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Theorem List for Metamath Proof Explorer - 17601-17700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempmtrfconj 17601 Any conjugate of a transposition is a transposition. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇       ((𝐹𝑅𝐺:𝐷1-1-onto𝐷) → ((𝐺𝐹) ∘ 𝐺) ∈ 𝑅)
 
Theoremsymgsssg 17602* The symmetric group has subgroups restricting the set of non-fixed points. (Contributed by Stefan O'Rear, 24-Aug-2015.)
𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)       (𝐷𝑉 → {𝑥𝐵 ∣ dom (𝑥 ∖ I ) ⊆ 𝑋} ∈ (SubGrp‘𝐺))
 
Theoremsymgfisg 17603* The symmetric group has a subgroup of permutations that move finitely many points. (Contributed by Stefan O'Rear, 24-Aug-2015.)
𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)       (𝐷𝑉 → {𝑥𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ∈ (SubGrp‘𝐺))
 
Theoremsymgtrf 17604 Transpositions are elements of the symmetric group. (Contributed by Stefan O'Rear, 23-Aug-2015.)
𝑇 = ran (pmTrsp‘𝐷)    &   𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)       𝑇𝐵
 
Theoremsymggen 17605* The span of the transpositions is the subgroup that moves finitely many points. (Contributed by Stefan O'Rear, 28-Aug-2015.)
𝑇 = ran (pmTrsp‘𝐷)    &   𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)    &   𝐾 = (mrCls‘(SubMnd‘𝐺))       (𝐷𝑉 → (𝐾𝑇) = {𝑥𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin})
 
Theoremsymggen2 17606 A finite permutation group is generated by the transpositions, see also Theorem 3.4 in [Rotman] p. 31. (Contributed by Stefan O'Rear, 28-Aug-2015.)
𝑇 = ran (pmTrsp‘𝐷)    &   𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)    &   𝐾 = (mrCls‘(SubMnd‘𝐺))       (𝐷 ∈ Fin → (𝐾𝑇) = 𝐵)
 
Theoremsymgtrinv 17607 To invert a permutation represented as a sequence of transpositions, reverse the sequence. (Contributed by Stefan O'Rear, 27-Aug-2015.)
𝑇 = ran (pmTrsp‘𝐷)    &   𝐺 = (SymGrp‘𝐷)    &   𝐼 = (invg𝐺)       ((𝐷𝑉𝑊 ∈ Word 𝑇) → (𝐼‘(𝐺 Σg 𝑊)) = (𝐺 Σg (reverse‘𝑊)))
 
Theorempmtr3ncomlem1 17608 Lemma 1 for pmtr3ncom 17610. (Contributed by AV, 17-Mar-2018.)
𝑇 = (pmTrsp‘𝐷)    &   𝐹 = (𝑇‘{𝑋, 𝑌})    &   𝐺 = (𝑇‘{𝑌, 𝑍})       ((𝐷𝑉 ∧ (𝑋𝐷𝑌𝐷𝑍𝐷) ∧ (𝑋𝑌𝑋𝑍𝑌𝑍)) → ((𝐺𝐹)‘𝑋) ≠ ((𝐹𝐺)‘𝑋))
 
Theorempmtr3ncomlem2 17609 Lemma 2 for pmtr3ncom 17610. (Contributed by AV, 17-Mar-2018.)
𝑇 = (pmTrsp‘𝐷)    &   𝐹 = (𝑇‘{𝑋, 𝑌})    &   𝐺 = (𝑇‘{𝑌, 𝑍})       ((𝐷𝑉 ∧ (𝑋𝐷𝑌𝐷𝑍𝐷) ∧ (𝑋𝑌𝑋𝑍𝑌𝑍)) → (𝐺𝐹) ≠ (𝐹𝐺))
 
