![]() |
Metamath
Proof Explorer Theorem List (p. 189 of 454) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-28701) |
![]() (28702-30224) |
![]() (30225-45333) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | lsmdisj3 18801 | Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 }) & ⊢ (𝜑 → (𝑆 ∩ 𝑇) = { 0 }) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝑆 ⊆ (𝑍‘𝑇)) ⇒ ⊢ (𝜑 → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 }) | ||
Theorem | lsmdisjr 18802 | Disjointness from a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 }) ⇒ ⊢ (𝜑 → ((𝑆 ∩ 𝑇) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })) | ||
Theorem | lsmdisj2r 18803 | Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 }) & ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) ⇒ ⊢ (𝜑 → ((𝑆 ⊕ 𝑈) ∩ 𝑇) = { 0 }) | ||
Theorem | lsmdisj3r 18804 | Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 }) & ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) ⇒ ⊢ (𝜑 → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 }) | ||
Theorem | lsmdisj2a 18805 | Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝜑 → ((((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 }) ↔ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 }))) | ||
Theorem | lsmdisj2b 18806 | Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝜑 → ((((𝑆 ⊕ 𝑈) ∩ 𝑇) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 }) ↔ ((𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 } ∧ (𝑇 ∩ 𝑈) = { 0 }))) | ||
Theorem | lsmdisj3a 18807 | Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝑆 ⊆ (𝑍‘𝑇)) ⇒ ⊢ (𝜑 → ((((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 }) ↔ ((𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 } ∧ (𝑇 ∩ 𝑈) = { 0 }))) | ||
Theorem | lsmdisj3b 18808 | Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) ⇒ ⊢ (𝜑 → ((((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 }) ↔ ((𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 } ∧ (𝑇 ∩ 𝑈) = { 0 }))) | ||
Theorem | subgdisj1 18809 | Vectors belonging to disjoint commuting subgroups are uniquely determined by their sum. (Contributed by NM, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) & ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) & ⊢ (𝜑 → 𝐴 ∈ 𝑇) & ⊢ (𝜑 → 𝐶 ∈ 𝑇) & ⊢ (𝜑 → 𝐵 ∈ 𝑈) & ⊢ (𝜑 → 𝐷 ∈ 𝑈) & ⊢ (𝜑 → (𝐴 + 𝐵) = (𝐶 + 𝐷)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐶) | ||
Theorem | subgdisj2 18810 | Vectors belonging to disjoint commuting subgroups are uniquely determined by their sum. (Contributed by NM, 12-Jul-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) & ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) & ⊢ (𝜑 → 𝐴 ∈ 𝑇) & ⊢ (𝜑 → 𝐶 ∈ 𝑇) & ⊢ (𝜑 → 𝐵 ∈ 𝑈) & ⊢ (𝜑 → 𝐷 ∈ 𝑈) & ⊢ (𝜑 → (𝐴 + 𝐵) = (𝐶 + 𝐷)) ⇒ ⊢ (𝜑 → 𝐵 = 𝐷) | ||
Theorem | subgdisjb 18811 | Vectors belonging to disjoint commuting subgroups are uniquely determined by their sum. Analogous to opth 5333, this theorem shows a way of representing a pair of vectors. (Contributed by NM, 5-Jul-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ + = (+g‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) & ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) & ⊢ (𝜑 → 𝐴 ∈ 𝑇) & ⊢ (𝜑 → 𝐶 ∈ 𝑇) & ⊢ (𝜑 → 𝐵 ∈ 𝑈) & ⊢ (𝜑 → 𝐷 ∈ 𝑈) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) = (𝐶 + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | ||
Theorem | pj1fval 18812* | The left projection function (for a direct product of group subspaces). (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ 𝑃 = (proj1‘𝐺) ⇒ ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇𝑃𝑈) = (𝑧 ∈ (𝑇 ⊕ 𝑈) ↦ (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦)))) | ||
Theorem | pj1val 18813* | The left projection function (for a direct product of group subspaces). (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ 𝑃 = (proj1‘𝐺) ⇒ ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → ((𝑇𝑃𝑈)‘𝑋) = (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦))) | ||
