P. 193 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 19201-19300 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	lsmcntzr 19201	The "subgroups commute" predicate applied to a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ ⊕ = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ 𝑍 = (Cntz‘𝐺) ⇒ ⊢ (𝜑 → (𝑆 ⊆ (𝑍‘(𝑇 ⊕ 𝑈)) ↔ (𝑆 ⊆ (𝑍‘𝑇) ∧ 𝑆 ⊆ (𝑍‘𝑈))))

Theorem	lsmdisj 19202	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 = (0_g‘𝐺) & ⊢ (𝜑 → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 }) ⇒ ⊢ (𝜑 → ((𝑆 ∩ 𝑈) = { 0 } ∧ (𝑇 ∩ 𝑈) = { 0 }))

Theorem	lsmdisj2 19203	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 = (0_g‘𝐺) & ⊢ (𝜑 → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 }) & ⊢ (𝜑 → (𝑆 ∩ 𝑇) = { 0 }) ⇒ ⊢ (𝜑 → (𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 })

Theorem	lsmdisj3 19204	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 = (0_g‘𝐺) & ⊢ (𝜑 → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 }) & ⊢ (𝜑 → (𝑆 ∩ 𝑇) = { 0 }) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝑆 ⊆ (𝑍‘𝑇)) ⇒ ⊢ (𝜑 → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 })

Theorem	lsmdisjr 19205	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 = (0_g‘𝐺) & ⊢ (𝜑 → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 }) ⇒ ⊢ (𝜑 → ((𝑆 ∩ 𝑇) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 }))

Theorem	lsmdisj2r 19206	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 = (0_g‘𝐺) & ⊢ (𝜑 → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 }) & ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) ⇒ ⊢ (𝜑 → ((𝑆 ⊕ 𝑈) ∩ 𝑇) = { 0 })

Theorem	lsmdisj3r 19207	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 = (0_g‘𝐺) & ⊢ (𝜑 → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 }) & ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) & ⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) ⇒ ⊢ (𝜑 → ((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 })

Theorem	lsmdisj2a 19208	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 = (0_g‘𝐺) ⇒ ⊢ (𝜑 → ((((𝑆 ⊕ 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 }) ↔ ((𝑇 ∩ (𝑆 ⊕ 𝑈)) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 })))

Theorem	lsmdisj2b 19209	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 = (0_g‘𝐺) ⇒ ⊢ (𝜑 → ((((𝑆 ⊕ 𝑈) ∩ 𝑇) = { 0 } ∧ (𝑆 ∩ 𝑈) = { 0 }) ↔ ((𝑆 ∩ (𝑇 ⊕ 𝑈)) = { 0 } ∧ (𝑇 ∩ 𝑈) = { 0 })))

Theorem	lsmdisj3a 19210	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 = (0_g‘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 19211	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 = (0_g‘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 19212	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 = (0_g‘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 19213	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 = (0_g‘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 19214	Vectors belonging to disjoint commuting subgroups are uniquely determined by their sum. Analogous to opth 5385, 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 = (0_g‘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 19215*	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 19216*	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 19217*	Uniqueness of a left projection. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ + = (+_g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ 0 = (0_g‘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 19218	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 = (0_g‘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 19219	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 = (0_g‘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 19220	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 = (0_g‘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 19221	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 = (0_g‘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 19222	The left projection function is the identity on the left subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ + = (+_g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ 0 = (0_g‘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 19223	The left projection function is the zero operator on the right subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ + = (+_g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ 0 = (0_g‘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 19224	The left projection function is a group homomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ + = (+_g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ 0 = (0_g‘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 19225	The left projection function is a group homomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ + = (+_g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ 0 = (0_g‘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 19226	The order of the direct product of groups. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ ⊕ = (LSSum‘𝐺) & ⊢ 0 = (0_g‘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 19227	Extend class notation with the free group equivalence relation.
class ~_FG

Syntax	cfrgp 19228	Extend class notation with the free group construction.
class freeGrp

Syntax	cvrgp 19229	Extend class notation with free group injection.
class var_FGrp

Definition	df-efg 19230*	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 inv_g𝑥, 𝐴𝐵 ≈ 𝐴𝑥(inv_g𝑥)𝐵. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ ~_FG = (𝑖 ∈ V ↦ ∩ {𝑟 ∣ (𝑟 Er Word (𝑖 × 2_o) ∧ ∀𝑥 ∈ Word (𝑖 × 2_o)∀𝑛 ∈ (0...(♯‘𝑥))∀𝑦 ∈ 𝑖 ∀𝑧 ∈ 2_o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1_o ∖ 𝑧)⟩”⟩⟩))})

