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Theorem List for Metamath Proof Explorer - 19501-19600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
10.2.12  Direct products
 
Syntaxclsm 19501 Extend class notation with subgroup sum.
class LSSum
 
Syntaxcpj1 19502 Extend class notation with left projection.
class proj1
 
Definitiondf-lsm 19503* Define subgroup sum (inner direct product of subgroups). (Contributed by NM, 28-Jan-2014.)
LSSum = (𝑀 ∈ V ↦ (𝑑 ∈ 𝒫 (Baseβ€˜π‘€), 𝑒 ∈ 𝒫 (Baseβ€˜π‘€) ↦ ran (π‘₯ ∈ 𝑑, 𝑦 ∈ 𝑒 ↦ (π‘₯(+gβ€˜π‘€)𝑦))))
 
Definitiondf-pj1 19504* Define the left projection function, which takes two subgroups 𝑑, 𝑒 with trivial intersection and returns a function mapping the elements of the subgroup sum 𝑑 + 𝑒 to their projections onto 𝑑. (The other projection function can be obtained by swapping the roles of 𝑑 and 𝑒.) (Contributed by Mario Carneiro, 15-Oct-2015.)
proj1 = (𝑀 ∈ V ↦ (𝑑 ∈ 𝒫 (Baseβ€˜π‘€), 𝑒 ∈ 𝒫 (Baseβ€˜π‘€) ↦ (𝑧 ∈ (𝑑(LSSumβ€˜π‘€)𝑒) ↦ (β„©π‘₯ ∈ 𝑑 βˆƒπ‘¦ ∈ 𝑒 𝑧 = (π‘₯(+gβ€˜π‘€)𝑦)))))
 
Theoremlsmfval 19505* The subgroup sum function (for a group or vector space). (Contributed by NM, 28-Jan-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐡 = (Baseβ€˜πΊ)    &    + = (+gβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    β‡’   (𝐺 ∈ 𝑉 β†’ βŠ• = (𝑑 ∈ 𝒫 𝐡, 𝑒 ∈ 𝒫 𝐡 ↦ ran (π‘₯ ∈ 𝑑, 𝑦 ∈ 𝑒 ↦ (π‘₯ + 𝑦))))
 
Theoremlsmvalx 19506* Subspace sum value (for a group or vector space). Extended domain version of lsmval 19515. (Contributed by NM, 28-Jan-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐡 = (Baseβ€˜πΊ)    &    + = (+gβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    β‡’   ((𝐺 ∈ 𝑉 ∧ 𝑇 βŠ† 𝐡 ∧ π‘ˆ βŠ† 𝐡) β†’ (𝑇 βŠ• π‘ˆ) = ran (π‘₯ ∈ 𝑇, 𝑦 ∈ π‘ˆ ↦ (π‘₯ + 𝑦)))
 
Theoremlsmelvalx 19507* Subspace sum membership (for a group or vector space). Extended domain version of lsmelval 19516. (Contributed by NM, 28-Jan-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐡 = (Baseβ€˜πΊ)    &    + = (+gβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    β‡’   ((𝐺 ∈ 𝑉 ∧ 𝑇 βŠ† 𝐡 ∧ π‘ˆ βŠ† 𝐡) β†’ (𝑋 ∈ (𝑇 βŠ• π‘ˆ) ↔ βˆƒπ‘¦ ∈ 𝑇 βˆƒπ‘§ ∈ π‘ˆ 𝑋 = (𝑦 + 𝑧)))
 
Theoremlsmelvalix 19508 Subspace sum membership (for a group or vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐡 = (Baseβ€˜πΊ)    &    + = (+gβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    β‡’   (((𝐺 ∈ 𝑉 ∧ 𝑇 βŠ† 𝐡 ∧ π‘ˆ βŠ† 𝐡) ∧ (𝑋 ∈ 𝑇 ∧ π‘Œ ∈ π‘ˆ)) β†’ (𝑋 + π‘Œ) ∈ (𝑇 βŠ• π‘ˆ))
 
Theoremoppglsm 19509 The subspace sum operation in the opposite group. (Contributed by Mario Carneiro, 19-Apr-2016.) (Proof shortened by AV, 2-Mar-2024.)
𝑂 = (oppgβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    β‡’   (𝑇(LSSumβ€˜π‘‚)π‘ˆ) = (π‘ˆ βŠ• 𝑇)
 
Theoremlsmssv 19510 Subgroup sum is a subset of the base. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝐡 = (Baseβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    β‡’   ((𝐺 ∈ Mnd ∧ 𝑇 βŠ† 𝐡 ∧ π‘ˆ βŠ† 𝐡) β†’ (𝑇 βŠ• π‘ˆ) βŠ† 𝐡)
 
Theoremlsmless1x 19511 Subset implies subgroup sum subset (extended domain version). (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐡 = (Baseβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    β‡’   (((𝐺 ∈ 𝑉 ∧ 𝑇 βŠ† 𝐡 ∧ π‘ˆ βŠ† 𝐡) ∧ 𝑅 βŠ† 𝑇) β†’ (𝑅 βŠ• π‘ˆ) βŠ† (𝑇 βŠ• π‘ˆ))
 
Theoremlsmless2x 19512 Subset implies subgroup sum subset (extended domain version). (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐡 = (Baseβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    β‡’   (((𝐺 ∈ 𝑉 ∧ 𝑅 βŠ† 𝐡 ∧ π‘ˆ βŠ† 𝐡) ∧ 𝑇 βŠ† π‘ˆ) β†’ (𝑅 βŠ• 𝑇) βŠ† (𝑅 βŠ• π‘ˆ))
 
Theoremlsmub1x 19513 Subgroup sum is an upper bound of its arguments. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝐡 = (Baseβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    β‡’   ((𝑇 βŠ† 𝐡 ∧ π‘ˆ ∈ (SubMndβ€˜πΊ)) β†’ 𝑇 βŠ† (𝑇 βŠ• π‘ˆ))
 
Theoremlsmub2x 19514 Subgroup sum is an upper bound of its arguments. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐡 = (Baseβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    β‡’   ((𝑇 ∈ (SubMndβ€˜πΊ) ∧ π‘ˆ βŠ† 𝐡) β†’ π‘ˆ βŠ† (𝑇 βŠ• π‘ˆ))
 
Theoremlsmval 19515* Subgroup sum value (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐡 = (Baseβ€˜πΊ)    &    + = (+gβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    β‡’   ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ)) β†’ (𝑇 βŠ• π‘ˆ) = ran (π‘₯ ∈ 𝑇, 𝑦 ∈ π‘ˆ ↦ (π‘₯ + 𝑦)))
 
Theoremlsmelval 19516* Subgroup sum membership (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
+ = (+gβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    β‡’   ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ)) β†’ (𝑋 ∈ (𝑇 βŠ• π‘ˆ) ↔ βˆƒπ‘¦ ∈ 𝑇 βˆƒπ‘§ ∈ π‘ˆ 𝑋 = (𝑦 + 𝑧)))
 
