P. 197 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 19601-19700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	subgdisj2 19601	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.)
⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑇)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑇)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑈)    &   ⊢ (𝜑 → (𝐴 + 𝐵) = (𝐶 + 𝐷))    ⇒   ⊢ (𝜑 → 𝐵 = 𝐷)
Theorem	subgdisjb 19602	Vectors belonging to disjoint commuting subgroups are uniquely determined by their sum. Analogous to opth 5475, this theorem shows a way of representing a pair of vectors. (Contributed by NM, 5-Jul-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑇)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑇)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵) = (𝐶 + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
Theorem	pj1fval 19603*	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)    &   ⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 𝑃 = (proj1‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) → (𝑇𝑃𝑈) = (𝑧 ∈ (𝑇 ⊕ 𝑈) ↦ (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑧 = (𝑥 + 𝑦))))
Theorem	pj1val 19604*	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)    &   ⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 𝑃 = (proj1‘𝐺)    ⇒   ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵) ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → ((𝑇𝑃𝑈)‘𝑋) = (℩𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦)))
Theorem	pj1eu 19605*	Uniqueness of a left projection. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → ∃!𝑥 ∈ 𝑇 ∃𝑦 ∈ 𝑈 𝑋 = (𝑥 + 𝑦))
Theorem	pj1f 19606	The left projection function maps a direct subspace sum onto the left factor. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))    &   ⊢ 𝑃 = (proj1‘𝐺)    ⇒   ⊢ (𝜑 → (𝑇𝑃𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)
Theorem	pj2f 19607	The right projection function maps a direct subspace sum onto the right factor. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))    &   ⊢ 𝑃 = (proj1‘𝐺)    ⇒   ⊢ (𝜑 → (𝑈𝑃𝑇):(𝑇 ⊕ 𝑈)⟶𝑈)
Theorem	pj1id 19608	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.)
⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))    &   ⊢ 𝑃 = (proj1‘𝐺)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑇 ⊕ 𝑈)) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)))
Theorem	pj1eq 19609	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.)
⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))    &   ⊢ 𝑃 = (proj1‘𝐺)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑇 ⊕ 𝑈))    &   ⊢ (𝜑 → 𝐵 ∈ 𝑇)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝑋 = (𝐵 + 𝐶) ↔ (((𝑇𝑃𝑈)‘𝑋) = 𝐵 ∧ ((𝑈𝑃𝑇)‘𝑋) = 𝐶)))
Theorem	pj1lid 19610	The left projection function is the identity on the left subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))    &   ⊢ 𝑃 = (proj1‘𝐺)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑇) → ((𝑇𝑃𝑈)‘𝑋) = 𝑋)
Theorem	pj1rid 19611	The left projection function is the zero operator on the right subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))    &   ⊢ 𝑃 = (proj1‘𝐺)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → ((𝑇𝑃𝑈)‘𝑋) = 0 )
Theorem	pj1ghm 19612	The left projection function is a group homomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))    &   ⊢ 𝑃 = (proj1‘𝐺)    ⇒   ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝐺 ↾s (𝑇 ⊕ 𝑈)) GrpHom 𝐺))
Theorem	pj1ghm2 19613	The left projection function is a group homomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ + = (+g‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))    &   ⊢ 𝑃 = (proj1‘𝐺)    ⇒   ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝐺 ↾s (𝑇 ⊕ 𝑈)) GrpHom (𝐺 ↾s 𝑇)))
Theorem	lsmhash 19614	The order of the direct product of groups. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))    &   ⊢ (𝜑 → 𝑇 ∈ Fin)    &   ⊢ (𝜑 → 𝑈 ∈ Fin)    ⇒   ⊢ (𝜑 → (♯‘(𝑇 ⊕ 𝑈)) = ((♯‘𝑇) · (♯‘𝑈)))
10.2.13  Free groups
Syntax	cefg 19615	Extend class notation with the free group equivalence relation.
class ~FG
Syntax	cfrgp 19616	Extend class notation with the free group construction.
class freeGrp
Syntax	cvrgp 19617	Extend class notation with free group injection.
