Step |
Hyp |
Ref |
Expression |
1 |
|
wrdt2ind.1 |
⊢ ( 𝑥 = ∅ → ( 𝜑 ↔ 𝜓 ) ) |
2 |
|
wrdt2ind.2 |
⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜒 ) ) |
3 |
|
wrdt2ind.3 |
⊢ ( 𝑥 = ( 𝑦 ++ 〈“ 𝑖 𝑗 ”〉 ) → ( 𝜑 ↔ 𝜃 ) ) |
4 |
|
wrdt2ind.4 |
⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜏 ) ) |
5 |
|
wrdt2ind.5 |
⊢ 𝜓 |
6 |
|
wrdt2ind.6 |
⊢ ( ( 𝑦 ∈ Word 𝐵 ∧ 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) → ( 𝜒 → 𝜃 ) ) |
7 |
|
oveq2 |
⊢ ( 𝑛 = 0 → ( 2 · 𝑛 ) = ( 2 · 0 ) ) |
8 |
7
|
eqeq1d |
⊢ ( 𝑛 = 0 → ( ( 2 · 𝑛 ) = ( ♯ ‘ 𝑥 ) ↔ ( 2 · 0 ) = ( ♯ ‘ 𝑥 ) ) ) |
9 |
8
|
imbi1d |
⊢ ( 𝑛 = 0 → ( ( ( 2 · 𝑛 ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) ↔ ( ( 2 · 0 ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) ) ) |
10 |
9
|
ralbidv |
⊢ ( 𝑛 = 0 → ( ∀ 𝑥 ∈ Word 𝐵 ( ( 2 · 𝑛 ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) ↔ ∀ 𝑥 ∈ Word 𝐵 ( ( 2 · 0 ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) ) ) |
11 |
|
oveq2 |
⊢ ( 𝑛 = 𝑘 → ( 2 · 𝑛 ) = ( 2 · 𝑘 ) ) |
12 |
11
|
eqeq1d |
⊢ ( 𝑛 = 𝑘 → ( ( 2 · 𝑛 ) = ( ♯ ‘ 𝑥 ) ↔ ( 2 · 𝑘 ) = ( ♯ ‘ 𝑥 ) ) ) |
13 |
12
|
imbi1d |
⊢ ( 𝑛 = 𝑘 → ( ( ( 2 · 𝑛 ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) ↔ ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) ) ) |
14 |
13
|
ralbidv |
⊢ ( 𝑛 = 𝑘 → ( ∀ 𝑥 ∈ Word 𝐵 ( ( 2 · 𝑛 ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) ↔ ∀ 𝑥 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) ) ) |
15 |
|
oveq2 |
⊢ ( 𝑛 = ( 𝑘 + 1 ) → ( 2 · 𝑛 ) = ( 2 · ( 𝑘 + 1 ) ) ) |
16 |
15
|
eqeq1d |
⊢ ( 𝑛 = ( 𝑘 + 1 ) → ( ( 2 · 𝑛 ) = ( ♯ ‘ 𝑥 ) ↔ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) |
17 |
16
|
imbi1d |
⊢ ( 𝑛 = ( 𝑘 + 1 ) → ( ( ( 2 · 𝑛 ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) ↔ ( ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) ) ) |
18 |
17
|
ralbidv |
⊢ ( 𝑛 = ( 𝑘 + 1 ) → ( ∀ 𝑥 ∈ Word 𝐵 ( ( 2 · 𝑛 ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) ↔ ∀ 𝑥 ∈ Word 𝐵 ( ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) ) ) |
19 |
|
oveq2 |
⊢ ( 𝑛 = 𝑚 → ( 2 · 𝑛 ) = ( 2 · 𝑚 ) ) |
20 |
19
|
eqeq1d |
⊢ ( 𝑛 = 𝑚 → ( ( 2 · 𝑛 ) = ( ♯ ‘ 𝑥 ) ↔ ( 2 · 𝑚 ) = ( ♯ ‘ 𝑥 ) ) ) |
21 |
20
|
imbi1d |
⊢ ( 𝑛 = 𝑚 → ( ( ( 2 · 𝑛 ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) ↔ ( ( 2 · 𝑚 ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) ) ) |
22 |
21
|
ralbidv |
⊢ ( 𝑛 = 𝑚 → ( ∀ 𝑥 ∈ Word 𝐵 ( ( 2 · 𝑛 ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) ↔ ∀ 𝑥 ∈ Word 𝐵 ( ( 2 · 𝑚 ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) ) ) |
23 |
|
2t0e0 |
⊢ ( 2 · 0 ) = 0 |
24 |
23
|
eqeq1i |
⊢ ( ( 2 · 0 ) = ( ♯ ‘ 𝑥 ) ↔ 0 = ( ♯ ‘ 𝑥 ) ) |
25 |
|
eqcom |
⊢ ( 0 = ( ♯ ‘ 𝑥 ) ↔ ( ♯ ‘ 𝑥 ) = 0 ) |
26 |
24 25
|
bitri |
⊢ ( ( 2 · 0 ) = ( ♯ ‘ 𝑥 ) ↔ ( ♯ ‘ 𝑥 ) = 0 ) |
27 |
|
hasheq0 |
⊢ ( 𝑥 ∈ Word 𝐵 → ( ( ♯ ‘ 𝑥 ) = 0 ↔ 𝑥 = ∅ ) ) |
28 |
26 27
|
syl5bb |
⊢ ( 𝑥 ∈ Word 𝐵 → ( ( 2 · 0 ) = ( ♯ ‘ 𝑥 ) ↔ 𝑥 = ∅ ) ) |
29 |
5 1
|
mpbiri |
⊢ ( 𝑥 = ∅ → 𝜑 ) |
30 |
28 29
|
syl6bi |
⊢ ( 𝑥 ∈ Word 𝐵 → ( ( 2 · 0 ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) ) |