Theorempmtr3ncom 17610* Transpositions over sets with at least 3 elements are not commutative, see also the remark in [Rotman] p. 28. (Contributed by AV, 21-Mar-2019.)
𝑇 = (pmTrsp‘𝐷)       ((𝐷𝑉 ∧ 3 ≤ (#‘𝐷)) → ∃𝑓 ∈ ran 𝑇𝑔 ∈ ran 𝑇(𝑔𝑓) ≠ (𝑓𝑔))
 
Theorempmtrdifellem1 17611 Lemma 1 for pmtrdifel 17615. (Contributed by AV, 15-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)    &   𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I ))       (𝑄𝑇𝑆𝑅)
 
Theorempmtrdifellem2 17612 Lemma 2 for pmtrdifel 17615. (Contributed by AV, 15-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)    &   𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I ))       (𝑄𝑇 → dom (𝑆 ∖ I ) = dom (𝑄 ∖ I ))
 
Theorempmtrdifellem3 17613* Lemma 3 for pmtrdifel 17615. (Contributed by AV, 15-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)    &   𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I ))       (𝑄𝑇 → ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑄𝑥) = (𝑆𝑥))
 
Theorempmtrdifellem4 17614 Lemma 4 for pmtrdifel 17615. (Contributed by AV, 28-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)    &   𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I ))       ((𝑄𝑇𝐾𝑁) → (𝑆𝐾) = 𝐾)
 
Theorempmtrdifel 17615* A transposition of elements of a set without a special element corresponds to a transposition of elements of the set. (Contributed by AV, 15-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)       𝑡𝑇𝑟𝑅𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡𝑥) = (𝑟𝑥)
 
Theorempmtrdifwrdellem1 17616* Lemma 1 for pmtrdifwrdel 17620. (Contributed by AV, 15-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)    &   𝑈 = (𝑥 ∈ (0..^(#‘𝑊)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑊𝑥) ∖ I )))       (𝑊 ∈ Word 𝑇𝑈 ∈ Word 𝑅)
 
Theorempmtrdifwrdellem2 17617* Lemma 2 for pmtrdifwrdel 17620. (Contributed by AV, 15-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)    &   𝑈 = (𝑥 ∈ (0..^(#‘𝑊)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑊𝑥) ∖ I )))       (𝑊 ∈ Word 𝑇 → (#‘𝑊) = (#‘𝑈))
 
Theorempmtrdifwrdellem3 17618* Lemma 3 for pmtrdifwrdel 17620. (Contributed by AV, 15-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)    &   𝑈 = (𝑥 ∈ (0..^(#‘𝑊)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑊𝑥) ∖ I )))       (𝑊 ∈ Word 𝑇 → ∀𝑖 ∈ (0..^(#‘𝑊))∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛))
 
Theorempmtrdifwrdel2lem1 17619* Lemma 1 for pmtrdifwrdel2 17621. (Contributed by AV, 31-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)    &   𝑈 = (𝑥 ∈ (0..^(#‘𝑊)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑊𝑥) ∖ I )))       ((𝑊 ∈ Word 𝑇𝐾𝑁) → ∀𝑖 ∈ (0..^(#‘𝑊))((𝑈𝑖)‘𝐾) = 𝐾)
 
Theorempmtrdifwrdel 17620* A sequence of transpositions of elements of a set without a special element corresponds to a sequence of transpositions of elements of the set. (Contributed by AV, 15-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)       𝑤 ∈ Word 𝑇𝑢 ∈ Word 𝑅((#‘𝑤) = (#‘𝑢) ∧ ∀𝑖 ∈ (0..^(#‘𝑤))∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤𝑖)‘𝑥) = ((𝑢𝑖)‘𝑥))
 
Theorempmtrdifwrdel2 17621* A sequence of transpositions of elements of a set without a special element corresponds to a sequence of transpositions of elements of the set not moving the special element. (Contributed by AV, 31-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)       (𝐾𝑁 → ∀𝑤 ∈ Word 𝑇𝑢 ∈ Word 𝑅((#‘𝑤) = (#‘𝑢) ∧ ∀𝑖 ∈ (0..^(#‘𝑤))(((𝑢𝑖)‘𝐾) = 𝐾 ∧ ∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤𝑖)‘𝑥) = ((𝑢𝑖)‘𝑥))))
 
