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Metamath Proof Explorer

Theorem efgredlemf

Description: Lemma for efgredleme . (Contributed by Mario Carneiro, 4-Jun-2016)

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 )
Assertion efgredlemf ( 𝜑 → ( ( 𝐴 ‘ 𝐾 ) ∈ 𝑊 ∧ ( 𝐵 ‘ 𝐿 ) ∈ 𝑊 ) )

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 1 2 3 4 5 6 efgsdm ⊢ ( 𝐴 ∈ dom 𝑆 ↔ ( 𝐴 ∈ ( Word 𝑊 ∖ { ∅ } ) ∧ ( 𝐴 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝐴 ) ) ( 𝐴 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝐴 ‘ ( 𝑖 − 1 ) ) ) ) )
15 14 simp1bi ⊢ ( 𝐴 ∈ dom 𝑆 → 𝐴 ∈ ( Word 𝑊 ∖ { ∅ } ) )
16 8 15 syl ⊢ ( 𝜑 → 𝐴 ∈ ( Word 𝑊 ∖ { ∅ } ) )
17 16 eldifad ⊢ ( 𝜑 → 𝐴 ∈ Word 𝑊 )
18 wrdf ⊢ ( 𝐴 ∈ Word 𝑊 → 𝐴 : ( 0 ..^ ( ♯ ‘ 𝐴 ) ) ⟶ 𝑊 )
19 17 18 syl ⊢ ( 𝜑 → 𝐴 : ( 0 ..^ ( ♯ ‘ 𝐴 ) ) ⟶ 𝑊 )
20 fzossfz ⊢ ( 0 ..^ ( ( ♯ ‘ 𝐴 ) − 1 ) ) ⊆ ( 0 ... ( ( ♯ ‘ 𝐴 ) − 1 ) )
21 lencl ⊢ ( 𝐴 ∈ Word 𝑊 → ( ♯ ‘ 𝐴 ) ∈ ℕ0 )
22 17 21 syl ⊢ ( 𝜑 → ( ♯ ‘ 𝐴 ) ∈ ℕ0 )
23 22 nn0zd ⊢ ( 𝜑 → ( ♯ ‘ 𝐴 ) ∈ ℤ )
24 fzoval ⊢ ( ( ♯ ‘ 𝐴 ) ∈ ℤ → ( 0 ..^ ( ♯ ‘ 𝐴 ) ) = ( 0 ... ( ( ♯ ‘ 𝐴 ) − 1 ) ) )
25 23 24 syl ⊢ ( 𝜑 → ( 0 ..^ ( ♯ ‘ 𝐴 ) ) = ( 0 ... ( ( ♯ ‘ 𝐴 ) − 1 ) ) )
26 20 25 sseqtrrid ⊢ ( 𝜑 → ( 0 ..^ ( ( ♯ ‘ 𝐴 ) − 1 ) ) ⊆ ( 0 ..^ ( ♯ ‘ 𝐴 ) ) )
27 1 2 3 4 5 6 7 8 9 10 11 efgredlema ⊢ ( 𝜑 → ( ( ( ♯ ‘ 𝐴 ) − 1 ) ∈ ℕ ∧ ( ( ♯ ‘ 𝐵 ) − 1 ) ∈ ℕ ) )
28 27 simpld ⊢ ( 𝜑 → ( ( ♯ ‘ 𝐴 ) − 1 ) ∈ ℕ )
29 fzo0end ⊢ ( ( ( ♯ ‘ 𝐴 ) − 1 ) ∈ ℕ → ( ( ( ♯ ‘ 𝐴 ) − 1 ) − 1 ) ∈ ( 0 ..^ ( ( ♯ ‘ 𝐴 ) − 1 ) ) )
30 28 29 syl ⊢ ( 𝜑 → ( ( ( ♯ ‘ 𝐴 ) − 1 ) − 1 ) ∈ ( 0 ..^ ( ( ♯ ‘ 𝐴 ) − 1 ) ) )