Theorem | pj1eu 18814* | Uniqueness of a left projection. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ + = (+g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) & ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → ∃!𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)) | ||
Theorem | pj1f 18815 | The left projection function maps a direct subspace sum onto the left factor. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ + = (+g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) & ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) & ⊢ 𝑃 = (proj1‘𝐺) ⇒ ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇) | ||
Theorem | pj2f 18816 | The right projection function maps a direct subspace sum onto the right factor. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ + = (+g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) & ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) & ⊢ 𝑃 = (proj1‘𝐺) ⇒ ⊢ (𝜑 → (𝑈𝑃𝑇):(𝑇 ⊕ 𝑈)⟶𝑈) | ||
Theorem | pj1id 18817 | Any element of a direct subspace sum can be decomposed into projections onto the left and right factors. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.) |
⊢ + = (+g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) & ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) & ⊢ 𝑃 = (proj1‘𝐺) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋))) | ||
Theorem | pj1eq 18818 | Any element of a direct subspace sum can be decomposed uniquely into projections onto the left and right factors. (Contributed by Mario Carneiro, 16-Oct-2015.) |
⊢ + = (+g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) & ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) & ⊢ 𝑃 = (proj1‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ (𝑇 ⊕ 𝑈)) & ⊢ (𝜑 → 𝐵 ∈ 𝑇) & ⊢ (𝜑 → 𝐶 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝑋 = (𝐵 + 𝐶) ↔ (((𝑇𝑃𝑈)‘𝑋) = 𝐵 ∧ ((𝑈𝑃𝑇)‘𝑋) = 𝐶))) | ||
Theorem | pj1lid 18819 | The left projection function is the identity on the left subspace. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ + = (+g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) & ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) & ⊢ 𝑃 = (proj1‘𝐺) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑇) → ((𝑇𝑃𝑈)‘𝑋) = 𝑋) | ||
Theorem | pj1rid 18820 | The left projection function is the zero operator on the right subspace. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ + = (+g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) & ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) & ⊢ 𝑃 = (proj1‘𝐺) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → ((𝑇𝑃𝑈)‘𝑋) = 0 ) | ||
Theorem | pj1ghm 18821 | The left projection function is a group homomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ + = (+g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) & ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) & ⊢ 𝑃 = (proj1‘𝐺) ⇒ ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝐺 ↾s (𝑇 ⊕ 𝑈)) GrpHom 𝐺)) | ||
Theorem | pj1ghm2 18822 | The left projection function is a group homomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ + = (+g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) & ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) & ⊢ 𝑃 = (proj1‘𝐺) ⇒ ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝐺 ↾s (𝑇 ⊕ 𝑈)) GrpHom (𝐺 ↾s 𝑇))) | ||
Theorem | lsmhash 18823 | The order of the direct product of groups. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ ⊕ = (LSSum‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) & ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) & ⊢ (𝜑 → 𝑇 ∈ Fin) & ⊢ (𝜑 → 𝑈 ∈ Fin) ⇒ ⊢ (𝜑 → (♯‘(𝑇 ⊕ 𝑈)) = ((♯‘𝑇) · (♯‘𝑈))) | ||
Syntax | cefg 18824 | Extend class notation with the free group equivalence relation. |
class ~FG | ||
Syntax | cfrgp 18825 | Extend class notation with the free group construction. |
class freeGrp | ||
Syntax | cvrgp 18826 | Extend class notation with free group injection. |
class varFGrp | ||
Definition | df-efg 18827* | Define the free group equivalence relation, which is the smallest equivalence relation ≈ such that for any words 𝐴, 𝐵 and formal symbol 𝑥 with inverse invg𝑥, 𝐴𝐵 ≈ 𝐴𝑥(invg𝑥)𝐵. (Contributed by Mario Carneiro, 1-Oct-2015.) |