Theoremlsmelvali 19517 Subgroup sum membership (for a left module or left vector space). (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
+ = (+gβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    β‡’   (((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ)) ∧ (𝑋 ∈ 𝑇 ∧ π‘Œ ∈ π‘ˆ)) β†’ (𝑋 + π‘Œ) ∈ (𝑇 βŠ• π‘ˆ))
 
Theoremlsmelvalm 19518* Subgroup sum membership analogue of lsmelval 19516 using vector subtraction. TODO: any way to shorten proof? (Contributed by NM, 16-Mar-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
βˆ’ = (-gβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    β‡’   (πœ‘ β†’ (𝑋 ∈ (𝑇 βŠ• π‘ˆ) ↔ βˆƒπ‘¦ ∈ 𝑇 βˆƒπ‘§ ∈ π‘ˆ 𝑋 = (𝑦 βˆ’ 𝑧)))
 
Theoremlsmelvalmi 19519 Membership of vector subtraction in subgroup sum. (Contributed by NM, 27-Apr-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
βˆ’ = (-gβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ 𝑋 ∈ 𝑇)    &   (πœ‘ β†’ π‘Œ ∈ π‘ˆ)    β‡’   (πœ‘ β†’ (𝑋 βˆ’ π‘Œ) ∈ (𝑇 βŠ• π‘ˆ))
 
Theoremlsmsubm 19520 The sum of two commuting submonoids is a submonoid. (Contributed by Mario Carneiro, 19-Apr-2016.)
βŠ• = (LSSumβ€˜πΊ)    &   π‘ = (Cntzβ€˜πΊ)    β‡’   ((𝑇 ∈ (SubMndβ€˜πΊ) ∧ π‘ˆ ∈ (SubMndβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) β†’ (𝑇 βŠ• π‘ˆ) ∈ (SubMndβ€˜πΊ))
 
Theoremlsmsubg 19521 The sum of two commuting subgroups is a subgroup. (Contributed by Mario Carneiro, 19-Apr-2016.)
βŠ• = (LSSumβ€˜πΊ)    &   π‘ = (Cntzβ€˜πΊ)    β‡’   ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) β†’ (𝑇 βŠ• π‘ˆ) ∈ (SubGrpβ€˜πΊ))
 
Theoremlsmcom2 19522 Subgroup sum commutes. (Contributed by Mario Carneiro, 22-Apr-2016.)
βŠ• = (LSSumβ€˜πΊ)    &   π‘ = (Cntzβ€˜πΊ)    β‡’   ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ)) β†’ (𝑇 βŠ• π‘ˆ) = (π‘ˆ βŠ• 𝑇))
 
10.2.12.1  Direct products (extension)
 
Theoremsmndlsmidm 19523 The direct product is idempotent for submonoids. (Contributed by AV, 27-Dec-2023.)
βŠ• = (LSSumβ€˜πΊ)    β‡’   (π‘ˆ ∈ (SubMndβ€˜πΊ) β†’ (π‘ˆ βŠ• π‘ˆ) = π‘ˆ)
 
Theoremlsmub1 19524 Subgroup sum is an upper bound of its arguments. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
βŠ• = (LSSumβ€˜πΊ)    β‡’   ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ)) β†’ 𝑇 βŠ† (𝑇 βŠ• π‘ˆ))
 
Theoremlsmub2 19525 Subgroup sum is an upper bound of its arguments. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
βŠ• = (LSSumβ€˜πΊ)    β‡’   ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ)) β†’ π‘ˆ βŠ† (𝑇 βŠ• π‘ˆ))
 
Theoremlsmunss 19526 Union of subgroups is a subset of subgroup sum. (Contributed by NM, 6-Feb-2014.) (Proof shortened by Mario Carneiro, 21-Jun-2014.)
βŠ• = (LSSumβ€˜πΊ)    β‡’   ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ)) β†’ (𝑇 βˆͺ π‘ˆ) βŠ† (𝑇 βŠ• π‘ˆ))
 
Theoremlsmless1 19527 Subset implies subgroup sum subset. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
βŠ• = (LSSumβ€˜πΊ)    β‡’   ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑆 βŠ† 𝑇) β†’ (𝑆 βŠ• π‘ˆ) βŠ† (𝑇 βŠ• π‘ˆ))
 
Theoremlsmless2 19528 Subset implies subgroup sum subset. (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
βŠ• = (LSSumβ€˜πΊ)    β‡’   ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† π‘ˆ) β†’ (𝑆 βŠ• 𝑇) βŠ† (𝑆 βŠ• π‘ˆ))
 
Theoremlsmless12 19529 Subset implies subgroup sum subset. (Contributed by NM, 14-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
βŠ• = (LSSumβ€˜πΊ)    β‡’   (((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ)) ∧ (𝑅 βŠ† 𝑆 ∧ 𝑇 βŠ† π‘ˆ)) β†’ (𝑅 βŠ• 𝑇) βŠ† (𝑆 βŠ• π‘ˆ))
 
Theoremlsmidm 19530 Subgroup sum is idempotent. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.) (Proof shortened by AV, 27-Dec-2023.)
βŠ• = (LSSumβ€˜πΊ)    β‡’   (π‘ˆ ∈ (SubGrpβ€˜πΊ) β†’ (π‘ˆ βŠ• π‘ˆ) = π‘ˆ)
 
Theoremlsmlub 19531 The least upper bound property of subgroup sum. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
βŠ• = (LSSumβ€˜πΊ)    β‡’   ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ)) β†’ ((𝑆 βŠ† π‘ˆ ∧ 𝑇 βŠ† π‘ˆ) ↔ (𝑆 βŠ• 𝑇) βŠ† π‘ˆ))
 
Theoremlsmss1 19532 Subgroup sum with a subset. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
βŠ• = (LSSumβ€˜πΊ)    β‡’   ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† π‘ˆ) β†’ (𝑇 βŠ• π‘ˆ) = π‘ˆ)
 
Theoremlsmss1b 19533 Subgroup sum with a subset. (Contributed by NM, 10-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
βŠ• = (LSSumβ€˜πΊ)    β‡’   ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ)) β†’ (𝑇 βŠ† π‘ˆ ↔ (𝑇 βŠ• π‘ˆ) = π‘ˆ))
 
Theoremlsmss2 19534 Subgroup sum with a subset. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
βŠ• = (LSSumβ€˜πΊ)    β‡’   ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ βŠ† 𝑇) β†’ (𝑇 βŠ• π‘ˆ) = 𝑇)
 
Theoremlsmss2b 19535 Subgroup sum with a subset. (Contributed by NM, 10-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
βŠ• = (LSSumβ€˜πΊ)    β‡’   ((𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ)) β†’ (π‘ˆ βŠ† 𝑇 ↔ (𝑇 βŠ• π‘ˆ) = 𝑇))
 
Theoremlsmass 19536 Subgroup sum is associative. (Contributed by NM, 2-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
βŠ• = (LSSumβ€˜πΊ)    β‡’   ((𝑅 ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ)) β†’ ((𝑅 βŠ• 𝑇) βŠ• π‘ˆ) = (𝑅 βŠ• (𝑇 βŠ• π‘ˆ)))
 