class varFGrp
Definition	df-efg 19618*	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, 𝐴𝐵 ≈ 𝐴𝑥(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))∀𝑦 ∈ 𝑖 ∀𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩⟩))})
Definition	df-frgp 19619	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 19618. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2o)) /s ( ~FG ‘𝑖)))
Definition	df-vrgp 19620*	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 ↦ [⟨“⟨𝑗, ∅⟩”⟩]( ~FG ‘𝑖)))
Theorem	efgmval 19621*	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 ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    ⇒   ⊢ ((𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 2o) → (𝐴𝑀𝐵) = ⟨𝐴, (1o ∖ 𝐵)⟩)
Theorem	efgmf 19622*	The formal inverse operation is an endofunction on the generating set. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    ⇒   ⊢ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)
Theorem	efgmnvl 19623*	The inversion function on the generators is an involution. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    ⇒   ⊢ (𝐴 ∈ (𝐼 × 2o) → (𝑀‘(𝑀‘𝐴)) = 𝐴)
Theorem	efgrcl 19624	Lemma for efgval 19626. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    ⇒   ⊢ (𝐴 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
Theorem	efglem 19625*	Lemma for efgval 19626. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    ⇒   ⊢ ∃𝑟(𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩⟩))
Theorem	efgval 19626*	Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    ⇒   ⊢ ∼ = ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ∀𝑛 ∈ (0...(♯‘𝑥))∀𝑦 ∈ 𝐼 ∀𝑧 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑦, 𝑧⟩⟨𝑦, (1o ∖ 𝑧)⟩”⟩⟩))}
Theorem	efger 19627	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))    &   ⊢ ∼ = ( ~FG ‘𝐼)    ⇒   ⊢ ∼ Er 𝑊
Theorem	efgi 19628	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))    &   ⊢ ∼ = ( ~FG ‘𝐼)    ⇒   ⊢ (((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴))) ∧ (𝐽 ∈ 𝐼 ∧ 𝐾 ∈ 2o)) → 𝐴 ∼ (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 𝐾⟩⟨𝐽, (1o ∖ 𝐾)⟩”⟩⟩))
Theorem	efgi0 19629	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))    &   ⊢ ∼ = ( ~FG ‘𝐼)    ⇒   ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴)) ∧ 𝐽 ∈ 𝐼) → 𝐴 ∼ (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, ∅⟩⟨𝐽, 1o⟩”⟩⟩))
Theorem	efgi1 19630	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))    &   ⊢ ∼ = ( ~FG ‘𝐼)    ⇒   ⊢ ((𝐴 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝐴)) ∧ 𝐽 ∈ 𝐼) → 𝐴 ∼ (𝐴 splice ⟨𝑁, 𝑁, ⟨“⟨𝐽, 1o⟩⟨𝐽, ∅⟩”⟩⟩))
Theorem	efgtf 19631*	Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    ⇒   ⊢ (𝑋 ∈ 𝑊 → ((𝑇‘𝑋) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ∧ (𝑇‘𝑋):((0...(♯‘𝑋)) × (𝐼 × 2o))⟶𝑊))
Theorem	efgtval 19632*	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))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    ⇒   ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝑋)) ∧ 𝐴 ∈ (𝐼 × 2o)) → (𝑁(𝑇‘𝑋)𝐴) = (𝑋 splice ⟨𝑁, 𝑁, ⟨“𝐴(𝑀‘𝐴)”⟩⟩))
Theorem	efgval2 19633*	Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    ⇒   ⊢ ∼ = ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥 ∈ 𝑊 ran (𝑇‘𝑥) ⊆ [𝑥]𝑟)}
Theorem	efgi2 19634*	Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    ⇒   ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ ran (𝑇‘𝐴)) → 𝐴 ∼ 𝐵)