31 |
30
|
rgen |
⊢ ∀ 𝑥 ∈ Word 𝐵 ( ( 2 · 0 ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) |
32 |
|
fveq2 |
⊢ ( 𝑥 = 𝑦 → ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑦 ) ) |
33 |
32
|
eqeq2d |
⊢ ( 𝑥 = 𝑦 → ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑥 ) ↔ ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) ) ) |
34 |
33 2
|
imbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) ↔ ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ) |
35 |
34
|
cbvralvw |
⊢ ( ∀ 𝑥 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) ↔ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) |
36 |
|
simprl |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → 𝑥 ∈ Word 𝐵 ) |
37 |
|
0zd |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → 0 ∈ ℤ ) |
38 |
|
lencl |
⊢ ( 𝑥 ∈ Word 𝐵 → ( ♯ ‘ 𝑥 ) ∈ ℕ0 ) |
39 |
36 38
|
syl |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( ♯ ‘ 𝑥 ) ∈ ℕ0 ) |
40 |
39
|
nn0zd |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( ♯ ‘ 𝑥 ) ∈ ℤ ) |
41 |
|
2z |
⊢ 2 ∈ ℤ |
42 |
41
|
a1i |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → 2 ∈ ℤ ) |
43 |
40 42
|
zsubcld |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( ( ♯ ‘ 𝑥 ) − 2 ) ∈ ℤ ) |
44 |
|
2re |
⊢ 2 ∈ ℝ |
45 |
44
|
a1i |
⊢ ( 𝑘 ∈ ℕ0 → 2 ∈ ℝ ) |
46 |
|
nn0re |
⊢ ( 𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ ) |
47 |
|
0le2 |
⊢ 0 ≤ 2 |
48 |
47
|
a1i |
⊢ ( 𝑘 ∈ ℕ0 → 0 ≤ 2 ) |
49 |
|
nn0ge0 |
⊢ ( 𝑘 ∈ ℕ0 → 0 ≤ 𝑘 ) |
50 |
45 46 48 49
|
mulge0d |
⊢ ( 𝑘 ∈ ℕ0 → 0 ≤ ( 2 · 𝑘 ) ) |
51 |
50
|
adantr |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → 0 ≤ ( 2 · 𝑘 ) ) |
52 |
|
2cnd |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → 2 ∈ ℂ ) |
53 |
|
simpl |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → 𝑘 ∈ ℕ0 ) |
54 |
53
|
nn0cnd |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → 𝑘 ∈ ℂ ) |
55 |
|
1cnd |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → 1 ∈ ℂ ) |
56 |
52 54 55
|
adddid |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( 2 · ( 𝑘 + 1 ) ) = ( ( 2 · 𝑘 ) + ( 2 · 1 ) ) ) |
57 |
|
simprr |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) |
58 |
|
2t1e2 |
⊢ ( 2 · 1 ) = 2 |
59 |
58
|
a1i |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( 2 · 1 ) = 2 ) |
60 |
59
|
oveq2d |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( ( 2 · 𝑘 ) + ( 2 · 1 ) ) = ( ( 2 · 𝑘 ) + 2 ) ) |
61 |
56 57 60
|
3eqtr3d |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( ♯ ‘ 𝑥 ) = ( ( 2 · 𝑘 ) + 2 ) ) |
62 |
61
|
oveq1d |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( ( ♯ ‘ 𝑥 ) − 2 ) = ( ( ( 2 · 𝑘 ) + 2 ) − 2 ) ) |
63 |
52 54
|
mulcld |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( 2 · 𝑘 ) ∈ ℂ ) |
64 |
63 52
|
pncand |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( ( ( 2 · 𝑘 ) + 2 ) − 2 ) = ( 2 · 𝑘 ) ) |
65 |
62 64
|
eqtrd |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( ( ♯ ‘ 𝑥 ) − 2 ) = ( 2 · 𝑘 ) ) |
66 |
51 65
|
breqtrrd |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → 0 ≤ ( ( ♯ ‘ 𝑥 ) − 2 ) ) |