Theoremmndlsmidm 19537 Subgroup sum is idempotent for monoids. This corresponds to the observation in [Lang] p. 6. (Contributed by AV, 27-Dec-2023.)
βŠ• = (LSSumβ€˜πΊ)    &   π΅ = (Baseβ€˜πΊ)    β‡’   (𝐺 ∈ Mnd β†’ (𝐡 βŠ• 𝐡) = 𝐡)
 
Theoremlsm01 19538 Subgroup sum with the zero subgroup. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
0 = (0gβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    β‡’   (𝑋 ∈ (SubGrpβ€˜πΊ) β†’ (𝑋 βŠ• { 0 }) = 𝑋)
 
Theoremlsm02 19539 Subgroup sum with the zero subgroup. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
0 = (0gβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    β‡’   (𝑋 ∈ (SubGrpβ€˜πΊ) β†’ ({ 0 } βŠ• 𝑋) = 𝑋)
 
Theoremsubglsm 19540 The subgroup sum evaluated within a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
𝐻 = (𝐺 β†Ύs 𝑆)    &    βŠ• = (LSSumβ€˜πΊ)    &   π΄ = (LSSumβ€˜π»)    β‡’   ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 βŠ† 𝑆 ∧ π‘ˆ βŠ† 𝑆) β†’ (𝑇 βŠ• π‘ˆ) = (π‘‡π΄π‘ˆ))
 
Theoremlssnle 19541 Equivalent expressions for "not less than". (chnlei 30733 analog.) (Contributed by NM, 10-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
βŠ• = (LSSumβ€˜πΊ)    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    β‡’   (πœ‘ β†’ (Β¬ π‘ˆ βŠ† 𝑇 ↔ 𝑇 ⊊ (𝑇 βŠ• π‘ˆ)))
 
Theoremlsmmod 19542 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.)
βŠ• = (LSSumβ€˜πΊ)    β‡’   (((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ)) ∧ 𝑆 βŠ† π‘ˆ) β†’ (𝑆 βŠ• (𝑇 ∩ π‘ˆ)) = ((𝑆 βŠ• 𝑇) ∩ π‘ˆ))
 
Theoremlsmmod2 19543 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.)
βŠ• = (LSSumβ€˜πΊ)    β‡’   (((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ 𝑇 ∈ (SubGrpβ€˜πΊ) ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ)) ∧ π‘ˆ βŠ† 𝑆) β†’ (𝑆 ∩ (𝑇 βŠ• π‘ˆ)) = ((𝑆 ∩ 𝑇) βŠ• π‘ˆ))
 
Theoremlsmpropd 19544* If two structures have the same components (properties), they have the same subspace structure. (Contributed by Mario Carneiro, 29-Jun-2015.) (Revised by AV, 25-Apr-2024.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝐿 ∈ π‘Š)    β‡’   (πœ‘ β†’ (LSSumβ€˜πΎ) = (LSSumβ€˜πΏ))
 
Theoremcntzrecd 19545 Commute the "subgroups commute" predicate. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝑍 = (Cntzβ€˜πΊ)    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ 𝑇 βŠ† (π‘β€˜π‘ˆ))    β‡’   (πœ‘ β†’ π‘ˆ βŠ† (π‘β€˜π‘‡))
 
Theoremlsmcntz 19546 The "subgroups commute" predicate applied to a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
βŠ• = (LSSumβ€˜πΊ)    &   (πœ‘ β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    &   π‘ = (Cntzβ€˜πΊ)    β‡’   (πœ‘ β†’ ((𝑆 βŠ• 𝑇) βŠ† (π‘β€˜π‘ˆ) ↔ (𝑆 βŠ† (π‘β€˜π‘ˆ) ∧ 𝑇 βŠ† (π‘β€˜π‘ˆ))))
 
Theoremlsmcntzr 19547 The "subgroups commute" predicate applied to a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
βŠ• = (LSSumβ€˜πΊ)    &   (πœ‘ β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    &   π‘ = (Cntzβ€˜πΊ)    β‡’   (πœ‘ β†’ (𝑆 βŠ† (π‘β€˜(𝑇 βŠ• π‘ˆ)) ↔ (𝑆 βŠ† (π‘β€˜π‘‡) ∧ 𝑆 βŠ† (π‘β€˜π‘ˆ))))
 
Theoremlsmdisj 19548 Disjointness from a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
βŠ• = (LSSumβ€˜πΊ)    &   (πœ‘ β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    &    0 = (0gβ€˜πΊ)    &   (πœ‘ β†’ ((𝑆 βŠ• 𝑇) ∩ π‘ˆ) = { 0 })    β‡’   (πœ‘ β†’ ((𝑆 ∩ π‘ˆ) = { 0 } ∧ (𝑇 ∩ π‘ˆ) = { 0 }))
 
Theoremlsmdisj2 19549 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016.)
βŠ• = (LSSumβ€˜πΊ)    &   (πœ‘ β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    &    0 = (0gβ€˜πΊ)    &   (πœ‘ β†’ ((𝑆 βŠ• 𝑇) ∩ π‘ˆ) = { 0 })    &   (πœ‘ β†’ (𝑆 ∩ 𝑇) = { 0 })    β‡’   (πœ‘ β†’ (𝑇 ∩ (𝑆 βŠ• π‘ˆ)) = { 0 })
 
Theoremlsmdisj3 19550 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
βŠ• = (LSSumβ€˜πΊ)    &   (πœ‘ β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    &    0 = (0gβ€˜πΊ)    &   (πœ‘ β†’ ((𝑆 βŠ• 𝑇) ∩ π‘ˆ) = { 0 })    &   (πœ‘ β†’ (𝑆 ∩ 𝑇) = { 0 })    &   π‘ = (Cntzβ€˜πΊ)    &   (πœ‘ β†’ 𝑆 βŠ† (π‘β€˜π‘‡))    β‡’   (πœ‘ β†’ (𝑆 ∩ (𝑇 βŠ• π‘ˆ)) = { 0 })
 
Theoremlsmdisjr 19551 Disjointness from a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
βŠ• = (LSSumβ€˜πΊ)    &   (πœ‘ β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    &    0 = (0gβ€˜πΊ)    &   (πœ‘ β†’ (𝑆 ∩ (𝑇 βŠ• π‘ˆ)) = { 0 })    β‡’   (πœ‘ β†’ ((𝑆 ∩ 𝑇) = { 0 } ∧ (𝑆 ∩ π‘ˆ) = { 0 }))
 
Theoremlsmdisj2r 19552 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016.)
βŠ• = (LSSumβ€˜πΊ)    &   (πœ‘ β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    &    0 = (0gβ€˜πΊ)    &   (πœ‘ β†’ (𝑆 ∩ (𝑇 βŠ• π‘ˆ)) = { 0 })    &   (πœ‘ β†’ (𝑇 ∩ π‘ˆ) = { 0 })    β‡’   (πœ‘ β†’ ((𝑆 βŠ• π‘ˆ) ∩ 𝑇) = { 0 })
 
