin ∧ 𝑃 ∈ 𝐵) → (𝑁‘𝑃) = (℩𝑠∃𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))
Theorem	psgnran 18319	The range of the permutation sign function for finite permutations. (Contributed by AV, 1-Jan-2019.)
⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑆 = (pmSgn‘𝑁)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑄 ∈ 𝑃) → (𝑆‘𝑄) ∈ {1, -1})
Theorem	gsmtrcl 18320	The group sum of transpositions of a finite set is a permutation, see also psgneldm2i 18309. (Contributed by AV, 19-Jan-2019.)
⊢ 𝑆 = (SymGrp‘𝑁)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝑇 = ran (pmTrsp‘𝑁)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑊 ∈ Word 𝑇) → (𝑆 Σg 𝑊) ∈ 𝐵)
Theorem	psgnfitr 18321*	A permutation of a finite set is generated by transpositions. (Contributed by AV, 13-Jan-2019.)
⊢ 𝐺 = (SymGrp‘𝑁)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑇 = ran (pmTrsp‘𝑁)    ⇒   ⊢ (𝑁 ∈ Fin → (𝑄 ∈ 𝐵 ↔ ∃𝑤 ∈ Word 𝑇𝑄 = (𝐺 Σg 𝑤)))
Theorem	psgnfieu 18322*	A permutation of a finite set has exactly one parity. (Contributed by AV, 13-Jan-2019.)
⊢ 𝐺 = (SymGrp‘𝑁)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑇 = ran (pmTrsp‘𝑁)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑄 ∈ 𝐵) → ∃!𝑠∃𝑤 ∈ Word 𝑇(𝑄 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))
Theorem	pmtrsn 18323	The value of the transposition generator function for a singleton is empty, i.e. there is no transposition for a singleton. This also holds for 𝐴 ∉ V, i.e. for the empty set {𝐴} = ∅ resulting in (pmTrsp‘∅) = ∅. (Contributed by AV, 6-Aug-2019.)
⊢ (pmTrsp‘{𝐴}) = ∅
Theorem	psgnsn 18324	The permutation sign function for a singleton. (Contributed by AV, 6-Aug-2019.)
⊢ 𝐷 = {𝐴}    &   ⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = 1)
Theorem	psgnprfval 18325*	The permutation sign function for a pair. (Contributed by AV, 10-Dec-2018.)
⊢ 𝐷 = {1, 2}    &   ⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ (𝑋 ∈ 𝐵 → (𝑁‘𝑋) = (℩𝑠∃𝑤 ∈ Word 𝑇(𝑋 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))))
Theorem	psgnprfval1 18326	The permutation sign of the identity for a pair. (Contributed by AV, 11-Dec-2018.)
⊢ 𝐷 = {1, 2}    &   ⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ (𝑁‘{⟨1, 1⟩, ⟨2, 2⟩}) = 1
Theorem	psgnprfval2 18327	The permutation sign of the transposition for a pair. (Contributed by AV, 10-Dec-2018.)
⊢ 𝐷 = {1, 2}    &   ⊢ 𝐺 = (SymGrp‘𝐷)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑇 = ran (pmTrsp‘𝐷)    &   ⊢ 𝑁 = (pmSgn‘𝐷)    ⇒   ⊢ (𝑁‘{⟨1, 2⟩, ⟨2, 1⟩}) = -1
10.2.10  p-Groups and Sylow groups; Sylow's theorems
Syntax	cod 18328	Extend class notation to include the order function on the elements of a group.
class od
Syntax	cgex 18329	Extend class notation to include the order function on the elements of a group.
class gEx
Syntax	cpgp 18330	Extend class notation to include the class of all p-groups.
class pGrp
Syntax	cslw 18331	Extend class notation to include the class of all Sylow p-subgroups of a group.
class pSyl
Definition	df-od 18332*	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 18333*	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 18334*	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 18335*	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 18336*	Value of the order function. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by AV, 5-Oct-2020.)
⊢ 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 18337*	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 18338*	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 18339	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 18340	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 18341	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 18342	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 18343	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 18344	Lemma for mndodcong 18345. (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 18345	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 18346	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 18347	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 18348	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 18349	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 18350	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 18351	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 18352	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 18353*	The oddvds 18350 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 18354*	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	odmulgid 18355	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 18356	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 18357	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 18358	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 18359*	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 18360	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 18361	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 18362	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 18363*	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 18364*	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 18365*	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 18366	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 18367	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	submod 18368	The order of an element is the same in a subgroup. (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 18369	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 18370	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 18371*	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 18372*	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 18373	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 18374	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 18375	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 18376*	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 18377*	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 18378*	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 18379	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 18380	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 18381*	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 18382	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 18383*	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 18384*	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 18385	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 18386*	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 18387	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 18388	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 18389	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 18390	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 18391*	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 18392	Reverse closure for the first argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.)
⊢ (𝑃 pGrp 𝐺 → 𝑃 ∈ ℙ)
Theorem	pgpgrp 18393	Reverse closure for the second argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.)
⊢ (𝑃 pGrp 𝐺 → 𝐺 ∈ Grp)
Theorem	pgpfi1 18394	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 18395	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 18396	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 18397*	Lemma for sylow1 18402. 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 18398*	Lemma for sylow1 18402. 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 18399*	Lemma for sylow1 18402. 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 18400*	Lemma for sylow1 18402. 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))