Theorempmtrprfval 17622* The transpositions on a pair. (Contributed by AV, 9-Dec-2018.)
(pmTrsp‘{1, 2}) = (𝑝 ∈ {{1, 2}} ↦ (𝑧 ∈ {1, 2} ↦ if(𝑧 = 1, 2, 1)))
 
Theorempmtrprfvalrn 17623 The range of the transpositions on a pair is actually a singleton: the transposition of the two elements of the pair. (Contributed by AV, 9-Dec-2018.)
ran (pmTrsp‘{1, 2}) = {{⟨1, 2⟩, ⟨2, 1⟩}}
 
10.2.9.5  The sign of a permutation
 
Syntaxcpsgn 17624 Syntax for the sign of a permutation.
class pmSgn
 
Syntaxcevpm 17625 Syntax for even permutations.
class pmEven
 
Definitiondf-psgn 17626* Define a function which takes the value 1 for even permutations and -1 for odd. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmSgn = (𝑑 ∈ V ↦ (𝑥 ∈ {𝑝 ∈ (Base‘(SymGrp‘𝑑)) ∣ dom (𝑝 ∖ I ) ∈ Fin} ↦ (℩𝑠𝑤 ∈ Word ran (pmTrsp‘𝑑)(𝑥 = ((SymGrp‘𝑑) Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤))))))
 
Definitiondf-evpm 17627 Define the set of even permutations on a given set. (Contributed by Stefan O'Rear, 9-Jul-2018.)
pmEven = (𝑑 ∈ V ↦ ((pmSgn‘𝑑) “ {1}))
 
Theorempsgnunilem1 17628* Lemma for psgnuni 17634. Given two consequtive transpositions in a representation of a permutation, either they are equal and therefore equivalent to the identity, or they are not and it is possible to commute them such that a chosen point in the left transposition is preserved in the right. By repeating this process, a point can be removed from a representation of the identity. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝑇 = ran (pmTrsp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑃𝑇)    &   (𝜑𝑄𝑇)    &   (𝜑𝐴 ∈ dom (𝑃 ∖ I ))       (𝜑 → ((𝑃𝑄) = ( I ↾ 𝐷) ∨ ∃𝑟𝑇𝑠𝑇 ((𝑃𝑄) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I ))))
 
Theorempsgnunilem5 17629* Lemma for psgnuni 17634. It is impossible to shift a transposition off the end because if the active transposition is at the right end, it is the only transposition moving 𝐴 in contradiction to this being a representation of the identity. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
𝐺 = (SymGrp‘𝐷)    &   𝑇 = ran (pmTrsp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝑇)    &   (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))    &   (𝜑 → (#‘𝑊) = 𝐿)    &   (𝜑𝐼 ∈ (0..^𝐿))    &   (𝜑𝐴 ∈ dom ((𝑊𝐼) ∖ I ))    &   (𝜑 → ∀𝑘 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊𝑘) ∖ I ))       (𝜑 → (𝐼 + 1) ∈ (0..^𝐿))
 
Theorempsgnunilem2 17630* Lemma for psgnuni 17634. Induction step for moving a transposition as far to the right as possible. (Contributed by Stefan O'Rear, 24-Aug-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
𝐺 = (SymGrp‘𝐷)    &   𝑇 = ran (pmTrsp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝑇)    &   (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))    &   (𝜑 → (#‘𝑊) = 𝐿)    &   (𝜑𝐼 ∈ (0..^𝐿))    &   (𝜑𝐴 ∈ dom ((𝑊𝐼) ∖ I ))    &   (𝜑 → ∀𝑘 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊𝑘) ∖ I ))    &   (𝜑 → ¬ ∃𝑥 ∈ Word 𝑇((#‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))       (𝜑 → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ ((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom ((𝑤‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom ((𝑤𝑗) ∖ I ))))
 