Theoremlsmdisj3r 19553 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016.)
βŠ• = (LSSumβ€˜πΊ)    &   (πœ‘ β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    &    0 = (0gβ€˜πΊ)    &   (πœ‘ β†’ (𝑆 ∩ (𝑇 βŠ• π‘ˆ)) = { 0 })    &   (πœ‘ β†’ (𝑇 ∩ π‘ˆ) = { 0 })    &   π‘ = (Cntzβ€˜πΊ)    &   (πœ‘ β†’ 𝑇 βŠ† (π‘β€˜π‘ˆ))    β‡’   (πœ‘ β†’ ((𝑆 βŠ• 𝑇) ∩ π‘ˆ) = { 0 })
 
Theoremlsmdisj2a 19554 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
βŠ• = (LSSumβ€˜πΊ)    &   (πœ‘ β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    &    0 = (0gβ€˜πΊ)    β‡’   (πœ‘ β†’ ((((𝑆 βŠ• 𝑇) ∩ π‘ˆ) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 }) ↔ ((𝑇 ∩ (𝑆 βŠ• π‘ˆ)) = { 0 } ∧ (𝑆 ∩ π‘ˆ) = { 0 })))
 
Theoremlsmdisj2b 19555 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
βŠ• = (LSSumβ€˜πΊ)    &   (πœ‘ β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    &    0 = (0gβ€˜πΊ)    β‡’   (πœ‘ β†’ ((((𝑆 βŠ• π‘ˆ) ∩ 𝑇) = { 0 } ∧ (𝑆 ∩ π‘ˆ) = { 0 }) ↔ ((𝑆 ∩ (𝑇 βŠ• π‘ˆ)) = { 0 } ∧ (𝑇 ∩ π‘ˆ) = { 0 })))
 
Theoremlsmdisj3a 19556 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
βŠ• = (LSSumβ€˜πΊ)    &   (πœ‘ β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    &    0 = (0gβ€˜πΊ)    &   π‘ = (Cntzβ€˜πΊ)    &   (πœ‘ β†’ 𝑆 βŠ† (π‘β€˜π‘‡))    β‡’   (πœ‘ β†’ ((((𝑆 βŠ• 𝑇) ∩ π‘ˆ) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 }) ↔ ((𝑆 ∩ (𝑇 βŠ• π‘ˆ)) = { 0 } ∧ (𝑇 ∩ π‘ˆ) = { 0 })))
 
Theoremlsmdisj3b 19557 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
βŠ• = (LSSumβ€˜πΊ)    &   (πœ‘ β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    &    0 = (0gβ€˜πΊ)    &   π‘ = (Cntzβ€˜πΊ)    &   (πœ‘ β†’ 𝑇 βŠ† (π‘β€˜π‘ˆ))    β‡’   (πœ‘ β†’ ((((𝑆 βŠ• 𝑇) ∩ π‘ˆ) = { 0 } ∧ (𝑆 ∩ 𝑇) = { 0 }) ↔ ((𝑆 ∩ (𝑇 βŠ• π‘ˆ)) = { 0 } ∧ (𝑇 ∩ π‘ˆ) = { 0 })))
 
Theoremsubgdisj1 19558 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.)
+ = (+gβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &   π‘ = (Cntzβ€˜πΊ)    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ (𝑇 ∩ π‘ˆ) = { 0 })    &   (πœ‘ β†’ 𝑇 βŠ† (π‘β€˜π‘ˆ))    &   (πœ‘ β†’ 𝐴 ∈ 𝑇)    &   (πœ‘ β†’ 𝐢 ∈ 𝑇)    &   (πœ‘ β†’ 𝐡 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝐷 ∈ π‘ˆ)    &   (πœ‘ β†’ (𝐴 + 𝐡) = (𝐢 + 𝐷))    β‡’   (πœ‘ β†’ 𝐴 = 𝐢)
 
Theoremsubgdisj2 19559 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.)
+ = (+gβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &   π‘ = (Cntzβ€˜πΊ)    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ (𝑇 ∩ π‘ˆ) = { 0 })    &   (πœ‘ β†’ 𝑇 βŠ† (π‘β€˜π‘ˆ))    &   (πœ‘ β†’ 𝐴 ∈ 𝑇)    &   (πœ‘ β†’ 𝐢 ∈ 𝑇)    &   (πœ‘ β†’ 𝐡 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝐷 ∈ π‘ˆ)    &   (πœ‘ β†’ (𝐴 + 𝐡) = (𝐢 + 𝐷))    β‡’   (πœ‘ β†’ 𝐡 = 𝐷)
 
Theoremsubgdisjb 19560 Vectors belonging to disjoint commuting subgroups are uniquely determined by their sum. Analogous to opth 5476, this theorem shows a way of representing a pair of vectors. (Contributed by NM, 5-Jul-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
+ = (+gβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &   π‘ = (Cntzβ€˜πΊ)    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ (𝑇 ∩ π‘ˆ) = { 0 })    &   (πœ‘ β†’ 𝑇 βŠ† (π‘β€˜π‘ˆ))    &   (πœ‘ β†’ 𝐴 ∈ 𝑇)    &   (πœ‘ β†’ 𝐢 ∈ 𝑇)    &   (πœ‘ β†’ 𝐡 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝐷 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ ((𝐴 + 𝐡) = (𝐢 + 𝐷) ↔ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷)))
 
Theorempj1fval 19561* 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.)
𝐡 = (Baseβ€˜πΊ)    &    + = (+gβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    &   π‘ƒ = (proj1β€˜πΊ)    β‡’   ((𝐺 ∈ 𝑉 ∧ 𝑇 βŠ† 𝐡 ∧ π‘ˆ βŠ† 𝐡) β†’ (π‘‡π‘ƒπ‘ˆ) = (𝑧 ∈ (𝑇 βŠ• π‘ˆ) ↦ (β„©π‘₯ ∈ 𝑇 βˆƒπ‘¦ ∈ π‘ˆ 𝑧 = (π‘₯ + 𝑦))))
 
Theorempj1val 19562* 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.)
𝐡 = (Baseβ€˜πΊ)    &    + = (+gβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    &   π‘ƒ = (proj1β€˜πΊ)    β‡’   (((𝐺 ∈ 𝑉 ∧ 𝑇 βŠ† 𝐡 ∧ π‘ˆ βŠ† 𝐡) ∧ 𝑋 ∈ (𝑇 βŠ• π‘ˆ)) β†’ ((π‘‡π‘ƒπ‘ˆ)β€˜π‘‹) = (β„©π‘₯ ∈ 𝑇 βˆƒπ‘¦ ∈ π‘ˆ 𝑋 = (π‘₯ + 𝑦)))
 
Theorempj1eu 19563* Uniqueness of a left projection. (Contributed by Mario Carneiro, 15-Oct-2015.)
+ = (+gβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &   π‘ = (Cntzβ€˜πΊ)    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ (𝑇 ∩ π‘ˆ) = { 0 })    &   (πœ‘ β†’ 𝑇 βŠ† (π‘β€˜π‘ˆ))    β‡’   ((πœ‘ ∧ 𝑋 ∈ (𝑇 βŠ• π‘ˆ)) β†’ βˆƒ!π‘₯ ∈ 𝑇 βˆƒπ‘¦ ∈ π‘ˆ 𝑋 = (π‘₯ + 𝑦))
 
