P. 199 of Theorem List - Metamath Proof Explorer

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**Theorem List for Metamath Proof Explorer -** 19801-19900 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	lsmss2b 19801	Subgroup sum with a subset. (Contributed by NM, 10-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑈 ⊆ 𝑇 ↔ (𝑇 ⊕ 𝑈) = 𝑇))

Theorem	lsmass 19802	Subgroup sum is associative. (Contributed by NM, 2-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ ((𝑅 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → ((𝑅 ⊕ 𝑇) ⊕ 𝑈) = (𝑅 ⊕ (𝑇 ⊕ 𝑈)))

Theorem	mndlsmidm 19803	Subgroup sum is idempotent for monoids. This corresponds to the observation in [Lang] p. 6. (Contributed by AV, 27-Dec-2023.)
⊢ ⊕ = (LSSum‘𝐺) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐺 ∈ Mnd → (𝐵 ⊕ 𝐵) = 𝐵)

Theorem	lsm01 19804	Subgroup sum with the zero subgroup. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
⊢ 0 = (0_g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ (𝑋 ∈ (SubGrp‘𝐺) → (𝑋 ⊕ { 0 }) = 𝑋)

Theorem	lsm02 19805	Subgroup sum with the zero subgroup. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
⊢ 0 = (0_g‘𝐺) & ⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ (𝑋 ∈ (SubGrp‘𝐺) → ({ 0 } ⊕ 𝑋) = 𝑋)

Theorem	subglsm 19806	The subgroup sum evaluated within a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
⊢ 𝐻 = (𝐺 ↾_s 𝑆) & ⊢ ⊕ = (LSSum‘𝐺) & ⊢ 𝐴 = (LSSum‘𝐻) ⇒ ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ 𝑆 ∧ 𝑈 ⊆ 𝑆) → (𝑇 ⊕ 𝑈) = (𝑇𝐴𝑈))

Theorem	lssnle 19807	Equivalent expressions for "not less than". (chnlei 31974 analog.) (Contributed by NM, 10-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
⊢ ⊕ = (LSSum‘𝐺) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) ⇒ ⊢ (𝜑 → (¬ 𝑈 ⊆ 𝑇 ↔ 𝑇 ⊊ (𝑇 ⊕ 𝑈)))

Theorem	lsmmod 19808	The modular law holds for subgroup sum. Similar to part of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑆 ⊆ 𝑈) → (𝑆 ⊕ (𝑇 ∩ 𝑈)) = ((𝑆 ⊕ 𝑇) ∩ 𝑈))

Theorem	lsmmod2 19809	Modular law dual for subgroup sum. Similar to part of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 8-Jan-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
⊢ ⊕ = (LSSum‘𝐺) ⇒ ⊢ (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ 𝑈 ⊆ 𝑆) → (𝑆 ∩ (𝑇 ⊕ 𝑈)) = ((𝑆 ∩ 𝑇) ⊕ 𝑈))

Theorem	lsmpropd 19810*	If two structures have the same components (properties), they have the same subspace structure. (Contributed by Mario Carneiro, 29-Jun-2015.) (Revised by AV, 25-Apr-2024.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+_g‘𝐾)𝑦) = (𝑥(+_g‘𝐿)𝑦)) & ⊢ (𝜑 → 𝐾 ∈ 𝑉) & ⊢ (𝜑 → 𝐿 ∈ 𝑊) ⇒ ⊢ (𝜑 → (LSSum‘𝐾) = (LSSum‘𝐿))

Theorem	cntzrecd 19811	Commute the "subgroups commute" predicate. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝑍 = (Cntz‘𝐺) & ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) & ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈)) ⇒ ⊢ (𝜑 → 𝑈 ⊆ (𝑍‘𝑇))

Theorem	lsmcntz 19812	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 ⊕ 𝑇) ⊆ (𝑍‘𝑈) ↔ (𝑆 ⊆ (𝑍‘𝑈) ∧ 𝑇 ⊆ (𝑍‘𝑈))))

Theorem	lsmcntzr 19813	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 19814	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 19815	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 19816	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 19817	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 19818	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 19819	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 19820	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 19821	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 19822	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 19823	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 19824	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 19825	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 19826	Vectors belonging to disjoint commuting subgroups are uniquely determined by their sum. Analogous to opth 5456, 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 19827*	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 19828*	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 19829*	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 19830	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 19831	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 19832	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 19833	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 19834	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 19835	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 19836	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 19837	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 19838	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 19839	Extend class notation with the free group equivalence relation.
class ~_FG

Syntax	cfrgp 19840	Extend class notation with the free group construction.
class freeGrp

Syntax	cvrgp 19841	Extend class notation with free group injection.
class var_FGrp

Definition	df-efg 19842*	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 19843	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 19842. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2_o)) /_s ( ~_FG ‘𝑖)))

Definition	df-vrgp 19844*	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 19845*	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 19846*	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 19847*	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 19848	Lemma for efgval 19850. (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 19849*	Lemma for efgval 19850. (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 19850*	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 19851	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 19852	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 19853	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 19854	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 19855*	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 19856*	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 19857*	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 19858*	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 19859*	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 19860*	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 19861*	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 19862*	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 19863*	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 19864*	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 19865*	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 19866*	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 19867*	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 19868*	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 19869*	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 19870*	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 19871*	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 19872*	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 19873*	The reduced word that forms the base of the sequence in efgsval 19864 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 19874*	Lemma for efgredleme 19876. (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 19875*	Lemma for efgred 19881. (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 19876*	Lemma for efgred 19881. (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 19877*	The reduced word that forms the base of the sequence in efgsval 19864 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 19878*	The reduced word that forms the base of the sequence in efgsval 19864 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 19879*	The reduced word that forms the base of the sequence in efgsval 19864 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 19880*	The reduced word that forms the base of the sequence in efgsval 19864 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 19881*	The reduced word that forms the base of the sequence in efgsval 19864 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 19882*	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 19883*	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 19884*	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 19885*	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 19886*	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 19887*	Lemma for efgrelex 19884. 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 19888*	Lemma for efgrelex 19884. 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 19889*	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 19890*	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 19891	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 19892	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 19893	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 19894	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 19895	The free group is a group. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝐺 = (freeGrp‘𝐼) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝐺 ∈ Grp)

Theorem	frgpadd 19896	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 19897*	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 19898*	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 19899*	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 19900	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, ∅⟩”⟩] ∼ )

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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 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-48000 481 48001-48100 482 48101-48200 483 48201-48300 484 48301-48400 485 48401-48500 486 48501-48600 487 48601-48700 488 48701-48800 489 48801-48900 490 48901-49000 491 49001-49100 492 49101-49200 493 49201-49300 494 49301-49400 495 49401-49500 496 49501-49600 497 49601-49700 498 49701-49800 499 49801-49900 500 49901-50000 501 50001-50100 502 50101-50200 503 50201-50300 504 50301-50400 505 50401-50500 506 50501-50600 507 50601-50700 508 50701-50800 509 50801-50831

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