Theorem	efgtlen 19635*	Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    ⇒   ⊢ ((𝑋 ∈ 𝑊 ∧ 𝐴 ∈ ran (𝑇‘𝑋)) → (♯‘𝐴) = ((♯‘𝑋) + 2))
Theorem	efginvrel2 19636*	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))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    ⇒   ⊢ (𝐴 ∈ 𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∼ ∅)
Theorem	efginvrel1 19637*	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))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    ⇒   ⊢ (𝐴 ∈ 𝑊 → ((𝑀 ∘ (reverse‘𝐴)) ++ 𝐴) ∼ ∅)
Theorem	efgsf 19638*	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))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))    ⇒   ⊢ 𝑆:{𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}⟶𝑊
Theorem	efgsdm 19639*	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))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))    ⇒   ⊢ (𝐹 ∈ dom 𝑆 ↔ (𝐹 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐹‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝐹))(𝐹‘𝑖) ∈ ran (𝑇‘(𝐹‘(𝑖 − 1)))))
Theorem	efgsval 19640*	Value of the auxiliary function 𝑆 defining a sequence of extensions. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))    ⇒   ⊢ (𝐹 ∈ dom 𝑆 → (𝑆‘𝐹) = (𝐹‘((♯‘𝐹) − 1)))
Theorem	efgsdmi 19641*	Property of the last link in the chain of extensions. (Contributed by Mario Carneiro, 29-Sep-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))    ⇒   ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((♯‘𝐹) − 1) ∈ ℕ) → (𝑆‘𝐹) ∈ ran (𝑇‘(𝐹‘(((♯‘𝐹) − 1) − 1))))
Theorem	efgsval2 19642*	Value of the auxiliary function 𝑆 defining a sequence of extensions. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))    ⇒   ⊢ ((𝐴 ∈ Word 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ++ ⟨“𝐵”⟩) ∈ dom 𝑆) → (𝑆‘(𝐴 ++ ⟨“𝐵”⟩)) = 𝐵)
Theorem	efgsrel 19643*	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))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))    ⇒   ⊢ (𝐹 ∈ dom 𝑆 → (𝐹‘0) ∼ (𝑆‘𝐹))
Theorem	efgs1 19644*	A singleton of an irreducible word is an extension sequence. (Contributed by Mario Carneiro, 27-Sep-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))    ⇒   ⊢ (𝐴 ∈ 𝐷 → ⟨“𝐴”⟩ ∈ dom 𝑆)
Theorem	efgs1b 19645*	Every extension sequence ending in an irreducible word is trivial. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))    ⇒   ⊢ (𝐴 ∈ dom 𝑆 → ((𝑆‘𝐴) ∈ 𝐷 ↔ (♯‘𝐴) = 1))
Theorem	efgsp1 19646*	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))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))    ⇒   ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝐴 ∈ ran (𝑇‘(𝑆‘𝐹))) → (𝐹 ++ ⟨“𝐴”⟩) ∈ dom 𝑆)
Theorem	efgsres 19647*	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))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))    ⇒   ⊢ ((𝐹 ∈ dom 𝑆 ∧ 𝑁 ∈ (1...(♯‘𝐹))) → (𝐹 ↾ (0..^𝑁)) ∈ dom 𝑆)
Theorem	efgsfo 19648*	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))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))    ⇒   ⊢ 𝑆:dom 𝑆–onto→𝑊
Theorem	efgredlema 19649*	The reduced word that forms the base of the sequence in efgsval 19640 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))    &   ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))    &   ⊢ (𝜑 → 𝐴 ∈ dom 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ dom 𝑆)    &   ⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵))    &   ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))    ⇒   ⊢ (𝜑 → (((♯‘𝐴) − 1) ∈ ℕ ∧ ((♯‘𝐵) − 1) ∈ ℕ))
