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

Theorempj1fval 18101* 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 18102* 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 18103* 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 18104 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 18105 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 18106 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 18107 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 18108 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 18109 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 18110 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 18111 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 18112 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 18113 Extend class notation with the free group equivalence relation.
class ~FG

Syntaxcfrgp 18114 Extend class notation with the free group construction.
class freeGrp

Syntaxcvrgp 18115 Extend class notation with free group injection.
class varFGrp

Definitiondf-efg 18116* 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 18117 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 18116. (Contributed by Mario Carneiro, 1-Oct-2015.)
freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2𝑜)) /s ( ~FG𝑖)))

Definitiondf-vrgp 18118* 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 18119* 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 18120* 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 18121* 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 18122 Lemma for efgval 18124. (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 18123* Lemma for efgval 18124. (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 18124* 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 18125 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 18126 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 18127 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 18128 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 18129* 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 18130* 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 18131* 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 18132* 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 18133* 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 18134* 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 18135* 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 18136* 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 18137* 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 18138* 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 18139* 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 18140* 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 18141* 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 18142* 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 18143* 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 18144* 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 18145* 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 18146* 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 18147* The reduced word that forms the base of the sequence in efgsval 18138 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 18148* Lemma for efgredleme 18150. (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 18149* Lemma for efgred 18155. (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 18150* Lemma for efgred 18155. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   (𝜑 → ∀𝑎 ∈ dom 𝑆𝑏 ∈ dom 𝑆((#‘(𝑆𝑎)) < (#‘(𝑆𝐴)) → ((𝑆𝑎) = (𝑆𝑏) → (𝑎‘0) = (𝑏‘0))))    &   (𝜑𝐴 ∈ dom 𝑆)    &   (𝜑𝐵 ∈ dom 𝑆)    &   (𝜑 → (𝑆𝐴) = (𝑆𝐵))    &   (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))    &   𝐾 = (((#‘𝐴) − 1) − 1)    &   𝐿 = (((#‘𝐵) − 1) − 1)    &   (𝜑𝑃 ∈ (0...(#‘(𝐴𝐾))))    &   (𝜑𝑄 ∈ (0...(#‘(𝐵𝐿))))    &   (𝜑𝑈 ∈ (𝐼 × 2𝑜))    &   (𝜑𝑉 ∈ (𝐼 × 2𝑜))    &   (𝜑 → (𝑆𝐴) = (𝑃(𝑇‘(𝐴𝐾))𝑈))    &   (𝜑 → (𝑆𝐵) = (𝑄(𝑇‘(𝐵𝐿))𝑉))    &   (𝜑 → ¬ (𝐴𝐾) = (𝐵𝐿))    &   (𝜑𝑃 ∈ (ℤ‘(𝑄 + 2)))    &   (𝜑𝐶 ∈ dom 𝑆)    &   (𝜑 → (𝑆𝐶) = (((𝐵𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴𝐾))⟩)))       (𝜑 → ((𝐴𝐾) ∈ ran (𝑇‘(𝑆𝐶)) ∧ (𝐵𝐿) ∈ ran (𝑇‘(𝑆𝐶))))

