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Metamath Proof Explorer

Theorem efgcpbl

Description: Two extension sequences have related endpoints iff they have the same base. (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 efgcpbl ( ( 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ 𝑋 ∼ 𝑌 ) → ( ( 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 eqid ⊢ { ⟨ 𝑖 , 𝑗 ⟩ ∣ ( { 𝑖 , 𝑗 } ⊆ 𝑊 ∧ ( ( 𝐴 ++ 𝑖 ) ++ 𝐵 ) ∼ ( ( 𝐴 ++ 𝑗 ) ++ 𝐵 ) ) } = { ⟨ 𝑖 , 𝑗 ⟩ ∣ ( { 𝑖 , 𝑗 } ⊆ 𝑊 ∧ ( ( 𝐴 ++ 𝑖 ) ++ 𝐵 ) ∼ ( ( 𝐴 ++ 𝑗 ) ++ 𝐵 ) ) }
8 1 2 3 4 5 6 7 efgcpbllemb ⊢ ( ( 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑊 ) → ∼ ⊆ { ⟨ 𝑖 , 𝑗 ⟩ ∣ ( { 𝑖 , 𝑗 } ⊆ 𝑊 ∧ ( ( 𝐴 ++ 𝑖 ) ++ 𝐵 ) ∼ ( ( 𝐴 ++ 𝑗 ) ++ 𝐵 ) ) } )
9 8 ssbrd ⊢ ( ( 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑊 ) → ( 𝑋 ∼ 𝑌 → 𝑋 { ⟨ 𝑖 , 𝑗 ⟩ ∣ ( { 𝑖 , 𝑗 } ⊆ 𝑊 ∧ ( ( 𝐴 ++ 𝑖 ) ++ 𝐵 ) ∼ ( ( 𝐴 ++ 𝑗 ) ++ 𝐵 ) ) } 𝑌 ) )
10 9 3impia ⊢ ( ( 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ 𝑋 ∼ 𝑌 ) → 𝑋 { ⟨ 𝑖 , 𝑗 ⟩ ∣ ( { 𝑖 , 𝑗 } ⊆ 𝑊 ∧ ( ( 𝐴 ++ 𝑖 ) ++ 𝐵 ) ∼ ( ( 𝐴 ++ 𝑗 ) ++ 𝐵 ) ) } 𝑌 )
11 1 2 3 4 5 6 7 efgcpbllema ⊢ ( 𝑋 { ⟨ 𝑖 , 𝑗 ⟩ ∣ ( { 𝑖 , 𝑗 } ⊆ 𝑊 ∧ ( ( 𝐴 ++ 𝑖 ) ++ 𝐵 ) ∼ ( ( 𝐴 ++ 𝑗 ) ++ 𝐵 ) ) } 𝑌 ↔ ( 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊 ∧ ( ( 𝐴 ++ 𝑋 ) ++ 𝐵 ) ∼ ( ( 𝐴 ++ 𝑌 ) ++ 𝐵 ) ) )
12 11 simp3bi ⊢ ( 𝑋 { ⟨ 𝑖 , 𝑗 ⟩ ∣ ( { 𝑖 , 𝑗 } ⊆ 𝑊 ∧ ( ( 𝐴 ++ 𝑖 ) ++ 𝐵 ) ∼ ( ( 𝐴 ++ 𝑗 ) ++ 𝐵 ) ) } 𝑌 → ( ( 𝐴 ++ 𝑋 ) ++ 𝐵 ) ∼ ( ( 𝐴 ++ 𝑌 ) ++ 𝐵 ) )
13 10 12 syl ⊢ ( ( 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑊 ∧ 𝑋 ∼ 𝑌 ) → ( ( 𝐴 ++ 𝑋 ) ++ 𝐵 ) ∼ ( ( 𝐴 ++ 𝑌 ) ++ 𝐵 ) )