67 |
43
|
zred |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( ( ♯ ‘ 𝑥 ) − 2 ) ∈ ℝ ) |
68 |
39
|
nn0red |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( ♯ ‘ 𝑥 ) ∈ ℝ ) |
69 |
|
2pos |
⊢ 0 < 2 |
70 |
44
|
a1i |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → 2 ∈ ℝ ) |
71 |
70 68
|
ltsubposd |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( 0 < 2 ↔ ( ( ♯ ‘ 𝑥 ) − 2 ) < ( ♯ ‘ 𝑥 ) ) ) |
72 |
69 71
|
mpbii |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( ( ♯ ‘ 𝑥 ) − 2 ) < ( ♯ ‘ 𝑥 ) ) |
73 |
67 68 72
|
ltled |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( ( ♯ ‘ 𝑥 ) − 2 ) ≤ ( ♯ ‘ 𝑥 ) ) |
74 |
|
elfz2 |
⊢ ( ( ( ♯ ‘ 𝑥 ) − 2 ) ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) ↔ ( ( 0 ∈ ℤ ∧ ( ♯ ‘ 𝑥 ) ∈ ℤ ∧ ( ( ♯ ‘ 𝑥 ) − 2 ) ∈ ℤ ) ∧ ( 0 ≤ ( ( ♯ ‘ 𝑥 ) − 2 ) ∧ ( ( ♯ ‘ 𝑥 ) − 2 ) ≤ ( ♯ ‘ 𝑥 ) ) ) ) |
75 |
74
|
biimpri |
⊢ ( ( ( 0 ∈ ℤ ∧ ( ♯ ‘ 𝑥 ) ∈ ℤ ∧ ( ( ♯ ‘ 𝑥 ) − 2 ) ∈ ℤ ) ∧ ( 0 ≤ ( ( ♯ ‘ 𝑥 ) − 2 ) ∧ ( ( ♯ ‘ 𝑥 ) − 2 ) ≤ ( ♯ ‘ 𝑥 ) ) ) → ( ( ♯ ‘ 𝑥 ) − 2 ) ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) ) |
76 |
37 40 43 66 73 75
|
syl32anc |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( ( ♯ ‘ 𝑥 ) − 2 ) ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) ) |
77 |
|
pfxlen |
⊢ ( ( 𝑥 ∈ Word 𝐵 ∧ ( ( ♯ ‘ 𝑥 ) − 2 ) ∈ ( 0 ... ( ♯ ‘ 𝑥 ) ) ) → ( ♯ ‘ ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ) = ( ( ♯ ‘ 𝑥 ) − 2 ) ) |
78 |
36 76 77
|
syl2anc |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( ♯ ‘ ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ) = ( ( ♯ ‘ 𝑥 ) − 2 ) ) |
79 |
78 65
|
eqtr2d |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( 2 · 𝑘 ) = ( ♯ ‘ ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ) ) |
80 |
79
|
adantlr |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( 2 · 𝑘 ) = ( ♯ ‘ ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ) ) |
81 |
|
fveq2 |
⊢ ( 𝑦 = ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) → ( ♯ ‘ 𝑦 ) = ( ♯ ‘ ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ) ) |
82 |
81
|
eqeq2d |
⊢ ( 𝑦 = ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) → ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) ↔ ( 2 · 𝑘 ) = ( ♯ ‘ ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ) ) ) |
83 |
|
vex |
⊢ 𝑦 ∈ V |
84 |
83 2
|
sbcie |
⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜒 ) |
85 |
|
dfsbcq |
⊢ ( 𝑦 = ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) → ( [ 𝑦 / 𝑥 ] 𝜑 ↔ [ ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) / 𝑥 ] 𝜑 ) ) |
86 |
84 85
|
bitr3id |
⊢ ( 𝑦 = ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) → ( 𝜒 ↔ [ ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) / 𝑥 ] 𝜑 ) ) |
87 |
82 86
|
imbi12d |
⊢ ( 𝑦 = ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) → ( ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ↔ ( ( 2 · 𝑘 ) = ( ♯ ‘ ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ) → [ ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) / 𝑥 ] 𝜑 ) ) ) |
88 |
|