Theorempj1f 19564 The left projection function maps a direct subspace sum onto the left factor. (Contributed by Mario Carneiro, 15-Oct-2015.)
+ = (+gβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &   π‘ = (Cntzβ€˜πΊ)    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ (𝑇 ∩ π‘ˆ) = { 0 })    &   (πœ‘ β†’ 𝑇 βŠ† (π‘β€˜π‘ˆ))    &   π‘ƒ = (proj1β€˜πΊ)    β‡’   (πœ‘ β†’ (π‘‡π‘ƒπ‘ˆ):(𝑇 βŠ• π‘ˆ)βŸΆπ‘‡)
 
Theorempj2f 19565 The right projection function maps a direct subspace sum onto the right factor. (Contributed by Mario Carneiro, 15-Oct-2015.)
+ = (+gβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &   π‘ = (Cntzβ€˜πΊ)    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ (𝑇 ∩ π‘ˆ) = { 0 })    &   (πœ‘ β†’ 𝑇 βŠ† (π‘β€˜π‘ˆ))    &   π‘ƒ = (proj1β€˜πΊ)    β‡’   (πœ‘ β†’ (π‘ˆπ‘ƒπ‘‡):(𝑇 βŠ• π‘ˆ)βŸΆπ‘ˆ)
 
Theorempj1id 19566 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.)
+ = (+gβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &   π‘ = (Cntzβ€˜πΊ)    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ (𝑇 ∩ π‘ˆ) = { 0 })    &   (πœ‘ β†’ 𝑇 βŠ† (π‘β€˜π‘ˆ))    &   π‘ƒ = (proj1β€˜πΊ)    β‡’   ((πœ‘ ∧ 𝑋 ∈ (𝑇 βŠ• π‘ˆ)) β†’ 𝑋 = (((π‘‡π‘ƒπ‘ˆ)β€˜π‘‹) + ((π‘ˆπ‘ƒπ‘‡)β€˜π‘‹)))
 
Theorempj1eq 19567 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.)
+ = (+gβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &   π‘ = (Cntzβ€˜πΊ)    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ (𝑇 ∩ π‘ˆ) = { 0 })    &   (πœ‘ β†’ 𝑇 βŠ† (π‘β€˜π‘ˆ))    &   π‘ƒ = (proj1β€˜πΊ)    &   (πœ‘ β†’ 𝑋 ∈ (𝑇 βŠ• π‘ˆ))    &   (πœ‘ β†’ 𝐡 ∈ 𝑇)    &   (πœ‘ β†’ 𝐢 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ (𝑋 = (𝐡 + 𝐢) ↔ (((π‘‡π‘ƒπ‘ˆ)β€˜π‘‹) = 𝐡 ∧ ((π‘ˆπ‘ƒπ‘‡)β€˜π‘‹) = 𝐢)))
 
Theorempj1lid 19568 The left projection function is the identity on the left subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)
+ = (+gβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &   π‘ = (Cntzβ€˜πΊ)    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ (𝑇 ∩ π‘ˆ) = { 0 })    &   (πœ‘ β†’ 𝑇 βŠ† (π‘β€˜π‘ˆ))    &   π‘ƒ = (proj1β€˜πΊ)    β‡’   ((πœ‘ ∧ 𝑋 ∈ 𝑇) β†’ ((π‘‡π‘ƒπ‘ˆ)β€˜π‘‹) = 𝑋)
 
Theorempj1rid 19569 The left projection function is the zero operator on the right subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)
+ = (+gβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &   π‘ = (Cntzβ€˜πΊ)    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ (𝑇 ∩ π‘ˆ) = { 0 })    &   (πœ‘ β†’ 𝑇 βŠ† (π‘β€˜π‘ˆ))    &   π‘ƒ = (proj1β€˜πΊ)    β‡’   ((πœ‘ ∧ 𝑋 ∈ π‘ˆ) β†’ ((π‘‡π‘ƒπ‘ˆ)β€˜π‘‹) = 0 )
 
Theorempj1ghm 19570 The left projection function is a group homomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
+ = (+gβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &   π‘ = (Cntzβ€˜πΊ)    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ (𝑇 ∩ π‘ˆ) = { 0 })    &   (πœ‘ β†’ 𝑇 βŠ† (π‘β€˜π‘ˆ))    &   π‘ƒ = (proj1β€˜πΊ)    β‡’   (πœ‘ β†’ (π‘‡π‘ƒπ‘ˆ) ∈ ((𝐺 β†Ύs (𝑇 βŠ• π‘ˆ)) GrpHom 𝐺))
 
Theorempj1ghm2 19571 The left projection function is a group homomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
+ = (+gβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &   π‘ = (Cntzβ€˜πΊ)    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ (𝑇 ∩ π‘ˆ) = { 0 })    &   (πœ‘ β†’ 𝑇 βŠ† (π‘β€˜π‘ˆ))    &   π‘ƒ = (proj1β€˜πΊ)    β‡’   (πœ‘ β†’ (π‘‡π‘ƒπ‘ˆ) ∈ ((𝐺 β†Ύs (𝑇 βŠ• π‘ˆ)) GrpHom (𝐺 β†Ύs 𝑇)))
 
Theoremlsmhash 19572 The order of the direct product of groups. (Contributed by Mario Carneiro, 21-Apr-2016.)
βŠ• = (LSSumβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &   π‘ = (Cntzβ€˜πΊ)    &   (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ (𝑇 ∩ π‘ˆ) = { 0 })    &   (πœ‘ β†’ 𝑇 βŠ† (π‘β€˜π‘ˆ))    &   (πœ‘ β†’ 𝑇 ∈ Fin)    &   (πœ‘ β†’ π‘ˆ ∈ Fin)    β‡’   (πœ‘ β†’ (β™―β€˜(𝑇 βŠ• π‘ˆ)) = ((β™―β€˜π‘‡) Β· (β™―β€˜π‘ˆ)))
 
10.2.13  Free groups
 
Syntaxcefg 19573 Extend class notation with the free group equivalence relation.
class ~FG
 
Syntaxcfrgp 19574 Extend class notation with the free group construction.
class freeGrp
 
Syntaxcvrgp 19575 Extend class notation with free group injection.
class varFGrp
 
Definitiondf-efg 19576* Define the free group equivalence relation, which is the smallest equivalence relation β‰ˆ such that for any words 𝐴, 𝐡 and formal symbol π‘₯ with inverse invgπ‘₯, 𝐴𝐡 β‰ˆ 𝐴π‘₯(invgπ‘₯)𝐡. (Contributed by Mario Carneiro, 1-Oct-2015.)
~FG = (𝑖 ∈ V ↦ ∩ {π‘Ÿ ∣ (π‘Ÿ Er Word (𝑖 Γ— 2o) ∧ βˆ€π‘₯ ∈ Word (𝑖 Γ— 2o)βˆ€π‘› ∈ (0...(β™―β€˜π‘₯))βˆ€π‘¦ ∈ 𝑖 βˆ€π‘§ ∈ 2o π‘₯π‘Ÿ(π‘₯ splice βŸ¨π‘›, 𝑛, βŸ¨β€œβŸ¨π‘¦, π‘§βŸ©βŸ¨π‘¦, (1o βˆ– 𝑧)βŸ©β€βŸ©βŸ©))})
 
