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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Definition | df-ogrp 30701 | Define class of all ordered groups. An ordered group is a group with a total ordering compatible with its operation. (Contributed by Thierry Arnoux, 13-Mar-2018.) |
| ⊢ oGrp = (Grp ∩ oMnd) | ||
| Theorem | isomnd 30702* | A (left) ordered monoid is a monoid with a total ordering compatible with its operation. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ ≤ = (le‘𝑀) ⇒ ⊢ (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐)))) | ||
| Theorem | isogrp 30703 | A (left-)ordered group is a group with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
| ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd)) | ||
| Theorem | ogrpgrp 30704 | A left-ordered group is a group. (Contributed by Thierry Arnoux, 9-Jul-2018.) |
| ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp) | ||
| Theorem | omndmnd 30705 | A left-ordered monoid is a monoid. (Contributed by Thierry Arnoux, 13-Mar-2018.) |
| ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Mnd) | ||
| Theorem | omndtos 30706 | A left-ordered monoid is a totally ordered set. (Contributed by Thierry Arnoux, 13-Mar-2018.) |
| ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset) | ||
| Theorem | omndadd 30707 | In an ordered monoid, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ ≤ = (le‘𝑀) & ⊢ + = (+g‘𝑀) ⇒ ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 + 𝑍) ≤ (𝑌 + 𝑍)) | ||
| Theorem | omndaddr 30708 | In a right ordered monoid, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ ≤ = (le‘𝑀) & ⊢ + = (+g‘𝑀) ⇒ ⊢ (((oppg‘𝑀) ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑍 + 𝑋) ≤ (𝑍 + 𝑌)) | ||
| Theorem | omndadd2d 30709 | In a commutative left ordered monoid, the ordering is compatible with monoid addition. Double addition version. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ ≤ = (le‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ oMnd) & ⊢ (𝜑 → 𝑊 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ≤ 𝑍) & ⊢ (𝜑 → 𝑌 ≤ 𝑊) & ⊢ (𝜑 → 𝑀 ∈ CMnd) ⇒ ⊢ (𝜑 → (𝑋 + 𝑌) ≤ (𝑍 + 𝑊)) | ||
| Theorem | omndadd2rd 30710 | In a left- and right- ordered monoid, the ordering is compatible with monoid addition. Double addition version. (Contributed by Thierry Arnoux, 2-May-2018.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ ≤ = (le‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ oMnd) & ⊢ (𝜑 → 𝑊 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ≤ 𝑍) & ⊢ (𝜑 → 𝑌 ≤ 𝑊) & ⊢ (𝜑 → (oppg‘𝑀) ∈ oMnd) ⇒ ⊢ (𝜑 → (𝑋 + 𝑌) ≤ (𝑍 + 𝑊)) | ||
| Theorem | submomnd 30711 | A submonoid of an ordered monoid is also ordered. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
| ⊢ ((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) → (𝑀 ↾s 𝐴) ∈ oMnd) | ||
| Theorem | xrge0omnd 30712 | The nonnegative extended real numbers form an ordered monoid. (Contributed by Thierry Arnoux, 22-Mar-2018.) |
| ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ oMnd | ||
| Theorem | omndmul2 30713 | In an ordered monoid, the ordering is compatible with group power. This version does not require the monoid to be commutative. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ ≤ = (le‘𝑀) & ⊢ · = (.g‘𝑀) & ⊢ 0 = (0g‘𝑀) ⇒ ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) → 0 ≤ (𝑁 · 𝑋)) | ||
| Theorem | omndmul3 30714 | In an ordered monoid, the ordering is compatible with group power. This version does not require the monoid to be commutative. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ ≤ = (le‘𝑀) & ⊢ · = (.g‘𝑀) & ⊢ 0 = (0g‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ oMnd) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑃 ∈ ℕ0) & ⊢ (𝜑 → 𝑁 ≤ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 0 ≤ 𝑋) ⇒ ⊢ (𝜑 → (𝑁 · 𝑋) ≤ (𝑃 · 𝑋)) | ||