Theoremefgredlemd 18151* The reduced word that forms the base of the sequence in efgsval 18138 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   (𝜑 → ∀𝑎 ∈ dom 𝑆𝑏 ∈ dom 𝑆((#‘(𝑆𝑎)) < (#‘(𝑆𝐴)) → ((𝑆𝑎) = (𝑆𝑏) → (𝑎‘0) = (𝑏‘0))))    &   (𝜑𝐴 ∈ dom 𝑆)    &   (𝜑𝐵 ∈ dom 𝑆)    &   (𝜑 → (𝑆𝐴) = (𝑆𝐵))    &   (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))    &   𝐾 = (((#‘𝐴) − 1) − 1)    &   𝐿 = (((#‘𝐵) − 1) − 1)    &   (𝜑𝑃 ∈ (0...(#‘(𝐴𝐾))))    &   (𝜑𝑄 ∈ (0...(#‘(𝐵𝐿))))    &   (𝜑𝑈 ∈ (𝐼 × 2𝑜))    &   (𝜑𝑉 ∈ (𝐼 × 2𝑜))    &   (𝜑 → (𝑆𝐴) = (𝑃(𝑇‘(𝐴𝐾))𝑈))    &   (𝜑 → (𝑆𝐵) = (𝑄(𝑇‘(𝐵𝐿))𝑉))    &   (𝜑 → ¬ (𝐴𝐾) = (𝐵𝐿))    &   (𝜑𝑃 ∈ (ℤ‘(𝑄 + 2)))    &   (𝜑𝐶 ∈ dom 𝑆)    &   (𝜑 → (𝑆𝐶) = (((𝐵𝐿) substr ⟨0, 𝑄⟩) ++ ((𝐴𝐾) substr ⟨(𝑄 + 2), (#‘(𝐴𝐾))⟩)))       (𝜑 → (𝐴‘0) = (𝐵‘0))

Theoremefgredlemc 18152* The reduced word that forms the base of the sequence in efgsval 18138 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   (𝜑 → ∀𝑎 ∈ dom 𝑆𝑏 ∈ dom 𝑆((#‘(𝑆𝑎)) < (#‘(𝑆𝐴)) → ((𝑆𝑎) = (𝑆𝑏) → (𝑎‘0) = (𝑏‘0))))    &   (𝜑𝐴 ∈ dom 𝑆)    &   (𝜑𝐵 ∈ dom 𝑆)    &   (𝜑 → (𝑆𝐴) = (𝑆𝐵))    &   (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))    &   𝐾 = (((#‘𝐴) − 1) − 1)    &   𝐿 = (((#‘𝐵) − 1) − 1)    &   (𝜑𝑃 ∈ (0...(#‘(𝐴𝐾))))    &   (𝜑𝑄 ∈ (0...(#‘(𝐵𝐿))))    &   (𝜑𝑈 ∈ (𝐼 × 2𝑜))    &   (𝜑𝑉 ∈ (𝐼 × 2𝑜))    &   (𝜑 → (𝑆𝐴) = (𝑃(𝑇‘(𝐴𝐾))𝑈))    &   (𝜑 → (𝑆𝐵) = (𝑄(𝑇‘(𝐵𝐿))𝑉))    &   (𝜑 → ¬ (𝐴𝐾) = (𝐵𝐿))       (𝜑 → (𝑃 ∈ (ℤ𝑄) → (𝐴‘0) = (𝐵‘0)))

Theoremefgredlemb 18153* The reduced word that forms the base of the sequence in efgsval 18138 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 30-Sep-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   (𝜑 → ∀𝑎 ∈ dom 𝑆𝑏 ∈ dom 𝑆((#‘(𝑆𝑎)) < (#‘(𝑆𝐴)) → ((𝑆𝑎) = (𝑆𝑏) → (𝑎‘0) = (𝑏‘0))))    &   (𝜑𝐴 ∈ dom 𝑆)    &   (𝜑𝐵 ∈ dom 𝑆)    &   (𝜑 → (𝑆𝐴) = (𝑆𝐵))    &   (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))    &   𝐾 = (((#‘𝐴) − 1) − 1)    &   𝐿 = (((#‘𝐵) − 1) − 1)    &   (𝜑𝑃 ∈ (0...(#‘(𝐴𝐾))))    &   (𝜑𝑄 ∈ (0...(#‘(𝐵𝐿))))    &   (𝜑𝑈 ∈ (𝐼 × 2𝑜))    &   (𝜑𝑉 ∈ (𝐼 × 2𝑜))    &   (𝜑 → (𝑆𝐴) = (𝑃(𝑇‘(𝐴𝐾))𝑈))    &   (𝜑 → (𝑆𝐵) = (𝑄(𝑇‘(𝐵𝐿))𝑉))    &   (𝜑 → ¬ (𝐴𝐾) = (𝐵𝐿))        ¬ 𝜑