⊢ ~FG = (𝑖 ∈ V ↦ ∩ {𝑟 ∣ (𝑟 Er Word (𝑖 × 2o) ∧ ∀𝑥 ∈ Word (𝑖 × 2o)∀𝑛 ∈ (0...(♯‘𝑥))∀𝑦 ∈ 𝑖 ∀𝑧 ∈ 2o 𝑥𝑟(𝑥 splice 〈𝑛, 𝑛, 〈“〈𝑦, 𝑧〉〈𝑦, (1o ∖ 𝑧)〉”〉〉))}) | ||
Definition | df-frgp 18828 | Define the free group on a set 𝐼 of generators, defined as the quotient of the free monoid on 𝐼 × 2o (representing the generator elements and their formal inverses) by the free group equivalence relation df-efg 18827. (Contributed by Mario Carneiro, 1-Oct-2015.) |
⊢ freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2o)) /s ( ~FG ‘𝑖))) | ||
Definition | df-vrgp 18829* | Define the canonical injection from the generating set 𝐼 into the base set of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ varFGrp = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ [〈“〈𝑗, ∅〉”〉]( ~FG ‘𝑖))) | ||
Theorem | efgmval 18830* | Value of the formal inverse operation for the generating set of a free group. (Contributed by Mario Carneiro, 27-Sep-2015.) |
⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) ⇒ ⊢ ((𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 2o) → (𝐴𝑀𝐵) = 〈𝐴, (1o ∖ 𝐵)〉) | ||
Theorem | efgmf 18831* | The formal inverse operation is an endofunction on the generating set. (Contributed by Mario Carneiro, 27-Sep-2015.) |
⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) ⇒ ⊢ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o) | ||
Theorem | efgmnvl 18832* | The inversion function on the generators is an involution. (Contributed by Mario Carneiro, 1-Oct-2015.) |
⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) ⇒ ⊢ (𝐴 ∈ (𝐼 × 2o) → (𝑀‘(𝑀‘𝐴)) = 𝐴) | ||
Theorem | efgrcl 18833 | Lemma for efgval 18835. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) ⇒ ⊢ (𝐴 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o))) | ||
Theorem | efglem 18834* | Lemma for efgval 18835. (Contributed by Mario Carneiro, 27-Sep-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) ⇒ ⊢ ∃𝑟(𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2o 𝑥𝑟(𝑥 splice 〈𝑛, 𝑛, 〈“〈𝑦, 𝑧〉〈𝑦, (1o ∖ 𝑧)〉”〉〉)) | ||
Theorem | efgval 18835* | Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) ⇒ ⊢ ∼ = ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2o 𝑥𝑟(𝑥 splice 〈𝑛, 𝑛, 〈“〈𝑦, 𝑧〉〈𝑦, (1o ∖ 𝑧)〉”〉〉))} | ||
Theorem | efger 18836 | Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) ⇒ ⊢ ∼ Er 𝑊 | ||
Theorem | efgi 18837 | Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) ⇒ ⊢ (((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴))) ∧ (𝐽 ∈ 𝐼 ∧ 𝐾 ∈ 2o)) → 𝐴 ∼ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, 𝐾〉〈𝐽, (1o ∖ 𝐾)〉”〉〉)) | ||
Theorem | efgi0 18838 | Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) ⇒ ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴)) ∧ 𝐽 ∈ 𝐼) → 𝐴 ∼ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, ∅〉〈𝐽, 1o〉”〉〉)) | ||
Theorem | efgi1 18839 | Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) ⇒ ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴)) ∧ 𝐽 ∈ 𝐼) → 𝐴 ∼ (𝐴 splice 〈𝑁, 𝑁, 〈“〈𝐽, 1o〉〈𝐽, ∅〉”〉〉)) | ||
Theorem | efgtf 18840* | Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) ⇒ ⊢ (𝑋 ∈ 𝑊 → ((𝑇‘𝑋) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice 〈𝑎, 𝑎, 〈“𝑏(𝑀‘𝑏)”〉〉)) ∧ (𝑇‘𝑋):((0...(♯‘𝑋)) × (𝐼 × 2o))⟶𝑊)) | ||
Theorem | efgtval 18841* | Value of the extension function, which maps a word (a representation of the group element as a sequence of elements and their inverses) to its direct extensions, defined as the original representation with an element and its inverse inserted somewhere in the string. (Contributed by Mario Carneiro, 29-Sep-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) ⇒ ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝑋)) ∧ 𝐴 ∈ (𝐼 × 2o)) → (𝑁(𝑇‘𝑋)𝐴) = (𝑋 splice 〈𝑁, 𝑁, 〈“𝐴(𝑀‘𝐴)”〉〉)) | ||
Theorem | efgval2 18842* | Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) ⇒ ⊢ ∼ = ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ⊆ [𝑥]𝑟)} | ||
Theorem | efgi2 18843* | Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) ⇒ ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ ran (𝑇‘𝐴)) → 𝐴 ∼ 𝐵) | ||
