Metamath Proof Explorer

Theorem efgs1b

Description: Every extension sequence ending in an irreducible word is trivial. (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 efgs1b ( 𝐴 ∈ dom 𝑆 → ( ( 𝑆 ‘ 𝐴 ) ∈ 𝐷 ↔ ( ♯ ‘ 𝐴 ) = 1 ) )

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 eldifn ⊢ ( ( 𝑆 ‘ 𝐴 ) ∈ ( 𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran ( 𝑇 ‘ 𝑥 ) ) → ¬ ( 𝑆 ‘ 𝐴 ) ∈ ∪ 𝑥 ∈ 𝑊 ran ( 𝑇 ‘ 𝑥 ) )
8 7 5 eleq2s ⊢ ( ( 𝑆 ‘ 𝐴 ) ∈ 𝐷 → ¬ ( 𝑆 ‘ 𝐴 ) ∈ ∪ 𝑥 ∈ 𝑊 ran ( 𝑇 ‘ 𝑥 ) )
9 1 2 3 4 5 6 efgsdm ⊢ ( 𝐴 ∈ dom 𝑆 ↔ ( 𝐴 ∈ ( Word 𝑊 ∖ { ∅ } ) ∧ ( 𝐴 ‘ 0 ) ∈ 𝐷 ∧ ∀ 𝑎 ∈ ( 1 ..^ ( ♯ ‘ 𝐴 ) ) ( 𝐴 ‘ 𝑎 ) ∈ ran ( 𝑇 ‘ ( 𝐴 ‘ ( 𝑎 − 1 ) ) ) ) )
10 9 simp1bi ⊢ ( 𝐴 ∈ dom 𝑆 → 𝐴 ∈ ( Word 𝑊 ∖ { ∅ } ) )
11 eldifsn ⊢ ( 𝐴 ∈ ( Word 𝑊 ∖ { ∅ } ) ↔ ( 𝐴 ∈ Word 𝑊 ∧ 𝐴 ≠ ∅ ) )
12 lennncl ⊢ ( ( 𝐴 ∈ Word 𝑊 ∧ 𝐴 ≠ ∅ ) → ( ♯ ‘ 𝐴 ) ∈ ℕ )
13 11 12 sylbi ⊢ ( 𝐴 ∈ ( Word 𝑊 ∖ { ∅ } ) → ( ♯ ‘ 𝐴 ) ∈ ℕ )
14 10 13 syl ⊢ ( 𝐴 ∈ dom 𝑆 → ( ♯ ‘ 𝐴 ) ∈ ℕ )
15 elnn1uz2 ⊢ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ↔ ( ( ♯ ‘ 𝐴 ) = 1 ∨ ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 2 ) ) )
16 14 15 sylib ⊢ ( 𝐴 ∈ dom 𝑆 → ( ( ♯ ‘ 𝐴 ) = 1 ∨ ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 2 ) ) )
17 16 ord ⊢ ( 𝐴 ∈ dom 𝑆 → ( ¬ ( ♯ ‘ 𝐴 ) = 1 → ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 2 ) ) )
18 10 eldifad ⊢ ( 𝐴 ∈ dom 𝑆 → 𝐴 ∈ Word 𝑊 )
19 18 adantr ⊢ ( ( 𝐴 ∈ dom 𝑆 ∧ ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 2 ) ) → 𝐴 ∈ Word 𝑊 )
20 wrdf ⊢ ( 𝐴 ∈ Word 𝑊 → 𝐴 : ( 0 ..^ ( ♯ ‘ 𝐴 ) ) ⟶ 𝑊 )
21 19 20 syl ⊢ ( ( 𝐴 ∈ dom 𝑆 ∧ ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 2 ) ) → 𝐴 : ( 0 ..^ ( ♯ ‘ 𝐴 ) ) ⟶ 𝑊 )
22 1z ⊢ 1 ∈ ℤ
23 simpr ⊢ ( ( 𝐴 ∈ dom 𝑆 ∧ ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 2 ) ) → ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 2 ) )
24 df-2 ⊢ 2 = ( 1 + 1 )
25 24 fveq2i ⊢ ( ℤ≥ ‘ 2 ) = ( ℤ≥ ‘ ( 1 + 1 ) )
26 23 25 eleqtrdi ⊢ ( ( 𝐴 ∈ dom 𝑆 ∧ ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 2 ) ) → ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ ( 1 + 1 ) ) )
27 eluzp1m1 ⊢ ( ( 1 ∈ ℤ ∧ ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ ( 1 + 1 ) ) ) → ( ( ♯ ‘ 𝐴 ) − 1 ) ∈ ( ℤ≥ ‘ 1 ) )
28 22 26 27 sylancr ⊢ ( ( 𝐴 ∈ dom 𝑆 ∧ ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 2 ) ) → ( ( ♯ ‘ 𝐴 ) − 1 ) ∈ ( ℤ≥ ‘ 1 ) )
29 nnuz ⊢ ℕ = ( ℤ≥ ‘ 1 )
30 28 29 eleqtrrdi ⊢ ( ( 𝐴 ∈ dom 𝑆 ∧ ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 2 ) ) → ( ( ♯ ‘ 𝐴 ) − 1 ) ∈ ℕ )
31 lbfzo0 ⊢ ( 0 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐴 ) − 1 ) ) ↔ ( ( ♯ ‘ 𝐴 ) − 1 ) ∈ ℕ )
32 30 31 sylibr ⊢ ( ( 𝐴 ∈ dom 𝑆 ∧ ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 2 ) ) → 0 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐴 ) − 1 ) ) )
33 fzoend ⊢ ( 0 ∈ ( 0 ..^ ( ( ♯ ‘ 𝐴 ) − 1 ) ) → ( ( ( ♯ ‘ 𝐴 ) − 1 ) − 1 ) ∈ ( 0 ..^ ( ( ♯ ‘ 𝐴 ) − 1 ) ) )
34 elfzofz ⊢ ( ( ( ( ♯ ‘ 𝐴 ) − 1 ) − 1 ) ∈ ( 0 ..^ ( ( ♯ ‘ 𝐴 ) − 1 ) ) → ( ( ( ♯ ‘ 𝐴 ) − 1 ) − 1 ) ∈ ( 0 ... ( ( ♯ ‘ 𝐴 ) − 1 ) ) )
35 32 33 34 3syl ⊢ ( ( 𝐴 ∈ dom 𝑆 ∧ ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 2 ) ) → ( ( ( ♯ ‘ 𝐴 ) − 1 ) − 1 ) ∈ ( 0 ... ( ( ♯ ‘ 𝐴 ) − 1 ) ) )
36 eluzelz ⊢ ( ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 2 ) → ( ♯ ‘ 𝐴 ) ∈ ℤ )
37 36 adantl ⊢ ( ( 𝐴 ∈ dom 𝑆 ∧ ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 2 ) ) → ( ♯ ‘ 𝐴 ) ∈ ℤ )
38 fzoval ⊢ ( ( ♯ ‘ 𝐴 ) ∈ ℤ → ( 0 ..^ ( ♯ ‘ 𝐴 ) ) = ( 0 ... ( ( ♯ ‘ 𝐴 ) − 1 ) ) )
39 37 38 syl ⊢ ( ( 𝐴 ∈ dom 𝑆 ∧ ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 2 ) ) → ( 0 ..^ ( ♯ ‘ 𝐴 ) ) = ( 0 ... ( ( ♯ ‘ 𝐴 ) − 1 ) ) )
40 35 39 eleqtrrd ⊢ ( ( 𝐴 ∈ dom 𝑆 ∧ ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 2 ) ) → ( ( ( ♯ ‘ 𝐴 ) − 1 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐴 ) ) )
41 21 40 ffvelcdmd ⊢ ( ( 𝐴 ∈ dom 𝑆 ∧ ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 2 ) ) → ( 𝐴 ‘ ( ( ( ♯ ‘ 𝐴 ) − 1 ) − 1 ) ) ∈ 𝑊 )
42 uz2m1nn ⊢ ( ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 2 ) → ( ( ♯ ‘ 𝐴 ) − 1 ) ∈ ℕ )
43 1 2 3 4 5 6 efgsdmi ⊢ ( ( 𝐴 ∈ dom 𝑆 ∧ ( ( ♯ ‘ 𝐴 ) − 1 ) ∈ ℕ ) → ( 𝑆 ‘ 𝐴 ) ∈ ran ( 𝑇 ‘ ( 𝐴 ‘ ( ( ( ♯ ‘ 𝐴 ) − 1 ) − 1 ) ) ) )
44 42 43 sylan2 ⊢ ( ( 𝐴 ∈ dom 𝑆 ∧ ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 2 ) ) → ( 𝑆 ‘ 𝐴 ) ∈ ran ( 𝑇 ‘ ( 𝐴 ‘ ( ( ( ♯ ‘ 𝐴 ) − 1 ) − 1 ) ) ) )
45 fveq2 ⊢ ( 𝑎 = ( 𝐴 ‘ ( ( ( ♯ ‘ 𝐴 ) − 1 ) − 1 ) ) → ( 𝑇 ‘ 𝑎 ) = ( 𝑇 ‘ ( 𝐴 ‘ ( ( ( ♯ ‘ 𝐴 ) − 1 ) − 1 ) ) ) )
46 45 rneqd ⊢ ( 𝑎 = ( 𝐴 ‘ ( ( ( ♯ ‘ 𝐴 ) − 1 ) − 1 ) ) → ran ( 𝑇 ‘ 𝑎 ) = ran ( 𝑇 ‘ ( 𝐴 ‘ ( ( ( ♯ ‘ 𝐴 ) − 1 ) − 1 ) ) ) )
47 46 eliuni ⊢ ( ( ( 𝐴 ‘ ( ( ( ♯ ‘ 𝐴 ) − 1 ) − 1 ) ) ∈ 𝑊 ∧ ( 𝑆 ‘ 𝐴 ) ∈ ran ( 𝑇 ‘ ( 𝐴 ‘ ( ( ( ♯ ‘ 𝐴 ) − 1 ) − 1 ) ) ) ) → ( 𝑆 ‘ 𝐴 ) ∈ ∪ 𝑎 ∈ 𝑊 ran ( 𝑇 ‘ 𝑎 ) )
48 41 44 47 syl2anc ⊢ ( ( 𝐴 ∈ dom 𝑆 ∧ ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 2 ) ) → ( 𝑆 ‘ 𝐴 ) ∈ ∪ 𝑎 ∈ 𝑊 ran ( 𝑇 ‘ 𝑎 ) )
49 fveq2 ⊢ ( 𝑎 = 𝑥 → ( 𝑇 ‘ 𝑎 ) = ( 𝑇 ‘ 𝑥 ) )
50 49 rneqd ⊢ ( 𝑎 = 𝑥 → ran ( 𝑇 ‘ 𝑎 ) = ran ( 𝑇 ‘ 𝑥 ) )
51 50 cbviunv ⊢ ∪ 𝑎 ∈ 𝑊 ran ( 𝑇 ‘ 𝑎 ) = ∪ 𝑥 ∈ 𝑊 ran ( 𝑇 ‘ 𝑥 )
52 48 51 eleqtrdi ⊢ ( ( 𝐴 ∈ dom 𝑆 ∧ ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 2 ) ) → ( 𝑆 ‘ 𝐴 ) ∈ ∪ 𝑥 ∈ 𝑊 ran ( 𝑇 ‘ 𝑥 ) )
53 52 ex ⊢ ( 𝐴 ∈ dom 𝑆 → ( ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 2 ) → ( 𝑆 ‘ 𝐴 ) ∈ ∪ 𝑥 ∈ 𝑊 ran ( 𝑇 ‘ 𝑥 ) ) )
54 17 53 syld ⊢ ( 𝐴 ∈ dom 𝑆 → ( ¬ ( ♯ ‘ 𝐴 ) = 1 → ( 𝑆 ‘ 𝐴 ) ∈ ∪ 𝑥 ∈ 𝑊 ran ( 𝑇 ‘ 𝑥 ) ) )
55 54 con1d ⊢ ( 𝐴 ∈ dom 𝑆 → ( ¬ ( 𝑆 ‘ 𝐴 ) ∈ ∪ 𝑥 ∈ 𝑊 ran ( 𝑇 ‘ 𝑥 ) → ( ♯ ‘ 𝐴 ) = 1 ) )
56 8 55 syl5 ⊢ ( 𝐴 ∈ dom 𝑆 → ( ( 𝑆 ‘ 𝐴 ) ∈ 𝐷 → ( ♯ ‘ 𝐴 ) = 1 ) )
57 9 simp2bi ⊢ ( 𝐴 ∈ dom 𝑆 → ( 𝐴 ‘ 0 ) ∈ 𝐷 )
58 oveq1 ⊢ ( ( ♯ ‘ 𝐴 ) = 1 → ( ( ♯ ‘ 𝐴 ) − 1 ) = ( 1 − 1 ) )
59 1m1e0 ⊢ ( 1 − 1 ) = 0
60 58 59 eqtrdi ⊢ ( ( ♯ ‘ 𝐴 ) = 1 → ( ( ♯ ‘ 𝐴 ) − 1 ) = 0 )
61 60 fveq2d ⊢ ( ( ♯ ‘ 𝐴 ) = 1 → ( 𝐴 ‘ ( ( ♯ ‘ 𝐴 ) − 1 ) ) = ( 𝐴 ‘ 0 ) )
62 61 eleq1d ⊢ ( ( ♯ ‘ 𝐴 ) = 1 → ( ( 𝐴 ‘ ( ( ♯ ‘ 𝐴 ) − 1 ) ) ∈ 𝐷 ↔ ( 𝐴 ‘ 0 ) ∈ 𝐷 ) )
63 57 62 syl5ibrcom ⊢ ( 𝐴 ∈ dom 𝑆 → ( ( ♯ ‘ 𝐴 ) = 1 → ( 𝐴 ‘ ( ( ♯ ‘ 𝐴 ) − 1 ) ) ∈ 𝐷 ) )
64 1 2 3 4 5 6 efgsval ⊢ ( 𝐴 ∈ dom 𝑆 → ( 𝑆 ‘ 𝐴 ) = ( 𝐴 ‘ ( ( ♯ ‘ 𝐴 ) − 1 ) ) )
65 64 eleq1d ⊢ ( 𝐴 ∈ dom 𝑆 → ( ( 𝑆 ‘ 𝐴 ) ∈ 𝐷 ↔ ( 𝐴 ‘ ( ( ♯ ‘ 𝐴 ) − 1 ) ) ∈ 𝐷 ) )
66 63 65 sylibrd ⊢ ( 𝐴 ∈ dom 𝑆 → ( ( ♯ ‘ 𝐴 ) = 1 → ( 𝑆 ‘ 𝐴 ) ∈ 𝐷 ) )
67 56 66 impbid ⊢ ( 𝐴 ∈ dom 𝑆 → ( ( 𝑆 ‘ 𝐴 ) ∈ 𝐷 ↔ ( ♯ ‘ 𝐴 ) = 1 ) )