Theoremefgredlem 18154* The reduced word that forms the base of the sequence in efgsval 18138 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 30-Sep-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   (𝜑 → ∀𝑎 ∈ dom 𝑆𝑏 ∈ dom 𝑆((#‘(𝑆𝑎)) < (#‘(𝑆𝐴)) → ((𝑆𝑎) = (𝑆𝑏) → (𝑎‘0) = (𝑏‘0))))    &   (𝜑𝐴 ∈ dom 𝑆)    &   (𝜑𝐵 ∈ dom 𝑆)    &   (𝜑 → (𝑆𝐴) = (𝑆𝐵))    &   (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))        ¬ 𝜑

Theoremefgred 18155* The reduced word that forms the base of the sequence in efgsval 18138 is uniquely determined, given the terminal point. (Contributed by Mario Carneiro, 28-Sep-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       ((𝐴 ∈ dom 𝑆𝐵 ∈ dom 𝑆 ∧ (𝑆𝐴) = (𝑆𝐵)) → (𝐴‘0) = (𝐵‘0))

Theoremefgrelexlema 18156* If two words 𝐴, 𝐵 are related under the free group equivalence, then there exist two extension sequences 𝑎, 𝑏 such that 𝑎 ends at 𝐴, 𝑏 ends at 𝐵, and 𝑎 and 𝐵 have the same starting point. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   𝐿 = {⟨𝑖, 𝑗⟩ ∣ ∃𝑐 ∈ (𝑆 “ {𝑖})∃𝑑 ∈ (𝑆 “ {𝑗})(𝑐‘0) = (𝑑‘0)}       (𝐴𝐿𝐵 ↔ ∃𝑎 ∈ (𝑆 “ {𝐴})∃𝑏 ∈ (𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0))

Theoremefgrelexlemb 18157* If two words 𝐴, 𝐵 are related under the free group equivalence, then there exist two extension sequences 𝑎, 𝑏 such that 𝑎 ends at 𝐴, 𝑏 ends at 𝐵, and 𝑎 and 𝐵 have the same starting point. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   𝐿 = {⟨𝑖, 𝑗⟩ ∣ ∃𝑐 ∈ (𝑆 “ {𝑖})∃𝑑 ∈ (𝑆 “ {𝑗})(𝑐‘0) = (𝑑‘0)}        𝐿

Theoremefgrelex 18158* If two words 𝐴, 𝐵 are related under the free group equivalence, then there exist two extension sequences 𝑎, 𝑏 such that 𝑎 ends at 𝐴, 𝑏 ends at 𝐵, and 𝑎 and 𝐵 have the same starting point. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       (𝐴 𝐵 → ∃𝑎 ∈ (𝑆 “ {𝐴})∃𝑏 ∈ (𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0))

Theoremefgredeu 18159* There is a unique reduced word equivalent to a given word. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       (𝐴𝑊 → ∃!𝑑𝐷 𝑑 𝐴)

Theoremefgred2 18160* Two extension sequences have related endpoints iff they have the same base. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       ((𝐴 ∈ dom 𝑆𝐵 ∈ dom 𝑆) → ((𝑆𝐴) (𝑆𝐵) ↔ (𝐴‘0) = (𝐵‘0)))

Theoremefgcpbllema 18161* Lemma for efgrelex 18158. Define an auxiliary equivalence relation 𝐿 such that 𝐴𝐿𝐵 if there are sequences from 𝐴 to 𝐵 passing through the same reduced word. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   𝐿 = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑊 ∧ ((𝐴 ++ 𝑖) ++ 𝐵) ((𝐴 ++ 𝑗) ++ 𝐵))}       (𝑋𝐿𝑌 ↔ (𝑋𝑊𝑌𝑊 ∧ ((𝐴 ++ 𝑋) ++ 𝐵) ((𝐴 ++ 𝑌) ++ 𝐵)))