| Theorem | omndmul 30715 | In a commutative ordered monoid, the ordering is compatible with group power. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ ≤ = (le‘𝑀) & ⊢ · = (.g‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ oMnd) & ⊢ (𝜑 → 𝑀 ∈ CMnd) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) ⇒ ⊢ (𝜑 → (𝑁 · 𝑋) ≤ (𝑁 · 𝑌)) | ||
| Theorem | ogrpinv0le 30716 | In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ ≤ = (le‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) → ( 0 ≤ 𝑋 ↔ (𝐼‘𝑋) ≤ 0 )) | ||
| Theorem | ogrpsub 30717 | In an ordered group, the ordering is compatible with group subtraction. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ ≤ = (le‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 − 𝑍) ≤ (𝑌 − 𝑍)) | ||
| Theorem | ogrpaddlt 30718 | In an ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 20-Jan-2018.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ < = (lt‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍)) | ||
| Theorem | ogrpaddltbi 30719 | In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ < = (lt‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 < 𝑌 ↔ (𝑋 + 𝑍) < (𝑌 + 𝑍))) | ||
| Theorem | ogrpaddltrd 30720 | In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ < = (lt‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → (oppg‘𝐺) ∈ oGrp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 < 𝑌) ⇒ ⊢ (𝜑 → (𝑍 + 𝑋) < (𝑍 + 𝑌)) | ||
| Theorem | ogrpaddltrbid 30721 | In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 4-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ < = (lt‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → (oppg‘𝐺) ∈ oGrp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 < 𝑌 ↔ (𝑍 + 𝑋) < (𝑍 + 𝑌))) | ||
| Theorem | ogrpsublt 30722 | In an ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ < = (lt‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 < 𝑌) → (𝑋 − 𝑍) < (𝑌 − 𝑍)) | ||
| Theorem | ogrpinv0lt 30723 | In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ < = (lt‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ (𝐼‘𝑋) < 0 )) | ||
| Theorem | ogrpinvlt 30724 | In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ < = (lt‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ (((𝐺 ∈ oGrp ∧ (oppg‘𝐺) ∈ oGrp) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ (𝐼‘𝑌) < (𝐼‘𝑋))) | ||
| Theorem | gsumle 30725 | A finite sum in an ordered monoid is monotonic. This proof would be much easier in an ordered group, where an inverse element would be available. (Contributed by Thierry Arnoux, 13-Mar-2018.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ ≤ = (le‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ oMnd) & ⊢ (𝜑 → 𝑀 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐹 ∘r ≤ 𝐺) ⇒ ⊢ (𝜑 → (𝑀 Σg 𝐹) ≤ (𝑀 Σg 𝐺)) | ||
| Theorem | symgfcoeu 30726* | Uniqueness property of permutations. (Contributed by Thierry Arnoux, 22-Aug-2020.) |
| ⊢ 𝐺 = (Base‘(SymGrp‘𝐷)) ⇒ ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝐺 ∧ 𝑄 ∈ 𝐺) → ∃!𝑝 ∈ 𝐺 𝑄 = (𝑃 ∘ 𝑝)) | ||
| Theorem | symgcom 30727 | Two permutations 𝑋 and 𝑌 commute if their orbits are disjoint. (Contributed by Thierry Arnoux, 15-Oct-2023.) |
| ⊢ 𝐺 = (SymGrp‘𝐴) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → (𝑋 ↾ 𝐸) = ( I ↾ 𝐸)) & ⊢ (𝜑 → (𝑌 ↾ 𝐹) = ( I ↾ 𝐹)) & ⊢ (𝜑 → (𝐸 ∩ 𝐹) = ∅) & ⊢ (𝜑 → (𝐸 ∪ 𝐹) = 𝐴) ⇒ ⊢ (𝜑 → (𝑋 ∘ 𝑌) = (𝑌 ∘ 𝑋)) | ||
| Theorem | symgcom2 30728 | Two permutations 𝑋 and 𝑌 commute if their orbits are disjoint. (Contributed by Thierry Arnoux, 17-Nov-2023.) |
