Step | Hyp | Ref
| Expression |
1 | | psgnunilem2.w |
. . . . . . 7
⊢ (𝜑 → 𝑊 ∈ Word 𝑇) |
2 | | wrd0 13627 |
. . . . . . 7
⊢ ∅
∈ Word 𝑇 |
3 | | splcl 13892 |
. . . . . . 7
⊢ ((𝑊 ∈ Word 𝑇 ∧ ∅ ∈ Word 𝑇) → (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉) ∈ Word 𝑇) |
4 | 1, 2, 3 | sylancl 580 |
. . . . . 6
⊢ (𝜑 → (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉) ∈ Word 𝑇) |
5 | 4 | adantr 474 |
. . . . 5
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉) ∈ Word 𝑇) |
6 | | fzossfz 12807 |
. . . . . . . . . . 11
⊢
(0..^𝐿) ⊆
(0...𝐿) |
7 | | psgnunilem2.ix |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐼 ∈ (0..^𝐿)) |
8 | 6, 7 | sseldi 3819 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐼 ∈ (0...𝐿)) |
9 | | elfznn0 12751 |
. . . . . . . . . 10
⊢ (𝐼 ∈ (0...𝐿) → 𝐼 ∈
ℕ0) |
10 | 8, 9 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐼 ∈
ℕ0) |
11 | | 2nn0 11661 |
. . . . . . . . . 10
⊢ 2 ∈
ℕ0 |
12 | | nn0addcl 11679 |
. . . . . . . . . 10
⊢ ((𝐼 ∈ ℕ0
∧ 2 ∈ ℕ0) → (𝐼 + 2) ∈
ℕ0) |
13 | 10, 11, 12 | sylancl 580 |
. . . . . . . . 9
⊢ (𝜑 → (𝐼 + 2) ∈
ℕ0) |
14 | 10 | nn0red 11703 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐼 ∈ ℝ) |
15 | | nn0addge1 11690 |
. . . . . . . . . 10
⊢ ((𝐼 ∈ ℝ ∧ 2 ∈
ℕ0) → 𝐼 ≤ (𝐼 + 2)) |
16 | 14, 11, 15 | sylancl 580 |
. . . . . . . . 9
⊢ (𝜑 → 𝐼 ≤ (𝐼 + 2)) |
17 | | elfz2nn0 12749 |
. . . . . . . . 9
⊢ (𝐼 ∈ (0...(𝐼 + 2)) ↔ (𝐼 ∈ ℕ0 ∧ (𝐼 + 2) ∈ ℕ0
∧ 𝐼 ≤ (𝐼 + 2))) |
18 | 10, 13, 16, 17 | syl3anbrc 1400 |
. . . . . . . 8
⊢ (𝜑 → 𝐼 ∈ (0...(𝐼 + 2))) |
19 | | psgnunilem2.g |
. . . . . . . . . . 11
⊢ 𝐺 = (SymGrp‘𝐷) |
20 | | psgnunilem2.t |
. . . . . . . . . . 11
⊢ 𝑇 = ran (pmTrsp‘𝐷) |
21 | | psgnunilem2.d |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐷 ∈ 𝑉) |
22 | | psgnunilem2.id |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷)) |
23 | | psgnunilem2.l |
. . . . . . . . . . 11
⊢ (𝜑 → (♯‘𝑊) = 𝐿) |
24 | | psgnunilem2.a |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )) |
25 | | psgnunilem2.al |
. . . . . . . . . . 11
⊢ (𝜑 → ∀𝑘 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I )) |
26 | 19, 20, 21, 1, 22, 23, 7, 24, 25 | psgnunilem5 18297 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐼 + 1) ∈ (0..^𝐿)) |
27 | | fzofzp1 12884 |
. . . . . . . . . 10
⊢ ((𝐼 + 1) ∈ (0..^𝐿) → ((𝐼 + 1) + 1) ∈ (0...𝐿)) |
28 | 26, 27 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐼 + 1) + 1) ∈ (0...𝐿)) |