Theorempsgnunilem3 17631* Lemma for psgnuni 17634. Any nonempty representation of the identity can be incrementally transformed into a representation two shorter. (Contributed by Stefan O'Rear, 25-Aug-2015.)
𝐺 = (SymGrp‘𝐷)    &   𝑇 = ran (pmTrsp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝑇)    &   (𝜑 → (#‘𝑊) = 𝐿)    &   (𝜑 → (#‘𝑊) ∈ ℕ)    &   (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))    &   (𝜑 → ¬ ∃𝑥 ∈ Word 𝑇((#‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))        ¬ 𝜑
 
Theorempsgnunilem4 17632 Lemma for psgnuni 17634. 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)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝑇)    &   (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))       (𝜑 → (-1↑(#‘𝑊)) = 1)
 
Theoremm1expaddsub 17633 Addition and subtraction of parities are the same. (Contributed by Stefan O'Rear, 27-Aug-2015.)
((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (-1↑(𝑋𝑌)) = (-1↑(𝑋 + 𝑌)))
 
Theorempsgnuni 17634 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)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝑇)    &   (𝜑𝑋 ∈ Word 𝑇)    &   (𝜑 → (𝐺 Σg 𝑊) = (𝐺 Σg 𝑋))       (𝜑 → (-1↑(#‘𝑊)) = (-1↑(#‘𝑋)))
 
Theorempsgnfval 17635* Function definition of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)
𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)    &   𝐹 = {𝑝𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin}    &   𝑇 = ran (pmTrsp‘𝐷)    &   𝑁 = (pmSgn‘𝐷)       𝑁 = (𝑥𝐹 ↦ (℩𝑠𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤)))))
 
Theorempsgnfn 17636* Functionality and domain of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)
𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)    &   𝐹 = {𝑝𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin}    &   𝑁 = (pmSgn‘𝐷)       𝑁 Fn 𝐹
 
Theorempsgndmsubg 17637 The finitary permutations are a subgroup. (Contributed by Stefan O'Rear, 28-Aug-2015.)
𝐺 = (SymGrp‘𝐷)    &   𝑁 = (pmSgn‘𝐷)       (𝐷𝑉 → dom 𝑁 ∈ (SubGrp‘𝐺))
 
Theorempsgneldm 17638 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 ↔ (𝑃𝐵 ∧ dom (𝑃 ∖ I ) ∈ Fin))
 
Theorempsgneldm2 17639* 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)       (𝐷𝑉 → (𝑃 ∈ dom 𝑁 ↔ ∃𝑤 ∈ Word 𝑇𝑃 = (𝐺 Σg 𝑤)))
 
Theorempsgneldm2i 17640 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)       ((𝐷𝑉𝑊 ∈ Word 𝑇) → (𝐺 Σg 𝑊) ∈ dom 𝑁)
 
Theorempsgneu 17641* 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 → ∃!𝑠𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤))))
 
Theorempsgnval 17642* 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 → (𝑁𝑃) = (℩𝑠𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤)))))
 
Theorempsgnvali 17643* 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) ∧ (𝑁𝑃) = (-1↑(#‘𝑤))))
 
Theorempsgnvalii 17644 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)       ((𝐷𝑉𝑊 ∈ Word 𝑇) → (𝑁‘(𝐺 Σg 𝑊)) = (-1↑(#‘𝑊)))
 
Theorempsgnpmtr 17645 All transpositions are odd. (Contributed by Stefan O'Rear, 29-Aug-2015.)
𝐺 = (SymGrp‘𝐷)    &   𝑇 = ran (pmTrsp‘𝐷)    &   𝑁 = (pmSgn‘𝐷)       (𝑃𝑇 → (𝑁𝑃) = -1)
 
Theorempsgn0fv0 17646 The permutation sign function for an empty set at an empty set is 1. (Contributed by AV, 27-Feb-2019.)
((pmSgn‘∅)‘∅) = 1
 