Theoremefgcpbllemb 18162* Lemma for efgrelex 18158. Show that 𝐿 is an equivalence relation containing all direct extensions of a word, so is closed under . (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))    &   𝐿 = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑊 ∧ ((𝐴 ++ 𝑖) ++ 𝐵) ((𝐴 ++ 𝑗) ++ 𝐵))}       ((𝐴𝑊𝐵𝑊) → 𝐿)

Theoremefgcpbl 18163* Two extension sequences have related endpoints iff they have the same base. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       ((𝐴𝑊𝐵𝑊𝑋 𝑌) → ((𝐴 ++ 𝑋) ++ 𝐵) ((𝐴 ++ 𝑌) ++ 𝐵))

Theoremefgcpbl2 18164* Two extension sequences have related endpoints iff they have the same base. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)    &   𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))    &   𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))    &   𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))       ((𝐴 𝑋𝐵 𝑌) → (𝐴 ++ 𝐵) (𝑋 ++ 𝑌))

Theoremfrgpval 18165 Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝐺 = (freeGrp‘𝐼)    &   𝑀 = (freeMnd‘(𝐼 × 2𝑜))    &    = ( ~FG𝐼)       (𝐼𝑉𝐺 = (𝑀 /s ))

Theoremfrgpcpbl 18166 Compatibility of the group operation with the free group equivalence relation. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
𝐺 = (freeGrp‘𝐼)    &   𝑀 = (freeMnd‘(𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &    + = (+g𝑀)       ((𝐴 𝐶𝐵 𝐷) → (𝐴 + 𝐵) (𝐶 + 𝐷))

Theoremfrgp0 18167 The free group is a group. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
𝐺 = (freeGrp‘𝐼)    &    = ( ~FG𝐼)       (𝐼𝑉 → (𝐺 ∈ Grp ∧ [∅] = (0g𝐺)))

Theoremfrgpeccl 18168 Closure of the quotient map in a free group. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝐺 = (freeGrp‘𝐼)    &    = ( ~FG𝐼)    &   𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &   𝐵 = (Base‘𝐺)       (𝑋𝑊 → [𝑋] 𝐵)

Theoremfrgpgrp 18169 The free group is a group. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝐺 = (freeGrp‘𝐼)       (𝐼𝑉𝐺 ∈ Grp)

Theoremfrgpadd 18170 Addition in the free group is given by concatenation. (Contributed by Mario Carneiro, 1-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &   𝐺 = (freeGrp‘𝐼)    &    = ( ~FG𝐼)    &    + = (+g𝐺)       ((𝐴𝑊𝐵𝑊) → ([𝐴] + [𝐵] ) = [(𝐴 ++ 𝐵)] )

Theoremfrgpinv 18171* The inverse of an element of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &   𝐺 = (freeGrp‘𝐼)    &    = ( ~FG𝐼)    &   𝑁 = (invg𝐺)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)       (𝐴𝑊 → (𝑁‘[𝐴] ) = [(𝑀 ∘ (reverse‘𝐴))] )

Theoremfrgpmhm 18172* The "natural map" from words of the free monoid to their cosets in the free group is a surjective monoid homomorphism. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑀 = (freeMnd‘(𝐼 × 2𝑜))    &   𝑊 = (Base‘𝑀)    &   𝐺 = (freeGrp‘𝐼)    &    = ( ~FG𝐼)    &   𝐹 = (𝑥𝑊 ↦ [𝑥] )       (𝐼𝑉𝐹 ∈ (𝑀 MndHom 𝐺))

Theoremvrgpfval 18173* The canonical injection from the generating set 𝐼 to the base set of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
= ( ~FG𝐼)    &   𝑈 = (varFGrp𝐼)       (𝐼𝑉𝑈 = (𝑗𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ))

Theoremvrgpval 18174 The value of the generating elements of a free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
= ( ~FG𝐼)    &   𝑈 = (varFGrp𝐼)       ((𝐼𝑉𝐴𝐼) → (𝑈𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] )