Definitiondf-frgp 19577 Define the free group on a set 𝐼 of generators, defined as the quotient of the free monoid on 𝐼 Γ— 2o (representing the generator elements and their formal inverses) by the free group equivalence relation df-efg 19576. (Contributed by Mario Carneiro, 1-Oct-2015.)
freeGrp = (𝑖 ∈ V ↦ ((freeMndβ€˜(𝑖 Γ— 2o)) /s ( ~FG β€˜π‘–)))
 
Definitiondf-vrgp 19578* 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 ↦ (𝑗 ∈ 𝑖 ↦ [βŸ¨β€œβŸ¨π‘—, βˆ…βŸ©β€βŸ©]( ~FG β€˜π‘–)))
 
Theoremefgmval 19579* Value of the formal inverse operation for the generating set of a free group. (Contributed by Mario Carneiro, 27-Sep-2015.)
𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ βŸ¨π‘¦, (1o βˆ– 𝑧)⟩)    β‡’   ((𝐴 ∈ 𝐼 ∧ 𝐡 ∈ 2o) β†’ (𝐴𝑀𝐡) = ⟨𝐴, (1o βˆ– 𝐡)⟩)
 
Theoremefgmf 19580* The formal inverse operation is an endofunction on the generating set. (Contributed by Mario Carneiro, 27-Sep-2015.)
𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ βŸ¨π‘¦, (1o βˆ– 𝑧)⟩)    β‡’   π‘€:(𝐼 Γ— 2o)⟢(𝐼 Γ— 2o)
 
Theoremefgmnvl 19581* The inversion function on the generators is an involution. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ βŸ¨π‘¦, (1o βˆ– 𝑧)⟩)    β‡’   (𝐴 ∈ (𝐼 Γ— 2o) β†’ (π‘€β€˜(π‘€β€˜π΄)) = 𝐴)
 
Theoremefgrcl 19582 Lemma for efgval 19584. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
π‘Š = ( I β€˜Word (𝐼 Γ— 2o))    β‡’   (𝐴 ∈ π‘Š β†’ (𝐼 ∈ V ∧ π‘Š = Word (𝐼 Γ— 2o)))
 
Theoremefglem 19583* Lemma for efgval 19584. (Contributed by Mario Carneiro, 27-Sep-2015.)
π‘Š = ( I β€˜Word (𝐼 Γ— 2o))    β‡’   βˆƒπ‘Ÿ(π‘Ÿ Er π‘Š ∧ βˆ€π‘₯ ∈ π‘Š βˆ€π‘› ∈ (0...(β™―β€˜π‘₯))βˆ€π‘¦ ∈ 𝐼 βˆ€π‘§ ∈ 2o π‘₯π‘Ÿ(π‘₯ splice βŸ¨π‘›, 𝑛, βŸ¨β€œβŸ¨π‘¦, π‘§βŸ©βŸ¨π‘¦, (1o βˆ– 𝑧)βŸ©β€βŸ©βŸ©))
 
Theoremefgval 19584* Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
π‘Š = ( I β€˜Word (𝐼 Γ— 2o))    &    ∼ = ( ~FG β€˜πΌ)    β‡’    ∼ = ∩ {π‘Ÿ ∣ (π‘Ÿ Er π‘Š ∧ βˆ€π‘₯ ∈ π‘Š βˆ€π‘› ∈ (0...(β™―β€˜π‘₯))βˆ€π‘¦ ∈ 𝐼 βˆ€π‘§ ∈ 2o π‘₯π‘Ÿ(π‘₯ splice βŸ¨π‘›, 𝑛, βŸ¨β€œβŸ¨π‘¦, π‘§βŸ©βŸ¨π‘¦, (1o βˆ– 𝑧)βŸ©β€βŸ©βŸ©))}
 
Theoremefger 19585 Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
π‘Š = ( I β€˜Word (𝐼 Γ— 2o))    &    ∼ = ( ~FG β€˜πΌ)    β‡’    ∼ Er π‘Š
 
Theoremefgi 19586 Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
π‘Š = ( I β€˜Word (𝐼 Γ— 2o))    &    ∼ = ( ~FG β€˜πΌ)    β‡’   (((𝐴 ∈ π‘Š ∧ 𝑁 ∈ (0...(β™―β€˜π΄))) ∧ (𝐽 ∈ 𝐼 ∧ 𝐾 ∈ 2o)) β†’ 𝐴 ∼ (𝐴 splice βŸ¨π‘, 𝑁, βŸ¨β€œβŸ¨π½, 𝐾⟩⟨𝐽, (1o βˆ– 𝐾)βŸ©β€βŸ©βŸ©))
 
Theoremefgi0 19587 Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
π‘Š = ( I β€˜Word (𝐼 Γ— 2o))    &    ∼ = ( ~FG β€˜πΌ)    β‡’   ((𝐴 ∈ π‘Š ∧ 𝑁 ∈ (0...(β™―β€˜π΄)) ∧ 𝐽 ∈ 𝐼) β†’ 𝐴 ∼ (𝐴 splice βŸ¨π‘, 𝑁, βŸ¨β€œβŸ¨π½, βˆ…βŸ©βŸ¨π½, 1oβŸ©β€βŸ©βŸ©))
 
Theoremefgi1 19588 Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
π‘Š = ( I β€˜Word (𝐼 Γ— 2o))    &    ∼ = ( ~FG β€˜πΌ)    β‡’   ((𝐴 ∈ π‘Š ∧ 𝑁 ∈ (0...(β™―β€˜π΄)) ∧ 𝐽 ∈ 𝐼) β†’ 𝐴 ∼ (𝐴 splice βŸ¨π‘, 𝑁, βŸ¨β€œβŸ¨π½, 1o⟩⟨𝐽, βˆ…βŸ©β€βŸ©βŸ©))
 
Theoremefgtf 19589* Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
π‘Š = ( I β€˜Word (𝐼 Γ— 2o))    &    ∼ = ( ~FG β€˜πΌ)    &   π‘€ = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ βŸ¨π‘¦, (1o βˆ– 𝑧)⟩)    &   π‘‡ = (𝑣 ∈ π‘Š ↦ (𝑛 ∈ (0...(β™―β€˜π‘£)), 𝑀 ∈ (𝐼 Γ— 2o) ↦ (𝑣 splice βŸ¨π‘›, 𝑛, βŸ¨β€œπ‘€(π‘€β€˜π‘€)β€βŸ©βŸ©)))    β‡’   (𝑋 ∈ π‘Š β†’ ((π‘‡β€˜π‘‹) = (π‘Ž ∈ (0...(β™―β€˜π‘‹)), 𝑏 ∈ (𝐼 Γ— 2o) ↦ (𝑋 splice βŸ¨π‘Ž, π‘Ž, βŸ¨β€œπ‘(π‘€β€˜π‘)β€βŸ©βŸ©)) ∧ (π‘‡β€˜π‘‹):((0...(β™―β€˜π‘‹)) Γ— (𝐼 Γ— 2o))βŸΆπ‘Š))
 