Definition	df-frgp 19231	Define the free group on a set 𝐼 of generators, defined as the quotient of the free monoid on 𝐼 × 2_o (representing the generator elements and their formal inverses) by the free group equivalence relation df-efg 19230. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2_o)) /_s ( ~_FG ‘𝑖)))

Definition	df-vrgp 19232*	Define the canonical injection from the generating set 𝐼 into the base set of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ var_FGrp = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~_FG ‘𝑖)))

Theorem	efgmval 19233*	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 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) ⇒ ⊢ ((𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 2_o) → (𝐴𝑀𝐵) = ⟨𝐴, (1_o ∖ 𝐵)⟩)

Theorem	efgmf 19234*	The formal inverse operation is an endofunction on the generating set. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) ⇒ ⊢ 𝑀:(𝐼 × 2_o)⟶(𝐼 × 2_o)

Theorem	efgmnvl 19235*	The inversion function on the generators is an involution. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) ⇒ ⊢ (𝐴 ∈ (𝐼 × 2_o) → (𝑀‘(𝑀‘𝐴)) = 𝐴)

Theorem	efgrcl 19236	Lemma for efgval 19238. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) ⇒ ⊢ (𝐴 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2_o)))

Theorem	efglem 19237*	Lemma for efgval 19238. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) ⇒ ⊢ ∃𝑟(𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2_o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1_o ∖ 𝑧)⟩”⟩⟩))

Theorem	efgval 19238*	Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) ⇒ ⊢ ∼ = ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2_o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1_o ∖ 𝑧)⟩”⟩⟩))}

Theorem	efger 19239	Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) ⇒ ⊢ ∼ Er 𝑊

Theorem	efgi 19240	Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) ⇒ ⊢ (((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴))) ∧ (𝐽 ∈ 𝐼 ∧ 𝐾 ∈ 2_o)) → 𝐴 ∼ (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1_o ∖ 𝐾)⟩”⟩⟩))

Theorem	efgi0 19241	Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) ⇒ ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴)) ∧ 𝐽 ∈ 𝐼) → 𝐴 ∼ (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, ∅⟩⟨𝐽, 1_o⟩”⟩⟩))

Theorem	efgi1 19242	Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) ⇒ ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴)) ∧ 𝐽 ∈ 𝐼) → 𝐴 ∼ (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 1_o⟩⟨𝐽, ∅⟩”⟩⟩))

Theorem	efgtf 19243*	Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩))) ⇒ ⊢ (𝑋 ∈ 𝑊 → ((𝑇‘𝑋) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2_o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ∧ (𝑇‘𝑋):((0...(♯‘𝑋)) × (𝐼 × 2_o))⟶𝑊))

Theorem	efgtval 19244*	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 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩))) ⇒ ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝑋)) ∧ 𝐴 ∈ (𝐼 × 2_o)) → (𝑁(𝑇‘𝑋)𝐴) = (𝑋 splice ⟨𝑁, 𝑁, ⟨“𝐴(𝑀‘𝐴)”⟩⟩))

Theorem	efgval2 19245*	Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩))) ⇒ ⊢ ∼ = ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ⊆ [𝑥]𝑟)}

Theorem	efgi2 19246*	Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩))) ⇒ ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ ran (𝑇‘𝐴)) → 𝐴 ∼ 𝐵)

Theorem	efgtlen 19247*	Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩))) ⇒ ⊢ ((𝑋 ∈ 𝑊 ∧ 𝐴 ∈ ran (𝑇‘𝑋)) → (♯‘𝐴) = ((♯‘𝑋) + 2))

Theorem	efginvrel2 19248*	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 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩))) ⇒ ⊢ (𝐴 ∈ 𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∼ ∅)

Theorem	efginvrel1 19249*	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 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩))) ⇒ ⊢ (𝐴 ∈ 𝑊 → ((𝑀 ∘ (reverse‘𝐴)) ++ 𝐴) ∼ ∅)