Theoremvrgpf 18175 The mapping from the index set to the generators is a function into the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
= ( ~FG𝐼)    &   𝑈 = (varFGrp𝐼)    &   𝐺 = (freeGrp‘𝐼)    &   𝑋 = (Base‘𝐺)       (𝐼𝑉𝑈:𝐼𝑋)

Theoremvrgpinv 18176 The inverse of a generating element is represented by 𝐴, 1⟩ instead of 𝐴, 0⟩. (Contributed by Mario Carneiro, 2-Oct-2015.)
= ( ~FG𝐼)    &   𝑈 = (varFGrp𝐼)    &   𝐺 = (freeGrp‘𝐼)    &   𝑁 = (invg𝐺)       ((𝐼𝑉𝐴𝐼) → (𝑁‘(𝑈𝐴)) = [⟨“⟨𝐴, 1𝑜⟩”⟩] )

Theoremfrgpuptf 18177* Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝐵 = (Base‘𝐻)    &   𝑁 = (invg𝐻)    &   𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))    &   (𝜑𝐻 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼𝐵)       (𝜑𝑇:(𝐼 × 2𝑜)⟶𝐵)

Theoremfrgpuptinv 18178* Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝐵 = (Base‘𝐻)    &   𝑁 = (invg𝐻)    &   𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))    &   (𝜑𝐻 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼𝐵)    &   𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)       ((𝜑𝐴 ∈ (𝐼 × 2𝑜)) → (𝑇‘(𝑀𝐴)) = (𝑁‘(𝑇𝐴)))

Theoremfrgpuplem 18179* Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝐵 = (Base‘𝐻)    &   𝑁 = (invg𝐻)    &   𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))    &   (𝜑𝐻 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼𝐵)    &   𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)       ((𝜑𝐴 𝐶) → (𝐻 Σg (𝑇𝐴)) = (𝐻 Σg (𝑇𝐶)))

Theoremfrgpupf 18180* Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝐵 = (Base‘𝐻)    &   𝑁 = (invg𝐻)    &   𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))    &   (𝜑𝐻 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼𝐵)    &   𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝐺 = (freeGrp‘𝐼)    &   𝑋 = (Base‘𝐺)    &   𝐸 = ran (𝑔𝑊 ↦ ⟨[𝑔] , (𝐻 Σg (𝑇𝑔))⟩)       (𝜑𝐸:𝑋𝐵)

Theoremfrgpupval 18181* Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝐵 = (Base‘𝐻)    &   𝑁 = (invg𝐻)    &   𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))    &   (𝜑𝐻 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼𝐵)    &   𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝐺 = (freeGrp‘𝐼)    &   𝑋 = (Base‘𝐺)    &   𝐸 = ran (𝑔𝑊 ↦ ⟨[𝑔] , (𝐻 Σg (𝑇𝑔))⟩)       ((𝜑𝐴𝑊) → (𝐸‘[𝐴] ) = (𝐻 Σg (𝑇𝐴)))

Theoremfrgpup1 18182* Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
𝐵 = (Base‘𝐻)    &   𝑁 = (invg𝐻)    &   𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))    &   (𝜑𝐻 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼𝐵)    &   𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝐺 = (freeGrp‘𝐼)    &   𝑋 = (Base‘𝐺)    &   𝐸 = ran (𝑔𝑊 ↦ ⟨[𝑔] , (𝐻 Σg (𝑇𝑔))⟩)       (𝜑𝐸 ∈ (𝐺 GrpHom 𝐻))

Theoremfrgpup2 18183* The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
𝐵 = (Base‘𝐻)    &   𝑁 = (invg𝐻)    &   𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))    &   (𝜑𝐻 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼𝐵)    &   𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝐺 = (freeGrp‘𝐼)    &   𝑋 = (Base‘𝐺)    &   𝐸 = ran (𝑔𝑊 ↦ ⟨[𝑔] , (𝐻 Σg (𝑇𝑔))⟩)    &   𝑈 = (varFGrp𝐼)    &   (𝜑𝐴𝐼)       (𝜑 → (𝐸‘(𝑈𝐴)) = (𝐹𝐴))