simplr |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) |
89 |
|
pfxcl |
⊢ ( 𝑥 ∈ Word 𝐵 → ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ∈ Word 𝐵 ) |
90 |
89
|
ad2antrl |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ∈ Word 𝐵 ) |
91 |
87 88 90
|
rspcdva |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( ( 2 · 𝑘 ) = ( ♯ ‘ ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ) → [ ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) / 𝑥 ] 𝜑 ) ) |
92 |
80 91
|
mpd |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → [ ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) / 𝑥 ] 𝜑 ) |
93 |
|
2nn0 |
⊢ 2 ∈ ℕ0 |
94 |
93
|
a1i |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → 2 ∈ ℕ0 ) |
95 |
52
|
addid2d |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( 0 + 2 ) = 2 ) |
96 |
|
0red |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → 0 ∈ ℝ ) |
97 |
65 67
|
eqeltrrd |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( 2 · 𝑘 ) ∈ ℝ ) |
98 |
96 97 70 51
|
leadd1dd |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( 0 + 2 ) ≤ ( ( 2 · 𝑘 ) + 2 ) ) |
99 |
95 98
|
eqbrtrrd |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → 2 ≤ ( ( 2 · 𝑘 ) + 2 ) ) |
100 |
99 61
|
breqtrrd |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → 2 ≤ ( ♯ ‘ 𝑥 ) ) |
101 |
|
nn0sub |
⊢ ( ( 2 ∈ ℕ0 ∧ ( ♯ ‘ 𝑥 ) ∈ ℕ0 ) → ( 2 ≤ ( ♯ ‘ 𝑥 ) ↔ ( ( ♯ ‘ 𝑥 ) − 2 ) ∈ ℕ0 ) ) |
102 |
101
|
biimpa |
⊢ ( ( ( 2 ∈ ℕ0 ∧ ( ♯ ‘ 𝑥 ) ∈ ℕ0 ) ∧ 2 ≤ ( ♯ ‘ 𝑥 ) ) → ( ( ♯ ‘ 𝑥 ) − 2 ) ∈ ℕ0 ) |
103 |
94 39 100 102
|
syl21anc |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( ( ♯ ‘ 𝑥 ) − 2 ) ∈ ℕ0 ) |
104 |
68
|
recnd |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( ♯ ‘ 𝑥 ) ∈ ℂ ) |
105 |
104 52 55
|
subsubd |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( ( ♯ ‘ 𝑥 ) − ( 2 − 1 ) ) = ( ( ( ♯ ‘ 𝑥 ) − 2 ) + 1 ) ) |
106 |
|
2m1e1 |
⊢ ( 2 − 1 ) = 1 |
107 |
106
|
a1i |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( 2 − 1 ) = 1 ) |
108 |
107
|
oveq2d |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( ( ♯ ‘ 𝑥 ) − ( 2 − 1 ) ) = ( ( ♯ ‘ 𝑥 ) − 1 ) ) |
109 |
105 108
|
eqtr3d |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( ( ( ♯ ‘ 𝑥 ) − 2 ) + 1 ) = ( ( ♯ ‘ 𝑥 ) − 1 ) ) |
110 |
68
|
lem1d |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( ( ♯ ‘ 𝑥 ) − 1 ) ≤ ( ♯ ‘ 𝑥 ) ) |
111 |
109 110
|
eqbrtrd |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( ( ( ♯ ‘ 𝑥 ) − 2 ) + 1 ) ≤ ( ♯ ‘ 𝑥 ) ) |
112 |
|
nn0p1elfzo |
⊢ ( ( ( ( ♯ ‘ 𝑥 ) − 2 ) ∈ ℕ0 ∧ ( ♯ ‘ 𝑥 ) ∈ ℕ0 ∧ ( ( ( ♯ ‘ 𝑥 ) − 2 ) + 1 ) ≤ ( ♯ ‘ 𝑥 ) ) → ( ( ♯ ‘ 𝑥 ) − 2 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑥 ) ) ) |
113 |
103 39 111 112
|
syl3anc |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( ( ♯ ‘ 𝑥 ) − 2 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑥 ) ) ) |
114 |
|
wrdsymbcl |
⊢ ( ( 𝑥 ∈ Word 𝐵 ∧ ( ( ♯ ‘ 𝑥 ) − 2 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑥 ) ) ) → ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) ∈ 𝐵 ) |
115 |
36 113 114