Theoremsygbasnfpfi 17647 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 ∧ 𝑃𝐵) → dom (𝑃 ∖ I ) ∈ Fin)
 
Theorempsgnfvalfi 17648* 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 = (𝑥𝐵 ↦ (℩𝑠𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤))))))
 
Theorempsgnvalfi 17649* 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 ∧ 𝑃𝐵) → (𝑁𝑃) = (℩𝑠𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤)))))
 
Theorempsgnran 17650 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 ∧ 𝑄𝑃) → (𝑆𝑄) ∈ {1, -1})
 
Theoremgsmtrcl 17651 The group sum of transpositions of a finite set is a permutation, see also psgneldm2i 17640. (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)
 
Theorempsgnfitr 17652* 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 → (𝑄𝐵 ↔ ∃𝑤 ∈ Word 𝑇𝑄 = (𝐺 Σg 𝑤)))
 
Theorempsgnfieu 17653* 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 ∧ 𝑄𝐵) → ∃!𝑠𝑤 ∈ Word 𝑇(𝑄 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤))))
 
Theorempmtrsn 17654 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}) = ∅
 
Theorempsgnsn 17655 The permutation sign function for a singleton. (Contributed by AV, 6-Aug-2019.)
𝐷 = {𝐴}    &   𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)    &   𝑁 = (pmSgn‘𝐷)       ((𝐴𝑉𝑋𝐵) → (𝑁𝑋) = 1)
 
Theorempsgnprfval 17656* 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)       (𝑋𝐵 → (𝑁𝑋) = (℩𝑠𝑤 ∈ Word 𝑇(𝑋 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤)))))
 
Theorempsgnprfval1 17657 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
 
Theorempsgnprfval2 17658 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
 
Syntaxcod 17659 Extend class notation to include the order function on the elements of a group.
class od
 
Syntaxcodold 17660 Extend class notation to include the order function on the elements of a group (old version).
class od
 
Syntaxcgex 17661 Extend class notation to include the order function on the elements of a group.
class gEx
 
Syntaxcgexold 17662 Extend class notation to include the order function on the elements of a group (old version).
class gEx
 
Syntaxcpgp 17663 Extend class notation to include the class of all p-groups.
class pGrp
 
Syntaxcpgpold 17664 Extend class notation to include the class of all p-groups (old version).
class pGrp
 
Syntaxcslw 17665 Extend class notation to include the class of all Sylow p-subgroups of a group.
class pSyl
 
Definitiondf-od 17666* 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(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
 
Definitiondf-odOLD 17667* Define the order of an element in a group. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by Stefan O'Rear, 4-Sep-2015.) Obsolete version of df-od 17666 as of 5-Oct-2020. (New usage is discouraged.)
od = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ {𝑛 ∈ ℕ ∣ (𝑛(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, sup(𝑖, ℝ, < ))))
 
Definitiondf-gex 17668* 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(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
 
Definitiondf-gexOLD 17669* Define the exponent of a group. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by Stefan O'Rear, 4-Sep-2015.) Obsolete version of df-gex 17668 as of 26-Sep-2020. (New usage is discouraged.)
gEx = (𝑔 ∈ V ↦ {𝑛 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑛(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, sup(𝑖, ℝ, < )))
 
Definitiondf-pgp 17670* 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) = (𝑝𝑛))}
 
Definitiondf-pgpOLD 17671* 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.) Obsolete version of df-pgp 17670 as of 5-Oct-2020. (New usage is discouraged.)
pGrp = {⟨𝑝, 𝑔⟩ ∣ ((𝑝 ∈ ℙ ∧ 𝑔 ∈ Grp) ∧ ∀𝑥 ∈ (Base‘𝑔)∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝𝑛))}
 
Definitiondf-slw 17672* 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 ↦ { ∈ (SubGrp‘𝑔) ∣ ∀𝑘 ∈ (SubGrp‘𝑔)((𝑘𝑝 pGrp (𝑔s 𝑘)) ↔ = 𝑘)})
 