Theorem	efgredlemf 19650*	Lemma for efgredleme 19652. (Contributed by Mario Carneiro, 4-Jun-2016.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))    &   ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))    &   ⊢ (𝜑 → 𝐴 ∈ dom 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ dom 𝑆)    &   ⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵))    &   ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))    &   ⊢ 𝐾 = (((♯‘𝐴) − 1) − 1)    &   ⊢ 𝐿 = (((♯‘𝐵) − 1) − 1)    ⇒   ⊢ (𝜑 → ((𝐴‘𝐾) ∈ 𝑊 ∧ (𝐵‘𝐿) ∈ 𝑊))
Theorem	efgredlemg 19651*	Lemma for efgred 19657. (Contributed by Mario Carneiro, 4-Jun-2016.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))    &   ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))    &   ⊢ (𝜑 → 𝐴 ∈ dom 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ dom 𝑆)    &   ⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵))    &   ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))    &   ⊢ 𝐾 = (((♯‘𝐴) − 1) − 1)    &   ⊢ 𝐿 = (((♯‘𝐵) − 1) − 1)    &   ⊢ (𝜑 → 𝑃 ∈ (0...(♯‘(𝐴‘𝐾))))    &   ⊢ (𝜑 → 𝑄 ∈ (0...(♯‘(𝐵‘𝐿))))    &   ⊢ (𝜑 → 𝑈 ∈ (𝐼 × 2o))    &   ⊢ (𝜑 → 𝑉 ∈ (𝐼 × 2o))    &   ⊢ (𝜑 → (𝑆‘𝐴) = (𝑃(𝑇‘(𝐴‘𝐾))𝑈))    &   ⊢ (𝜑 → (𝑆‘𝐵) = (𝑄(𝑇‘(𝐵‘𝐿))𝑉))    ⇒   ⊢ (𝜑 → (♯‘(𝐴‘𝐾)) = (♯‘(𝐵‘𝐿)))
Theorem	efgredleme 19652*	Lemma for efgred 19657. (Contributed by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 15-Oct-2022.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))    &   ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))    &   ⊢ (𝜑 → 𝐴 ∈ dom 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ dom 𝑆)    &   ⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵))    &   ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))    &   ⊢ 𝐾 = (((♯‘𝐴) − 1) − 1)    &   ⊢ 𝐿 = (((♯‘𝐵) − 1) − 1)    &   ⊢ (𝜑 → 𝑃 ∈ (0...(♯‘(𝐴‘𝐾))))    &   ⊢ (𝜑 → 𝑄 ∈ (0...(♯‘(𝐵‘𝐿))))    &   ⊢ (𝜑 → 𝑈 ∈ (𝐼 × 2o))    &   ⊢ (𝜑 → 𝑉 ∈ (𝐼 × 2o))    &   ⊢ (𝜑 → (𝑆‘𝐴) = (𝑃(𝑇‘(𝐴‘𝐾))𝑈))    &   ⊢ (𝜑 → (𝑆‘𝐵) = (𝑄(𝑇‘(𝐵‘𝐿))𝑉))    &   ⊢ (𝜑 → ¬ (𝐴‘𝐾) = (𝐵‘𝐿))    &   ⊢ (𝜑 → 𝑃 ∈ (ℤ≥‘(𝑄 + 2)))    &   ⊢ (𝜑 → 𝐶 ∈ dom 𝑆)    &   ⊢ (𝜑 → (𝑆‘𝐶) = (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩)))    ⇒   ⊢ (𝜑 → ((𝐴‘𝐾) ∈ ran (𝑇‘(𝑆‘𝐶)) ∧ (𝐵‘𝐿) ∈ ran (𝑇‘(𝑆‘𝐶))))
Theorem	efgredlemd 19653*	The reduced word that forms the base of the sequence in efgsval 19640 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 15-Oct-2022.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))    &   ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))    &   ⊢ (𝜑 → 𝐴 ∈ dom 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ dom 𝑆)    &   ⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵))    &   ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))    &   ⊢ 𝐾 = (((♯‘𝐴) − 1) − 1)    &   ⊢ 𝐿 = (((♯‘𝐵) − 1) − 1)    &   ⊢ (𝜑 → 𝑃 ∈ (0...(♯‘(𝐴‘𝐾))))    &   ⊢ (𝜑 → 𝑄 ∈ (0...(♯‘(𝐵‘𝐿))))    &   ⊢ (𝜑 → 𝑈 ∈ (𝐼 × 2o))    &   ⊢ (𝜑 → 𝑉 ∈ (𝐼 × 2o))    &   ⊢ (𝜑 → (𝑆‘𝐴) = (𝑃(𝑇‘(𝐴‘𝐾))𝑈))    &   ⊢ (𝜑 → (𝑆‘𝐵) = (𝑄(𝑇‘(𝐵‘𝐿))𝑉))    &   ⊢ (𝜑 → ¬ (𝐴‘𝐾) = (𝐵‘𝐿))    &   ⊢ (𝜑 → 𝑃 ∈ (ℤ≥‘(𝑄 + 2)))    &   ⊢ (𝜑 → 𝐶 ∈ dom 𝑆)    &   ⊢ (𝜑 → (𝑆‘𝐶) = (((𝐵‘𝐿) prefix 𝑄) ++ ((𝐴‘𝐾) substr ⟨(𝑄 + 2), (♯‘(𝐴‘𝐾))⟩)))    ⇒   ⊢ (𝜑 → (𝐴‘0) = (𝐵‘0))