| ⊢ 𝐺 = (SymGrp‘𝐴) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → (dom (𝑋 ∖ I ) ∩ dom (𝑌 ∖ I )) = ∅) ⇒ ⊢ (𝜑 → (𝑋 ∘ 𝑌) = (𝑌 ∘ 𝑋)) | ||
| Theorem | symgcntz 30729* | All elements of a (finite) set of permutations commute if their orbits are disjoint. (Contributed by Thierry Arnoux, 20-Nov-2023.) |
| ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑍 = (Cntz‘𝑆) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 dom (𝑥 ∖ I )) ⇒ ⊢ (𝜑 → 𝐴 ⊆ (𝑍‘𝐴)) | ||
| Theorem | odpmco 30730 | The composition of two odd permutations is even. (Contributed by Thierry Arnoux, 15-Oct-2023.) |
| ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐴 = (pmEven‘𝐷) ⇒ ⊢ ((𝐷 ∈ Fin ∧ 𝑋 ∈ (𝐵 ∖ 𝐴) ∧ 𝑌 ∈ (𝐵 ∖ 𝐴)) → (𝑋 ∘ 𝑌) ∈ 𝐴) | ||
| Theorem | symgsubg 30731 | The value of the group subtraction operation of the symmetric group. (Contributed by Thierry Arnoux, 15-Oct-2023.) |
| ⊢ 𝐺 = (SymGrp‘𝐴) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 ∘ ◡𝑌)) | ||
| Theorem | pmtrprfv2 30732 | In a transposition of two given points, each maps to the other. (Contributed by Thierry Arnoux, 22-Aug-2020.) |
| ⊢ 𝑇 = (pmTrsp‘𝐷) ⇒ ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → ((𝑇‘{𝑋, 𝑌})‘𝑌) = 𝑋) | ||
| Theorem | pmtrcnel 30733 | Composing a permutation 𝐹 with a transposition which results in moving at least one less point. Here the set of points moved by a permutation 𝐹 is expressed as dom (𝐹 ∖ I ). (Contributed by Thierry Arnoux, 16-Nov-2023.) |
| ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑇 = (pmTrsp‘𝐷) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐽 = (𝐹‘𝐼) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐼 ∈ dom (𝐹 ∖ I )) ⇒ ⊢ (𝜑 → dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ⊆ (dom (𝐹 ∖ I ) ∖ {𝐼})) | ||
| Theorem | pmtrcnel2 30734 | Variation on pmtrcnel 30733. (Contributed by Thierry Arnoux, 16-Nov-2023.) |
| ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑇 = (pmTrsp‘𝐷) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐽 = (𝐹‘𝐼) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐼 ∈ dom (𝐹 ∖ I )) ⇒ ⊢ (𝜑 → (dom (𝐹 ∖ I ) ∖ {𝐼, 𝐽}) ⊆ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )) | ||
| Theorem | pmtrcnelor 30735 | Composing a permutation 𝐹 with a transposition which results in moving one or two less points. (Contributed by Thierry Arnoux, 16-Nov-2023.) |
| ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑇 = (pmTrsp‘𝐷) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐽 = (𝐹‘𝐼) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐼 ∈ dom (𝐹 ∖ I )) & ⊢ 𝐸 = dom (𝐹 ∖ I ) & ⊢ 𝐴 = dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ⇒ ⊢ (𝜑 → (𝐴 = (𝐸 ∖ {𝐼, 𝐽}) ∨ 𝐴 = (𝐸 ∖ {𝐼}))) | ||
| Theorem | pmtridf1o 30736 | Transpositions of 𝑋 and 𝑌 (understood to be the identity when 𝑋 = 𝑌), are bijections. (Contributed by Thierry Arnoux, 1-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → 𝑇:𝐴–1-1-onto→𝐴) | ||
| Theorem | pmtridfv1 30737 | Value at X of the transposition of 𝑋 and 𝑌 (understood to be the identity when X = Y ). (Contributed by Thierry Arnoux, 3-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → (𝑇‘𝑋) = 𝑌) | ||
| Theorem | pmtridfv2 30738 | Value at Y of the transposition of 𝑋 and 𝑌 (understood to be the identity when X = Y ). (Contributed by Thierry Arnoux, 3-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → (𝑇‘𝑌) = 𝑋) | ||
| Theorem | psgnid 30739 | Permutation sign of the identity. (Contributed by Thierry Arnoux, 21-Aug-2020.) |
| ⊢ 𝑆 = (pmSgn‘𝐷) ⇒ ⊢ (𝐷 ∈ Fin → (𝑆‘( I ↾ 𝐷)) = 1) | ||
| Theorem | psgndmfi 30740 | For a finite base set, the permutation sign is defined for all permutations. (Contributed by Thierry Arnoux, 22-Aug-2020.) |
