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Theorem List for Metamath Proof Explorer - 17901-18000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlsmlub 17901 The least upper bound property of subgroup sum. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
= (LSSum‘𝐺)       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → ((𝑆𝑈𝑇𝑈) ↔ (𝑆 𝑇) ⊆ 𝑈))

Theoremlsmss1 17902 Subgroup sum with a subset. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑈) → (𝑇 𝑈) = 𝑈)

Theoremlsmss1b 17903 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)) → (𝑇𝑈 ↔ (𝑇 𝑈) = 𝑈))

Theoremlsmss2 17904 Subgroup sum with a subset. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑈𝑇) → (𝑇 𝑈) = 𝑇)

Theoremlsmss2b 17905 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)) → (𝑈𝑇 ↔ (𝑇 𝑈) = 𝑇))

Theoremlsmass 17906 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)))

Theoremlsm01 17907 Subgroup sum with the zero subgroup. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
0 = (0g𝐺)    &    = (LSSum‘𝐺)       (𝑋 ∈ (SubGrp‘𝐺) → (𝑋 { 0 }) = 𝑋)

Theoremlsm02 17908 Subgroup sum with the zero subgroup. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
0 = (0g𝐺)    &    = (LSSum‘𝐺)       (𝑋 ∈ (SubGrp‘𝐺) → ({ 0 } 𝑋) = 𝑋)

Theoremsubglsm 17909 The subgroup sum evaluated within a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝐻 = (𝐺s 𝑆)    &    = (LSSum‘𝐺)    &   𝐴 = (LSSum‘𝐻)       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → (𝑇 𝑈) = (𝑇𝐴𝑈))

Theoremlssnle 17910 Equivalent expressions for "not less than". (chnlei 27728 analog.) (Contributed by NM, 10-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))       (𝜑 → (¬ 𝑈𝑇𝑇 ⊊ (𝑇 𝑈)))

Theoremlsmmod 17911 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)) ∧ 𝑆𝑈) → (𝑆 (𝑇𝑈)) = ((𝑆 𝑇) ∩ 𝑈))

Theoremlsmmod2 17912 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)) ∧ 𝑈𝑆) → (𝑆 ∩ (𝑇 𝑈)) = ((𝑆𝑇) 𝑈))

Theoremlsmpropd 17913* If two structures have the same components (properties), they have the same subspace structure. (Contributed by Mario Carneiro, 29-Jun-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   (𝜑𝐾 ∈ V)    &   (𝜑𝐿 ∈ V)       (𝜑 → (LSSum‘𝐾) = (LSSum‘𝐿))

Theoremcntzrecd 17914 Commute the "subgroups commute" predicate. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ⊆ (𝑍𝑈))       (𝜑𝑈 ⊆ (𝑍𝑇))

Theoremlsmcntz 17915 The "subgroups commute" predicate applied to a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
= (LSSum‘𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   𝑍 = (Cntz‘𝐺)       (𝜑 → ((𝑆 𝑇) ⊆ (𝑍𝑈) ↔ (𝑆 ⊆ (𝑍𝑈) ∧ 𝑇 ⊆ (𝑍𝑈))))

Theoremlsmcntzr 17916 The "subgroups commute" predicate applied to a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
= (LSSum‘𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   𝑍 = (Cntz‘𝐺)       (𝜑 → (𝑆 ⊆ (𝑍‘(𝑇 𝑈)) ↔ (𝑆 ⊆ (𝑍𝑇) ∧ 𝑆 ⊆ (𝑍𝑈))))

Theoremlsmdisj 17917 Disjointness from a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
= (LSSum‘𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &    0 = (0g𝐺)    &   (𝜑 → ((𝑆 𝑇) ∩ 𝑈) = { 0 })       (𝜑 → ((𝑆𝑈) = { 0 } ∧ (𝑇𝑈) = { 0 }))

Theoremlsmdisj2 17918 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016.)
= (LSSum‘𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &    0 = (0g𝐺)    &   (𝜑 → ((𝑆 𝑇) ∩ 𝑈) = { 0 })    &   (𝜑 → (𝑆𝑇) = { 0 })       (𝜑 → (𝑇 ∩ (𝑆 𝑈)) = { 0 })

Theoremlsmdisj3 17919 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
= (LSSum‘𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &    0 = (0g𝐺)    &   (𝜑 → ((𝑆 𝑇) ∩ 𝑈) = { 0 })    &   (𝜑 → (𝑆𝑇) = { 0 })    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑆 ⊆ (𝑍𝑇))       (𝜑 → (𝑆 ∩ (𝑇 𝑈)) = { 0 })

Theoremlsmdisjr 17920 Disjointness from a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
= (LSSum‘𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &    0 = (0g𝐺)    &   (𝜑 → (𝑆 ∩ (𝑇 𝑈)) = { 0 })       (𝜑 → ((𝑆𝑇) = { 0 } ∧ (𝑆𝑈) = { 0 }))