|
syl2anc |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) ∈ 𝐵 ) |
116 |
115
|
adantlr |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) ∈ 𝐵 ) |
117 |
|
nn0ge2m1nn0 |
⊢ ( ( ( ♯ ‘ 𝑥 ) ∈ ℕ0 ∧ 2 ≤ ( ♯ ‘ 𝑥 ) ) → ( ( ♯ ‘ 𝑥 ) − 1 ) ∈ ℕ0 ) |
118 |
39 100 117
|
syl2anc |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( ( ♯ ‘ 𝑥 ) − 1 ) ∈ ℕ0 ) |
119 |
104 55
|
npcand |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( ( ( ♯ ‘ 𝑥 ) − 1 ) + 1 ) = ( ♯ ‘ 𝑥 ) ) |
120 |
68
|
leidd |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( ♯ ‘ 𝑥 ) ≤ ( ♯ ‘ 𝑥 ) ) |
121 |
119 120
|
eqbrtrd |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( ( ( ♯ ‘ 𝑥 ) − 1 ) + 1 ) ≤ ( ♯ ‘ 𝑥 ) ) |
122 |
|
nn0p1elfzo |
⊢ ( ( ( ( ♯ ‘ 𝑥 ) − 1 ) ∈ ℕ0 ∧ ( ♯ ‘ 𝑥 ) ∈ ℕ0 ∧ ( ( ( ♯ ‘ 𝑥 ) − 1 ) + 1 ) ≤ ( ♯ ‘ 𝑥 ) ) → ( ( ♯ ‘ 𝑥 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑥 ) ) ) |
123 |
118 39 121 122
|
syl3anc |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( ( ♯ ‘ 𝑥 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑥 ) ) ) |
124 |
|
wrdsymbcl |
⊢ ( ( 𝑥 ∈ Word 𝐵 ∧ ( ( ♯ ‘ 𝑥 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑥 ) ) ) → ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 1 ) ) ∈ 𝐵 ) |
125 |
36 123 124
|
syl2anc |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 1 ) ) ∈ 𝐵 ) |
126 |
125
|
adantlr |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 1 ) ) ∈ 𝐵 ) |
127 |
|
oveq1 |
⊢ ( 𝑦 = ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) → ( 𝑦 ++ 〈“ 𝑖 𝑗 ”〉 ) = ( ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ++ 〈“ 𝑖 𝑗 ”〉 ) ) |
128 |
127
|
sbceq1d |
⊢ ( 𝑦 = ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) → ( [ ( 𝑦 ++ 〈“ 𝑖 𝑗 ”〉 ) / 𝑥 ] 𝜑 ↔ [ ( ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ++ 〈“ 𝑖 𝑗 ”〉 ) / 𝑥 ] 𝜑 ) ) |
129 |
85 128
|
imbi12d |
⊢ ( 𝑦 = ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) → ( ( [ 𝑦 / 𝑥 ] 𝜑 → [ ( 𝑦 ++ 〈“ 𝑖 𝑗 ”〉 ) / 𝑥 ] 𝜑 ) ↔ ( [ ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) / 𝑥 ] 𝜑 → [ ( ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ++ 〈“ 𝑖 𝑗 ”〉 ) / 𝑥 ] 𝜑 ) ) ) |
130 |
|
id |
⊢ ( 𝑖 = ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) → 𝑖 = ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) ) |
131 |
|
eqidd |
⊢ ( 𝑖 = ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) → 𝑗 = 𝑗 ) |
132 |
130 131
|
s2eqd |
⊢ ( 𝑖 = ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) → 〈“ 𝑖 𝑗 ”〉 = 〈“ ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) 𝑗 ”〉 ) |
133 |
132
|
oveq2d |
⊢ ( 𝑖 = ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) → ( ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ++ 〈“ 𝑖 𝑗 ”〉 ) = ( ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ++ 〈“ ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) 𝑗 ”〉 ) ) |
134 |
133
|
sbceq1d |
⊢ ( 𝑖 = ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) → ( [ ( ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ++ 〈“ 𝑖 𝑗 ”〉 ) / 𝑥 ] 𝜑 ↔ [ ( ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ++ 〈“ ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) 𝑗 ”〉 ) / 𝑥 ] 𝜑 ) ) |