| ⊢ 𝑆 = (pmSgn‘𝐷) & ⊢ 𝐺 = (Base‘(SymGrp‘𝐷)) ⇒ ⊢ (𝐷 ∈ Fin → 𝑆 Fn 𝐺) | ||
| Theorem | pmtrto1cl 30741 | Useful lemma for the following theorems. (Contributed by Thierry Arnoux, 21-Aug-2020.) |
| ⊢ 𝐷 = (1...𝑁) & ⊢ 𝑇 = (pmTrsp‘𝐷) ⇒ ⊢ ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → (𝑇‘{𝐾, (𝐾 + 1)}) ∈ ran 𝑇) | ||
| Theorem | psgnfzto1stlem 30742* | Lemma for psgnfzto1st 30747. Our permutation of rank (𝑛 + 1) can be written as a permutation of rank 𝑛 composed with a transposition. (Contributed by Thierry Arnoux, 21-Aug-2020.) |
| ⊢ 𝐷 = (1...𝑁) ⇒ ⊢ ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, (𝐾 + 1), if(𝑖 ≤ (𝐾 + 1), (𝑖 − 1), 𝑖))) = (((pmTrsp‘𝐷)‘{𝐾, (𝐾 + 1)}) ∘ (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐾, if(𝑖 ≤ 𝐾, (𝑖 − 1), 𝑖))))) | ||
| Theorem | fzto1stfv1 30743* | Value of our permutation 𝑃 at 1. (Contributed by Thierry Arnoux, 23-Aug-2020.) |
| ⊢ 𝐷 = (1...𝑁) & ⊢ 𝑃 = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) ⇒ ⊢ (𝐼 ∈ 𝐷 → (𝑃‘1) = 𝐼) | ||
| Theorem | fzto1st1 30744* | Special case where the permutation defined in psgnfzto1st 30747 is the identity. (Contributed by Thierry Arnoux, 21-Aug-2020.) |
| ⊢ 𝐷 = (1...𝑁) & ⊢ 𝑃 = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) ⇒ ⊢ (𝐼 = 1 → 𝑃 = ( I ↾ 𝐷)) | ||
| Theorem | fzto1st 30745* | The function moving one element to the first position (and shifting all elements before it) is a permutation. (Contributed by Thierry Arnoux, 21-Aug-2020.) |
| ⊢ 𝐷 = (1...𝑁) & ⊢ 𝑃 = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) & ⊢ 𝐺 = (SymGrp‘𝐷) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐼 ∈ 𝐷 → 𝑃 ∈ 𝐵) | ||
| Theorem | fzto1stinvn 30746* | Value of the inverse of our permutation 𝑃 at 𝐼 (Contributed by Thierry Arnoux, 23-Aug-2020.) |
| ⊢ 𝐷 = (1...𝑁) & ⊢ 𝑃 = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) & ⊢ 𝐺 = (SymGrp‘𝐷) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐼 ∈ 𝐷 → (◡𝑃‘𝐼) = 1) | ||
| Theorem | psgnfzto1st 30747* | The permutation sign for moving one element to the first position. (Contributed by Thierry Arnoux, 21-Aug-2020.) |
| ⊢ 𝐷 = (1...𝑁) & ⊢ 𝑃 = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) & ⊢ 𝐺 = (SymGrp‘𝐷) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑆 = (pmSgn‘𝐷) ⇒ ⊢ (𝐼 ∈ 𝐷 → (𝑆‘𝑃) = (-1↑(𝐼 + 1))) | ||
| Syntax | ctocyc 30748 | Extend class notation with the permutation cycle builder. |
| class toCyc | ||
| Definition | df-tocyc 30749* | Define a convenience permutation cycle builder. Given a list of elements to be cycled, in the form of a word, this function produces the corresponding permutation cycle. See definition in [Lang] p. 30. (Contributed by Thierry Arnoux, 19-Sep-2023.) |
| ⊢ toCyc = (𝑑 ∈ V ↦ (𝑤 ∈ {𝑢 ∈ Word 𝑑 ∣ 𝑢:dom 𝑢–1-1→𝑑} ↦ (( I ↾ (𝑑 ∖ ran 𝑤)) ∪ ((𝑤 cyclShift 1) ∘ ◡𝑤)))) | ||
| Theorem | tocycval 30750* | Value of the cycle builder. (Contributed by Thierry Arnoux, 22-Sep-2023.) |
| ⊢ 𝐶 = (toCyc‘𝐷) ⇒ ⊢ (𝐷 ∈ 𝑉 → 𝐶 = (𝑤 ∈ {𝑢 ∈ Word 𝐷 ∣ 𝑢:dom 𝑢–1-1→𝐷} ↦ (( I ↾ (𝐷 ∖ ran 𝑤)) ∪ ((𝑤 cyclShift 1) ∘ ◡𝑤)))) | ||
| Theorem | tocycfv 30751 | Function value of a permutation cycle built from a word. (Contributed by Thierry Arnoux, 18-Sep-2023.) |
| ⊢ 𝐶 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) & ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) ⇒ ⊢ (𝜑 → (𝐶‘𝑊) = (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊))) | ||
| Theorem | tocycfvres1 30752 | A cyclic permutation is a cyclic shift on its orbit. (Contributed by Thierry Arnoux, 15-Oct-2023.) |
| ⊢ 𝐶 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) & ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) ⇒ ⊢ (𝜑 → ((𝐶‘𝑊) ↾ ran 𝑊) = ((𝑊 cyclShift 1) ∘ ◡𝑊)) | ||