Theoremlsmdisj2r 17921 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016.)
= (LSSum‘𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &    0 = (0g𝐺)    &   (𝜑 → (𝑆 ∩ (𝑇 𝑈)) = { 0 })    &   (𝜑 → (𝑇𝑈) = { 0 })       (𝜑 → ((𝑆 𝑈) ∩ 𝑇) = { 0 })

Theoremlsmdisj3r 17922 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016.)
= (LSSum‘𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &    0 = (0g𝐺)    &   (𝜑 → (𝑆 ∩ (𝑇 𝑈)) = { 0 })    &   (𝜑 → (𝑇𝑈) = { 0 })    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ⊆ (𝑍𝑈))       (𝜑 → ((𝑆 𝑇) ∩ 𝑈) = { 0 })

Theoremlsmdisj2a 17923 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
= (LSSum‘𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &    0 = (0g𝐺)       (𝜑 → ((((𝑆 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆𝑇) = { 0 }) ↔ ((𝑇 ∩ (𝑆 𝑈)) = { 0 } ∧ (𝑆𝑈) = { 0 })))

Theoremlsmdisj2b 17924 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
= (LSSum‘𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &    0 = (0g𝐺)       (𝜑 → ((((𝑆 𝑈) ∩ 𝑇) = { 0 } ∧ (𝑆𝑈) = { 0 }) ↔ ((𝑆 ∩ (𝑇 𝑈)) = { 0 } ∧ (𝑇𝑈) = { 0 })))

Theoremlsmdisj3a 17925 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
= (LSSum‘𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑆 ⊆ (𝑍𝑇))       (𝜑 → ((((𝑆 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆𝑇) = { 0 }) ↔ ((𝑆 ∩ (𝑇 𝑈)) = { 0 } ∧ (𝑇𝑈) = { 0 })))

Theoremlsmdisj3b 17926 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
= (LSSum‘𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ⊆ (𝑍𝑈))       (𝜑 → ((((𝑆 𝑇) ∩ 𝑈) = { 0 } ∧ (𝑆𝑇) = { 0 }) ↔ ((𝑆 ∩ (𝑇 𝑈)) = { 0 } ∧ (𝑇𝑈) = { 0 })))

Theoremsubgdisj1 17927 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.)
+ = (+g𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑇𝑈) = { 0 })    &   (𝜑𝑇 ⊆ (𝑍𝑈))    &   (𝜑𝐴𝑇)    &   (𝜑𝐶𝑇)    &   (𝜑𝐵𝑈)    &   (𝜑𝐷𝑈)    &   (𝜑 → (𝐴 + 𝐵) = (𝐶 + 𝐷))       (𝜑𝐴 = 𝐶)

Theoremsubgdisj2 17928 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.)
+ = (+g𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑇𝑈) = { 0 })    &   (𝜑𝑇 ⊆ (𝑍𝑈))    &   (𝜑𝐴𝑇)    &   (𝜑𝐶𝑇)    &   (𝜑𝐵𝑈)    &   (𝜑𝐷𝑈)    &   (𝜑 → (𝐴 + 𝐵) = (𝐶 + 𝐷))       (𝜑𝐵 = 𝐷)

Theoremsubgdisjb 17929 Vectors belonging to disjoint commuting subgroups are uniquely determined by their sum. Analogous to opth 4871, this theorem shows a way of representing a pair of vectors. (Contributed by NM, 5-Jul-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
+ = (+g𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑇𝑈) = { 0 })    &   (𝜑𝑇 ⊆ (𝑍𝑈))    &   (𝜑𝐴𝑇)    &   (𝜑𝐶𝑇)    &   (𝜑𝐵𝑈)    &   (𝜑𝐷𝑈)       (𝜑 → ((𝐴 + 𝐵) = (𝐶 + 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Theorempj1fval 17930* 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)    &    + = (+g𝐺)    &    = (LSSum‘𝐺)    &   𝑃 = (proj1𝐺)       ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑇𝑃𝑈) = (𝑧 ∈ (𝑇 𝑈) ↦ (𝑥𝑇𝑦𝑈 𝑧 = (𝑥 + 𝑦))))

Theorempj1val 17931* 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)    &    + = (+g𝐺)    &    = (LSSum‘𝐺)    &   𝑃 = (proj1𝐺)       (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑋 ∈ (𝑇 𝑈)) → ((𝑇𝑃𝑈)‘𝑋) = (𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦)))

Theorempj1eu 17932* Uniqueness of a left projection. (Contributed by Mario Carneiro, 15-Oct-2015.)
+ = (+g𝐺)    &    = (LSSum‘𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑇𝑈) = { 0 })    &   (𝜑𝑇 ⊆ (𝑍𝑈))       ((𝜑𝑋 ∈ (𝑇 𝑈)) → ∃!𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦))