Theoremodfval 17673* Value of the order function. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by AV, 5-Oct-2020.)
𝑋 = (Base‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)    &   𝑂 = (od‘𝐺)       𝑂 = (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
 
Theoremodval 17674* 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)    &    · = (.g𝐺)    &    0 = (0g𝐺)    &   𝑂 = (od‘𝐺)    &   𝐼 = {𝑦 ∈ ℕ ∣ (𝑦 · 𝐴) = 0 }       (𝐴𝑋 → (𝑂𝐴) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
 
Theoremodlem1 17675* 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)    &    · = (.g𝐺)    &    0 = (0g𝐺)    &   𝑂 = (od‘𝐺)    &   𝐼 = {𝑦 ∈ ℕ ∣ (𝑦 · 𝐴) = 0 }       (𝐴𝑋 → (((𝑂𝐴) = 0 ∧ 𝐼 = ∅) ∨ (𝑂𝐴) ∈ 𝐼))
 
TheoremodfvalOLD 17676* Value of the order function. (Contributed by Mario Carneiro, 13-Jul-2014.) Obsolete version of odfval 17673 as of 5-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = (Base‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)    &   𝑂 = (od‘𝐺)       𝑂 = (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, sup(𝑖, ℝ, < )))
 
TheoremodvalOLD 17677* Second substitution for the group order definition. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by Stefan O'Rear, 5-Sep-2015.) Obsolete version of odval 17674 as of 5-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = (Base‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)    &   𝑂 = (od‘𝐺)    &   𝐼 = {𝑦 ∈ ℕ ∣ (𝑦 · 𝐴) = 0 }       (𝐴𝑋 → (𝑂𝐴) = if(𝐼 = ∅, 0, sup(𝐼, ℝ, < )))
 
Theoremodlem1OLD 17678* 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.) Obsolete version of odlem1 17675 as of 5-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = (Base‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)    &   𝑂 = (od‘𝐺)    &   𝐼 = {𝑦 ∈ ℕ ∣ (𝑦 · 𝐴) = 0 }       (𝐴𝑋 → (((𝑂𝐴) = 0 ∧ 𝐼 = ∅) ∨ (𝑂𝐴) ∈ 𝐼))
 
Theoremodcl 17679 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)       (𝐴𝑋 → (𝑂𝐴) ∈ ℕ0)
 
Theoremodf 17680 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
 
Theoremodid 17681 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)    &    · = (.g𝐺)    &    0 = (0g𝐺)       (𝐴𝑋 → ((𝑂𝐴) · 𝐴) = 0 )
 
Theoremodlem2 17682 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)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐴𝑋𝑁 ∈ ℕ ∧ (𝑁 · 𝐴) = 0 ) → (𝑂𝐴) ∈ (1...𝑁))
 
Theoremodmodnn0 17683 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)    &    · = (.g𝐺)    &    0 = (0g𝐺)       (((𝐺 ∈ Mnd ∧ 𝐴𝑋𝑁 ∈ ℕ0) ∧ (𝑂𝐴) ∈ ℕ) → ((𝑁 mod (𝑂𝐴)) · 𝐴) = (𝑁 · 𝐴))
 
Theoremmndodconglem 17684 Lemma for mndodcong 17685. (Contributed by Mario Carneiro, 23-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝑋)    &   (𝜑 → (𝑂𝐴) ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑀 < (𝑂𝐴))    &   (𝜑𝑁 < (𝑂𝐴))    &   (𝜑 → (𝑀 · 𝐴) = (𝑁 · 𝐴))       ((𝜑𝑀𝑁) → 𝑀 = 𝑁)
 
Theoremmndodcong 17685 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)    &    · = (.g𝐺)    &    0 = (0g𝐺)       (((𝐺 ∈ Mnd ∧ 𝐴𝑋) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑂𝐴) ∈ ℕ) → ((𝑂𝐴) ∥ (𝑀𝑁) ↔ (𝑀 · 𝐴) = (𝑁 · 𝐴)))
 