| Theorem | tocycfvres2 30753 | A cyclic permutation is the identity outside of its orbit. (Contributed by Thierry Arnoux, 15-Oct-2023.) |
| ⊢ 𝐶 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) & ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) ⇒ ⊢ (𝜑 → ((𝐶‘𝑊) ↾ (𝐷 ∖ ran 𝑊)) = ( I ↾ (𝐷 ∖ ran 𝑊))) | ||
| Theorem | cycpmfvlem 30754 | Lemma for cycpmfv1 30755 and cycpmfv2 30756. (Contributed by Thierry Arnoux, 22-Sep-2023.) |
| ⊢ 𝐶 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) & ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) & ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝑊))) ⇒ ⊢ (𝜑 → ((𝐶‘𝑊)‘(𝑊‘𝑁)) = (((𝑊 cyclShift 1) ∘ ◡𝑊)‘(𝑊‘𝑁))) | ||
| Theorem | cycpmfv1 30755 | Value of a cycle function for any element but the last. (Contributed by Thierry Arnoux, 22-Sep-2023.) |
| ⊢ 𝐶 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) & ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) & ⊢ (𝜑 → 𝑁 ∈ (0..^((♯‘𝑊) − 1))) ⇒ ⊢ (𝜑 → ((𝐶‘𝑊)‘(𝑊‘𝑁)) = (𝑊‘(𝑁 + 1))) | ||
| Theorem | cycpmfv2 30756 | Value of a cycle function for the last element of the orbit. (Contributed by Thierry Arnoux, 22-Sep-2023.) |
| ⊢ 𝐶 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) & ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) & ⊢ (𝜑 → 0 < (♯‘𝑊)) & ⊢ (𝜑 → 𝑁 = ((♯‘𝑊) − 1)) ⇒ ⊢ (𝜑 → ((𝐶‘𝑊)‘(𝑊‘𝑁)) = (𝑊‘0)) | ||
| Theorem | cycpmfv3 30757 | Values outside of the orbit are unchanged by a cycle. (Contributed by Thierry Arnoux, 22-Sep-2023.) |
| ⊢ 𝐶 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) & ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → ¬ 𝑋 ∈ ran 𝑊) ⇒ ⊢ (𝜑 → ((𝐶‘𝑊)‘𝑋) = 𝑋) | ||
| Theorem | cycpmcl 30758 | Cyclic permutations are permutations. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
| ⊢ 𝐶 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) & ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) ⇒ ⊢ (𝜑 → (𝐶‘𝑊) ∈ (Base‘𝑆)) | ||
| Theorem | tocycf 30759* | The permutation cycle builder as a function. (Contributed by Thierry Arnoux, 25-Sep-2023.) |
| ⊢ 𝐶 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝐵 = (Base‘𝑆) ⇒ ⊢ (𝐷 ∈ 𝑉 → 𝐶:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶𝐵) | ||
| Theorem | tocyc01 30760 | Permutation cycles built from the empty set or a singleton are the identity. (Contributed by Thierry Arnoux, 21-Nov-2023.) |
| ⊢ 𝐶 = (toCyc‘𝐷) ⇒ ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑊 ∈ (dom 𝐶 ∩ (◡♯ “ {0, 1}))) → (𝐶‘𝑊) = ( I ↾ 𝐷)) | ||
| Theorem | cycpm2tr 30761 | A cyclic permutation of 2 elements is a transposition. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
| ⊢ 𝐶 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ 𝑇 = (pmTrsp‘𝐷) ⇒ ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽”⟩) = (𝑇‘{𝐼, 𝐽})) | ||
| Theorem | cycpm2cl 30762 | Closure for the 2-cycles. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
| ⊢ 𝐶 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ 𝑆 = (SymGrp‘𝐷) ⇒ ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆)) | ||
| Theorem | cyc2fv1 30763 | Function value of a 2-cycle at the first point. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
| ⊢ 𝐶 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ 𝑆 = (SymGrp‘𝐷) ⇒ ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐼) = 𝐽) | ||
| Theorem | cyc2fv2 30764 | Function value of a 2-cycle at the second point. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
| ⊢ 𝐶 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ 𝑆 = (SymGrp‘𝐷) ⇒ ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐽) = 𝐼) | ||
| Theorem | trsp2cyc 30765* | Exhibit the word a transposition corresponds to, as a cycle. (Contributed by Thierry Arnoux, 25-Sep-2023.) |