Theorem	efgredlemc 19654*	The reduced word that forms the base of the sequence in efgsval 19640 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 15-Oct-2022.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))    &   ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))    &   ⊢ (𝜑 → 𝐴 ∈ dom 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ dom 𝑆)    &   ⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵))    &   ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))    &   ⊢ 𝐾 = (((♯‘𝐴) − 1) − 1)    &   ⊢ 𝐿 = (((♯‘𝐵) − 1) − 1)    &   ⊢ (𝜑 → 𝑃 ∈ (0...(♯‘(𝐴‘𝐾))))    &   ⊢ (𝜑 → 𝑄 ∈ (0...(♯‘(𝐵‘𝐿))))    &   ⊢ (𝜑 → 𝑈 ∈ (𝐼 × 2o))    &   ⊢ (𝜑 → 𝑉 ∈ (𝐼 × 2o))    &   ⊢ (𝜑 → (𝑆‘𝐴) = (𝑃(𝑇‘(𝐴‘𝐾))𝑈))    &   ⊢ (𝜑 → (𝑆‘𝐵) = (𝑄(𝑇‘(𝐵‘𝐿))𝑉))    &   ⊢ (𝜑 → ¬ (𝐴‘𝐾) = (𝐵‘𝐿))    ⇒   ⊢ (𝜑 → (𝑃 ∈ (ℤ≥‘𝑄) → (𝐴‘0) = (𝐵‘0)))
Theorem	efgredlemb 19655*	The reduced word that forms the base of the sequence in efgsval 19640 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 30-Sep-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))    &   ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))    &   ⊢ (𝜑 → 𝐴 ∈ dom 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ dom 𝑆)    &   ⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵))    &   ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))    &   ⊢ 𝐾 = (((♯‘𝐴) − 1) − 1)    &   ⊢ 𝐿 = (((♯‘𝐵) − 1) − 1)    &   ⊢ (𝜑 → 𝑃 ∈ (0...(♯‘(𝐴‘𝐾))))    &   ⊢ (𝜑 → 𝑄 ∈ (0...(♯‘(𝐵‘𝐿))))    &   ⊢ (𝜑 → 𝑈 ∈ (𝐼 × 2o))    &   ⊢ (𝜑 → 𝑉 ∈ (𝐼 × 2o))    &   ⊢ (𝜑 → (𝑆‘𝐴) = (𝑃(𝑇‘(𝐴‘𝐾))𝑈))    &   ⊢ (𝜑 → (𝑆‘𝐵) = (𝑄(𝑇‘(𝐵‘𝐿))𝑉))    &   ⊢ (𝜑 → ¬ (𝐴‘𝐾) = (𝐵‘𝐿))    ⇒   ⊢ ¬ 𝜑
Theorem	efgredlem 19656*	The reduced word that forms the base of the sequence in efgsval 19640 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 30-Sep-2015.) (Proof shortened by AV, 3-Nov-2022.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))    &   ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0))))    &   ⊢ (𝜑 → 𝐴 ∈ dom 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ dom 𝑆)    &   ⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵))    &   ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))    ⇒   ⊢ ¬ 𝜑
Theorem	efgred 19657*	The reduced word that forms the base of the sequence in efgsval 19640 is uniquely determined, given the terminal point. (Contributed by Mario Carneiro, 28-Sep-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))    ⇒   ⊢ ((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆 ∧ (𝑆‘𝐴) = (𝑆‘𝐵)) → (𝐴‘0) = (𝐵‘0))
Theorem	efgrelexlema 19658*	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))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ 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))
Theorem	efgrelexlemb 19659*	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))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))    &   ⊢ 𝐿 = {⟨𝑖, 𝑗⟩ ∣ ∃𝑐 ∈ (◡𝑆 “ {𝑖})∃𝑑 ∈ (◡𝑆 “ {𝑗})(𝑐‘0) = (𝑑‘0)}    ⇒   ⊢ ∼ ⊆ 𝐿
Theorem	efgrelex 19660*	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))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))    ⇒   ⊢ (𝐴 ∼ 𝐵 → ∃𝑎 ∈ (◡𝑆 “ {𝐴})∃𝑏 ∈ (◡𝑆 “ {𝐵})(𝑎‘0) = (𝑏‘0))
Theorem	efgredeu 19661*	There is a unique reduced word equivalent to a given word. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))    ⇒   ⊢ (𝐴 ∈ 𝑊 → ∃!𝑑 ∈ 𝐷 𝑑 ∼ 𝐴)