Theoremefgtval 19590* 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.)
π‘Š = ( I β€˜Word (𝐼 Γ— 2o))    &    ∼ = ( ~FG β€˜πΌ)    &   π‘€ = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ βŸ¨π‘¦, (1o βˆ– 𝑧)⟩)    &   π‘‡ = (𝑣 ∈ π‘Š ↦ (𝑛 ∈ (0...(β™―β€˜π‘£)), 𝑀 ∈ (𝐼 Γ— 2o) ↦ (𝑣 splice βŸ¨π‘›, 𝑛, βŸ¨β€œπ‘€(π‘€β€˜π‘€)β€βŸ©βŸ©)))    β‡’   ((𝑋 ∈ π‘Š ∧ 𝑁 ∈ (0...(β™―β€˜π‘‹)) ∧ 𝐴 ∈ (𝐼 Γ— 2o)) β†’ (𝑁(π‘‡β€˜π‘‹)𝐴) = (𝑋 splice βŸ¨π‘, 𝑁, βŸ¨β€œπ΄(π‘€β€˜π΄)β€βŸ©βŸ©))
 
Theoremefgval2 19591* Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
π‘Š = ( I β€˜Word (𝐼 Γ— 2o))    &    ∼ = ( ~FG β€˜πΌ)    &   π‘€ = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ βŸ¨π‘¦, (1o βˆ– 𝑧)⟩)    &   π‘‡ = (𝑣 ∈ π‘Š ↦ (𝑛 ∈ (0...(β™―β€˜π‘£)), 𝑀 ∈ (𝐼 Γ— 2o) ↦ (𝑣 splice βŸ¨π‘›, 𝑛, βŸ¨β€œπ‘€(π‘€β€˜π‘€)β€βŸ©βŸ©)))    β‡’    ∼ = ∩ {π‘Ÿ ∣ (π‘Ÿ Er π‘Š ∧ βˆ€π‘₯ ∈ π‘Š ran (π‘‡β€˜π‘₯) βŠ† [π‘₯]π‘Ÿ)}
 
Theoremefgi2 19592* Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
π‘Š = ( I β€˜Word (𝐼 Γ— 2o))    &    ∼ = ( ~FG β€˜πΌ)    &   π‘€ = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ βŸ¨π‘¦, (1o βˆ– 𝑧)⟩)    &   π‘‡ = (𝑣 ∈ π‘Š ↦ (𝑛 ∈ (0...(β™―β€˜π‘£)), 𝑀 ∈ (𝐼 Γ— 2o) ↦ (𝑣 splice βŸ¨π‘›, 𝑛, βŸ¨β€œπ‘€(π‘€β€˜π‘€)β€βŸ©βŸ©)))    β‡’   ((𝐴 ∈ π‘Š ∧ 𝐡 ∈ ran (π‘‡β€˜π΄)) β†’ 𝐴 ∼ 𝐡)
 
Theoremefgtlen 19593* Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
π‘Š = ( I β€˜Word (𝐼 Γ— 2o))    &    ∼ = ( ~FG β€˜πΌ)    &   π‘€ = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ βŸ¨π‘¦, (1o βˆ– 𝑧)⟩)    &   π‘‡ = (𝑣 ∈ π‘Š ↦ (𝑛 ∈ (0...(β™―β€˜π‘£)), 𝑀 ∈ (𝐼 Γ— 2o) ↦ (𝑣 splice βŸ¨π‘›, 𝑛, βŸ¨β€œπ‘€(π‘€β€˜π‘€)β€βŸ©βŸ©)))    β‡’   ((𝑋 ∈ π‘Š ∧ 𝐴 ∈ ran (π‘‡β€˜π‘‹)) β†’ (β™―β€˜π΄) = ((β™―β€˜π‘‹) + 2))
 
Theoremefginvrel2 19594* 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.)
π‘Š = ( I β€˜Word (𝐼 Γ— 2o))    &    ∼ = ( ~FG β€˜πΌ)    &   π‘€ = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ βŸ¨π‘¦, (1o βˆ– 𝑧)⟩)    &   π‘‡ = (𝑣 ∈ π‘Š ↦ (𝑛 ∈ (0...(β™―β€˜π‘£)), 𝑀 ∈ (𝐼 Γ— 2o) ↦ (𝑣 splice βŸ¨π‘›, 𝑛, βŸ¨β€œπ‘€(π‘€β€˜π‘€)β€βŸ©βŸ©)))    β‡’   (𝐴 ∈ π‘Š β†’ (𝐴 ++ (𝑀 ∘ (reverseβ€˜π΄))) ∼ βˆ…)
 
Theoremefginvrel1 19595* 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.)
π‘Š = ( I β€˜Word (𝐼 Γ— 2o))    &    ∼ = ( ~FG β€˜πΌ)    &   π‘€ = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ βŸ¨π‘¦, (1o βˆ– 𝑧)⟩)    &   π‘‡ = (𝑣 ∈ π‘Š ↦ (𝑛 ∈ (0...(β™―β€˜π‘£)), 𝑀 ∈ (𝐼 Γ— 2o) ↦ (𝑣 splice βŸ¨π‘›, 𝑛, βŸ¨β€œπ‘€(π‘€β€˜π‘€)β€βŸ©βŸ©)))    β‡’   (𝐴 ∈ π‘Š β†’ ((𝑀 ∘ (reverseβ€˜π΄)) ++ 𝐴) ∼ βˆ…)
 
Theoremefgsf 19596* Value of the auxiliary function 𝑆 defining a sequence of extensions starting at some irreducible word. (Contributed by Mario Carneiro, 1-Oct-2015.)
π‘Š = ( I β€˜Word (𝐼 Γ— 2o))    &    ∼ = ( ~FG β€˜πΌ)    &   π‘€ = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ βŸ¨π‘¦, (1o βˆ– 𝑧)⟩)    &   π‘‡ = (𝑣 ∈ π‘Š ↦ (𝑛 ∈ (0...(β™―β€˜π‘£)), 𝑀 ∈ (𝐼 Γ— 2o) ↦ (𝑣 splice βŸ¨π‘›, 𝑛, βŸ¨β€œπ‘€(π‘€β€˜π‘€)β€βŸ©βŸ©)))    &   π· = (π‘Š βˆ– βˆͺ π‘₯ ∈ π‘Š ran (π‘‡β€˜π‘₯))    &   π‘† = (π‘š ∈ {𝑑 ∈ (Word π‘Š βˆ– {βˆ…}) ∣ ((π‘‘β€˜0) ∈ 𝐷 ∧ βˆ€π‘˜ ∈ (1..^(β™―β€˜π‘‘))(π‘‘β€˜π‘˜) ∈ ran (π‘‡β€˜(π‘‘β€˜(π‘˜ βˆ’ 1))))} ↦ (π‘šβ€˜((β™―β€˜π‘š) βˆ’ 1)))    β‡’   π‘†:{𝑑 ∈ (Word π‘Š βˆ– {βˆ…}) ∣ ((π‘‘β€˜0) ∈ 𝐷 ∧ βˆ€π‘˜ ∈ (1..^(β™―β€˜π‘‘))(π‘‘β€˜π‘˜) ∈ ran (π‘‡β€˜(π‘‘β€˜(π‘˜ βˆ’ 1))))}βŸΆπ‘Š
 