| ⊢ 𝑇 = ran (pmTrsp‘𝐷) & ⊢ 𝐶 = (toCyc‘𝐷) ⇒ ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) → ∃𝑖 ∈ 𝐷 ∃𝑗 ∈ 𝐷 (𝑖 ≠ 𝑗 ∧ 𝑃 = (𝐶‘⟨“𝑖𝑗”⟩))) | ||
| Theorem | cycpmco2f1 30766 | The word U used in cycpmco2 30775 is injective, so it can represent a cycle and form a cyclic permutation (𝑀‘𝑈). (Contributed by Thierry Arnoux, 4-Jan-2024.) |
| ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ⇒ ⊢ (𝜑 → 𝑈:dom 𝑈–1-1→𝐷) | ||
| Theorem | cycpmco2rn 30767 | The orbit of the composition of a cyclic permutation and a well-chosen transposition is one element more than the orbit of the original permutation. (Contributed by Thierry Arnoux, 4-Jan-2024.) |
| ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ⇒ ⊢ (𝜑 → ran 𝑈 = (ran 𝑊 ∪ {𝐼})) | ||
| Theorem | cycpmco2lem1 30768 | Lemma for cycpmco2 30775. (Contributed by Thierry Arnoux, 4-Jan-2024.) |
| ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ⇒ ⊢ (𝜑 → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼)) = ((𝑀‘𝑊)‘𝐽)) | ||
| Theorem | cycpmco2lem2 30769 | Lemma for cycpmco2 30775. (Contributed by Thierry Arnoux, 4-Jan-2024.) |
| ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ⇒ ⊢ (𝜑 → (𝑈‘𝐸) = 𝐼) | ||
| Theorem | cycpmco2lem3 30770 | Lemma for cycpmco2 30775. (Contributed by Thierry Arnoux, 4-Jan-2024.) |
| ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ⇒ ⊢ (𝜑 → ((♯‘𝑈) − 1) = (♯‘𝑊)) | ||
| Theorem | cycpmco2lem4 30771 | Lemma for cycpmco2 30775. (Contributed by Thierry Arnoux, 4-Jan-2024.) |
| ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ⇒ ⊢ (𝜑 → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼)) = ((𝑀‘𝑈)‘𝐼)) | ||
| Theorem | cycpmco2lem5 30772 | Lemma for cycpmco2 30775. (Contributed by Thierry Arnoux, 4-Jan-2024.) |
| ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) & ⊢ (𝜑 → 𝐾 ∈ ran 𝑊) & ⊢ (𝜑 → (◡𝑈‘𝐾) = ((♯‘𝑈) − 1)) ⇒ ⊢ (𝜑 → ((𝑀‘𝑈)‘𝐾) = ((𝑀‘𝑊)‘𝐾)) | ||
| Theorem | cycpmco2lem6 30773 | Lemma for cycpmco2 30775. (Contributed by Thierry Arnoux, 4-Jan-2024.) |
| ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) & ⊢ (𝜑 → 𝐾 ∈ ran 𝑊) & ⊢ (𝜑 → 𝐾 ≠ 𝐼) & ⊢ (𝜑 → (◡𝑈‘𝐾) ∈ (𝐸..^((♯‘𝑈) − 1))) ⇒ ⊢ (𝜑 → ((𝑀‘𝑈)‘𝐾) = ((𝑀‘𝑊)‘𝐾)) | ||
| Theorem | cycpmco2lem7 30774 | Lemma for cycpmco2 30775. (Contributed by Thierry Arnoux, 4-Jan-2024.) |
| ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) & ⊢ (𝜑 → 𝐾 ∈ ran 𝑊) & ⊢ (𝜑 → 𝐾 ≠ 𝐽) & ⊢ (𝜑 → (◡𝑈‘𝐾) ∈ (0..^𝐸)) ⇒ ⊢ (𝜑 → ((𝑀‘𝑈)‘𝐾) = ((𝑀‘𝑊)‘𝐾)) | ||
| Theorem | cycpmco2 30775 | The composition of a cyclic permutation and a transposition of one element in the cycle and one outside the cycle results in a cyclic permutation with one more element in its orbit. (Contributed by Thierry Arnoux, 2-Jan-2024.) |
| ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ⇒ ⊢ (𝜑 → ((𝑀‘𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩)) = (𝑀‘𝑈)) | ||
| Theorem | cyc2fvx 30776 | Function value of a 2-cycle outside of its orbit. (Contributed by Thierry Arnoux, 19-Sep-2023.) |
| ⊢ 𝐶 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐾 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ (𝜑 → 𝐽 ≠ 𝐾) & ⊢ (𝜑 → 𝐾 ≠ 𝐼) ⇒ ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐾) = 𝐾) | ||
| Theorem | cycpm3cl 30777 | Closure of the 3-cycles in the permutations. (Contributed by Thierry Arnoux, 19-Sep-2023.) |
| ⊢ 𝐶 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐾 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ (𝜑 → 𝐽 ≠ 𝐾) & ⊢ (𝜑 → 𝐾 ≠ 𝐼) ⇒ ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) ∈ (Base‘𝑆)) | ||
| Theorem | cycpm3cl2 30778 | Closure of the 3-cycles in the class of 3-cycles. (Contributed by Thierry Arnoux, 19-Sep-2023.) |