135 |
134
|
imbi2d |
⊢ ( 𝑖 = ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) → ( ( [ ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) / 𝑥 ] 𝜑 → [ ( ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ++ 〈“ 𝑖 𝑗 ”〉 ) / 𝑥 ] 𝜑 ) ↔ ( [ ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) / 𝑥 ] 𝜑 → [ ( ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ++ 〈“ ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) 𝑗 ”〉 ) / 𝑥 ] 𝜑 ) ) ) |
136 |
|
eqidd |
⊢ ( 𝑗 = ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 1 ) ) → ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) = ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) ) |
137 |
|
id |
⊢ ( 𝑗 = ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 1 ) ) → 𝑗 = ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 1 ) ) ) |
138 |
136 137
|
s2eqd |
⊢ ( 𝑗 = ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 1 ) ) → 〈“ ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) 𝑗 ”〉 = 〈“ ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 1 ) ) ”〉 ) |
139 |
138
|
oveq2d |
⊢ ( 𝑗 = ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 1 ) ) → ( ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ++ 〈“ ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) 𝑗 ”〉 ) = ( ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ++ 〈“ ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 1 ) ) ”〉 ) ) |
140 |
139
|
sbceq1d |
⊢ ( 𝑗 = ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 1 ) ) → ( [ ( ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ++ 〈“ ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) 𝑗 ”〉 ) / 𝑥 ] 𝜑 ↔ [ ( ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ++ 〈“ ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 1 ) ) ”〉 ) / 𝑥 ] 𝜑 ) ) |
141 |
140
|
imbi2d |
⊢ ( 𝑗 = ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 1 ) ) → ( ( [ ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) / 𝑥 ] 𝜑 → [ ( ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ++ 〈“ ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) 𝑗 ”〉 ) / 𝑥 ] 𝜑 ) ↔ ( [ ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) / 𝑥 ] 𝜑 → [ ( ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ++ 〈“ ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 1 ) ) ”〉 ) / 𝑥 ] 𝜑 ) ) ) |
142 |
|
ovex |
⊢ ( 𝑦 ++ 〈“ 𝑖 𝑗 ”〉 ) ∈ V |
143 |
142 3
|
sbcie |
⊢ ( [ ( 𝑦 ++ 〈“ 𝑖 𝑗 ”〉 ) / 𝑥 ] 𝜑 ↔ 𝜃 ) |
144 |
6 84 143
|
3imtr4g |
⊢ ( ( 𝑦 ∈ Word 𝐵 ∧ 𝑖 ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) → ( [ 𝑦 / 𝑥 ] 𝜑 → [ ( 𝑦 ++ 〈“ 𝑖 𝑗 ”〉 ) / 𝑥 ] 𝜑 ) ) |
145 |
129 135 141 144
|
vtocl3ga |
⊢ ( ( ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ∈ Word 𝐵 ∧ ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) ∈ 𝐵 ∧ ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 1 ) ) ∈ 𝐵 ) → ( [ ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) / 𝑥 ] 𝜑 → [ ( ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ++ 〈“ ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 1 ) ) ”〉 ) / 𝑥 ] 𝜑 ) ) |
