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Theorem List for Metamath Proof Explorer - 17701-17800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempmtrfrn 17701 A transposition (as a kind of function) is the function transposing the two points it moves. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇    &   𝑃 = dom (𝐹 ∖ I )       (𝐹𝑅 → ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝐹 = (𝑇𝑃)))

Theorempmtrffv 17702 Mapping of a point under a transposition function. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇    &   𝑃 = dom (𝐹 ∖ I )       ((𝐹𝑅𝑍𝐷) → (𝐹𝑍) = if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍))

Theorempmtrrn2 17703* For any transposition there are two points it is transposing. (Contributed by SO, 15-Jul-2018.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇       (𝐹𝑅 → ∃𝑥𝐷𝑦𝐷 (𝑥𝑦𝐹 = (𝑇‘{𝑥, 𝑦})))

Theorempmtrfinv 17704 A transposition function is an involution. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇       (𝐹𝑅 → (𝐹𝐹) = ( I ↾ 𝐷))

Theorempmtrfmvdn0 17705 A transposition moves at least one point. (Contributed by Stefan O'Rear, 23-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇       (𝐹𝑅 → dom (𝐹 ∖ I ) ≠ ∅)

Theorempmtrff1o 17706 A transposition function is a permutation. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇       (𝐹𝑅𝐹:𝐷1-1-onto𝐷)

Theorempmtrfcnv 17707 A transposition function is its own inverse. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇       (𝐹𝑅𝐹 = 𝐹)

Theorempmtrfb 17708 An intrinsic characterization of the transposition permutations. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇       (𝐹𝑅 ↔ (𝐷 ∈ V ∧ 𝐹:𝐷1-1-onto𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2𝑜))

Theorempmtrfconj 17709 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 17710* 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 17711* 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 17712 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 17713* 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 17714 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 17715 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 17716 Lemma 1 for pmtr3ncom 17718. (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 17717 Lemma 2 for pmtr3ncom 17718. (Contributed by AV, 17-Mar-2018.)
𝑇 = (pmTrsp‘𝐷)    &   𝐹 = (𝑇‘{𝑋, 𝑌})    &   𝐺 = (𝑇‘{𝑌, 𝑍})       ((𝐷𝑉 ∧ (𝑋𝐷𝑌𝐷𝑍𝐷) ∧ (𝑋𝑌𝑋𝑍𝑌𝑍)) → (𝐺𝐹) ≠ (𝐹𝐺))

Theorempmtr3ncom 17718* 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 17719 Lemma 1 for pmtrdifel 17723. (Contributed by AV, 15-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)    &   𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I ))       (𝑄𝑇𝑆𝑅)

Theorempmtrdifellem2 17720 Lemma 2 for pmtrdifel 17723. (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 17721* Lemma 3 for pmtrdifel 17723. (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 17722 Lemma 4 for pmtrdifel 17723. (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 17723* 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 17724* Lemma 1 for pmtrdifwrdel 17728. (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 17725* Lemma 2 for pmtrdifwrdel 17728. (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 17726* Lemma 3 for pmtrdifwrdel 17728. (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 17727* Lemma 1 for pmtrdifwrdel2 17729. (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 17728* 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 17729* 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 17730* 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 17731 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 17732 Syntax for the sign of a permutation.
class pmSgn

Syntaxcevpm 17733 Syntax for even permutations.
class pmEven

Definitiondf-psgn 17734* 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 17735 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 17736* Lemma for psgnuni 17742. 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 17737* Lemma for psgnuni 17742. 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 17738* Lemma for psgnuni 17742. 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 17739* Lemma for psgnuni 17742. 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 17740 Lemma for psgnuni 17742. 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 17741 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 17742 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 17743* 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 17744* 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 17745 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 17746 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 17747* 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 17748 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 17749* 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 17750* 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 17751* 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 17752 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 17753 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 17754 The permutation sign function for an empty set at an empty set is 1. (Contributed by AV, 27-Feb-2019.)
((pmSgn‘∅)‘∅) = 1

Theoremsygbasnfpfi 17755 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 17756* 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 17757* 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 17758 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 17759 The group sum of transpositions of a finite set is a permutation, see also psgneldm2i 17748. (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 17760* 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 17761* 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 17762 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 17763 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 17764* 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 17765 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 17766 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 17767 Extend class notation to include the order function on the elements of a group.
class od

Syntaxcgex 17768 Extend class notation to include the order function on the elements of a group.
class gEx

Syntaxcpgp 17769 Extend class notation to include the class of all p-groups.
class pGrp

Syntaxcslw 17770 Extend class notation to include the class of all Sylow p-subgroups of a group.
class pSyl

Definitiondf-od 17771* 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-gex 17772* 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-pgp 17773* 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-slw 17774* 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 17775* 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 17776* 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 17777* 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))

Theoremodcl 17778 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 17779 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 17780 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 17781 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 17782 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 17783 Lemma for mndodcong 17784. (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 17784 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 17785 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 17786 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 17787 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 17788 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 17789 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 17790 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 17791 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 17792* The oddvds 17789 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 17793* 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 )))

Theoremodmulgid 17794 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 17795 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))

Theoremodmulg 17796 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)    &    · = (.g𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝑋𝑁 ∈ ℤ) → (𝑂𝐴) = ((𝑁 gcd (𝑂𝐴)) · (𝑂‘(𝑁 · 𝐴))))

Theoremodmulgeq 17797 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)    &    · = (.g𝐺)       (((𝐺 ∈ Grp ∧ 𝐴𝑋𝑁 ∈ ℤ) ∧ (𝑂𝐴) ∈ ℕ) → ((𝑂‘(𝑁 · 𝐴)) = (𝑂𝐴) ↔ (𝑁 gcd (𝑂𝐴)) = 1))

Theoremodbezout 17798* 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)    &    · = (.g𝐺)       (((𝐺 ∈ Grp ∧ 𝐴𝑋𝑁 ∈ ℤ) ∧ (𝑁 gcd (𝑂𝐴)) = 1) → ∃𝑥 ∈ ℤ (𝑥 · (𝑁 · 𝐴)) = 𝐴)

Theoremod1 17799 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 = (0g𝐺)       (𝐺 ∈ Grp → (𝑂0 ) = 1)

Theoremodeq1 17800 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 = (0g𝐺)    &   𝑋 = (Base‘𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑂𝐴) = 1 ↔ 𝐴 = 0 ))

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