| ⊢ 𝐶 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐾 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ (𝜑 → 𝐽 ≠ 𝐾) & ⊢ (𝜑 → 𝐾 ≠ 𝐼) ⇒ ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) ∈ (𝐶 “ (◡♯ “ {3}))) | ||
| Theorem | cyc3fv1 30779 | Function value of a 3-cycle at the first point. (Contributed by Thierry Arnoux, 19-Sep-2023.) |
| ⊢ 𝐶 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐾 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ (𝜑 → 𝐽 ≠ 𝐾) & ⊢ (𝜑 → 𝐾 ≠ 𝐼) ⇒ ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐼) = 𝐽) | ||
| Theorem | cyc3fv2 30780 | Function value of a 3-cycle at the second point. (Contributed by Thierry Arnoux, 19-Sep-2023.) |
| ⊢ 𝐶 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐾 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ (𝜑 → 𝐽 ≠ 𝐾) & ⊢ (𝜑 → 𝐾 ≠ 𝐼) ⇒ ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐽) = 𝐾) | ||
| Theorem | cyc3fv3 30781 | Function value of a 3-cycle at the third point. (Contributed by Thierry Arnoux, 19-Sep-2023.) |
| ⊢ 𝐶 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐾 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ (𝜑 → 𝐽 ≠ 𝐾) & ⊢ (𝜑 → 𝐾 ≠ 𝐼) ⇒ ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐾) = 𝐼) | ||
| Theorem | cyc3co2 30782 | Represent a 3-cycle as a composition of two 2-cycles. (Contributed by Thierry Arnoux, 19-Sep-2023.) |
| ⊢ 𝐶 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐾 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ (𝜑 → 𝐽 ≠ 𝐾) & ⊢ (𝜑 → 𝐾 ≠ 𝐼) & ⊢ · = (+g‘𝑆) ⇒ ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) = ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))) | ||
| Theorem | cycpmconjvlem 30783 | Lemma for cycpmconjv 30784 (Contributed by Thierry Arnoux, 9-Oct-2023.) |
| ⊢ (𝜑 → 𝐹:𝐷–1-1-onto→𝐷) & ⊢ (𝜑 → 𝐵 ⊆ 𝐷) ⇒ ⊢ (𝜑 → ((𝐹 ↾ (𝐷 ∖ 𝐵)) ∘ ◡𝐹) = ( I ↾ (𝐷 ∖ ran (𝐹 ↾ 𝐵)))) | ||
| Theorem | cycpmconjv 30784 | A formula for computing conjugacy classes of cyclic permutations. Formula in property (b) of [Lang] p. 32. (Contributed by Thierry Arnoux, 9-Oct-2023.) |
| ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ + = (+g‘𝑆) & ⊢ − = (-g‘𝑆) & ⊢ 𝐵 = (Base‘𝑆) ⇒ ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 + (𝑀‘𝑊)) − 𝐺) = (𝑀‘(𝐺 ∘ 𝑊))) | ||
| Theorem | cycpmrn 30785 | The range of the word used to build a cycle is the cycle's orbit, i.e. the set of points it moves. (Contributed by Thierry Arnoux, 20-Nov-2023.) |
| ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) & ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) & ⊢ (𝜑 → 1 < (♯‘𝑊)) ⇒ ⊢ (𝜑 → ran 𝑊 = dom ((𝑀‘𝑊) ∖ I )) | ||
| Theorem | tocyccntz 30786* | All elements of a (finite) set of cycles commute if their orbits are disjoint. (Contributed by Thierry Arnoux, 27-Nov-2023.) |
| ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑍 = (Cntz‘𝑆) & ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 ran 𝑥) & ⊢ (𝜑 → 𝐴 ⊆ dom 𝑀) ⇒ ⊢ (𝜑 → (𝑀 “ 𝐴) ⊆ (𝑍‘(𝑀 “ 𝐴))) | ||
| Theorem | evpmval 30787 | Value of the set of even permutations, the alternating group. (Contributed by Thierry Arnoux, 1-Nov-2023.) |
| ⊢ 𝐴 = (pmEven‘𝐷) ⇒ ⊢ (𝐷 ∈ 𝑉 → 𝐴 = (◡(pmSgn‘𝐷) “ {1})) | ||
| Theorem | cnmsgn0g 30788 | The neutral element of the sign subgroup of the complex numbers. (Contributed by Thierry Arnoux, 1-Nov-2023.) |
| ⊢ 𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1}) ⇒ ⊢ 1 = (0g‘𝑈) | ||
| Theorem | evpmsubg 30789 | The alternating group is a subgroup of the symmetric group. (Contributed by Thierry Arnoux, 1-Nov-2023.) |
| ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝐴 = (pmEven‘𝐷) ⇒ ⊢ (𝐷 ∈ Fin → 𝐴 ∈ (SubGrp‘𝑆)) | ||
| Theorem | evpmid 30790 | The identity is an even permutation. (Contributed by Thierry Arnoux, 18-Sep-2023.) |
| ⊢ 𝑆 = (SymGrp‘𝐷) ⇒ ⊢ (𝐷 ∈ Fin → ( I ↾ 𝐷) ∈ (pmEven‘𝐷)) | ||