146 |
90 116 126 145
|
syl3anc |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( [ ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) / 𝑥 ] 𝜑 → [ ( ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ++ 〈“ ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 1 ) ) ”〉 ) / 𝑥 ] 𝜑 ) ) |
147 |
92 146
|
mpd |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → [ ( ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ++ 〈“ ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 1 ) ) ”〉 ) / 𝑥 ] 𝜑 ) |
148 |
|
simprl |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → 𝑥 ∈ Word 𝐵 ) |
149 |
|
1red |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → 1 ∈ ℝ ) |
150 |
|
simpll |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → 𝑘 ∈ ℕ0 ) |
151 |
150
|
nn0red |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → 𝑘 ∈ ℝ ) |
152 |
151 149
|
readdcld |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( 𝑘 + 1 ) ∈ ℝ ) |
153 |
44
|
a1i |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → 2 ∈ ℝ ) |
154 |
47
|
a1i |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → 0 ≤ 2 ) |
155 |
|
0p1e1 |
⊢ ( 0 + 1 ) = 1 |
156 |
|
0red |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → 0 ∈ ℝ ) |
157 |
150
|
nn0ge0d |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → 0 ≤ 𝑘 ) |
158 |
149
|
leidd |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → 1 ≤ 1 ) |
159 |
156 149 151 149 157 158
|
le2addd |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( 0 + 1 ) ≤ ( 𝑘 + 1 ) ) |
160 |
155 159
|
eqbrtrrid |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → 1 ≤ ( 𝑘 + 1 ) ) |
161 |
149 152 153 154 160
|
lemul2ad |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( 2 · 1 ) ≤ ( 2 · ( 𝑘 + 1 ) ) ) |
162 |
58 161
|
eqbrtrrid |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → 2 ≤ ( 2 · ( 𝑘 + 1 ) ) ) |
163 |
|
simprr |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) |
164 |
162 163
|
breqtrd |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → 2 ≤ ( ♯ ‘ 𝑥 ) ) |
165 |
|
eqid |
⊢ ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝑥 ) |
166 |
165
|
pfxlsw2ccat |
⊢ ( ( 𝑥 ∈ Word 𝐵 ∧ 2 ≤ ( ♯ ‘ 𝑥 ) ) → 𝑥 = ( ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ++ 〈“ ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 1 ) ) ”〉 ) ) |
167 |
166
|
eqcomd |
⊢ ( ( 𝑥 ∈ Word 𝐵 ∧ 2 ≤ ( ♯ ‘ 𝑥 ) ) → ( ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ++ 〈“ ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 1 ) ) ”〉 ) = 𝑥 ) |
168 |
167
|
eqcomd |
⊢ ( ( 𝑥 ∈ Word 𝐵 ∧ 2 ≤ ( ♯ ‘ 𝑥 ) ) → 𝑥 = ( ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ++ 〈“ ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 1 ) ) ”〉 ) ) |
169 |
148 164 168
|
syl2anc |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → 𝑥 = ( ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ++ 〈“ ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 1 ) ) ”〉 ) ) |
170 |
|
sbceq1a |
