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Theorem List for Metamath Proof Explorer - 18701-18800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlsmless2x 18701 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 18702 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 18703 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 18704* 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 18705* 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 18706 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 18707* Subgroup sum membership analogue of lsmelval 18705 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 18708 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 18709 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 18710 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 18711 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))
 
10.2.12.1  Direct products (extension)
 
Theoremsmndlsmidm 18712 The direct product is idempotent for submonoids. (Contributed by AV, 27-Dec-2023.)
= (LSSum‘𝐺)       (𝑈 ∈ (SubMnd‘𝐺) → (𝑈 𝑈) = 𝑈)
 
Theoremlsmub1 18713 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 18714 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 18715 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 18716 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 18717 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 18718 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 18719 Subgroup sum is idempotent. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.) (Proof shortened by AV, 27-Dec-2023.)
= (LSSum‘𝐺)       (𝑈 ∈ (SubGrp‘𝐺) → (𝑈 𝑈) = 𝑈)
 
TheoremlsmidmOLD 18720 Obsolete proof of lsmidm 18719 as of 13-Jan-2024. Subgroup sum is idempotent. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
= (LSSum‘𝐺)       (𝑈 ∈ (SubGrp‘𝐺) → (𝑈 𝑈) = 𝑈)
 
Theoremlsmlub 18721 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 18722 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 18723 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 18724 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 18725 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 18726 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)))
 
Theoremmndlsmidm 18727 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)
 
Theoremlsm01 18728 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 18729 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 18730 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 18731 Equivalent expressions for "not less than". (chnlei 29190 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 18732 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 18733 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 18734* 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 18735 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 18736 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 18737 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 18738 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 18739 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 18740 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 18741 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 18742 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 18743 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 18744 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 18745 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 18746 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 18747 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 18748 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 18749 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 18750 Vectors belonging to disjoint commuting subgroups are uniquely determined by their sum. Analogous to opth 5360, 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 18751* 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 18752* 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 18753* 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 18754 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 18755 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 18756 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 18757 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 18758 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 18759 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 18760 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 18761 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 18762 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)))
 
10.2.13  Free groups
 
Syntaxcefg 18763 Extend class notation with the free group equivalence relation.
class ~FG
 
Syntaxcfrgp 18764 Extend class notation with the free group construction.
class freeGrp
 
Syntaxcvrgp 18765 Extend class notation with free group injection.
class varFGrp
 
Definitiondf-efg 18766* 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 18767 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 18766. (Contributed by Mario Carneiro, 1-Oct-2015.)
freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2o)) /s ( ~FG𝑖)))
 
Definitiondf-vrgp 18768* 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 18769* 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 18770* 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 18771* 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 18772 Lemma for efgval 18774. (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 18773* Lemma for efgval 18774. (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 18774* 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 18775 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 18776 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 18777 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 18778 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 18779* 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 18780* 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 18781* 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 18782* 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 18783* 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 18784* 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 18785* 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 18786* 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 18787* 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 18788* 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 18789* 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 18790* 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 18791* 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 18792* 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 18793* 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 18794* 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 18795* An initial segment of an extension sequence is an extension sequence. (Contributed by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 3-Nov-2022.)
𝑊 = ( I ‘Word (𝐼 × 2o))    &    = ( ~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 18796* For any word, there is a sequence of extensions starting at a reduced word and ending at the target word, such that each word in the chain is an extension of the previous (inserting an element and its inverse at adjacent indices somewhere in the sequence). (Contributed by Mario Carneiro, 27-Sep-2015.)
𝑊 = ( I ‘Word (𝐼 × 2o))    &    = ( ~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 18797* The reduced word that forms the base of the sequence in efgsval 18788 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 18798* Lemma for efgredleme 18800. (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 18799* Lemma for efgred 18805. (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 18800* Lemma for efgred 18805. (Contributed by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 15-Oct-2022.)
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) prefix 𝑄) ++ ((𝐴𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴𝐾))⟩)))       (𝜑 → ((𝐴𝐾) ∈ ran (𝑇‘(𝑆𝐶)) ∧ (𝐵𝐿) ∈ ran (𝑇‘(𝑆𝐶))))
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