Theorem	efgsf 19250*	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 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 19251*	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 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 19252*	Value of the auxiliary function 𝑆 defining a sequence of extensions. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 19253*	Property of the last link in the chain of extensions. (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 19254*	Value of the auxiliary function 𝑆 defining a sequence of extensions. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 19255*	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 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 19256*	A singleton of an irreducible word is an extension sequence. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 19257*	Every extension sequence ending in an irreducible word is trivial. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 19258*	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 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 19259*	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 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 19260*	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 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 19261*	The reduced word that forms the base of the sequence in efgsval 19252 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 19262*	Lemma for efgredleme 19264. (Contributed by Mario Carneiro, 4-Jun-2016.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 19263*	Lemma for efgred 19269. (Contributed by Mario Carneiro, 4-Jun-2016.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 × 2_o)) & ⊢ (𝜑 → 𝑉 ∈ (𝐼 × 2_o)) & ⊢ (𝜑 → (𝑆‘𝐴) = (𝑃(𝑇‘(𝐴‘𝐾))𝑈)) & ⊢ (𝜑 → (𝑆‘𝐵) = (𝑄(𝑇‘(𝐵‘𝐿))𝑉)) ⇒ ⊢ (𝜑 → (♯‘(𝐴‘𝐾)) = (♯‘(𝐵‘𝐿)))

Theorem	efgredleme 19264*	Lemma for efgred 19269. (Contributed by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 15-Oct-2022.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 × 2_o)) & ⊢ (𝜑 → 𝑉 ∈ (𝐼 × 2_o)) & ⊢ (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 19265*	The reduced word that forms the base of the sequence in efgsval 19252 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 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 × 2_o)) & ⊢ (𝜑 → 𝑉 ∈ (𝐼 × 2_o)) & ⊢ (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 19266*	The reduced word that forms the base of the sequence in efgsval 19252 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 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 × 2_o)) & ⊢ (𝜑 → 𝑉 ∈ (𝐼 × 2_o)) & ⊢ (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 19267*	The reduced word that forms the base of the sequence in efgsval 19252 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 30-Sep-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 × 2_o)) & ⊢ (𝜑 → 𝑉 ∈ (𝐼 × 2_o)) & ⊢ (𝜑 → (𝑆‘𝐴) = (𝑃(𝑇‘(𝐴‘𝐾))𝑈)) & ⊢ (𝜑 → (𝑆‘𝐵) = (𝑄(𝑇‘(𝐵‘𝐿))𝑉)) & ⊢ (𝜑 → ¬ (𝐴‘𝐾) = (𝐵‘𝐿)) ⇒ ⊢ ¬ 𝜑

Theorem	efgredlem 19268*	The reduced word that forms the base of the sequence in efgsval 19252 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 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 19269*	The reduced word that forms the base of the sequence in efgsval 19252 is uniquely determined, given the terminal point. (Contributed by Mario Carneiro, 28-Sep-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 19270*	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 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 19271*	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 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 19272*	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 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 19273*	There is a unique reduced word equivalent to a given word. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 19274*	Two extension sequences have related endpoints iff they have the same base. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 19275*	Lemma for efgrelex 19272. 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 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 19276*	Lemma for efgrelex 19272. 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 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 19277*	Two extension sequences have related endpoints iff they have the same base. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 19278*	Two extension sequences have related endpoints iff they have the same base. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) & ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 19279	Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ 𝑀 = (freeMnd‘(𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝐺 = (𝑀 /_s ∼ ))

Theorem	frgpcpbl 19280	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 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ + = (+_g‘𝑀) ⇒ ⊢ ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → (𝐴 + 𝐵) ∼ (𝐶 + 𝐷))

Theorem	frgp0 19281	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 ∧ [∅] ∼ = (0_g‘𝐺)))

Theorem	frgpeccl 19282	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 × 2_o)) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝑋 ∈ 𝑊 → [𝑋] ∼ ∈ 𝐵)

Theorem	frgpgrp 19283	The free group is a group. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝐺 = (freeGrp‘𝐼) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝐺 ∈ Grp)

Theorem	frgpadd 19284	Addition in the free group is given by concatenation. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ + = (+_g‘𝐺) ⇒ ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑊) → ([𝐴] ∼ + [𝐵] ∼ ) = [(𝐴 ++ 𝐵)] ∼ )

Theorem	frgpinv 19285*	The inverse of an element of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑁 = (inv_g‘𝐺) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) ⇒ ⊢ (𝐴 ∈ 𝑊 → (𝑁‘[𝐴] ∼ ) = [(𝑀 ∘ (reverse‘𝐴))] ∼ )

Theorem	frgpmhm 19286*	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 × 2_o)) & ⊢ 𝑊 = (Base‘𝑀) & ⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝐹 = (𝑥 ∈ 𝑊 ↦ [𝑥] ∼ ) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝐹 ∈ (𝑀 MndHom 𝐺))

Theorem	vrgpfval 19287*	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 = (var_FGrp‘𝐼) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ ))

Theorem	vrgpval 19288	The value of the generating elements of a free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝑈 = (var_FGrp‘𝐼) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )

Theorem	vrgpf 19289	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 = (var_FGrp‘𝐼) & ⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ 𝑋 = (Base‘𝐺) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝑈:𝐼⟶𝑋)

Theorem	vrgpinv 19290	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 = (var_FGrp‘𝐼) & ⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ 𝑁 = (inv_g‘𝐺) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑁‘(𝑈‘𝐴)) = [⟨“⟨𝐴, 1_o⟩”⟩] ∼ )

Theorem	frgpuptf 19291*	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 = (inv_g‘𝐻) & ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦)))) & ⊢ (𝜑 → 𝐻 ∈ Grp) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) ⇒ ⊢ (𝜑 → 𝑇:(𝐼 × 2_o)⟶𝐵)

Theorem	frgpuptinv 19292*	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 = (inv_g‘𝐻) & ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦)))) & ⊢ (𝜑 → 𝐻 ∈ Grp) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) & ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐼 × 2_o)) → (𝑇‘(𝑀‘𝐴)) = (𝑁‘(𝑇‘𝐴)))

Theorem	frgpuplem 19293*	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 = (inv_g‘𝐻) & ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦)))) & ⊢ (𝜑 → 𝐻 ∈ Grp) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) & ⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∼ 𝐶) → (𝐻 Σ_g (𝑇 ∘ 𝐴)) = (𝐻 Σ_g (𝑇 ∘ 𝐶)))

Theorem	frgpupf 19294*	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 = (inv_g‘𝐻) & ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦)))) & ⊢ (𝜑 → 𝐻 ∈ Grp) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) & ⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐸 = ran (𝑔 ∈ 𝑊 ↦ ⟨[𝑔] ∼ , (𝐻 Σ_g (𝑇 ∘ 𝑔))⟩) ⇒ ⊢ (𝜑 → 𝐸:𝑋⟶𝐵)

Theorem	frgpupval 19295*	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 = (inv_g‘𝐻) & ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦)))) & ⊢ (𝜑 → 𝐻 ∈ Grp) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) & ⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐸 = ran (𝑔 ∈ 𝑊 ↦ ⟨[𝑔] ∼ , (𝐻 Σ_g (𝑇 ∘ 𝑔))⟩) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑊) → (𝐸‘[𝐴] ∼ ) = (𝐻 Σ_g (𝑇 ∘ 𝐴)))

Theorem	frgpup1 19296*	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 = (inv_g‘𝐻) & ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦)))) & ⊢ (𝜑 → 𝐻 ∈ Grp) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) & ⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐸 = ran (𝑔 ∈ 𝑊 ↦ ⟨[𝑔] ∼ , (𝐻 Σ_g (𝑇 ∘ 𝑔))⟩) ⇒ ⊢ (𝜑 → 𝐸 ∈ (𝐺 GrpHom 𝐻))

Theorem	frgpup2 19297*	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 = (inv_g‘𝐻) & ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦)))) & ⊢ (𝜑 → 𝐻 ∈ Grp) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) & ⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐸 = ran (𝑔 ∈ 𝑊 ↦ ⟨[𝑔] ∼ , (𝐻 Σ_g (𝑇 ∘ 𝑔))⟩) & ⊢ 𝑈 = (var_FGrp‘𝐼) & ⊢ (𝜑 → 𝐴 ∈ 𝐼) ⇒ ⊢ (𝜑 → (𝐸‘(𝑈‘𝐴)) = (𝐹‘𝐴))

Theorem	frgpup3lem 19298*	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 = (inv_g‘𝐻) & ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦)))) & ⊢ (𝜑 → 𝐻 ∈ Grp) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) & ⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o)) & ⊢ ∼ = ( ~_FG ‘𝐼) & ⊢ 𝐺 = (freeGrp‘𝐼) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐸 = ran (𝑔 ∈ 𝑊 ↦ ⟨[𝑔] ∼ , (𝐻 Σ_g (𝑇 ∘ 𝑔))⟩) & ⊢ 𝑈 = (var_FGrp‘𝐼) & ⊢ (𝜑 → 𝐾 ∈ (𝐺 GrpHom 𝐻)) & ⊢ (𝜑 → (𝐾 ∘ 𝑈) = 𝐹) ⇒ ⊢ (𝜑 → 𝐾 = 𝐸)

Theorem	frgpup3 19299*	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 = (var_FGrp‘𝐼) ⇒ ⊢ ((𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝐵) → ∃!𝑚 ∈ (𝐺 GrpHom 𝐻)(𝑚 ∘ 𝑈) = 𝐹)

Theorem	0frgp 19300	The free group on zero generators is trivial. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐺 = (freeGrp‘∅) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ 𝐵 ≈ 1_o

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7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46395

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