Theoremefgsdm 19597* Elementhood in the domain of 𝑆, the set of sequences of extensions starting at an irreducible word. (Contributed by Mario Carneiro, 27-Sep-2015.)
π‘Š = ( I β€˜Word (𝐼 Γ— 2o))    &    ∼ = ( ~FG β€˜πΌ)    &   π‘€ = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ βŸ¨π‘¦, (1o βˆ– 𝑧)⟩)    &   π‘‡ = (𝑣 ∈ π‘Š ↦ (𝑛 ∈ (0...(β™―β€˜π‘£)), 𝑀 ∈ (𝐼 Γ— 2o) ↦ (𝑣 splice βŸ¨π‘›, 𝑛, βŸ¨β€œπ‘€(π‘€β€˜π‘€)β€βŸ©βŸ©)))    &   π· = (π‘Š βˆ– βˆͺ π‘₯ ∈ π‘Š ran (π‘‡β€˜π‘₯))    &   π‘† = (π‘š ∈ {𝑑 ∈ (Word π‘Š βˆ– {βˆ…}) ∣ ((π‘‘β€˜0) ∈ 𝐷 ∧ βˆ€π‘˜ ∈ (1..^(β™―β€˜π‘‘))(π‘‘β€˜π‘˜) ∈ ran (π‘‡β€˜(π‘‘β€˜(π‘˜ βˆ’ 1))))} ↦ (π‘šβ€˜((β™―β€˜π‘š) βˆ’ 1)))    β‡’   (𝐹 ∈ dom 𝑆 ↔ (𝐹 ∈ (Word π‘Š βˆ– {βˆ…}) ∧ (πΉβ€˜0) ∈ 𝐷 ∧ βˆ€π‘– ∈ (1..^(β™―β€˜πΉ))(πΉβ€˜π‘–) ∈ ran (π‘‡β€˜(πΉβ€˜(𝑖 βˆ’ 1)))))
 
Theoremefgsval 19598* Value of the auxiliary function 𝑆 defining a sequence of extensions. (Contributed by Mario Carneiro, 27-Sep-2015.)
π‘Š = ( I β€˜Word (𝐼 Γ— 2o))    &    ∼ = ( ~FG β€˜πΌ)    &   π‘€ = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ βŸ¨π‘¦, (1o βˆ– 𝑧)⟩)    &   π‘‡ = (𝑣 ∈ π‘Š ↦ (𝑛 ∈ (0...(β™―β€˜π‘£)), 𝑀 ∈ (𝐼 Γ— 2o) ↦ (𝑣 splice βŸ¨π‘›, 𝑛, βŸ¨β€œπ‘€(π‘€β€˜π‘€)β€βŸ©βŸ©)))    &   π· = (π‘Š βˆ– βˆͺ π‘₯ ∈ π‘Š ran (π‘‡β€˜π‘₯))    &   π‘† = (π‘š ∈ {𝑑 ∈ (Word π‘Š βˆ– {βˆ…}) ∣ ((π‘‘β€˜0) ∈ 𝐷 ∧ βˆ€π‘˜ ∈ (1..^(β™―β€˜π‘‘))(π‘‘β€˜π‘˜) ∈ ran (π‘‡β€˜(π‘‘β€˜(π‘˜ βˆ’ 1))))} ↦ (π‘šβ€˜((β™―β€˜π‘š) βˆ’ 1)))    β‡’   (𝐹 ∈ dom 𝑆 β†’ (π‘†β€˜πΉ) = (πΉβ€˜((β™―β€˜πΉ) βˆ’ 1)))
 
Theoremefgsdmi 19599* Property of the last link in the chain of extensions. (Contributed by Mario Carneiro, 29-Sep-2015.)
π‘Š = ( I β€˜Word (𝐼 Γ— 2o))    &    ∼ = ( ~FG β€˜πΌ)    &   π‘€ = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ βŸ¨π‘¦, (1o βˆ– 𝑧)⟩)    &   π‘‡ = (𝑣 ∈ π‘Š ↦ (𝑛 ∈ (0...(β™―β€˜π‘£)), 𝑀 ∈ (𝐼 Γ— 2o) ↦ (𝑣 splice βŸ¨π‘›, 𝑛, βŸ¨β€œπ‘€(π‘€β€˜π‘€)β€βŸ©βŸ©)))    &   π· = (π‘Š βˆ– βˆͺ π‘₯ ∈ π‘Š ran (π‘‡β€˜π‘₯))    &   π‘† = (π‘š ∈ {𝑑 ∈ (Word π‘Š βˆ– {βˆ…}) ∣ ((π‘‘β€˜0) ∈ 𝐷 ∧ βˆ€π‘˜ ∈ (1..^(β™―β€˜π‘‘))(π‘‘β€˜π‘˜) ∈ ran (π‘‡β€˜(π‘‘β€˜(π‘˜ βˆ’ 1))))} ↦ (π‘šβ€˜((β™―β€˜π‘š) βˆ’ 1)))    β‡’   ((𝐹 ∈ dom 𝑆 ∧ ((β™―β€˜πΉ) βˆ’ 1) ∈ β„•) β†’ (π‘†β€˜πΉ) ∈ ran (π‘‡β€˜(πΉβ€˜(((β™―β€˜πΉ) βˆ’ 1) βˆ’ 1))))
 
Theoremefgsval2 19600* Value of the auxiliary function 𝑆 defining a sequence of extensions. (Contributed by Mario Carneiro, 1-Oct-2015.)
π‘Š = ( I β€˜Word (𝐼 Γ— 2o))    &    ∼ = ( ~FG β€˜πΌ)    &   π‘€ = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ βŸ¨π‘¦, (1o βˆ– 𝑧)⟩)    &   π‘‡ = (𝑣 ∈ π‘Š ↦ (𝑛 ∈ (0...(β™―β€˜π‘£)), 𝑀 ∈ (𝐼 Γ— 2o) ↦ (𝑣 splice βŸ¨π‘›, 𝑛, βŸ¨β€œπ‘€(π‘€β€˜π‘€)β€βŸ©βŸ©)))    &   π· = (π‘Š βˆ– βˆͺ π‘₯ ∈ π‘Š ran (π‘‡β€˜π‘₯))    &   π‘† = (π‘š ∈ {𝑑 ∈ (Word π‘Š βˆ– {βˆ…}) ∣ ((π‘‘β€˜0) ∈ 𝐷 ∧ βˆ€π‘˜ ∈ (1..^(β™―β€˜π‘‘))(π‘‘β€˜π‘˜) ∈ ran (π‘‡β€˜(π‘‘β€˜(π‘˜ βˆ’ 1))))} ↦ (π‘šβ€˜((β™―β€˜π‘š) βˆ’ 1)))    β‡’   ((𝐴 ∈ Word π‘Š ∧ 𝐡 ∈ π‘Š ∧ (𝐴 ++ βŸ¨β€œπ΅β€βŸ©) ∈ dom 𝑆) β†’ (π‘†β€˜(𝐴 ++ βŸ¨β€œπ΅β€βŸ©)) = 𝐡)
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