Theorempj1f 17933 The left projection function maps a direct subspace sum onto the left factor. (Contributed by Mario Carneiro, 15-Oct-2015.)
+ = (+g𝐺)    &    = (LSSum‘𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑇𝑈) = { 0 })    &   (𝜑𝑇 ⊆ (𝑍𝑈))    &   𝑃 = (proj1𝐺)       (𝜑 → (𝑇𝑃𝑈):(𝑇 𝑈)⟶𝑇)

Theorempj2f 17934 The right projection function maps a direct subspace sum onto the right factor. (Contributed by Mario Carneiro, 15-Oct-2015.)
+ = (+g𝐺)    &    = (LSSum‘𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑇𝑈) = { 0 })    &   (𝜑𝑇 ⊆ (𝑍𝑈))    &   𝑃 = (proj1𝐺)       (𝜑 → (𝑈𝑃𝑇):(𝑇 𝑈)⟶𝑈)

Theorempj1id 17935 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.)
+ = (+g𝐺)    &    = (LSSum‘𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑇𝑈) = { 0 })    &   (𝜑𝑇 ⊆ (𝑍𝑈))    &   𝑃 = (proj1𝐺)       ((𝜑𝑋 ∈ (𝑇 𝑈)) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)))

Theorempj1eq 17936 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.)
+ = (+g𝐺)    &    = (LSSum‘𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑇𝑈) = { 0 })    &   (𝜑𝑇 ⊆ (𝑍𝑈))    &   𝑃 = (proj1𝐺)    &   (𝜑𝑋 ∈ (𝑇 𝑈))    &   (𝜑𝐵𝑇)    &   (𝜑𝐶𝑈)       (𝜑 → (𝑋 = (𝐵 + 𝐶) ↔ (((𝑇𝑃𝑈)‘𝑋) = 𝐵 ∧ ((𝑈𝑃𝑇)‘𝑋) = 𝐶)))

Theorempj1lid 17937 The left projection function is the identity on the left subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)
+ = (+g𝐺)    &    = (LSSum‘𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑇𝑈) = { 0 })    &   (𝜑𝑇 ⊆ (𝑍𝑈))    &   𝑃 = (proj1𝐺)       ((𝜑𝑋𝑇) → ((𝑇𝑃𝑈)‘𝑋) = 𝑋)

Theorempj1rid 17938 The left projection function is the zero operator on the right subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)
+ = (+g𝐺)    &    = (LSSum‘𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑇𝑈) = { 0 })    &   (𝜑𝑇 ⊆ (𝑍𝑈))    &   𝑃 = (proj1𝐺)       ((𝜑𝑋𝑈) → ((𝑇𝑃𝑈)‘𝑋) = 0 )

Theorempj1ghm 17939 The left projection function is a group homomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
+ = (+g𝐺)    &    = (LSSum‘𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑇𝑈) = { 0 })    &   (𝜑𝑇 ⊆ (𝑍𝑈))    &   𝑃 = (proj1𝐺)       (𝜑 → (𝑇𝑃𝑈) ∈ ((𝐺s (𝑇 𝑈)) GrpHom 𝐺))

Theorempj1ghm2 17940 The left projection function is a group homomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
+ = (+g𝐺)    &    = (LSSum‘𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑇𝑈) = { 0 })    &   (𝜑𝑇 ⊆ (𝑍𝑈))    &   𝑃 = (proj1𝐺)       (𝜑 → (𝑇𝑃𝑈) ∈ ((𝐺s (𝑇 𝑈)) GrpHom (𝐺s 𝑇)))

Theoremlsmhash 17941 The order of the direct product of groups. (Contributed by Mario Carneiro, 21-Apr-2016.)
= (LSSum‘𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑 → (𝑇𝑈) = { 0 })    &   (𝜑𝑇 ⊆ (𝑍𝑈))    &   (𝜑𝑇 ∈ Fin)    &   (𝜑𝑈 ∈ Fin)       (𝜑 → (#‘(𝑇 𝑈)) = ((#‘𝑇) · (#‘𝑈)))

10.2.12  Free groups

Syntaxcefg 17942 Extend class notation with the free group equivalence relation.
class ~FG

Syntaxcfrgp 17943 Extend class notation with the free group construction.
class freeGrp

Syntaxcvrgp 17944 Extend class notation with free group injection.
class varFGrp

Definitiondf-efg 17945* Define the free group equivalence relation, which is the smallest equivalence relation such that for any words 𝐴, 𝐵 and formal symbol 𝑥 with inverse invg𝑥, 𝐴𝐵𝐴𝑥(invg𝑥)𝐵. (Contributed by Mario Carneiro, 1-Oct-2015.)
~FG = (𝑖 ∈ V ↦ {𝑟 ∣ (𝑟 Er Word (𝑖 × 2𝑜) ∧ ∀𝑥 ∈ Word (𝑖 × 2𝑜)∀𝑛 ∈ (0...(#‘𝑥))∀𝑦𝑖𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))})

Definitiondf-frgp 17946 Define the free group on a set 𝐼 of generators, defined as the quotient of the free monoid on 𝐼 × 2𝑜 (representing the generator elements and their formal inverses) by the free group equivalence relation df-efg 17945. (Contributed by Mario Carneiro, 1-Oct-2015.)
freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2𝑜)) /s ( ~FG𝑖)))

Definitiondf-vrgp 17947* Define the canonical injection from the generating set 𝐼 into the base set of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
varFGrp = (𝑖 ∈ V ↦ (𝑗𝑖 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG𝑖)))

Theoremefgmval 17948* Value of the formal inverse operation for the generating set of a free group. (Contributed by Mario Carneiro, 27-Sep-2015.)
𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)       ((𝐴𝐼𝐵 ∈ 2𝑜) → (𝐴𝑀𝐵) = ⟨𝐴, (1𝑜𝐵)⟩)

Theoremefgmf 17949* The formal inverse operation is an endofunction on the generating set. (Contributed by Mario Carneiro, 27-Sep-2015.)
𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)       𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)

Theoremefgmnvl 17950* The inversion function on the generators is an involution. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)       (𝐴 ∈ (𝐼 × 2𝑜) → (𝑀‘(𝑀𝐴)) = 𝐴)

Theoremefgrcl 17951 Lemma for efgval 17953. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))       (𝐴𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2𝑜)))

Theoremefglem 17952* Lemma for efgval 17953. (Contributed by Mario Carneiro, 27-Sep-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))       𝑟(𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(#‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))

Theoremefgval 17953* Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)        = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(#‘𝑥))∀𝑦𝐼𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1𝑜𝑧)⟩”⟩⟩))}

Theoremefger 17954 Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)        Er 𝑊

Theoremefgi 17955 Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)       (((𝐴𝑊𝑁 ∈ (0...(#‘𝐴))) ∧ (𝐽𝐼𝐾 ∈ 2𝑜)) → 𝐴 (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1𝑜𝐾)⟩”⟩⟩))

Theoremefgi0 17956 Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)       ((𝐴𝑊𝑁 ∈ (0...(#‘𝐴)) ∧ 𝐽𝐼) → 𝐴 (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, ∅⟩⟨𝐽, 1𝑜⟩”⟩⟩))

Theoremefgi1 17957 Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)       ((𝐴𝑊𝑁 ∈ (0...(#‘𝐴)) ∧ 𝐽𝐼) → 𝐴 (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 1𝑜⟩⟨𝐽, ∅⟩”⟩⟩))

Theoremefgtf 17958* Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))       (𝑋𝑊 → ((𝑇𝑋) = (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) ∧ (𝑇𝑋):((0...(#‘𝑋)) × (𝐼 × 2𝑜))⟶𝑊))

Theoremefgtval 17959* Value of the extension function, which maps a word (a representation of the group element as a sequence of elements and their inverses) to its direct extensions, defined as the original representation with an element and its inverse inserted somewhere in the string. (Contributed by Mario Carneiro, 29-Sep-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))       ((𝑋𝑊𝑁 ∈ (0...(#‘𝑋)) ∧ 𝐴 ∈ (𝐼 × 2𝑜)) → (𝑁(𝑇𝑋)𝐴) = (𝑋 splice ⟨𝑁, 𝑁, ⟨“𝐴(𝑀𝐴)”⟩⟩))

Theoremefgval2 17960* Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))        = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊 ran (𝑇𝑥) ⊆ [𝑥]𝑟)}

Theoremefgi2 17961* Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))       ((𝐴𝑊𝐵 ∈ ran (𝑇𝐴)) → 𝐴 𝐵)

Theoremefgtlen 17962* Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))       ((𝑋𝑊𝐴 ∈ ran (𝑇𝑋)) → (#‘𝐴) = ((#‘𝑋) + 2))

Theoremefginvrel2 17963* The inverse of the reverse of a word composed with the word relates to the identity. (This provides an explicit expression for the representation of the group inverse, given a representative of the free group equivalence class.) (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))       (𝐴𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∅)

Theoremefginvrel1 17964* The inverse of the reverse of a word composed with the word relates to the identity. (This provides an explicit expression for the representation of the group inverse, given a representative of the free group equivalence class.) (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))       (𝐴𝑊 → ((𝑀 ∘ (reverse‘𝐴)) ++ 𝐴) ∅)

Theoremefgsf 17965* Value of the auxiliary function 𝑆 defining a sequence of extensions starting at some irreducible word. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       𝑆:{𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}⟶𝑊

Theoremefgsdm 17966* Elementhood in the domain of 𝑆, the set of sequences of extensions starting at an irreducible word. (Contributed by Mario Carneiro, 27-Sep-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       (𝐹 ∈ dom 𝑆 ↔ (𝐹 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐹‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(#‘𝐹))(𝐹𝑖) ∈ ran (𝑇‘(𝐹‘(𝑖 − 1)))))

Theoremefgsval 17967* Value of the auxiliary function 𝑆 defining a sequence of extensions. (Contributed by Mario Carneiro, 27-Sep-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       (𝐹 ∈ dom 𝑆 → (𝑆𝐹) = (𝐹‘((#‘𝐹) − 1)))

Theoremefgsdmi 17968* Property of the last link in the chain of extensions. (Contributed by Mario Carneiro, 29-Sep-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       ((𝐹 ∈ dom 𝑆 ∧ ((#‘𝐹) − 1) ∈ ℕ) → (𝑆𝐹) ∈ ran (𝑇‘(𝐹‘(((#‘𝐹) − 1) − 1))))

Theoremefgsval2 17969* Value of the auxiliary function 𝑆 defining a sequence of extensions. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       ((𝐴 ∈ Word 𝑊𝐵𝑊 ∧ (𝐴 ++ ⟨“𝐵”⟩) ∈ dom 𝑆) → (𝑆‘(𝐴 ++ ⟨“𝐵”⟩)) = 𝐵)

Theoremefgsrel 17970* The start and end of any extension sequence are related (i.e. evaluate to the same element of the quotient group to be created). (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       (𝐹 ∈ dom 𝑆 → (𝐹‘0) (𝑆𝐹))

Theoremefgs1 17971* A singleton of an irreducible word is an extension sequence. (Contributed by Mario Carneiro, 27-Sep-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       (𝐴𝐷 → ⟨“𝐴”⟩ ∈ dom 𝑆)

Theoremefgs1b 17972* Every extension sequence ending in an irreducible word is trivial. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       (𝐴 ∈ dom 𝑆 → ((𝑆𝐴) ∈ 𝐷 ↔ (#‘𝐴) = 1))

Theoremefgsp1 17973* If 𝐹 is an extension sequence and 𝐴 is an extension of the last element of 𝐹, then 𝐹 + ⟨“𝐴”⟩ is an extension sequence. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       ((𝐹 ∈ dom 𝑆𝐴 ∈ ran (𝑇‘(𝑆𝐹))) → (𝐹 ++ ⟨“𝐴”⟩) ∈ dom 𝑆)

Theoremefgsres 17974* An initial segment of an extension sequence is an extension sequence. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       ((𝐹 ∈ dom 𝑆𝑁 ∈ (1...(#‘𝐹))) → (𝐹 ↾ (0..^𝑁)) ∈ dom 𝑆)

Theoremefgsfo 17975* 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 indexes somewhere in the sequence). (Contributed by Mario Carneiro, 27-Sep-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       𝑆:dom 𝑆onto𝑊

Theoremefgredlema 17976* The reduced word that forms the base of the sequence in efgsval 17967 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   (𝜑 → ∀𝑎 ∈ dom 𝑆𝑏 ∈ dom 𝑆((#‘(𝑆𝑎)) < (#‘(𝑆𝐴)) → ((𝑆𝑎) = (𝑆𝑏) → (𝑎‘0) = (𝑏‘0))))    &   (𝜑𝐴 ∈ dom 𝑆)    &   (𝜑𝐵 ∈ dom 𝑆)    &   (𝜑 → (𝑆𝐴) = (𝑆𝐵))    &   (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))       (𝜑 → (((#‘𝐴) − 1) ∈ ℕ ∧ ((#‘𝐵) − 1) ∈ ℕ))

Theoremefgredlemf 17977* Lemma for efgredleme 17979. (Contributed by Mario Carneiro, 4-Jun-2016.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   (𝜑 → ∀𝑎 ∈ dom 𝑆𝑏 ∈ dom 𝑆((#‘(𝑆𝑎)) < (#‘(𝑆𝐴)) → ((𝑆𝑎) = (𝑆𝑏) → (𝑎‘0) = (𝑏‘0))))    &   (𝜑𝐴 ∈ dom 𝑆)    &   (𝜑𝐵 ∈ dom 𝑆)    &   (𝜑 → (𝑆𝐴) = (𝑆𝐵))    &   (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))    &   𝐾 = (((#‘𝐴) − 1) − 1)    &   𝐿 = (((#‘𝐵) − 1) − 1)       (𝜑 → ((𝐴𝐾) ∈ 𝑊 ∧ (𝐵𝐿) ∈ 𝑊))

Theoremefgredlemg 17978* Lemma for efgred 17984. (Contributed by Mario Carneiro, 4-Jun-2016.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   (𝜑 → ∀𝑎 ∈ dom 𝑆𝑏 ∈ dom 𝑆((#‘(𝑆𝑎)) < (#‘(𝑆𝐴)) → ((𝑆𝑎) = (𝑆𝑏) → (𝑎‘0) = (𝑏‘0))))    &   (𝜑𝐴 ∈ dom 𝑆)    &   (𝜑𝐵 ∈ dom 𝑆)    &   (𝜑 → (𝑆𝐴) = (𝑆𝐵))    &   (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))    &   𝐾 = (((#‘𝐴) − 1) − 1)    &   𝐿 = (((#‘𝐵) − 1) − 1)    &   (𝜑𝑃 ∈ (0...(#‘(𝐴𝐾))))    &   (𝜑𝑄 ∈ (0...(#‘(𝐵𝐿))))    &   (𝜑𝑈 ∈ (𝐼 × 2𝑜))    &   (𝜑𝑉 ∈ (𝐼 × 2𝑜))    &   (𝜑 → (𝑆𝐴) = (𝑃(𝑇‘(𝐴𝐾))𝑈))    &   (𝜑 → (𝑆𝐵) = (𝑄(𝑇‘(𝐵𝐿))𝑉))       (𝜑 → (#‘(𝐴𝐾)) = (#‘(𝐵𝐿)))

Theoremefgredleme 17979* Lemma for efgred 17984. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   (𝜑 → ∀𝑎 ∈ dom 𝑆𝑏 ∈ dom 𝑆((#‘(𝑆𝑎)) < (#‘(𝑆𝐴)) → ((𝑆𝑎) = (𝑆𝑏) → (𝑎‘0) = (𝑏‘0))))    &   (𝜑𝐴 ∈ dom 𝑆)    &   (𝜑𝐵 ∈ dom 𝑆)    &   (𝜑 → (𝑆𝐴) = (𝑆𝐵))    &   (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))    &   𝐾 = (((#‘𝐴) − 1) − 1)    &   𝐿 = (((#‘𝐵) − 1) − 1)    &   (𝜑𝑃 ∈ (0...(#‘(𝐴𝐾))))    &   (𝜑𝑄 ∈ (0...(#‘(𝐵𝐿))))    &   (𝜑𝑈 ∈ (𝐼 × 2𝑜))    &   (𝜑𝑉 ∈ (𝐼 × 2𝑜))    &   (𝜑 → (𝑆𝐴) = (𝑃(𝑇‘(𝐴𝐾))𝑈))    &   (𝜑 → (𝑆𝐵) = (𝑄(𝑇‘(𝐵𝐿))𝑉))    &   (𝜑 → ¬ (𝐴𝐾) = (𝐵𝐿))    &   (𝜑𝑃 ∈ (ℤ‘(𝑄 + 2)))    &   (𝜑𝐶 ∈ dom 𝑆)    &   (𝜑 → (𝑆𝐶) = (((𝐵𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴𝐾))⟩)))       (𝜑 → ((𝐴𝐾) ∈ ran (𝑇‘(𝑆𝐶)) ∧ (𝐵𝐿) ∈ ran (𝑇‘(𝑆𝐶))))

Theoremefgredlemd 17980* The reduced word that forms the base of the sequence in efgsval 17967 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   (𝜑 → ∀𝑎 ∈ dom 𝑆𝑏 ∈ dom 𝑆((#‘(𝑆𝑎)) < (#‘(𝑆𝐴)) → ((𝑆𝑎) = (𝑆𝑏) → (𝑎‘0) = (𝑏‘0))))    &   (𝜑𝐴 ∈ dom 𝑆)    &   (𝜑𝐵 ∈ dom 𝑆)    &   (𝜑 → (𝑆𝐴) = (𝑆𝐵))    &   (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))    &   𝐾 = (((#‘𝐴) − 1) − 1)    &   𝐿 = (((#‘𝐵) − 1) − 1)    &   (𝜑𝑃 ∈ (0...(#‘(𝐴𝐾))))    &   (𝜑𝑄 ∈ (0...(#‘(𝐵𝐿))))    &   (𝜑𝑈 ∈ (𝐼 × 2𝑜))    &   (𝜑𝑉 ∈ (𝐼 × 2𝑜))    &   (𝜑 → (𝑆𝐴) = (𝑃(𝑇‘(𝐴𝐾))𝑈))    &   (𝜑 → (𝑆𝐵) = (𝑄(𝑇‘(𝐵𝐿))𝑉))    &   (𝜑 → ¬ (𝐴𝐾) = (𝐵𝐿))    &   (𝜑𝑃 ∈ (ℤ‘(𝑄 + 2)))    &   (𝜑𝐶 ∈ dom 𝑆)    &   (𝜑 → (𝑆𝐶) = (((𝐵𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴𝐾))⟩)))       (𝜑 → (𝐴‘0) = (𝐵‘0))

Theoremefgredlemc 17981* The reduced word that forms the base of the sequence in efgsval 17967 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   (𝜑 → ∀𝑎 ∈ dom 𝑆𝑏 ∈ dom 𝑆((#‘(𝑆𝑎)) < (#‘(𝑆𝐴)) → ((𝑆𝑎) = (𝑆𝑏) → (𝑎‘0) = (𝑏‘0))))    &   (𝜑𝐴 ∈ dom 𝑆)    &   (𝜑𝐵 ∈ dom 𝑆)    &   (𝜑 → (𝑆𝐴) = (𝑆𝐵))    &   (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))    &   𝐾 = (((#‘𝐴) − 1) − 1)    &   𝐿 = (((#‘𝐵) − 1) − 1)    &   (𝜑𝑃 ∈ (0...(#‘(𝐴𝐾))))    &   (𝜑𝑄 ∈ (0...(#‘(𝐵𝐿))))    &   (𝜑𝑈 ∈ (𝐼 × 2𝑜))    &   (𝜑𝑉 ∈ (𝐼 × 2𝑜))    &   (𝜑 → (𝑆𝐴) = (𝑃(𝑇‘(𝐴𝐾))𝑈))    &   (𝜑 → (𝑆𝐵) = (𝑄(𝑇‘(𝐵𝐿))𝑉))    &   (𝜑 → ¬ (𝐴𝐾) = (𝐵𝐿))       (𝜑 → (𝑃 ∈ (ℤ𝑄) → (𝐴‘0) = (𝐵‘0)))

Theoremefgredlemb 17982* The reduced word that forms the base of the sequence in efgsval 17967 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 30-Sep-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   (𝜑 → ∀𝑎 ∈ dom 𝑆𝑏 ∈ dom 𝑆((#‘(𝑆𝑎)) < (#‘(𝑆𝐴)) → ((𝑆𝑎) = (𝑆𝑏) → (𝑎‘0) = (𝑏‘0))))    &   (𝜑𝐴 ∈ dom 𝑆)    &   (𝜑𝐵 ∈ dom 𝑆)    &   (𝜑 → (𝑆𝐴) = (𝑆𝐵))    &   (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))    &   𝐾 = (((#‘𝐴) − 1) − 1)    &   𝐿 = (((#‘𝐵) − 1) − 1)    &   (𝜑𝑃 ∈ (0...(#‘(𝐴𝐾))))    &   (𝜑𝑄 ∈ (0...(#‘(𝐵𝐿))))    &   (𝜑𝑈 ∈ (𝐼 × 2𝑜))    &   (𝜑𝑉 ∈ (𝐼 × 2𝑜))    &   (𝜑 → (𝑆𝐴) = (𝑃(𝑇‘(𝐴𝐾))𝑈))    &   (𝜑 → (𝑆𝐵) = (𝑄(𝑇‘(𝐵𝐿))𝑉))    &   (𝜑 → ¬ (𝐴𝐾) = (𝐵𝐿))        ¬ 𝜑

Theoremefgredlem 17983* The reduced word that forms the base of the sequence in efgsval 17967 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 30-Sep-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   (𝜑 → ∀𝑎 ∈ dom 𝑆𝑏 ∈ dom 𝑆((#‘(𝑆𝑎)) < (#‘(𝑆𝐴)) → ((𝑆𝑎) = (𝑆𝑏) → (𝑎‘0) = (𝑏‘0))))    &   (𝜑𝐴 ∈ dom 𝑆)    &   (𝜑𝐵 ∈ dom 𝑆)    &   (𝜑 → (𝑆𝐴) = (𝑆𝐵))    &   (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))        ¬ 𝜑

Theoremefgred 17984* The reduced word that forms the base of the sequence in efgsval 17967 is uniquely determined, given the terminal point. (Contributed by Mario Carneiro, 28-Sep-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       ((𝐴 ∈ dom 𝑆𝐵 ∈ dom 𝑆 ∧ (𝑆𝐴) = (𝑆𝐵)) → (𝐴‘0) = (𝐵‘0))

Theoremefgrelexlema 17985* If two words 𝐴, 𝐵 are related under the free group equivalence, then there exist two extension sequences 𝑎, 𝑏 such that 𝑎 ends at 𝐴, 𝑏 ends at 𝐵, and 𝑎 and 𝐵 have the same starting point. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ 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))

Theoremefgrelexlemb 17986* If two words 𝐴, 𝐵 are related under the free group equivalence, then there exist two extension sequences 𝑎, 𝑏 such that 𝑎 ends at 𝐴, 𝑏 ends at 𝐵, and 𝑎 and 𝐵 have the same starting point. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   𝐿 = {⟨𝑖, 𝑗⟩ ∣ ∃𝑐 ∈ (𝑆 “ {𝑖})∃𝑑 ∈ (𝑆 “ {𝑗})(𝑐‘0) = (𝑑‘0)}        𝐿

Theoremefgrelex 17987* If two words 𝐴, 𝐵 are related under the free group equivalence, then there exist two extension sequences 𝑎, 𝑏 such that 𝑎 ends at 𝐴, 𝑏 ends at 𝐵, and 𝑎 and 𝐵 have the same starting point. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       (𝐴 𝐵 → ∃𝑎 ∈ (𝑆 “ {𝐴})∃𝑏 ∈ (𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0))

Theoremefgredeu 17988* There is a unique reduced word equivalent to a given word. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       (𝐴𝑊 → ∃!𝑑𝐷 𝑑 𝐴)

Theoremefgred2 17989* Two extension sequences have related endpoints iff they have the same base. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       ((𝐴 ∈ dom 𝑆𝐵 ∈ dom 𝑆) → ((𝑆𝐴) (𝑆𝐵) ↔ (𝐴‘0) = (𝐵‘0)))

Theoremefgcpbllema 17990* Lemma for efgrelex 17987. Define an auxiliary equivalence relation 𝐿 such that 𝐴𝐿𝐵 if there are sequences from 𝐴 to 𝐵 passing through the same reduced word. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   𝐿 = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑊 ∧ ((𝐴 ++ 𝑖) ++ 𝐵) ((𝐴 ++ 𝑗) ++ 𝐵))}       (𝑋𝐿𝑌 ↔ (𝑋𝑊𝑌𝑊 ∧ ((𝐴 ++ 𝑋) ++ 𝐵) ((𝐴 ++ 𝑌) ++ 𝐵)))

Theoremefgcpbllemb 17991* Lemma for efgrelex 17987. Show that 𝐿 is an equivalence relation containing all direct extensions of a word, so is closed under . (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   𝐿 = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑊 ∧ ((𝐴 ++ 𝑖) ++ 𝐵) ((𝐴 ++ 𝑗) ++ 𝐵))}       ((𝐴𝑊𝐵𝑊) → 𝐿)

Theoremefgcpbl 17992* Two extension sequences have related endpoints iff they have the same base. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       ((𝐴𝑊𝐵𝑊𝑋 𝑌) → ((𝐴 ++ 𝑋) ++ 𝐵) ((𝐴 ++ 𝑌) ++ 𝐵))

Theoremefgcpbl2 17993* Two extension sequences have related endpoints iff they have the same base. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       ((𝐴 𝑋𝐵 𝑌) → (𝐴 ++ 𝐵) (𝑋 ++ 𝑌))

Theoremfrgpval 17994 Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝐺 = (freeGrp‘𝐼)    &   𝑀 = (freeMnd‘(𝐼 × 2𝑜))    &    = ( ~FG𝐼)       (𝐼𝑉𝐺 = (𝑀 /s ))

Theoremfrgpcpbl 17995 Compatibility of the group operation with the free group equivalence relation. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
𝐺 = (freeGrp‘𝐼)    &   𝑀 = (freeMnd‘(𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &    + = (+g𝑀)       ((𝐴 𝐶𝐵 𝐷) → (𝐴 + 𝐵) (𝐶 + 𝐷))

Theoremfrgp0 17996 The free group is a group. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
𝐺 = (freeGrp‘𝐼)    &    = ( ~FG𝐼)       (𝐼𝑉 → (𝐺 ∈ Grp ∧ [∅] = (0g𝐺)))

Theoremfrgpeccl 17997 Closure of the quotient map in a free group. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝐺 = (freeGrp‘𝐼)    &    = ( ~FG𝐼)    &   𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &   𝐵 = (Base‘𝐺)       (𝑋𝑊 → [𝑋] 𝐵)

Theoremfrgpgrp 17998 The free group is a group. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝐺 = (freeGrp‘𝐼)       (𝐼𝑉𝐺 ∈ Grp)

Theoremfrgpadd 17999 Addition in the free group is given by concatenation. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &   𝐺 = (freeGrp‘𝐼)    &    = ( ~FG𝐼)    &    + = (+g𝐺)       ((𝐴𝑊𝐵𝑊) → ([𝐴] + [𝐵] ) = [(𝐴 ++ 𝐵)] )

Theoremfrgpinv 18000* The inverse of an element of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &   𝐺 = (freeGrp‘𝐼)    &    = ( ~FG𝐼)    &   𝑁 = (invg𝐺)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)       (𝐴𝑊 → (𝑁‘[𝐴] ) = [(𝑀 ∘ (reverse‘𝐴))] )

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