Theorem	efgred2 19662*	Two extension sequences have related endpoints iff they have the same base. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))    ⇒   ⊢ ((𝐴 ∈ dom 𝑆 ∧ 𝐵 ∈ dom 𝑆) → ((𝑆‘𝐴) ∼ (𝑆‘𝐵) ↔ (𝐴‘0) = (𝐵‘0)))
Theorem	efgcpbllema 19663*	Lemma for efgrelex 19660. 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))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))    &   ⊢ 𝐿 = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑊 ∧ ((𝐴 ++ 𝑖) ++ 𝐵) ∼ ((𝐴 ++ 𝑗) ++ 𝐵))}    ⇒   ⊢ (𝑋𝐿𝑌 ↔ (𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊 ∧ ((𝐴 ++ 𝑋) ++ 𝐵) ∼ ((𝐴 ++ 𝑌) ++ 𝐵)))
Theorem	efgcpbllemb 19664*	Lemma for efgrelex 19660. 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))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))    &   ⊢ 𝐿 = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑊 ∧ ((𝐴 ++ 𝑖) ++ 𝐵) ∼ ((𝐴 ++ 𝑗) ++ 𝐵))}    ⇒   ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑊) → ∼ ⊆ 𝐿)
Theorem	efgcpbl 19665*	Two extension sequences have related endpoints iff they have the same base. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))    ⇒   ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ 𝑋 ∼ 𝑌) → ((𝐴 ++ 𝑋) ++ 𝐵) ∼ ((𝐴 ++ 𝑌) ++ 𝐵))
Theorem	efgcpbl2 19666*	Two extension sequences have related endpoints iff they have the same base. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1)))    ⇒   ⊢ ((𝐴 ∼ 𝑋 ∧ 𝐵 ∼ 𝑌) → (𝐴 ++ 𝐵) ∼ (𝑋 ++ 𝑌))
Theorem	frgpval 19667	Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝐺 = (freeGrp‘𝐼)    &   ⊢ 𝑀 = (freeMnd‘(𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝐺 = (𝑀 /s ∼ ))
Theorem	frgpcpbl 19668	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))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ + = (+g‘𝑀)    ⇒   ⊢ ((𝐴 ∼ 𝐶 ∧ 𝐵 ∼ 𝐷) → (𝐴 + 𝐵) ∼ (𝐶 + 𝐷))
Theorem	frgp0 19669	The free group is a group. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
⊢ 𝐺 = (freeGrp‘𝐼)    &   ⊢ ∼ = ( ~FG ‘𝐼)    ⇒   ⊢ (𝐼 ∈ 𝑉 → (𝐺 ∈ Grp ∧ [∅] ∼ = (0g‘𝐺)))
Theorem	frgpeccl 19670	Closure of the quotient map in a free group. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝐺 = (freeGrp‘𝐼)    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝑋 ∈ 𝑊 → [𝑋] ∼ ∈ 𝐵)
Theorem	frgpgrp 19671	The free group is a group. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝐺 = (freeGrp‘𝐼)    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝐺 ∈ Grp)
Theorem	frgpadd 19672	Addition in the free group is given by concatenation. (Contributed by Mario Carneiro, 1-Oct-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ 𝐺 = (freeGrp‘𝐼)    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑊) → ([𝐴] ∼ + [𝐵] ∼ ) = [(𝐴 ++ 𝐵)] ∼ )
Theorem	frgpinv 19673*	The inverse of an element of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ 𝐺 = (freeGrp‘𝐼)    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑁 = (invg‘𝐺)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    ⇒   ⊢ (𝐴 ∈ 𝑊 → (𝑁‘[𝐴] ∼ ) = [(𝑀 ∘ (reverse‘𝐴))] ∼ )
Theorem	frgpmhm 19674*	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)    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑊 ↦ [𝑥] ∼ )    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝐹 ∈ (𝑀 MndHom 𝐺))