Theorem | efgtlen 18844* | Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) ⇒ ⊢ ((𝑋 ∈ 𝑊 ∧ 𝐴 ∈ ran (𝑇‘𝑋)) → (♯‘𝐴) = ((♯‘𝑋) + 2)) | ||
Theorem | efginvrel2 18845* | The inverse of the reverse of a word composed with the word relates to the identity. (This provides an explicit expression for the representation of the group inverse, given a representative of the free group equivalence class.) (Contributed by Mario Carneiro, 1-Oct-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) ⇒ ⊢ (𝐴 ∈ 𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∼ ∅) | ||
Theorem | efginvrel1 18846* | The inverse of the reverse of a word composed with the word relates to the identity. (This provides an explicit expression for the representation of the group inverse, given a representative of the free group equivalence class.) (Contributed by Mario Carneiro, 1-Oct-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) ⇒ ⊢ (𝐴 ∈ 𝑊 → ((𝑀 ∘ (reverse‘𝐴)) ++ 𝐴) ∼ ∅) | ||
Theorem | efgsf 18847* | Value of the auxiliary function 𝑆 defining a sequence of extensions starting at some irreducible word. (Contributed by Mario Carneiro, 1-Oct-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) ⇒ ⊢ 𝑆:{𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}⟶𝑊 | ||
Theorem | efgsdm 18848* | Elementhood in the domain of 𝑆, the set of sequences of extensions starting at an irreducible word. (Contributed by Mario Carneiro, 27-Sep-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) ⇒ ⊢ (𝐹 ∈ dom 𝑆 ↔ (𝐹 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐹‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝐹))(𝐹‘𝑖) ∈ ran (𝑇‘(𝐹‘(𝑖 − 1))))) | ||
Theorem | efgsval 18849* | Value of the auxiliary function 𝑆 defining a sequence of extensions. (Contributed by Mario Carneiro, 27-Sep-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) ⇒ ⊢ (𝐹 ∈ dom 𝑆 → (𝑆‘𝐹) = (𝐹‘((♯‘𝐹) − 1))) | ||
Theorem | efgsdmi 18850* | Property of the last link in the chain of extensions. (Contributed by Mario Carneiro, 29-Sep-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) ⇒ ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((♯‘𝐹) − 1) ∈ ℕ) → (𝑆‘𝐹) ∈ ran (𝑇‘(𝐹‘(((♯‘𝐹) − 1) − 1)))) | ||
Theorem | efgsval2 18851* | Value of the auxiliary function 𝑆 defining a sequence of extensions. (Contributed by Mario Carneiro, 1-Oct-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) ⇒ ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ++ 〈“𝐵”〉) ∈ dom 𝑆) → (𝑆‘(𝐴 ++ 〈“𝐵”〉)) = 𝐵) | ||
Theorem | efgsrel 18852* | The start and end of any extension sequence are related (i.e. evaluate to the same element of the quotient group to be created). (Contributed by Mario Carneiro, 1-Oct-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) ⇒ ⊢ (𝐹 ∈ dom 𝑆 → (𝐹‘0) ∼ (𝑆‘𝐹)) | ||
Theorem | efgs1 18853* | A singleton of an irreducible word is an extension sequence. (Contributed by Mario Carneiro, 27-Sep-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) ⇒ ⊢ (𝐴 ∈ 𝐷 → 〈“𝐴”〉 ∈ dom 𝑆) | ||
Theorem | efgs1b 18854* | Every extension sequence ending in an irreducible word is trivial. (Contributed by Mario Carneiro, 1-Oct-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) ⇒ ⊢ (𝐴 ∈ dom 𝑆 → ((𝑆‘𝐴) ∈ 𝐷 ↔ (♯‘𝐴) = 1)) | ||
Theorem | efgsp1 18855* | If 𝐹 is an extension sequence and 𝐴 is an extension of the last element of 𝐹, then 𝐹 + 〈“𝐴”〉 is an extension sequence. (Contributed by Mario Carneiro, 1-Oct-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) ⇒ ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (𝐹 ++ 〈“𝐴”〉) ∈ dom 𝑆) | ||
Theorem | efgsres 18856* | An initial segment of an extension sequence is an extension sequence. (Contributed by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 3-Nov-2022.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) ⇒ ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝑁 ∈ (1...(♯‘𝐹))) → (𝐹 ↾ (0..^𝑁)) ∈ dom 𝑆) | ||
Theorem | efgsfo 18857* | For any word, there is a sequence of extensions starting at a reduced word and ending at the target word, such that each word in the chain is an extension of the previous (inserting an element and its inverse at adjacent indices somewhere in the sequence). (Contributed by Mario Carneiro, 27-Sep-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) ⇒ ⊢ 𝑆:dom 𝑆–onto→𝑊 | ||