31 12 30 eqeltrid ⊢ ( 𝜑 → 𝐾 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐴 ) − 1 ) ) )
32 26 31 sseldd ⊢ ( 𝜑 → 𝐾 ∈ ( 0 ..^ ( ♯ ‘ 𝐴 ) ) )
33 19 32 ffvelcdmd ⊢ ( 𝜑 → ( 𝐴 ‘ 𝐾 ) ∈ 𝑊 )
34 1 2 3 4 5 6 efgsdm ⊢ ( 𝐵 ∈ dom 𝑆 ↔ ( 𝐵 ∈ ( Word 𝑊 ∖ { ∅ } ) ∧ ( 𝐵 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑖 ∈ ( 1 ..^ ( ♯ ‘ 𝐵 ) ) ( 𝐵 ‘ 𝑖 ) ∈ ran ( 𝑇 ‘ ( 𝐵 ‘ ( 𝑖 − 1 ) ) ) ) )
35 34 simp1bi ⊢ ( 𝐵 ∈ dom 𝑆 → 𝐵 ∈ ( Word 𝑊 ∖ { ∅ } ) )
36 9 35 syl ⊢ ( 𝜑 → 𝐵 ∈ ( Word 𝑊 ∖ { ∅ } ) )
37 36 eldifad ⊢ ( 𝜑 → 𝐵 ∈ Word 𝑊 )
38 wrdf ⊢ ( 𝐵 ∈ Word 𝑊 → 𝐵 : ( 0 ..^ ( ♯ ‘ 𝐵 ) ) ⟶ 𝑊 )
39 37 38 syl ⊢ ( 𝜑 → 𝐵 : ( 0 ..^ ( ♯ ‘ 𝐵 ) ) ⟶ 𝑊 )
40 fzossfz ⊢ ( 0 ..^ ( ( ♯ ‘ 𝐵 ) − 1 ) ) ⊆ ( 0 ... ( ( ♯ ‘ 𝐵 ) − 1 ) )
41 lencl ⊢ ( 𝐵 ∈ Word 𝑊 → ( ♯ ‘ 𝐵 ) ∈ ℕ0 )
42 37 41 syl ⊢ ( 𝜑 → ( ♯ ‘ 𝐵 ) ∈ ℕ0 )
43 42 nn0zd ⊢ ( 𝜑 → ( ♯ ‘ 𝐵 ) ∈ ℤ )
44 fzoval ⊢ ( ( ♯ ‘ 𝐵 ) ∈ ℤ → ( 0 ..^ ( ♯ ‘ 𝐵 ) ) = ( 0 ... ( ( ♯ ‘ 𝐵 ) − 1 ) ) )
45 43 44 syl ⊢ ( 𝜑 → ( 0 ..^ ( ♯ ‘ 𝐵 ) ) = ( 0 ... ( ( ♯ ‘ 𝐵 ) − 1 ) ) )
46 40 45 sseqtrrid ⊢ ( 𝜑 → ( 0 ..^ ( ( ♯ ‘ 𝐵 ) − 1 ) ) ⊆ ( 0 ..^ ( ♯ ‘ 𝐵 ) ) )
47 fzo0end ⊢ ( ( ( ♯ ‘ 𝐵 ) − 1 ) ∈ ℕ → ( ( ( ♯ ‘ 𝐵 ) − 1 ) − 1 ) ∈ ( 0 ..^ ( ( ♯ ‘ 𝐵 ) − 1 ) ) )
48 27 47 simpl2im ⊢ ( 𝜑 → ( ( ( ♯ ‘ 𝐵 ) − 1 ) − 1 ) ∈ ( 0 ..^ ( ( ♯ ‘ 𝐵 ) − 1 ) ) )
49 13 48 eqeltrid ⊢ ( 𝜑 → 𝐿 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐵 ) − 1 ) ) )
50 46 49 sseldd ⊢ ( 𝜑 → 𝐿 ∈ ( 0 ..^ ( ♯ ‘ 𝐵 ) ) )
51 39 50 ffvelcdmd ⊢ ( 𝜑 → ( 𝐵 ‘ 𝐿 ) ∈ 𝑊 )
52 33 51 jca ⊢ ( 𝜑 → ( ( 𝐴 ‘ 𝐾 ) ∈ 𝑊 ∧ ( 𝐵 ‘ 𝐿 ) ∈ 𝑊 ) )