Theorem	vrgpfval 19675*	The canonical injection from the generating set 𝐼 to the base set of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑈 = (varFGrp‘𝐼)    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝑈 = (𝑗 ∈ 𝐼 ↦ [⟨“⟨𝑗, ∅⟩”⟩] ∼ ))
Theorem	vrgpval 19676	The value of the generating elements of a free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑈 = (varFGrp‘𝐼)    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )
Theorem	vrgpf 19677	The mapping from the index set to the generators is a function into the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑈 = (varFGrp‘𝐼)    &   ⊢ 𝐺 = (freeGrp‘𝐼)    &   ⊢ 𝑋 = (Base‘𝐺)    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝑈:𝐼⟶𝑋)
Theorem	vrgpinv 19678	The inverse of a generating element is represented by ⟨𝐴, 1⟩ instead of ⟨𝐴, 0⟩. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝑈 = (varFGrp‘𝐼)    &   ⊢ 𝐺 = (freeGrp‘𝐼)    &   ⊢ 𝑁 = (invg‘𝐺)    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑁‘(𝑈‘𝐴)) = [⟨“⟨𝐴, 1o⟩”⟩] ∼ )
Theorem	frgpuptf 19679*	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 = (invg‘𝐻)    &   ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))))    &   ⊢ (𝜑 → 𝐻 ∈ Grp)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐼⟶𝐵)    ⇒   ⊢ (𝜑 → 𝑇:(𝐼 × 2o)⟶𝐵)
Theorem	frgpuptinv 19680*	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 = (invg‘𝐻)    &   ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))))    &   ⊢ (𝜑 → 𝐻 ∈ Grp)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐼⟶𝐵)    &   ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐼 × 2o)) → (𝑇‘(𝑀‘𝐴)) = (𝑁‘(𝑇‘𝐴)))
Theorem	frgpuplem 19681*	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 = (invg‘𝐻)    &   ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))))    &   ⊢ (𝜑 → 𝐻 ∈ Grp)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐼⟶𝐵)    &   ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∼ 𝐶) → (𝐻 Σg (𝑇 ∘ 𝐴)) = (𝐻 Σg (𝑇 ∘ 𝐶)))
Theorem	frgpupf 19682*	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 = (invg‘𝐻)    &   ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))))    &   ⊢ (𝜑 → 𝐻 ∈ Grp)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐼⟶𝐵)    &   ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝐺 = (freeGrp‘𝐼)    &   ⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐸 = ran (𝑔 ∈ 𝑊 ↦ ⟨[𝑔] ∼ , (𝐻 Σg (𝑇 ∘ 𝑔))⟩)    ⇒   ⊢ (𝜑 → 𝐸:𝑋⟶𝐵)
Theorem	frgpupval 19683*	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 = (invg‘𝐻)    &   ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))))    &   ⊢ (𝜑 → 𝐻 ∈ Grp)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐼⟶𝐵)    &   ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝐺 = (freeGrp‘𝐼)    &   ⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐸 = ran (𝑔 ∈ 𝑊 ↦ ⟨[𝑔] ∼ , (𝐻 Σg (𝑇 ∘ 𝑔))⟩)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑊) → (𝐸‘[𝐴] ∼ ) = (𝐻 Σg (𝑇 ∘ 𝐴)))
Theorem	frgpup1 19684*	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 = (invg‘𝐻)    &   ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))))    &   ⊢ (𝜑 → 𝐻 ∈ Grp)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐼⟶𝐵)    &   ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝐺 = (freeGrp‘𝐼)    &   ⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐸 = ran (𝑔 ∈ 𝑊 ↦ ⟨[𝑔] ∼ , (𝐻 Σg (𝑇 ∘ 𝑔))⟩)    ⇒   ⊢ (𝜑 → 𝐸 ∈ (𝐺 GrpHom 𝐻))