Theoremfrgpup3lem 18184* The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
𝐵 = (Base‘𝐻)    &   𝑁 = (invg𝐻)    &   𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))    &   (𝜑𝐻 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼𝐵)    &   𝑊 = ( I ‘Word (𝐼 × 2𝑜))    &    = ( ~FG𝐼)    &   𝐺 = (freeGrp‘𝐼)    &   𝑋 = (Base‘𝐺)    &   𝐸 = ran (𝑔𝑊 ↦ ⟨[𝑔] , (𝐻 Σg (𝑇𝑔))⟩)    &   𝑈 = (varFGrp𝐼)    &   (𝜑𝐾 ∈ (𝐺 GrpHom 𝐻))    &   (𝜑 → (𝐾𝑈) = 𝐹)       (𝜑𝐾 = 𝐸)

Theoremfrgpup3 18185* Universal property of the free monoid by existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
𝐺 = (freeGrp‘𝐼)    &   𝐵 = (Base‘𝐻)    &   𝑈 = (varFGrp𝐼)       ((𝐻 ∈ Grp ∧ 𝐼𝑉𝐹:𝐼𝐵) → ∃!𝑚 ∈ (𝐺 GrpHom 𝐻)(𝑚𝑈) = 𝐹)

Theorem0frgp 18186 The free group on zero generators is trivial. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝐺 = (freeGrp‘∅)    &   𝐵 = (Base‘𝐺)       𝐵 ≈ 1𝑜

10.3  Abelian groups

10.3.1  Definition and basic properties

Syntaxccmn 18187 Extend class notation with class of all commutative monoids.
class CMnd

Syntaxcabl 18188 Extend class notation with class of all Abelian groups.
class Abel

Definitiondf-cmn 18189* Define class of all commutative monoids. (Contributed by Mario Carneiro, 6-Jan-2015.)
CMnd = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)(𝑎(+g𝑔)𝑏) = (𝑏(+g𝑔)𝑎)}

Definitiondf-abl 18190 Define class of all Abelian groups. (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
Abel = (Grp ∩ CMnd)

Theoremisabl 18191 The predicate "is an Abelian (commutative) group." (Contributed by NM, 17-Oct-2011.)
(𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))

Theoremablgrp 18192 An Abelian group is a group. (Contributed by NM, 26-Aug-2011.)
(𝐺 ∈ Abel → 𝐺 ∈ Grp)

Theoremablcmn 18193 An Abelian group is a commutative monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
(𝐺 ∈ Abel → 𝐺 ∈ CMnd)

Theoremiscmn 18194* The predicate "is a commutative monoid." (Contributed by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))

Theoremisabl2 18195* The predicate "is an Abelian (commutative) group." (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))

Theoremcmnpropd 18196* If two structures have the same group components (properties), one is a commutative monoid iff the other one is. (Contributed by Mario Carneiro, 6-Jan-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))       (𝜑 → (𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd))

Theoremablpropd 18197* If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 6-Dec-2014.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))       (𝜑 → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel))

Theoremablprop 18198 If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 11-Oct-2013.)
(Base‘𝐾) = (Base‘𝐿)    &   (+g𝐾) = (+g𝐿)       (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel)

Theoremiscmnd 18199* Properties that determine a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
(𝜑𝐵 = (Base‘𝐺))    &   (𝜑+ = (+g𝐺))    &   (𝜑𝐺 ∈ Mnd)    &   ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))       (𝜑𝐺 ∈ CMnd)

Theoremisabld 18200* Properties that determine an Abelian group. (Contributed by NM, 6-Aug-2013.)
(𝜑𝐵 = (Base‘𝐺))    &   (𝜑+ = (+g𝐺))    &   (𝜑𝐺 ∈ Grp)    &   ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))       (𝜑𝐺 ∈ Abel)

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