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Type | Label | Description |
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Statement | ||
Definition | df-evpm 19401 | Define the set of even permutations on a given set. (Contributed by Stefan O'Rear, 9-Jul-2018.) |
⊢ pmEven = (𝑑 ∈ V ↦ (◡(pmSgn‘𝑑) “ {1})) | ||
Theorem | psgnunilem1 19402* | Lemma for psgnuni 19408. 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) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑃 ∈ 𝑇) & ⊢ (𝜑 → 𝑄 ∈ 𝑇) & ⊢ (𝜑 → 𝐴 ∈ dom (𝑃 ∖ I )) ⇒ ⊢ (𝜑 → ((𝑃 ∘ 𝑄) = ( I ↾ 𝐷) ∨ ∃𝑟 ∈ 𝑇 ∃𝑠 ∈ 𝑇 ((𝑃 ∘ 𝑄) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) | ||
Theorem | psgnunilem5 19403* | Lemma for psgnuni 19408. 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.) (Proof shortened by AV, 12-Oct-2022.) |
⊢ 𝐺 = (SymGrp‘𝐷) & ⊢ 𝑇 = ran (pmTrsp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ Word 𝑇) & ⊢ (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷)) & ⊢ (𝜑 → (♯‘𝑊) = 𝐿) & ⊢ (𝜑 → 𝐼 ∈ (0..^𝐿)) & ⊢ (𝜑 → 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )) & ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I )) ⇒ ⊢ (𝜑 → (𝐼 + 1) ∈ (0..^𝐿)) | ||
Theorem | psgnunilem2 19404* | Lemma for psgnuni 19408. 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) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ Word 𝑇) & ⊢ (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷)) & ⊢ (𝜑 → (♯‘𝑊) = 𝐿) & ⊢ (𝜑 → 𝐼 ∈ (0..^𝐿)) & ⊢ (𝜑 → 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )) & ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ 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 ((𝑤‘𝑗) ∖ I )))) | ||
Theorem | psgnunilem3 19405* | Lemma for psgnuni 19408. 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) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ 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 | ||
Theorem | psgnunilem4 19406 | Lemma for psgnuni 19408. 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 19407 | 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 19408 | 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 19409* | 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 19410* | 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 19411 | 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 19412 | 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 19413* | 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 19414 | 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 19415* | 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 19416* | 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 19417* | 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 19418 | 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 19419 | 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 19420 | The permutation sign function for an empty set at an empty set is 1. (Contributed by AV, 27-Feb-2019.) |
⊢ ((pmSgn‘∅)‘∅) = 1 | ||
Theorem | sygbasnfpfi 19421 | 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 19422* | 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 19423* | 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 19424 | 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 19425 | The group sum of transpositions of a finite set is a permutation, see also psgneldm2i 19414. (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 19426* | 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 19427* | 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 19428 | 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 19429 | 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 19430* | 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 19431 | 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 19432 | 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 | ||
Syntax | cod 19433 | Extend class notation to include the order function on the elements of a group. |
class od | ||
Syntax | cgex 19434 | Extend class notation to include the order function on the elements of a group. |
class gEx | ||
Syntax | cpgp 19435 | Extend class notation to include the class of all p-groups. |
class pGrp | ||
Syntax | cslw 19436 | Extend class notation to include the class of all Sylow p-subgroups of a group. |
class pSyl | ||
Definition | df-od 19437* | 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 19438* | 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 19439* | 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 19440* | 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 19441* | Value of the order function. For a shorter proof using ax-rep 5284, see odfvalALT 19442. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by AV, 5-Oct-2020.) Remove dependency on ax-rep 5284. (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 19442* | Shorter proof of odfval 19441 using ax-rep 5284. (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 19443* | 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 19444* | 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 19445 | 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 19446 | 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 19447 | 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 19448 | 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 19449 | 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 19450 | Lemma for mndodcong 19451. (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 19451 | 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 19452 | 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 19453 | 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 19454 | 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 19455 | 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 19456 | 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 19457 | 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 19458 | 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 19459* | The oddvds 19456 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 19460* | 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 19461 | The order of a group element is always a nonnegative integer, deduction form of odcl 19445. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑂 = (od‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℕ0) | ||
Theorem | odm1inv 19462 | 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 19463 | 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 19464 | 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 19465 | 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 19466 | 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 19467* | 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 19468 | 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 19469 | 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 19470 | 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 19471* | 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 19472* | 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 19473* | 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 19474 | 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 19475 | 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 19476* | 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 19477 | A shorter proof of 0subg 19067 using df-od 19437. (Contributed by SN, 31-Jan-2025.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝐺 ∈ Grp → { 0 } ∈ (SubGrp‘𝐺)) | ||
Theorem | submod 19478 | 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 19479 | 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 19480 | 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 19481* | 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 19482* | 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 19483 | 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 19484 | 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 19485 | 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 19486* | 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 19487* | 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 19488* | 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 19489 | 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 19490 | 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 19491* | 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 19492 | 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 19493* | 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 19494* | 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 19495 | 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 19496* | 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 19497 | 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 19498 | 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 19499 | 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 19500 | 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)) |
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