Theoremmndodcongi 17686 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)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Mnd ∧ 𝐴𝑋 ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0)) → ((𝑂𝐴) ∥ (𝑀𝑁) → (𝑀 · 𝐴) = (𝑁 · 𝐴)))
 
Theoremoddvdsnn0 17687 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)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Mnd ∧ 𝐴𝑋𝑁 ∈ ℕ0) → ((𝑂𝐴) ∥ 𝑁 ↔ (𝑁 · 𝐴) = 0 ))
 
Theoremodnncl 17688 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)    &    · = (.g𝐺)    &    0 = (0g𝐺)       (((𝐺 ∈ Grp ∧ 𝐴𝑋𝑁 ∈ ℤ) ∧ (𝑁 ≠ 0 ∧ (𝑁 · 𝐴) = 0 )) → (𝑂𝐴) ∈ ℕ)
 
Theoremodmod 17689 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)    &    · = (.g𝐺)    &    0 = (0g𝐺)       (((𝐺 ∈ Grp ∧ 𝐴𝑋𝑁 ∈ ℤ) ∧ (𝑂𝐴) ∈ ℕ) → ((𝑁 mod (𝑂𝐴)) · 𝐴) = (𝑁 · 𝐴))
 
Theoremoddvds 17690 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)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝑋𝑁 ∈ ℤ) → ((𝑂𝐴) ∥ 𝑁 ↔ (𝑁 · 𝐴) = 0 ))
 
Theoremoddvdsi 17691 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)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑂𝐴) ∥ 𝑁) → (𝑁 · 𝐴) = 0 )
 
Theoremodcong 17692 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)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝑋 ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑂𝐴) ∥ (𝑀𝑁) ↔ (𝑀 · 𝐴) = (𝑁 · 𝐴)))
 
Theoremodeq 17693* The oddvds 17690 property uniquely defines the group order. (Contributed by Stefan O'Rear, 6-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝑋𝑁 ∈ ℕ0) → (𝑁 = (𝑂𝐴) ↔ ∀𝑦 ∈ ℕ0 (𝑁𝑦 ↔ (𝑦 · 𝐴) = 0 )))
 
Theoremodval2 17694* A non-conditional definition of the group order. (Contributed by Stefan O'Rear, 6-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑂𝐴) = (𝑥 ∈ ℕ0𝑦 ∈ ℕ0 (𝑥𝑦 ↔ (𝑦 · 𝐴) = 0 )))
 
TheoremodclOLD 17695 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.) Obsolete version of odcl 17679 as of 5-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)       (𝐴𝑋 → (𝑂𝐴) ∈ ℕ0)
 
TheoremodfOLD 17696 Functionality of the group element order. (Contributed by Stefan O'Rear, 5-Sep-2015.) Obsolete version of odf 17680 as of 5-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)       𝑂:𝑋⟶ℕ0
 
TheoremodidOLD 17697 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.) Obsolete version of odid 17681 as of 5-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       (𝐴𝑋 → ((𝑂𝐴) · 𝐴) = 0 )
 
Theoremodlem2OLD 17698 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.) Obsolete version of odlem2 17682 as of 5-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐴𝑋𝑁 ∈ ℕ ∧ (𝑁 · 𝐴) = 0 ) → (𝑂𝐴) ∈ (1...𝑁))
 
Theoremodmulgid 17699 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)    &    · = (.g𝐺)       (((𝐺 ∈ Grp ∧ 𝐴𝑋𝑁 ∈ ℤ) ∧ 𝐾 ∈ ℤ) → ((𝑂‘(𝑁 · 𝐴)) ∥ 𝐾 ↔ (𝑂𝐴) ∥ (𝐾 · 𝑁)))
 
Theoremodmulg2 17700 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)    &    · = (.g𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝑋𝑁 ∈ ℤ) → (𝑂‘(𝑁 · 𝐴)) ∥ (𝑂𝐴))
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