Theorem | efgredlema 18858* | The reduced word that forms the base of the sequence in efgsval 18849 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 1-Oct-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) & ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))) & ⊢ (𝜑 → 𝐴 ∈ dom 𝑆) & ⊢ (𝜑 → 𝐵 ∈ dom 𝑆) & ⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵)) & ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0)) ⇒ ⊢ (𝜑 → (((♯‘𝐴) − 1) ∈ ℕ ∧ ((♯‘𝐵) − 1) ∈ ℕ)) | ||
Theorem | efgredlemf 18859* | Lemma for efgredleme 18861. (Contributed by Mario Carneiro, 4-Jun-2016.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) & ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))) & ⊢ (𝜑 → 𝐴 ∈ dom 𝑆) & ⊢ (𝜑 → 𝐵 ∈ dom 𝑆) & ⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵)) & ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0)) & ⊢ 𝐾 = (((♯‘𝐴) − 1) − 1) & ⊢ 𝐿 = (((♯‘𝐵) − 1) − 1) ⇒ ⊢ (𝜑 → ((𝐴‘𝐾) ∈ 𝑊 ∧ (𝐵‘𝐿) ∈ 𝑊)) | ||
Theorem | efgredlemg 18860* | Lemma for efgred 18866. (Contributed by Mario Carneiro, 4-Jun-2016.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) & ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))) & ⊢ (𝜑 → 𝐴 ∈ dom 𝑆) & ⊢ (𝜑 → 𝐵 ∈ dom 𝑆) & ⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵)) & ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0)) & ⊢ 𝐾 = (((♯‘𝐴) − 1) − 1) & ⊢ 𝐿 = (((♯‘𝐵) − 1) − 1) & ⊢ (𝜑 → 𝑃 ∈ (0...(♯‘(𝐴‘𝐾)))) & ⊢ (𝜑 → 𝑄 ∈ (0...(♯‘(𝐵‘𝐿)))) & ⊢ (𝜑 → 𝑈 ∈ (𝐼 × 2o)) & ⊢ (𝜑 → 𝑉 ∈ (𝐼 × 2o)) & ⊢ (𝜑 → (𝑆‘𝐴) = (𝑃(𝑇‘(𝐴‘𝐾))𝑈)) & ⊢ (𝜑 → (𝑆‘𝐵) = (𝑄(𝑇‘(𝐵‘𝐿))𝑉)) ⇒ ⊢ (𝜑 → (♯‘(𝐴‘𝐾)) = (♯‘(𝐵‘𝐿))) | ||
Theorem | efgredleme 18861* | Lemma for efgred 18866. (Contributed by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 15-Oct-2022.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) & ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))) & ⊢ (𝜑 → 𝐴 ∈ dom 𝑆) & ⊢ (𝜑 → 𝐵 ∈ dom 𝑆) & ⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵)) & ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0)) & ⊢ 𝐾 = (((♯‘𝐴) − 1) − 1) & ⊢ 𝐿 = (((♯‘𝐵) − 1) − 1) & ⊢ (𝜑 → 𝑃 ∈ (0...(♯‘(𝐴‘𝐾)))) & ⊢ (𝜑 → 𝑄 ∈ (0...(♯‘(𝐵‘𝐿)))) & ⊢ (𝜑 → 𝑈 ∈ (𝐼 × 2o)) & ⊢ (𝜑 → 𝑉 ∈ (𝐼 × 2o)) & ⊢ (𝜑 → (𝑆‘𝐴) = (𝑃(𝑇‘(𝐴‘𝐾))𝑈)) & ⊢ (𝜑 → (𝑆‘𝐵) = (𝑄(𝑇‘(𝐵‘𝐿))𝑉)) & ⊢ (𝜑 → ¬ (𝐴‘𝐾) = (𝐵‘𝐿)) & ⊢ (𝜑 → 𝑃 ∈ (ℤ≥‘(𝑄 + 2))) & ⊢ (𝜑 → 𝐶 ∈ dom 𝑆) & ⊢ (𝜑 → (𝑆‘𝐶) = (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr 〈(𝑄 + 2), (♯‘(𝐴‘𝐾))〉))) ⇒ ⊢ (𝜑 → ((𝐴‘𝐾) ∈ ran (𝑇‘(𝑆‘𝐶)) ∧ (𝐵‘𝐿) ∈ ran (𝑇‘(𝑆‘𝐶)))) | ||
Theorem | efgredlemd 18862* | The reduced word that forms the base of the sequence in efgsval 18849 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 15-Oct-2022.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) & ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))) & ⊢ (𝜑 → 𝐴 ∈ dom 𝑆) & ⊢ (𝜑 → 𝐵 ∈ dom 𝑆) & ⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵)) & ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0)) & ⊢ 𝐾 = (((♯‘𝐴) − 1) − 1) & ⊢ 𝐿 = (((♯‘𝐵) − 1) − 1) & ⊢ (𝜑 → 𝑃 ∈ (0...(♯‘(𝐴‘𝐾)))) & ⊢ (𝜑 → 𝑄 ∈ (0...(♯‘(𝐵‘𝐿)))) & ⊢ (𝜑 → 𝑈 ∈ (𝐼 × 2o)) & ⊢ (𝜑 → 𝑉 ∈ (𝐼 × 2o)) & ⊢ (𝜑 → (𝑆‘𝐴) = (𝑃(𝑇‘(𝐴‘𝐾))𝑈)) & ⊢ (𝜑 → (𝑆‘𝐵) = (𝑄(𝑇‘(𝐵‘𝐿))𝑉)) & ⊢ (𝜑 → ¬ (𝐴‘𝐾) = (𝐵‘𝐿)) & ⊢ (𝜑 → 𝑃 ∈ (ℤ≥‘(𝑄 + 2))) & ⊢ (𝜑 → 𝐶 ∈ dom 𝑆) & ⊢ (𝜑 → (𝑆‘𝐶) = (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr 〈(𝑄 + 2), (♯‘(𝐴‘𝐾))〉))) ⇒ ⊢ (𝜑 → (𝐴‘0) = (𝐵‘0)) | ||
Theorem | efgredlemc 18863* | The reduced word that forms the base of the sequence in efgsval 18849 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 15-Oct-2022.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) & ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))) & ⊢ (𝜑 → 𝐴 ∈ dom 𝑆) & ⊢ (𝜑 → 𝐵 ∈ dom 𝑆) & ⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵)) & ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0)) & ⊢ 𝐾 = (((♯‘𝐴) − 1) − 1) & ⊢ 𝐿 = (((♯‘𝐵) − 1) − 1) & ⊢ (𝜑 → 𝑃 ∈ (0...(♯‘(𝐴‘𝐾)))) & ⊢ (𝜑 → 𝑄 ∈ (0...(♯‘(𝐵‘𝐿)))) & ⊢ (𝜑 → 𝑈 ∈ (𝐼 × 2o)) & ⊢ (𝜑 → 𝑉 ∈ (𝐼 × 2o)) & ⊢ (𝜑 → (𝑆‘𝐴) = (𝑃(𝑇‘(𝐴‘𝐾))𝑈)) & ⊢ (𝜑 → (𝑆‘𝐵) = (𝑄(𝑇‘(𝐵‘𝐿))𝑉)) & ⊢ (𝜑 → ¬ (𝐴‘𝐾) = (𝐵‘𝐿)) ⇒ ⊢ (𝜑 → (𝑃 ∈ (ℤ≥‘𝑄) → (𝐴‘0) = (𝐵‘0))) | ||