29 | 10 | nn0cnd 11704 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐼 ∈ ℂ) |
30 | | 1cnd 10371 |
. . . . . . . . . . 11
⊢ (𝜑 → 1 ∈
ℂ) |
31 | 29, 30, 30 | addassd 10399 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐼 + 1) + 1) = (𝐼 + (1 + 1))) |
32 | | df-2 11438 |
. . . . . . . . . . 11
⊢ 2 = (1 +
1) |
33 | 32 | oveq2i 6933 |
. . . . . . . . . 10
⊢ (𝐼 + 2) = (𝐼 + (1 + 1)) |
34 | 31, 33 | syl6reqr 2833 |
. . . . . . . . 9
⊢ (𝜑 → (𝐼 + 2) = ((𝐼 + 1) + 1)) |
35 | 23 | oveq2d 6938 |
. . . . . . . . 9
⊢ (𝜑 → (0...(♯‘𝑊)) = (0...𝐿)) |
36 | 28, 34, 35 | 3eltr4d 2874 |
. . . . . . . 8
⊢ (𝜑 → (𝐼 + 2) ∈ (0...(♯‘𝑊))) |
37 | 2 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → ∅ ∈ Word 𝑇) |
38 | 1, 18, 36, 37 | spllen 13896 |
. . . . . . 7
⊢ (𝜑 → (♯‘(𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉)) =
((♯‘𝑊) +
((♯‘∅) − ((𝐼 + 2) − 𝐼)))) |
39 | | hash0 13473 |
. . . . . . . . . . 11
⊢
(♯‘∅) = 0 |
40 | 39 | oveq1i 6932 |
. . . . . . . . . 10
⊢
((♯‘∅) − ((𝐼 + 2) − 𝐼)) = (0 − ((𝐼 + 2) − 𝐼)) |
41 | | df-neg 10609 |
. . . . . . . . . 10
⊢ -((𝐼 + 2) − 𝐼) = (0 − ((𝐼 + 2) − 𝐼)) |
42 | 40, 41 | eqtr4i 2805 |
. . . . . . . . 9
⊢
((♯‘∅) − ((𝐼 + 2) − 𝐼)) = -((𝐼 + 2) − 𝐼) |
43 | | 2cn 11450 |
. . . . . . . . . . 11
⊢ 2 ∈
ℂ |
44 | | pncan2 10629 |
. . . . . . . . . . 11
⊢ ((𝐼 ∈ ℂ ∧ 2 ∈
ℂ) → ((𝐼 + 2)
− 𝐼) =
2) |
45 | 29, 43, 44 | sylancl 580 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐼 + 2) − 𝐼) = 2) |
46 | 45 | negeqd 10616 |
. . . . . . . . 9
⊢ (𝜑 → -((𝐼 + 2) − 𝐼) = -2) |
47 | 42, 46 | syl5eq 2826 |
. . . . . . . 8
⊢ (𝜑 → ((♯‘∅)
− ((𝐼 + 2) −
𝐼)) = -2) |
48 | 23, 47 | oveq12d 6940 |
. . . . . . 7
⊢ (𝜑 → ((♯‘𝑊) + ((♯‘∅)
− ((𝐼 + 2) −
𝐼))) = (𝐿 + -2)) |
49 | | elfzel2 12657 |
. . . . . . . . . 10
⊢ (𝐼 ∈ (0...𝐿) → 𝐿 ∈ ℤ) |
50 | 8, 49 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐿 ∈ ℤ) |
51 | 50 | zcnd 11835 |
. . . . . . . 8
⊢ (𝜑 → 𝐿 ∈ ℂ) |
52 | | negsub 10671 |
. . . . . . . 8
⊢ ((𝐿 ∈ ℂ ∧ 2 ∈
ℂ) → (𝐿 + -2) =
(𝐿 −
2)) |
53 | 51, 43, 52 | sylancl 580 |
. . . . . . 7
⊢ (𝜑 → (𝐿 + -2) = (𝐿 − 2)) |
54 | 38, 48, 53 | 3eqtrd 2818 |
. . . . . 6
⊢ (𝜑 → (♯‘(𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉)) = (𝐿 − 2)) |
55 | 54 | adantr 474 |
. . . . 5
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (♯‘(𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉)) = (𝐿 − 2)) |
56 | | splid 13894 |
. . . . . . . . 9
⊢ ((𝑊 ∈ Word 𝑇 ∧ (𝐼 ∈ (0...(𝐼 + 2)) ∧ (𝐼 + 2) ∈ (0...(♯‘𝑊)))) → (𝑊 splice 〈𝐼, (𝐼 + 2), (𝑊 substr 〈𝐼, (𝐼 + 2)〉)〉) = 𝑊) |
