Theorem efgcpbllema

Description: Lemma for efgrelex . Define an auxiliary equivalence relation L such that A L B if there are sequences from A to B passing through the same reduced word. (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 ) ) )
	efgcpbllem.1	⊢ 𝐿 = { ⟨ 𝑖 , 𝑗 ⟩ ∣ ( { 𝑖 , 𝑗 } ⊆ 𝑊 ∧ ( ( 𝐴 ++ 𝑖 ) ++ 𝐵 ) ∼ ( ( 𝐴 ++ 𝑗 ) ++ 𝐵 ) ) }
Assertion	efgcpbllema	⊢ ( 𝑋 𝐿 𝑌 ↔ ( 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊 ∧ ( ( 𝐴 ++ 𝑋 ) ++ 𝐵 ) ∼ ( ( 𝐴 ++ 𝑌 ) ++ 𝐵 ) ) )

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		efgcpbllem.1	⊢ 𝐿 = { ⟨ 𝑖 , 𝑗 ⟩ ∣ ( { 𝑖 , 𝑗 } ⊆ 𝑊 ∧ ( ( 𝐴 ++ 𝑖 ) ++ 𝐵 ) ∼ ( ( 𝐴 ++ 𝑗 ) ++ 𝐵 ) ) }
8		oveq2	⊢ ( 𝑖 = 𝑋 → ( 𝐴 ++ 𝑖 ) = ( 𝐴 ++ 𝑋 ) )
9	8	oveq1d	⊢ ( 𝑖 = 𝑋 → ( ( 𝐴 ++ 𝑖 ) ++ 𝐵 ) = ( ( 𝐴 ++ 𝑋 ) ++ 𝐵 ) )
10		oveq2	⊢ ( 𝑗 = 𝑌 → ( 𝐴 ++ 𝑗 ) = ( 𝐴 ++ 𝑌 ) )
11	10	oveq1d	⊢ ( 𝑗 = 𝑌 → ( ( 𝐴 ++ 𝑗 ) ++ 𝐵 ) = ( ( 𝐴 ++ 𝑌 ) ++ 𝐵 ) )
12	9 11	breqan12d	⊢ ( ( 𝑖 = 𝑋 ∧ 𝑗 = 𝑌 ) → ( ( ( 𝐴 ++ 𝑖 ) ++ 𝐵 ) ∼ ( ( 𝐴 ++ 𝑗 ) ++ 𝐵 ) ↔ ( ( 𝐴 ++ 𝑋 ) ++ 𝐵 ) ∼ ( ( 𝐴 ++ 𝑌 ) ++ 𝐵 ) ) )
13		vex	⊢ 𝑖 ∈ V
14		vex	⊢ 𝑗 ∈ V
15	13 14	prss	⊢ ( ( 𝑖 ∈ 𝑊 ∧ 𝑗 ∈ 𝑊 ) ↔ { 𝑖 , 𝑗 } ⊆ 𝑊 )
16	15	anbi1i	⊢ ( ( ( 𝑖 ∈ 𝑊 ∧ 𝑗 ∈ 𝑊 ) ∧ ( ( 𝐴 ++ 𝑖 ) ++ 𝐵 ) ∼ ( ( 𝐴 ++ 𝑗 ) ++ 𝐵 ) ) ↔ ( { 𝑖 , 𝑗 } ⊆ 𝑊 ∧ ( ( 𝐴 ++ 𝑖 ) ++ 𝐵 ) ∼ ( ( 𝐴 ++ 𝑗 ) ++ 𝐵 ) ) )
17	16	opabbii	⊢ { ⟨ 𝑖 , 𝑗 ⟩ ∣ ( ( 𝑖 ∈ 𝑊 ∧ 𝑗 ∈ 𝑊 ) ∧ ( ( 𝐴 ++ 𝑖 ) ++ 𝐵 ) ∼ ( ( 𝐴 ++ 𝑗 ) ++ 𝐵 ) ) } = { ⟨ 𝑖 , 𝑗 ⟩ ∣ ( { 𝑖 , 𝑗 } ⊆ 𝑊 ∧ ( ( 𝐴 ++ 𝑖 ) ++ 𝐵 ) ∼ ( ( 𝐴 ++ 𝑗 ) ++ 𝐵 ) ) }
18	7 17	eqtr4i	⊢ 𝐿 = { ⟨ 𝑖 , 𝑗 ⟩ ∣ ( ( 𝑖 ∈ 𝑊 ∧ 𝑗 ∈ 𝑊 ) ∧ ( ( 𝐴 ++ 𝑖 ) ++ 𝐵 ) ∼ ( ( 𝐴 ++ 𝑗 ) ++ 𝐵 ) ) }
19	12 18	brab2a	⊢ ( 𝑋 𝐿 𝑌 ↔ ( ( 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊 ) ∧ ( ( 𝐴 ++ 𝑋 ) ++ 𝐵 ) ∼ ( ( 𝐴 ++ 𝑌 ) ++ 𝐵 ) ) )
20		df-3an	⊢ ( ( 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊 ∧ ( ( 𝐴 ++ 𝑋 ) ++ 𝐵 ) ∼ ( ( 𝐴 ++ 𝑌 ) ++ 𝐵 ) ) ↔ ( ( 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊 ) ∧ ( ( 𝐴 ++ 𝑋 ) ++ 𝐵 ) ∼ ( ( 𝐴 ++ 𝑌 ) ++ 𝐵 ) ) )
21	19 20	bitr4i	⊢ ( 𝑋 𝐿 𝑌 ↔ ( 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊 ∧ ( ( 𝐴 ++ 𝑋 ) ++ 𝐵 ) ∼ ( ( 𝐴 ++ 𝑌 ) ++ 𝐵 ) ) )