Metamath Proof Explorer

Theorem efgredlemb

Description: The reduced word that forms the base of the sequence in efgsval is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 30-Sep-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 ) ) )
efgredlem.1 ⊢ ( 𝜑 → ∀ 𝑎 ∈ dom 𝑆 ∀ 𝑏 ∈ dom 𝑆 ( ( ♯ ‘ ( 𝑆 ‘ 𝑎 ) ) < ( ♯ ‘ ( 𝑆 ‘ 𝐴 ) ) → ( ( 𝑆 ‘ 𝑎 ) = ( 𝑆 ‘ 𝑏 ) → ( 𝑎 ‘ 0 ) = ( 𝑏 ‘ 0 ) ) ) )
efgredlem.2 ⊢ ( 𝜑 → 𝐴 ∈ dom 𝑆 )
efgredlem.3 ⊢ ( 𝜑 → 𝐵 ∈ dom 𝑆 )
efgredlem.4 ⊢ ( 𝜑 → ( 𝑆 ‘ 𝐴 ) = ( 𝑆 ‘ 𝐵 ) )
efgredlem.5 ⊢ ( 𝜑 → ¬ ( 𝐴 ‘ 0 ) = ( 𝐵 ‘ 0 ) )
efgredlemb.k ⊢ 𝐾 = ( ( ( ♯ ‘ 𝐴 ) − 1 ) − 1 )
efgredlemb.l ⊢ 𝐿 = ( ( ( ♯ ‘ 𝐵 ) − 1 ) − 1 )
efgredlemb.p ⊢ ( 𝜑 → 𝑃 ∈ ( 0 ... ( ♯ ‘ ( 𝐴 ‘ 𝐾 ) ) ) )
efgredlemb.q ⊢ ( 𝜑 → 𝑄 ∈ ( 0 ... ( ♯ ‘ ( 𝐵 ‘ 𝐿 ) ) ) )
efgredlemb.u ⊢ ( 𝜑 → 𝑈 ∈ ( 𝐼 × 2o ) )
efgredlemb.v ⊢ ( 𝜑 → 𝑉 ∈ ( 𝐼 × 2o ) )
efgredlemb.6 ⊢ ( 𝜑 → ( 𝑆 ‘ 𝐴 ) = ( 𝑃 ( 𝑇 ‘ ( 𝐴 ‘ 𝐾 ) ) 𝑈 ) )
efgredlemb.7 ⊢ ( 𝜑 → ( 𝑆 ‘ 𝐵 ) = ( 𝑄 ( 𝑇 ‘ ( 𝐵 ‘ 𝐿 ) ) 𝑉 ) )
efgredlemb.8 ⊢ ( 𝜑 → ¬ ( 𝐴 ‘ 𝐾 ) = ( 𝐵 ‘ 𝐿 ) )
Assertion efgredlemb ¬ 𝜑

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 efgredlem.1 ⊢ ( 𝜑 → ∀ 𝑎 ∈ dom 𝑆 ∀ 𝑏 ∈ dom 𝑆 ( ( ♯ ‘ ( 𝑆 ‘ 𝑎 ) ) < ( ♯ ‘ ( 𝑆 ‘ 𝐴 ) ) → ( ( 𝑆 ‘ 𝑎 ) = ( 𝑆 ‘ 𝑏 ) → ( 𝑎 ‘ 0 ) = ( 𝑏 ‘ 0 ) ) ) )
8 efgredlem.2 ⊢ ( 𝜑 → 𝐴 ∈ dom 𝑆 )
9 efgredlem.3 ⊢ ( 𝜑 → 𝐵 ∈ dom 𝑆 )
10 efgredlem.4 ⊢ ( 𝜑 → ( 𝑆 ‘ 𝐴 ) = ( 𝑆 ‘ 𝐵 ) )
11 efgredlem.5 ⊢ ( 𝜑 → ¬ ( 𝐴 ‘ 0 ) = ( 𝐵 ‘ 0 ) )
12 efgredlemb.k ⊢ 𝐾 = ( ( ( ♯ ‘ 𝐴 ) − 1 ) − 1 )
13 efgredlemb.l ⊢ 𝐿 = ( ( ( ♯ ‘ 𝐵 ) − 1 ) − 1 )
14 efgredlemb.p ⊢ ( 𝜑 → 𝑃 ∈ ( 0 ... ( ♯ ‘ ( 𝐴 ‘ 𝐾 ) ) ) )
15 efgredlemb.q ⊢ ( 𝜑 → 𝑄 ∈ ( 0 ... ( ♯ ‘ ( 𝐵 ‘ 𝐿 ) ) ) )
16 efgredlemb.u ⊢ ( 𝜑 → 𝑈 ∈ ( 𝐼 × 2o ) )
17 efgredlemb.v ⊢ ( 𝜑 → 𝑉 ∈ ( 𝐼 × 2o ) )
18 efgredlemb.6 ⊢ ( 𝜑 → ( 𝑆 ‘ 𝐴 ) = ( 𝑃 ( 𝑇 ‘ ( 𝐴 ‘ 𝐾 ) ) 𝑈 ) )
19 efgredlemb.7 ⊢ ( 𝜑 → ( 𝑆 ‘ 𝐵 ) = ( 𝑄 ( 𝑇 ‘ ( 𝐵 ‘ 𝐿 ) ) 𝑉 ) )
20 efgredlemb.8 ⊢ ( 𝜑 → ¬ ( 𝐴 ‘ 𝐾 ) = ( 𝐵 ‘ 𝐿 ) )
21 fveq2 ⊢ ( ( 𝑆 ‘ 𝐴 ) = ( 𝑆 ‘ 𝐵 ) → ( ♯ ‘ ( 𝑆 ‘ 𝐴 ) ) = ( ♯ ‘ ( 𝑆 ‘ 𝐵 ) ) )
22 21 breq2d ⊢ ( ( 𝑆 ‘ 𝐴 ) = ( 𝑆 ‘ 𝐵 ) → ( ( ♯ ‘ ( 𝑆 ‘ 𝑎 ) ) < ( ♯ ‘ ( 𝑆 ‘ 𝐴 ) ) ↔ ( ♯ ‘ ( 𝑆 ‘ 𝑎 ) ) < ( ♯ ‘ ( 𝑆 ‘ 𝐵 ) ) ) )
23 22 imbi1d ⊢ ( ( 𝑆 ‘ 𝐴 ) = ( 𝑆 ‘ 𝐵 ) → ( ( ( ♯ ‘ ( 𝑆 ‘ 𝑎 ) ) < ( ♯ ‘ ( 𝑆 ‘ 𝐴 ) ) → ( ( 𝑆 ‘ 𝑎 ) = ( 𝑆 ‘ 𝑏 ) → ( 𝑎 ‘ 0 ) = ( 𝑏 ‘ 0 ) ) ) ↔ ( ( ♯ ‘ ( 𝑆 ‘ 𝑎 ) ) < ( ♯ ‘ ( 𝑆 ‘ 𝐵 ) ) → ( ( 𝑆 ‘ 𝑎 ) = ( 𝑆 ‘ 𝑏 ) → ( 𝑎 ‘ 0 ) = ( 𝑏 ‘ 0 ) ) ) ) )
24 23 2ralbidv ⊢ ( ( 𝑆 ‘ 𝐴 ) = ( 𝑆 ‘ 𝐵 ) → ( ∀ 𝑎 ∈ dom 𝑆 ∀ 𝑏 ∈ dom 𝑆 ( ( ♯ ‘ ( 𝑆 ‘ 𝑎 ) ) < ( ♯ ‘ ( 𝑆 ‘ 𝐴 ) ) → ( ( 𝑆 ‘ 𝑎 ) = ( 𝑆 ‘ 𝑏 ) → ( 𝑎 ‘ 0 ) = ( 𝑏 ‘ 0 ) ) ) ↔ ∀ 𝑎 ∈ dom 𝑆 ∀ 𝑏 ∈ dom 𝑆 ( ( ♯ ‘ ( 𝑆 ‘ 𝑎 ) ) < ( ♯ ‘ ( 𝑆 ‘ 𝐵 ) ) → ( ( 𝑆 ‘ 𝑎 ) = ( 𝑆 ‘ 𝑏 ) → ( 𝑎 ‘ 0 ) = ( 𝑏 ‘ 0 ) ) ) ) )
25 10 24 syl ⊢ ( 𝜑 → ( ∀ 𝑎 ∈ dom 𝑆 ∀ 𝑏 ∈ dom 𝑆 ( ( ♯ ‘ ( 𝑆 ‘ 𝑎 ) ) < ( ♯ ‘ ( 𝑆 ‘ 𝐴 ) ) → ( ( 𝑆 ‘ 𝑎 ) = ( 𝑆 ‘ 𝑏 ) → ( 𝑎 ‘ 0 ) = ( 𝑏 ‘ 0 ) ) ) ↔ ∀ 𝑎 ∈ dom 𝑆 ∀ 𝑏 ∈ dom 𝑆 ( ( ♯ ‘ ( 𝑆 ‘ 𝑎 ) ) < ( ♯ ‘ ( 𝑆 ‘ 𝐵 ) ) → ( ( 𝑆 ‘ 𝑎 ) = ( 𝑆 ‘ 𝑏 ) → ( 𝑎 ‘ 0 ) = ( 𝑏 ‘ 0 ) ) ) ) )
26 7 25 mpbid ⊢ ( 𝜑 → ∀ 𝑎 ∈ dom 𝑆 ∀ 𝑏 ∈ dom 𝑆 ( ( ♯ ‘ ( 𝑆 ‘ 𝑎 ) ) < ( ♯ ‘ ( 𝑆 ‘ 𝐵 ) ) → ( ( 𝑆 ‘ 𝑎 ) = ( 𝑆 ‘ 𝑏 ) → ( 𝑎 ‘ 0 ) = ( 𝑏 ‘ 0 ) ) ) )
27 10 eqcomd ⊢ ( 𝜑 → ( 𝑆 ‘ 𝐵 ) = ( 𝑆 ‘ 𝐴 ) )
28 eqcom ⊢ ( ( 𝐴 ‘ 0 ) = ( 𝐵 ‘ 0 ) ↔ ( 𝐵 ‘ 0 ) = ( 𝐴 ‘ 0 ) )
29 11 28 sylnib ⊢ ( 𝜑 → ¬ ( 𝐵 ‘ 0 ) = ( 𝐴 ‘ 0 ) )
30 eqcom ⊢ ( ( 𝐴 ‘ 𝐾 ) = ( 𝐵 ‘ 𝐿 ) ↔ ( 𝐵 ‘ 𝐿 ) = ( 𝐴 ‘ 𝐾 ) )
31 20 30 sylnib ⊢ ( 𝜑 → ¬ ( 𝐵 ‘ 𝐿 ) = ( 𝐴 ‘ 𝐾 ) )
32 1 2 3 4 5 6 26 9 8 27 29 13 12 15 14 17 16 19 18 31 efgredlemc ⊢ ( 𝜑 → ( 𝑄 ∈ ( ℤ≥ ‘ 𝑃 ) → ( 𝐵 ‘ 0 ) = ( 𝐴 ‘ 0 ) ) )
33 32 28 imbitrrdi ⊢ ( 𝜑 → ( 𝑄 ∈ ( ℤ≥ ‘ 𝑃 ) → ( 𝐴 ‘ 0 ) = ( 𝐵 ‘ 0 ) ) )
34 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 efgredlemc ⊢ ( 𝜑 → ( 𝑃 ∈ ( ℤ≥ ‘ 𝑄 ) → ( 𝐴 ‘ 0 ) = ( 𝐵 ‘ 0 ) ) )
35 14 elfzelzd ⊢ ( 𝜑 → 𝑃 ∈ ℤ )
36 15 elfzelzd ⊢ ( 𝜑 → 𝑄 ∈ ℤ )
37 uztric ⊢ ( ( 𝑃 ∈ ℤ ∧ 𝑄 ∈ ℤ ) → ( 𝑄 ∈ ( ℤ≥ ‘ 𝑃 ) ∨ 𝑃 ∈ ( ℤ≥ ‘ 𝑄 ) ) )
38 35 36 37 syl2anc ⊢ ( 𝜑 → ( 𝑄 ∈ ( ℤ≥ ‘ 𝑃 ) ∨ 𝑃 ∈ ( ℤ≥ ‘ 𝑄 ) ) )
39 33 34 38 mpjaod ⊢ ( 𝜑 → ( 𝐴 ‘ 0 ) = ( 𝐵 ‘ 0 ) )
40 39 11 pm2.65i ⊢ ¬ 𝜑