| Theorem | altgnsg 30791 | The alternating group (pmEven‘𝐷) is a normal subgroup of the symmetric group. (Contributed by Thierry Arnoux, 18-Sep-2023.) |
| ⊢ 𝑆 = (SymGrp‘𝐷) ⇒ ⊢ (𝐷 ∈ Fin → (pmEven‘𝐷) ∈ (NrmSGrp‘𝑆)) | ||
| Theorem | cyc3evpm 30792 | 3-Cycles are even permutations. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
| ⊢ 𝐶 = ((toCyc‘𝐷) “ (◡♯ “ {3})) & ⊢ 𝐴 = (pmEven‘𝐷) ⇒ ⊢ (𝐷 ∈ Fin → 𝐶 ⊆ 𝐴) | ||
| Theorem | cyc3genpmlem 30793* | Lemma for cyc3genpm 30794. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
| ⊢ 𝐶 = (𝑀 “ (◡♯ “ {3})) & ⊢ 𝐴 = (pmEven‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑁 = (♯‘𝐷) & ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ · = (+g‘𝑆) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐾 ∈ 𝐷) & ⊢ (𝜑 → 𝐿 ∈ 𝐷) & ⊢ (𝜑 → 𝐸 = (𝑀‘⟨“𝐼𝐽”⟩)) & ⊢ (𝜑 → 𝐹 = (𝑀‘⟨“𝐾𝐿”⟩)) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ (𝜑 → 𝐾 ≠ 𝐿) ⇒ ⊢ (𝜑 → ∃𝑐 ∈ Word 𝐶(𝐸 · 𝐹) = (𝑆 Σg 𝑐)) | ||
| Theorem | cyc3genpm 30794* | The alternating group 𝐴 is generated by 3-cycles. Property (a) of [Lang] p. 32 . (Contributed by Thierry Arnoux, 27-Sep-2023.) |
| ⊢ 𝐶 = (𝑀 “ (◡♯ “ {3})) & ⊢ 𝐴 = (pmEven‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑁 = (♯‘𝐷) & ⊢ 𝑀 = (toCyc‘𝐷) ⇒ ⊢ (𝐷 ∈ Fin → (𝑄 ∈ 𝐴 ↔ ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤))) | ||
| Theorem | cycpmgcl 30795 | Cyclic permutations are permutations, similar to cycpmcl 30758, but where the set of cyclic permutations of length 𝑃 is expressed in terms of a preimage. (Contributed by Thierry Arnoux, 13-Oct-2023.) |
| ⊢ 𝐶 = (𝑀 “ (◡♯ “ {𝑃})) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑁 = (♯‘𝐷) & ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝐵 = (Base‘𝑆) ⇒ ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ (0...𝑁)) → 𝐶 ⊆ 𝐵) | ||
| Theorem | cycpmconjslem1 30796 | Lemma for cycpmconjs 30798 (Contributed by Thierry Arnoux, 14-Oct-2023.) |
| ⊢ 𝐶 = (𝑀 “ (◡♯ “ {𝑃})) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑁 = (♯‘𝐷) & ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) & ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) & ⊢ (𝜑 → (♯‘𝑊) = 𝑃) ⇒ ⊢ (𝜑 → ((◡𝑊 ∘ (𝑀‘𝑊)) ∘ 𝑊) = (( I ↾ (0..^𝑃)) cyclShift 1)) | ||
| Theorem | cycpmconjslem2 30797* | Lemma for cycpmconjs 30798 (Contributed by Thierry Arnoux, 14-Oct-2023.) |
| ⊢ 𝐶 = (𝑀 “ (◡♯ “ {𝑃})) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑁 = (♯‘𝐷) & ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ + = (+g‘𝑆) & ⊢ − = (-g‘𝑆) & ⊢ (𝜑 → 𝑃 ∈ (0...𝑁)) & ⊢ (𝜑 → 𝐷 ∈ Fin) & ⊢ (𝜑 → 𝑄 ∈ 𝐶) ⇒ ⊢ (𝜑 → ∃𝑞(𝑞:(0..^𝑁)–1-1-onto→𝐷 ∧ ((◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁))))) | ||
| Theorem | cycpmconjs 30798* | All cycles of the same length are conjugate in the symmetric group. (Contributed by Thierry Arnoux, 14-Oct-2023.) |
| ⊢ 𝐶 = (𝑀 “ (◡♯ “ {𝑃})) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑁 = (♯‘𝐷) & ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ + = (+g‘𝑆) & ⊢ − = (-g‘𝑆) & ⊢ (𝜑 → 𝑃 ∈ (0...𝑁)) & ⊢ (𝜑 → 𝐷 ∈ Fin) & ⊢ (𝜑 → 𝑄 ∈ 𝐶) & ⊢ (𝜑 → 𝑇 ∈ 𝐶) ⇒ ⊢ (𝜑 → ∃𝑝 ∈ 𝐵 𝑄 = ((𝑝 + 𝑇) − 𝑝)) | ||
| Theorem | cyc3conja 30799* | All 3-cycles are conjugate in the alternating group An for n>= 5. Property (b) of [Lang] p. 32. (Contributed by Thierry Arnoux, 15-Oct-2023.) |
| ⊢ 𝐶 = (𝑀 “ (◡♯ “ {3})) & ⊢ 𝐴 = (pmEven‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑁 = (♯‘𝐷) & ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ + = (+g‘𝑆) & ⊢ − = (-g‘𝑆) & ⊢ (𝜑 → 5 ≤ 𝑁) & ⊢ (𝜑 → 𝐷 ∈ Fin) & ⊢ (𝜑 → 𝑄 ∈ 𝐶) & ⊢ (𝜑 → 𝑇 ∈ 𝐶) ⇒ ⊢ (𝜑 → ∃𝑝 ∈ 𝐴 𝑄 = ((𝑝 + 𝑇) − 𝑝)) | ||
| Syntax | csgns 30800 | Extend class notation to include the Signum function. |
| class sgns | ||
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