P. 198 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 19701-19800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	lsmdisj3 19701	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 19702	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 19703	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 19704	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 19705	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 19706	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 19707	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 19708	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 19709	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 19710	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 19711	Vectors belonging to disjoint commuting subgroups are uniquely determined by their sum. Analogous to opth 5481, 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 19712*	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 19713*	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 19714*	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 19715	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 19716	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 19717	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 19718	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 19719	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 19720	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 19721	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 19722	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 19723	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)))
10.2.13  Free groups
Syntax	cefg 19724	Extend class notation with the free group equivalence relation.
class ~FG
Syntax	cfrgp 19725	Extend class notation with the free group construction.
class freeGrp
Syntax	cvrgp 19726	Extend class notation with free group injection.
class varFGrp
Definition	df-efg 19727*	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 19728	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 19727. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2o)) /s ( ~FG ‘𝑖)))
Definition	df-vrgp 19729*	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 19730*	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 19731*	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 19732*	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 19733	Lemma for efgval 19735. (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 19734*	Lemma for efgval 19735. (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 19735*	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 19736	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 19737	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 19738	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 19739	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 19740*	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 19741*	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 19742*	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 19743*	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 19744*	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 19745*	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 19746*	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 19747*	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 19748*	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 19749*	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 19750*	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 19751*	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 19752*	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 19753*	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 19754*	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 19755*	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 19756*	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 19757*	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 19758*	The reduced word that forms the base of the sequence in efgsval 19749 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 19759*	Lemma for efgredleme 19761. (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 19760*	Lemma for efgred 19766. (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 19761*	Lemma for efgred 19766. (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 19762*	The reduced word that forms the base of the sequence in efgsval 19749 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 19763*	The reduced word that forms the base of the sequence in efgsval 19749 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 19764*	The reduced word that forms the base of the sequence in efgsval 19749 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 19765*	The reduced word that forms the base of the sequence in efgsval 19749 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 19766*	The reduced word that forms the base of the sequence in efgsval 19749 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 19767*	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 19768*	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 19769*	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 19770*	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 19771*	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 19772*	Lemma for efgrelex 19769. 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 19773*	Lemma for efgrelex 19769. 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 19774*	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 19775*	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 19776	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 19777	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 19778	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 19779	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 19780	The free group is a group. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝐺 = (freeGrp‘𝐼)    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝐺 ∈ Grp)
Theorem	frgpadd 19781	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 19782*	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 19783*	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 19784*	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 19785	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 19786	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 19787	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 19788*	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 19789*	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 19790*	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 19791*	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 19792*	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 19793*	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 19794*	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 19795*	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 19796*	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 19797	The free group on zero generators is trivial. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐺 = (freeGrp‘∅)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ 𝐵 ≈ 1o
10.2.14  Abelian groups
10.2.14.1  Definition and basic properties
Syntax	ccmn 19798	Extend class notation with class of all commutative monoids.
class CMnd
Syntax	cabl 19799	Extend class notation with class of all Abelian groups.
class Abel
Definition	df-cmn 19800*	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)}