Theorem | efgredlemb 18864* | The reduced word that forms the base of the sequence in efgsval 18849 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 30-Sep-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) & ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))) & ⊢ (𝜑 → 𝐴 ∈ dom 𝑆) & ⊢ (𝜑 → 𝐵 ∈ dom 𝑆) & ⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵)) & ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0)) & ⊢ 𝐾 = (((♯‘𝐴) − 1) − 1) & ⊢ 𝐿 = (((♯‘𝐵) − 1) − 1) & ⊢ (𝜑 → 𝑃 ∈ (0...(♯‘(𝐴‘𝐾)))) & ⊢ (𝜑 → 𝑄 ∈ (0...(♯‘(𝐵‘𝐿)))) & ⊢ (𝜑 → 𝑈 ∈ (𝐼 × 2o)) & ⊢ (𝜑 → 𝑉 ∈ (𝐼 × 2o)) & ⊢ (𝜑 → (𝑆‘𝐴) = (𝑃(𝑇‘(𝐴‘𝐾))𝑈)) & ⊢ (𝜑 → (𝑆‘𝐵) = (𝑄(𝑇‘(𝐵‘𝐿))𝑉)) & ⊢ (𝜑 → ¬ (𝐴‘𝐾) = (𝐵‘𝐿)) ⇒ ⊢ ¬ 𝜑 | ||
Theorem | efgredlem 18865* | The reduced word that forms the base of the sequence in efgsval 18849 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 30-Sep-2015.) (Proof shortened by AV, 3-Nov-2022.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) & ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))) & ⊢ (𝜑 → 𝐴 ∈ dom 𝑆) & ⊢ (𝜑 → 𝐵 ∈ dom 𝑆) & ⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵)) & ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0)) ⇒ ⊢ ¬ 𝜑 | ||
Theorem | efgred 18866* | The reduced word that forms the base of the sequence in efgsval 18849 is uniquely determined, given the terminal point. (Contributed by Mario Carneiro, 28-Sep-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) ⇒ ⊢ ((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆 ∧ (𝑆‘𝐴) = (𝑆‘𝐵)) → (𝐴‘0) = (𝐵‘0)) | ||
Theorem | efgrelexlema 18867* | If two words 𝐴, 𝐵 are related under the free group equivalence, then there exist two extension sequences 𝑎, 𝑏 such that 𝑎 ends at 𝐴, 𝑏 ends at 𝐵, and 𝑎 and 𝐵 have the same starting point. (Contributed by Mario Carneiro, 1-Oct-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) & ⊢ 𝐿 = {〈𝑖, 𝑗〉 ∣ ∃𝑐 ∈ (◡𝑆 “ {𝑖})∃𝑑 ∈ (◡𝑆 “ {𝑗})(𝑐‘0) = (𝑑‘0)} ⇒ ⊢ (𝐴𝐿𝐵 ↔ ∃𝑎 ∈ (◡𝑆 “ {𝐴})∃𝑏 ∈ (◡𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0)) | ||
Theorem | efgrelexlemb 18868* | If two words 𝐴, 𝐵 are related under the free group equivalence, then there exist two extension sequences 𝑎, 𝑏 such that 𝑎 ends at 𝐴, 𝑏 ends at 𝐵, and 𝑎 and 𝐵 have the same starting point. (Contributed by Mario Carneiro, 1-Oct-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) & ⊢ 𝐿 = {〈𝑖, 𝑗〉 ∣ ∃𝑐 ∈ (◡𝑆 “ {𝑖})∃𝑑 ∈ (◡𝑆 “ {𝑗})(𝑐‘0) = (𝑑‘0)} ⇒ ⊢ ∼ ⊆ 𝐿 | ||
Theorem | efgrelex 18869* | If two words 𝐴, 𝐵 are related under the free group equivalence, then there exist two extension sequences 𝑎, 𝑏 such that 𝑎 ends at 𝐴, 𝑏 ends at 𝐵, and 𝑎 and 𝐵 have the same starting point. (Contributed by Mario Carneiro, 1-Oct-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) ⇒ ⊢ (𝐴 ∼ 𝐵 → ∃𝑎 ∈ (◡𝑆 “ {𝐴})∃𝑏 ∈ (◡𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0)) | ||
Theorem | efgredeu 18870* | There is a unique reduced word equivalent to a given word. (Contributed by Mario Carneiro, 1-Oct-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) ⇒ ⊢ (𝐴 ∈ 𝑊 → ∃!𝑑 ∈ 𝐷 𝑑 ∼ 𝐴) | ||
Theorem | efgred2 18871* | Two extension sequences have related endpoints iff they have the same base. (Contributed by Mario Carneiro, 1-Oct-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) ⇒ ⊢ ((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) → ((𝑆‘𝐴) ∼ (𝑆‘𝐵) ↔ (𝐴‘0) = (𝐵‘0))) | ||
Theorem | efgcpbllema 18872* | Lemma for efgrelex 18869. Define an auxiliary equivalence relation 𝐿 such that 𝐴𝐿𝐵 if there are sequences from 𝐴 to 𝐵 passing through the same reduced word. (Contributed by Mario Carneiro, 1-Oct-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) & ⊢ 𝐿 = {〈𝑖, 𝑗〉 ∣ ({𝑖, 𝑗} ⊆ 𝑊 ∧ ((𝐴 ++ 𝑖) ++ 𝐵) ∼ ((𝐴 ++ 𝑗) ++ 𝐵))} ⇒ ⊢ (𝑋𝐿𝑌 ↔ (𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊 ∧ ((𝐴 ++ 𝑋) ++ 𝐵) ∼ ((𝐴 ++ 𝑌) ++ 𝐵))) | ||
Theorem | efgcpbllemb 18873* | Lemma for efgrelex 18869. Show that 𝐿 is an equivalence relation containing all direct extensions of a word, so is closed under ∼. (Contributed by Mario Carneiro, 1-Oct-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) & ⊢ 𝐿 = {〈𝑖, 𝑗〉 ∣ ({𝑖, 𝑗} ⊆ 𝑊 ∧ ((𝐴 ++ 𝑖) ++ 𝐵) ∼ ((𝐴 ++ 𝑗) ++ 𝐵))} ⇒ ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑊) → ∼ ⊆ 𝐿) | ||
Theorem | efgcpbl 18874* | Two extension sequences have related endpoints iff they have the same base. (Contributed by Mario Carneiro, 1-Oct-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) ⇒ ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ 𝑋 ∼ 𝑌) → ((𝐴 ++ 𝑋) ++ 𝐵) ∼ ((𝐴 ++ 𝑌) ++ 𝐵)) | ||
Theorem | efgcpbl2 18875* | Two extension sequences have related endpoints iff they have the same base. (Contributed by Mario Carneiro, 1-Oct-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) & ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) & ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) ⇒ ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐴 ++ 𝐵) ∼ (𝑋 ++ 𝑌)) | ||
Theorem | frgpval 18876 | Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.) |
⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ 𝑀 = (freeMnd‘(𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝐺 = (𝑀 /s ∼ )) | ||
Theorem | frgpcpbl 18877 | Compatibility of the group operation with the free group equivalence relation. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) |
⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ 𝑀 = (freeMnd‘(𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ + = (+g‘𝑀) ⇒ ⊢ ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → (𝐴 + 𝐵) ∼ (𝐶 + 𝐷)) | ||
Theorem | frgp0 18878 | The free group is a group. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.) |
⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ ∼ = ( ~FG ‘𝐼) ⇒ ⊢ (𝐼 ∈ 𝑉 → (𝐺 ∈ Grp ∧ [∅] ∼ = (0g‘𝐺))) | ||
Theorem | frgpeccl 18879 | Closure of the quotient map in a free group. (Contributed by Mario Carneiro, 1-Oct-2015.) |
⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝑋 ∈ 𝑊 → [𝑋] ∼ ∈ 𝐵) | ||
Theorem | frgpgrp 18880 | The free group is a group. (Contributed by Mario Carneiro, 1-Oct-2015.) |
⊢ 𝐺 = (freeGrp‘𝐼) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝐺 ∈ Grp) | ||
Theorem | frgpadd 18881 | Addition in the free group is given by concatenation. (Contributed by Mario Carneiro, 1-Oct-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑊) → ([𝐴] ∼ + [𝐵] ∼ ) = [(𝐴 ++ 𝐵)] ∼ ) | ||
Theorem | frgpinv 18882* | The inverse of an element of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑁 = (invg‘𝐺) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) ⇒ ⊢ (𝐴 ∈ 𝑊 → (𝑁‘[𝐴] ∼ ) = [(𝑀 ∘ (reverse‘𝐴))] ∼ ) | ||
Theorem | frgpmhm 18883* | The "natural map" from words of the free monoid to their cosets in the free group is a surjective monoid homomorphism. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑀 = (freeMnd‘(𝐼 × 2o)) & ⊢ 𝑊 = (Base‘𝑀) & ⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝐹 = (𝑥 ∈ 𝑊 ↦ [𝑥] ∼ ) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝐹 ∈ (𝑀 MndHom 𝐺)) | ||
Theorem | vrgpfval 18884* | The canonical injection from the generating set 𝐼 to the base set of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑈 = (varFGrp‘𝐼) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ [〈“〈𝑗, ∅〉”〉] ∼ )) | ||
Theorem | vrgpval 18885 | The value of the generating elements of a free group. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑈 = (varFGrp‘𝐼) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [〈“〈𝐴, ∅〉”〉] ∼ ) | ||
Theorem | vrgpf 18886 | The mapping from the index set to the generators is a function into the free group. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑈 = (varFGrp‘𝐼) & ⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ 𝑋 = (Base‘𝐺) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝑈:𝐼⟶𝑋) | ||
Theorem | vrgpinv 18887 | The inverse of a generating element is represented by 〈𝐴, 1〉 instead of 〈𝐴, 0〉. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝑈 = (varFGrp‘𝐼) & ⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ 𝑁 = (invg‘𝐺) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑁‘(𝑈‘𝐴)) = [〈“〈𝐴, 1o〉”〉] ∼ ) | ||