Theorem	frgpup2 19685*	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 = (invg‘𝐻)    &   ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))))    &   ⊢ (𝜑 → 𝐻 ∈ Grp)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐼⟶𝐵)    &   ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝐺 = (freeGrp‘𝐼)    &   ⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐸 = ran (𝑔 ∈ 𝑊 ↦ ⟨[𝑔] ∼ , (𝐻 Σg (𝑇 ∘ 𝑔))⟩)    &   ⊢ 𝑈 = (varFGrp‘𝐼)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐼)    ⇒   ⊢ (𝜑 → (𝐸‘(𝑈‘𝐴)) = (𝐹‘𝐴))
Theorem	frgpup3lem 19686*	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 = (invg‘𝐻)    &   ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))))    &   ⊢ (𝜑 → 𝐻 ∈ Grp)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐼⟶𝐵)    &   ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))    &   ⊢ ∼ = ( ~FG ‘𝐼)    &   ⊢ 𝐺 = (freeGrp‘𝐼)    &   ⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐸 = ran (𝑔 ∈ 𝑊 ↦ ⟨[𝑔] ∼ , (𝐻 Σg (𝑇 ∘ 𝑔))⟩)    &   ⊢ 𝑈 = (varFGrp‘𝐼)    &   ⊢ (𝜑 → 𝐾 ∈ (𝐺 GrpHom 𝐻))    &   ⊢ (𝜑 → (𝐾 ∘ 𝑈) = 𝐹)    ⇒   ⊢ (𝜑 → 𝐾 = 𝐸)
Theorem	frgpup3 19687*	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 = (varFGrp‘𝐼)    ⇒   ⊢ ((𝐻 ∈ Grp ∧ 𝐼 ∈ 𝑉 ∧ 𝐹:𝐼⟶𝐵) → ∃!𝑚 ∈ (𝐺 GrpHom 𝐻)(𝑚 ∘ 𝑈) = 𝐹)
Theorem	0frgp 19688	The free group on zero generators is trivial. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐺 = (freeGrp‘∅)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ 𝐵 ≈ 1o
10.2.14  Abelian groups
10.2.14.1  Definition and basic properties
Syntax	ccmn 19689	Extend class notation with class of all commutative monoids.
class CMnd
Syntax	cabl 19690	Extend class notation with class of all Abelian groups.
class Abel
Definition	df-cmn 19691*	Define class of all commutative monoids. (Contributed by Mario Carneiro, 6-Jan-2015.)
⊢ CMnd = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)(𝑎(+g‘𝑔)𝑏) = (𝑏(+g‘𝑔)𝑎)}
Definition	df-abl 19692	Define class of all Abelian groups. (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
⊢ Abel = (Grp ∩ CMnd)
Theorem	isabl 19693	The predicate "is an Abelian (commutative) group". (Contributed by NM, 17-Oct-2011.)
⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
Theorem	ablgrp 19694	An Abelian group is a group. (Contributed by NM, 26-Aug-2011.)
⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
Theorem	ablgrpd 19695	An Abelian group is a group, deduction form of ablgrp 19694. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ (𝜑 → 𝐺 ∈ Abel)    ⇒   ⊢ (𝜑 → 𝐺 ∈ Grp)
Theorem	ablcmn 19696	An Abelian group is a commutative monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd)
Theorem	ablcmnd 19697	An Abelian group is a commutative monoid. (Contributed by SN, 1-Jun-2024.)
⊢ (𝜑 → 𝐺 ∈ Abel)    ⇒   ⊢ (𝜑 → 𝐺 ∈ CMnd)
Theorem	iscmn 19698*	The predicate "is a commutative monoid". (Contributed by Mario Carneiro, 6-Jan-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
Theorem	isabl2 19699*	The predicate "is an Abelian (commutative) group". (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
Theorem	cmnpropd 19700*	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.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    ⇒   ⊢ (𝜑 → (𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd))