Theorem efgrelex

Description: 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)

	Ref	Expression
Hypotheses	efgval.w	⊢ 𝑊 = ( I ‘ Word ( 𝐼 × 2o ) )
	efgval.r	⊢ ∼ = ( ~FG ‘ 𝐼 )
	efgval2.m	⊢ 𝑀 = ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ ⟨ 𝑦 , ( 1o ∖ 𝑧 ) ⟩ )
	efgval2.t	⊢ 𝑇 = ( 𝑣 ∈ 𝑊 ↦ ( 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) , 𝑤 ∈ ( 𝐼 × 2o ) ↦ ( 𝑣 splice ⟨ 𝑛 , 𝑛 , ⟨“ 𝑤 ( 𝑀 ‘ 𝑤 ) ”⟩ ⟩ ) ) )
	efgred.d	⊢ 𝐷 = ( 𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran ( 𝑇 ‘ 𝑥 ) )
	efgred.s	⊢ 𝑆 = ( 𝑚 ∈ { 𝑡 ∈ ( Word 𝑊 ∖ { ∅ } ) ∣ ( ( 𝑡 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ) } ↦ ( 𝑚 ‘ ( ( ♯ ‘ 𝑚 ) − 1 ) ) )
Assertion	efgrelex	⊢ ( 𝐴 ∼ 𝐵 → ∃ 𝑎 ∈ ( ◡ 𝑆 “ { 𝐴 } ) ∃ 𝑏 ∈ ( ◡ 𝑆 “ { 𝐵 } ) ( 𝑎 ‘ 0 ) = ( 𝑏 ‘ 0 ) )

Proof

Step	Hyp	Ref	Expression
1		efgval.w	⊢ 𝑊 = ( I ‘ Word ( 𝐼 × 2o ) )
2		efgval.r	⊢ ∼ = ( ~FG ‘ 𝐼 )
3		efgval2.m	⊢ 𝑀 = ( 𝑦 ∈ 𝐼 , 𝑧 ∈ 2o ↦ ⟨ 𝑦 , ( 1o ∖ 𝑧 ) ⟩ )
4		efgval2.t	⊢ 𝑇 = ( 𝑣 ∈ 𝑊 ↦ ( 𝑛 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) , 𝑤 ∈ ( 𝐼 × 2o ) ↦ ( 𝑣 splice ⟨ 𝑛 , 𝑛 , ⟨“ 𝑤 ( 𝑀 ‘ 𝑤 ) ”⟩ ⟩ ) ) )
5		efgred.d	⊢ 𝐷 = ( 𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran ( 𝑇 ‘ 𝑥 ) )
6		efgred.s	⊢ 𝑆 = ( 𝑚 ∈ { 𝑡 ∈ ( Word 𝑊 ∖ { ∅ } ) ∣ ( ( 𝑡 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑡 ) ) ( 𝑡 ‘ 𝑘 ) ∈ ran ( 𝑇 ‘ ( 𝑡 ‘ ( 𝑘 − 1 ) ) ) ) } ↦ ( 𝑚 ‘ ( ( ♯ ‘ 𝑚 ) − 1 ) ) )
7		eqid	⊢ { ⟨ 𝑖 , 𝑗 ⟩ ∣ ∃ 𝑐 ∈ ( ◡ 𝑆 “ { 𝑖 } ) ∃ 𝑑 ∈ ( ◡ 𝑆 “ { 𝑗 } ) ( 𝑐 ‘ 0 ) = ( 𝑑 ‘ 0 ) } = { ⟨ 𝑖 , 𝑗 ⟩ ∣ ∃ 𝑐 ∈ ( ◡ 𝑆 “ { 𝑖 } ) ∃ 𝑑 ∈ ( ◡ 𝑆 “ { 𝑗 } ) ( 𝑐 ‘ 0 ) = ( 𝑑 ‘ 0 ) }
8	1 2 3 4 5 6 7	efgrelexlemb	⊢ ∼ ⊆ { ⟨ 𝑖 , 𝑗 ⟩ ∣ ∃ 𝑐 ∈ ( ◡ 𝑆 “ { 𝑖 } ) ∃ 𝑑 ∈ ( ◡ 𝑆 “ { 𝑗 } ) ( 𝑐 ‘ 0 ) = ( 𝑑 ‘ 0 ) }
9	8	ssbri	⊢ ( 𝐴 ∼ 𝐵 → 𝐴 { ⟨ 𝑖 , 𝑗 ⟩ ∣ ∃ 𝑐 ∈ ( ◡ 𝑆 “ { 𝑖 } ) ∃ 𝑑 ∈ ( ◡ 𝑆 “ { 𝑗 } ) ( 𝑐 ‘ 0 ) = ( 𝑑 ‘ 0 ) } 𝐵 )
10	1 2 3 4 5 6 7	efgrelexlema	⊢ ( 𝐴 { ⟨ 𝑖 , 𝑗 ⟩ ∣ ∃ 𝑐 ∈ ( ◡ 𝑆 “ { 𝑖 } ) ∃ 𝑑 ∈ ( ◡ 𝑆 “ { 𝑗 } ) ( 𝑐 ‘ 0 ) = ( 𝑑 ‘ 0 ) } 𝐵 ↔ ∃ 𝑎 ∈ ( ◡ 𝑆 “ { 𝐴 } ) ∃ 𝑏 ∈ ( ◡ 𝑆 “ { 𝐵 } ) ( 𝑎 ‘ 0 ) = ( 𝑏 ‘ 0 ) )
11	9 10	sylib	⊢ ( 𝐴 ∼ 𝐵 → ∃ 𝑎 ∈ ( ◡ 𝑆 “ { 𝐴 } ) ∃ 𝑏 ∈ ( ◡ 𝑆 “ { 𝐵 } ) ( 𝑎 ‘ 0 ) = ( 𝑏 ‘ 0 ) )