⊢ ( 𝑥 = ( ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ++ 〈“ ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 1 ) ) ”〉 ) → ( 𝜑 ↔ [ ( ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ++ 〈“ ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 1 ) ) ”〉 ) / 𝑥 ] 𝜑 ) ) |
171 |
169 170
|
syl |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → ( 𝜑 ↔ [ ( ( 𝑥 prefix ( ( ♯ ‘ 𝑥 ) − 2 ) ) ++ 〈“ ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 2 ) ) ( 𝑥 ‘ ( ( ♯ ‘ 𝑥 ) − 1 ) ) ”〉 ) / 𝑥 ] 𝜑 ) ) |
172 |
147 171
|
mpbird |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ ( 𝑥 ∈ Word 𝐵 ∧ ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) ) ) → 𝜑 ) |
173 |
172
|
expr |
⊢ ( ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) ∧ 𝑥 ∈ Word 𝐵 ) → ( ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) ) |
174 |
173
|
ralrimiva |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) ) → ∀ 𝑥 ∈ Word 𝐵 ( ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) ) |
175 |
174
|
ex |
⊢ ( 𝑘 ∈ ℕ0 → ( ∀ 𝑦 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑦 ) → 𝜒 ) → ∀ 𝑥 ∈ Word 𝐵 ( ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) ) ) |
176 |
35 175
|
syl5bi |
⊢ ( 𝑘 ∈ ℕ0 → ( ∀ 𝑥 ∈ Word 𝐵 ( ( 2 · 𝑘 ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) → ∀ 𝑥 ∈ Word 𝐵 ( ( 2 · ( 𝑘 + 1 ) ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) ) ) |
177 |
10 14 18 22 31 176
|
nn0ind |
⊢ ( 𝑚 ∈ ℕ0 → ∀ 𝑥 ∈ Word 𝐵 ( ( 2 · 𝑚 ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) ) |
178 |
177
|
adantl |
⊢ ( ( 𝐴 ∈ Word 𝐵 ∧ 𝑚 ∈ ℕ0 ) → ∀ 𝑥 ∈ Word 𝐵 ( ( 2 · 𝑚 ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) ) |
179 |
|
simpl |
⊢ ( ( 𝐴 ∈ Word 𝐵 ∧ 𝑚 ∈ ℕ0 ) → 𝐴 ∈ Word 𝐵 ) |
180 |
|
fveq2 |
⊢ ( 𝑥 = 𝐴 → ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝐴 ) ) |
181 |
180
|
eqeq2d |
⊢ ( 𝑥 = 𝐴 → ( ( 2 · 𝑚 ) = ( ♯ ‘ 𝑥 ) ↔ ( 2 · 𝑚 ) = ( ♯ ‘ 𝐴 ) ) ) |
182 |
181 4
|
imbi12d |
⊢ ( 𝑥 = 𝐴 → ( ( ( 2 · 𝑚 ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) ↔ ( ( 2 · 𝑚 ) = ( ♯ ‘ 𝐴 ) → 𝜏 ) ) ) |
183 |
182
|
adantl |
⊢ ( ( ( 𝐴 ∈ Word 𝐵 ∧ 𝑚 ∈ ℕ0 ) ∧ 𝑥 = 𝐴 ) → ( ( ( 2 · 𝑚 ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) ↔ ( ( 2 · 𝑚 ) = ( ♯ ‘ 𝐴 ) → 𝜏 ) ) ) |
184 |
179 183
|
rspcdv |
⊢ ( ( 𝐴 ∈ Word 𝐵 ∧ 𝑚 ∈ ℕ0 ) → ( ∀ 𝑥 ∈ Word 𝐵 ( ( 2 · 𝑚 ) = ( ♯ ‘ 𝑥 ) → 𝜑 ) → ( ( 2 · 𝑚 ) = ( ♯ ‘ 𝐴 ) → 𝜏 ) ) ) |
185 |
178 184
|
mpd |
⊢ ( ( 𝐴 ∈ Word 𝐵 ∧ 𝑚 ∈ ℕ0 ) → ( ( 2 · 𝑚 ) = ( ♯ ‘ 𝐴 ) → 𝜏 ) ) |
186 |
185
|
imp |
⊢ ( ( ( 𝐴 ∈ Word 𝐵 ∧ 𝑚 ∈ ℕ0 ) ∧ ( 2 · 𝑚 ) = ( ♯ ‘ 𝐴 ) ) → 𝜏 ) |
187 |
186
|
adantllr |
⊢ ( ( ( ( 𝐴 ∈ Word 𝐵 ∧ 2 ∥ ( ♯ ‘ 𝐴 ) ) ∧ 𝑚 ∈ ℕ0 ) ∧ ( 2 · 𝑚 ) = ( ♯ ‘ 𝐴 ) ) → 𝜏 ) |
188 |
|
lencl |
⊢ ( 𝐴 ∈ Word 𝐵 → ( ♯ ‘ 𝐴 ) ∈ ℕ0 ) |
189 |
|
evennn02n |
⊢ ( ( ♯ ‘ 𝐴 ) ∈ ℕ0 → ( 2 ∥ ( ♯ ‘ 𝐴 ) ↔ ∃ 𝑚 ∈ ℕ0 ( 2 · 𝑚 ) = ( ♯ ‘ 𝐴 ) ) ) |
190 |
189
|
biimpa |
⊢ ( ( ( ♯ ‘ 𝐴 ) ∈ ℕ0 ∧ 2 ∥ ( ♯ ‘ 𝐴 ) ) → ∃ 𝑚 ∈ ℕ0 ( 2 · 𝑚 ) = ( ♯ ‘ 𝐴 ) ) |
191 |
188 190
|
sylan |
⊢ ( ( 𝐴 ∈ Word 𝐵 ∧ 2 ∥ ( ♯ ‘ 𝐴 ) ) → ∃ 𝑚 ∈ ℕ0 ( 2 · 𝑚 ) = ( ♯ ‘ 𝐴 ) ) |
192 |
187 191
|
r19.29a |
⊢ ( ( 𝐴 ∈ Word 𝐵 ∧ 2 ∥ ( ♯ ‘ 𝐴 ) ) → 𝜏 ) |