Theorem | frgpuptf 18888* | Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝐵 = (Base‘𝐻) & ⊢ 𝑁 = (invg‘𝐻) & ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦)))) & ⊢ (𝜑 → 𝐻 ∈ Grp) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) ⇒ ⊢ (𝜑 → 𝑇:(𝐼 × 2o)⟶𝐵) | ||
Theorem | frgpuptinv 18889* | Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝐵 = (Base‘𝐻) & ⊢ 𝑁 = (invg‘𝐻) & ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦)))) & ⊢ (𝜑 → 𝐻 ∈ Grp) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐼 × 2o)) → (𝑇‘(𝑀‘𝐴)) = (𝑁‘(𝑇‘𝐴))) | ||
Theorem | frgpuplem 18890* | Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝐵 = (Base‘𝐻) & ⊢ 𝑁 = (invg‘𝐻) & ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦)))) & ⊢ (𝜑 → 𝐻 ∈ Grp) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) & ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∼ 𝐶) → (𝐻 Σg (𝑇 ∘ 𝐴)) = (𝐻 Σg (𝑇 ∘ 𝐶))) | ||
Theorem | frgpupf 18891* | Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝐵 = (Base‘𝐻) & ⊢ 𝑁 = (invg‘𝐻) & ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦)))) & ⊢ (𝜑 → 𝐻 ∈ Grp) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) & ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐸 = ran (𝑔 ∈ 𝑊 ↦ 〈[𝑔] ∼ , (𝐻 Σg (𝑇 ∘ 𝑔))〉) ⇒ ⊢ (𝜑 → 𝐸:𝑋⟶𝐵) | ||
Theorem | frgpupval 18892* | Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝐵 = (Base‘𝐻) & ⊢ 𝑁 = (invg‘𝐻) & ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦)))) & ⊢ (𝜑 → 𝐻 ∈ Grp) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) & ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐸 = ran (𝑔 ∈ 𝑊 ↦ 〈[𝑔] ∼ , (𝐻 Σg (𝑇 ∘ 𝑔))〉) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑊) → (𝐸‘[𝐴] ∼ ) = (𝐻 Σg (𝑇 ∘ 𝐴))) | ||
Theorem | frgpup1 18893* | Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) |
⊢ 𝐵 = (Base‘𝐻) & ⊢ 𝑁 = (invg‘𝐻) & ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦)))) & ⊢ (𝜑 → 𝐻 ∈ Grp) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) & ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐸 = ran (𝑔 ∈ 𝑊 ↦ 〈[𝑔] ∼ , (𝐻 Σg (𝑇 ∘ 𝑔))〉) ⇒ ⊢ (𝜑 → 𝐸 ∈ (𝐺 GrpHom 𝐻)) | ||
Theorem | frgpup2 18894* | The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) |
⊢ 𝐵 = (Base‘𝐻) & ⊢ 𝑁 = (invg‘𝐻) & ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦)))) & ⊢ (𝜑 → 𝐻 ∈ Grp) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) & ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐸 = ran (𝑔 ∈ 𝑊 ↦ 〈[𝑔] ∼ , (𝐻 Σg (𝑇 ∘ 𝑔))〉) & ⊢ 𝑈 = (varFGrp‘𝐼) & ⊢ (𝜑 → 𝐴 ∈ 𝐼) ⇒ ⊢ (𝜑 → (𝐸‘(𝑈‘𝐴)) = (𝐹‘𝐴)) | ||
Theorem | frgpup3lem 18895* | The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) |
⊢ 𝐵 = (Base‘𝐻) & ⊢ 𝑁 = (invg‘𝐻) & ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦)))) & ⊢ (𝜑 → 𝐻 ∈ Grp) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) & ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) & ⊢ ∼ = ( ~FG ‘𝐼) & ⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐸 = ran (𝑔 ∈ 𝑊 ↦ 〈[𝑔] ∼ , (𝐻 Σg (𝑇 ∘ 𝑔))〉) & ⊢ 𝑈 = (varFGrp‘𝐼) & ⊢ (𝜑 → 𝐾 ∈ (𝐺 GrpHom 𝐻)) & ⊢ (𝜑 → (𝐾 ∘ 𝑈) = 𝐹) ⇒ ⊢ (𝜑 → 𝐾 = 𝐸) | ||
Theorem | frgpup3 18896* | Universal property of the free monoid by existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) |
⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ 𝐵 = (Base‘𝐻) & ⊢ 𝑈 = (varFGrp‘𝐼) ⇒ ⊢ ((𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝐵) → ∃!𝑚 ∈ (𝐺 GrpHom 𝐻)(𝑚 ∘ 𝑈) = 𝐹) | ||
Theorem | 0frgp 18897 | The free group on zero generators is trivial. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐺 = (freeGrp‘∅) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ 𝐵 ≈ 1o | ||
Syntax | ccmn 18898 | Extend class notation with class of all commutative monoids. |
class CMnd | ||
Syntax | cabl 18899 | Extend class notation with class of all Abelian groups. |
class Abel | ||
Definition | df-cmn 18900* | Define class of all commutative monoids. (Contributed by Mario Carneiro, 6-Jan-2015.) |
⊢ CMnd = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)(𝑎(+g‘𝑔)𝑏) = (𝑏(+g‘𝑔)𝑎)} |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |