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Theorem List for Metamath Proof Explorer - 18101-18200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlsmelvalx 18101* Subspace sum membership (for a group or vector space). Extended domain version of lsmelval 18110. (Contributed by NM, 28-Jan-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (LSSum‘𝐺)       ((𝐺𝑉𝑇𝐵𝑈𝐵) → (𝑋 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑋 = (𝑦 + 𝑧)))

Theoremlsmelvalix 18102 Subspace sum membership (for a group or vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (LSSum‘𝐺)       (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ (𝑋𝑇𝑌𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 𝑈))

Theoremoppglsm 18103 The subspace sum operation in the opposite group. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝑂 = (oppg𝐺)    &    = (LSSum‘𝐺)       (𝑇(LSSum‘𝑂)𝑈) = (𝑈 𝑇)

Theoremlsmssv 18104 Subgroup sum is a subset of the base. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    = (LSSum‘𝐺)       ((𝐺 ∈ Mnd ∧ 𝑇𝐵𝑈𝐵) → (𝑇 𝑈) ⊆ 𝐵)

Theoremlsmless1x 18105 Subset implies subgroup sum subset (extended domain version). (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    = (LSSum‘𝐺)       (((𝐺𝑉𝑇𝐵𝑈𝐵) ∧ 𝑅𝑇) → (𝑅 𝑈) ⊆ (𝑇 𝑈))

Theoremlsmless2x 18106 Subset implies subgroup sum subset (extended domain version). (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    = (LSSum‘𝐺)       (((𝐺𝑉𝑅𝐵𝑈𝐵) ∧ 𝑇𝑈) → (𝑅 𝑇) ⊆ (𝑅 𝑈))

Theoremlsmub1x 18107 Subgroup sum is an upper bound of its arguments. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    = (LSSum‘𝐺)       ((𝑇𝐵𝑈 ∈ (SubMnd‘𝐺)) → 𝑇 ⊆ (𝑇 𝑈))

Theoremlsmub2x 18108 Subgroup sum is an upper bound of its arguments. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    = (LSSum‘𝐺)       ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈𝐵) → 𝑈 ⊆ (𝑇 𝑈))

Theoremlsmval 18109* Subgroup sum value (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (LSSum‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥 + 𝑦)))

Theoremlsmelval 18110* Subgroup sum membership (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
+ = (+g𝐺)    &    = (LSSum‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑋 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑋 = (𝑦 + 𝑧)))

Theoremlsmelvali 18111 Subgroup sum membership (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
+ = (+g𝐺)    &    = (LSSum‘𝐺)       (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑋𝑇𝑌𝑈)) → (𝑋 + 𝑌) ∈ (𝑇 𝑈))

Theoremlsmelvalm 18112* Subgroup sum membership analogue of lsmelval 18110 using vector subtraction. TODO: any way to shorten proof? (Contributed by NM, 16-Mar-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (-g𝐺)    &    = (LSSum‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))       (𝜑 → (𝑋 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑋 = (𝑦 𝑧)))

Theoremlsmelvalmi 18113 Membership of vector subtraction in subgroup sum. (Contributed by NM, 27-Apr-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (-g𝐺)    &    = (LSSum‘𝐺)    &   (𝜑𝑇 ∈ (SubGrp‘𝐺))    &   (𝜑𝑈 ∈ (SubGrp‘𝐺))    &   (𝜑𝑋𝑇)    &   (𝜑𝑌𝑈)       (𝜑 → (𝑋 𝑌) ∈ (𝑇 𝑈))

Theoremlsmsubm 18114 The sum of two commuting submonoids is a submonoid. (Contributed by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)    &   𝑍 = (Cntz‘𝐺)       ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ∈ (SubMnd‘𝐺))

Theoremlsmsubg 18115 The sum of two commuting subgroups is a subgroup. (Contributed by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)    &   𝑍 = (Cntz‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ∈ (SubGrp‘𝐺))

Theoremlsmcom2 18116 Subgroup sum commutes. (Contributed by Mario Carneiro, 22-Apr-2016.)
= (LSSum‘𝐺)    &   𝑍 = (Cntz‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) = (𝑈 𝑇))

Theoremlsmub1 18117 Subgroup sum is an upper bound of its arguments. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑇 ⊆ (𝑇 𝑈))

Theoremlsmub2 18118 Subgroup sum is an upper bound of its arguments. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → 𝑈 ⊆ (𝑇 𝑈))

Theoremlsmunss 18119 Union of subgroups is a subset of subgroup sum. (Contributed by NM, 6-Feb-2014.) (Proof shortened by Mario Carneiro, 21-Jun-2014.)
= (LSSum‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇𝑈) ⊆ (𝑇 𝑈))

Theoremlsmless1 18120 Subset implies subgroup sum subset. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)       ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑆𝑇) → (𝑆 𝑈) ⊆ (𝑇 𝑈))

Theoremlsmless2 18121 Subset implies subgroup sum subset. (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)       ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑈) → (𝑆 𝑇) ⊆ (𝑆 𝑈))

Theoremlsmless12 18122 Subset implies subgroup sum subset. (Contributed by NM, 14-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
= (LSSum‘𝐺)       (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (𝑅𝑆𝑇𝑈)) → (𝑅 𝑇) ⊆ (𝑆 𝑈))

Theoremlsmidm 18123 Subgroup sum is idempotent. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
= (LSSum‘𝐺)       (𝑈 ∈ (SubGrp‘𝐺) → (𝑈 𝑈) = 𝑈)

Theoremlsmlub 18124 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 18125 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 18126 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 18127 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 18128 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 18129 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 18130 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 18131 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 18132 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 18133 Equivalent expressions for "not less than". (chnlei 28472 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 18134 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 18135 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 18136* 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 18137 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 18138 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 18139 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 18140 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 18141 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 18142 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 18143 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 18144 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 18145 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 18146 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 18147 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 18148 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 18149 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 18150 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 18151 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 18152 Vectors belonging to disjoint commuting subgroups are uniquely determined by their sum. Analogous to opth 4974, 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 18153* 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 18154* 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 18155* 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 18156 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 18157 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 18158 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 18159 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 18160 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 18161 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 18162 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 18163 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 18164 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 18165 Extend class notation with the free group equivalence relation.
class ~FG

Syntaxcfrgp 18166 Extend class notation with the free group construction.
class freeGrp

Syntaxcvrgp 18167 Extend class notation with free group injection.
class varFGrp

Definitiondf-efg 18168* 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 18169 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 18168. (Contributed by Mario Carneiro, 1-Oct-2015.)
freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2𝑜)) /s ( ~FG𝑖)))

Definitiondf-vrgp 18170* 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 18171* 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 18172* 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 18173* 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 18174 Lemma for efgval 18176. (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 18175* Lemma for efgval 18176. (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 18176* 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 18177 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 18178 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 18179 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 18180 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 18181* 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 18182* 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 18183* 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 18184* 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 18185* 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 18186* 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 18187* 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 18188* 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 18189* 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 18190* 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 18191* 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 18192* 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 18193* 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 18194* 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 18195* 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 18196* 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 18197* 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 18198* 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 18199* The reduced word that forms the base of the sequence in efgsval 18190 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 18200* Lemma for efgredleme 18202. (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))

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