57 | 1, 18, 36, 56 | syl12anc 827 |
. . . . . . . 8
⊢ (𝜑 → (𝑊 splice 〈𝐼, (𝐼 + 2), (𝑊 substr 〈𝐼, (𝐼 + 2)〉)〉) = 𝑊) |
58 | 57 | oveq2d 6938 |
. . . . . . 7
⊢ (𝜑 → (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), (𝑊 substr 〈𝐼, (𝐼 + 2)〉)〉)) = (𝐺 Σg 𝑊)) |
59 | 58 | adantr 474 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), (𝑊 substr 〈𝐼, (𝐼 + 2)〉)〉)) = (𝐺 Σg 𝑊)) |
60 | | eqid 2778 |
. . . . . . 7
⊢
(Base‘𝐺) =
(Base‘𝐺) |
61 | 19 | symggrp 18203 |
. . . . . . . . . 10
⊢ (𝐷 ∈ 𝑉 → 𝐺 ∈ Grp) |
62 | 21, 61 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐺 ∈ Grp) |
63 | | grpmnd 17816 |
. . . . . . . . 9
⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) |
64 | 62, 63 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐺 ∈ Mnd) |
65 | 64 | adantr 474 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → 𝐺 ∈ Mnd) |
66 | 20, 19, 60 | symgtrf 18272 |
. . . . . . . . . 10
⊢ 𝑇 ⊆ (Base‘𝐺) |
67 | | sswrd 13607 |
. . . . . . . . . 10
⊢ (𝑇 ⊆ (Base‘𝐺) → Word 𝑇 ⊆ Word (Base‘𝐺)) |
68 | 66, 67 | ax-mp 5 |
. . . . . . . . 9
⊢ Word
𝑇 ⊆ Word
(Base‘𝐺) |
69 | 68, 1 | sseldi 3819 |
. . . . . . . 8
⊢ (𝜑 → 𝑊 ∈ Word (Base‘𝐺)) |
70 | 69 | adantr 474 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → 𝑊 ∈ Word (Base‘𝐺)) |
71 | 18 | adantr 474 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → 𝐼 ∈ (0...(𝐼 + 2))) |
72 | 36 | adantr 474 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐼 + 2) ∈ (0...(♯‘𝑊))) |
73 | | swrdcl 13735 |
. . . . . . . . 9
⊢ (𝑊 ∈ Word (Base‘𝐺) → (𝑊 substr 〈𝐼, (𝐼 + 2)〉) ∈ Word (Base‘𝐺)) |
74 | 69, 73 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (𝑊 substr 〈𝐼, (𝐼 + 2)〉) ∈ Word (Base‘𝐺)) |
75 | 74 | adantr 474 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝑊 substr 〈𝐼, (𝐼 + 2)〉) ∈ Word (Base‘𝐺)) |
76 | | wrd0 13627 |
. . . . . . . 8
⊢ ∅
∈ Word (Base‘𝐺) |
77 | 76 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ∅ ∈ Word
(Base‘𝐺)) |
78 | 23 | oveq2d 6938 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (0..^(♯‘𝑊)) = (0..^𝐿)) |
79 | 26, 78 | eleqtrrd 2862 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐼 + 1) ∈ (0..^(♯‘𝑊))) |
80 | | swrds2 14091 |
. . . . . . . . . . . 12
⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝐼 ∈ ℕ0 ∧ (𝐼 + 1) ∈
(0..^(♯‘𝑊)))
→ (𝑊 substr
〈𝐼, (𝐼 + 2)〉) = 〈“(𝑊‘𝐼)(𝑊‘(𝐼 + 1))”〉) |
81 | 1, 10, 79, 80 | syl3anc 1439 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑊 substr 〈𝐼, (𝐼 + 2)〉) = 〈“(𝑊‘𝐼)(𝑊‘(𝐼 + 1))”〉) |
82 | 81 | oveq2d 6938 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐺 Σg (𝑊 substr 〈𝐼, (𝐼 + 2)〉)) = (𝐺 Σg
〈“(𝑊‘𝐼)(𝑊‘(𝐼 + 1))”〉)) |
83 | | wrdf 13604 |
. . . . . . . . . . . . . . 15
⊢ (𝑊 ∈ Word 𝑇 → 𝑊:(0..^(♯‘𝑊))⟶𝑇) |
84 | 1, 83 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑊:(0..^(♯‘𝑊))⟶𝑇) |
85 | 78 | feq2d 6277 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑊:(0..^(♯‘𝑊))⟶𝑇 ↔ 𝑊:(0..^𝐿)⟶𝑇)) |
86 | 84, 85 | mpbid 224 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑊:(0..^𝐿)⟶𝑇) |
87 | 86, 7 | ffvelrnd 6624 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑊‘𝐼) ∈ 𝑇) |
88 | 66, 87 | sseldi 3819 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑊‘𝐼) ∈ (Base‘𝐺)) |
89 | 86, 26 | ffvelrnd 6624 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑊‘(𝐼 + 1)) ∈ 𝑇) |
90 | 66, 89 | sseldi 3819 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑊‘(𝐼 + 1)) ∈ (Base‘𝐺)) |
91 | | eqid 2778 |
. . . . . . . . . . . 12
⊢
(+g‘𝐺) = (+g‘𝐺) |
92 | 60, 91 | gsumws2 17765 |
. . . . . . . . . . 11
⊢ ((𝐺 ∈ Mnd ∧ (𝑊‘𝐼) ∈ (Base‘𝐺) ∧ (𝑊‘(𝐼 + 1)) ∈ (Base‘𝐺)) → (𝐺 Σg
〈“(𝑊‘𝐼)(𝑊‘(𝐼 + 1))”〉) = ((𝑊‘𝐼)(+g‘𝐺)(𝑊‘(𝐼 + 1)))) |
93 | 64, 88, 90, 92 | syl3anc 1439 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐺 Σg
〈“(𝑊‘𝐼)(𝑊‘(𝐼 + 1))”〉) = ((𝑊‘𝐼)(+g‘𝐺)(𝑊‘(𝐼 + 1)))) |
94 | 19, 60, 91 | symgov 18193 |
. . . . . . . . . . 11
⊢ (((𝑊‘𝐼) ∈ (Base‘𝐺) ∧ (𝑊‘(𝐼 + 1)) ∈ (Base‘𝐺)) → ((𝑊‘𝐼)(+g‘𝐺)(𝑊‘(𝐼 + 1))) = ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1)))) |
95 | 88, 90, 94 | syl2anc 579 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑊‘𝐼)(+g‘𝐺)(𝑊‘(𝐼 + 1))) = ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1)))) |
96 | 82, 93, 95 | 3eqtrd 2818 |
. . . . . . . . 9
⊢ (𝜑 → (𝐺 Σg (𝑊 substr 〈𝐼, (𝐼 + 2)〉)) = ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1)))) |
97 | 96 | adantr 474 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐺 Σg (𝑊 substr 〈𝐼, (𝐼 + 2)〉)) = ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1)))) |
98 | | simpr 479 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) |
99 | 19 | symgid 18204 |
. . . . . . . . . . 11
⊢ (𝐷 ∈ 𝑉 → ( I ↾ 𝐷) = (0g‘𝐺)) |
100 | 21, 99 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → ( I ↾ 𝐷) = (0g‘𝐺)) |
101 | | eqid 2778 |
. . . . . . . . . . 11
⊢
(0g‘𝐺) = (0g‘𝐺) |
102 | 101 | gsum0 17664 |
. . . . . . . . . 10
⊢ (𝐺 Σg
∅) = (0g‘𝐺) |
103 | 100, 102 | syl6eqr 2832 |
. . . . . . . . 9
⊢ (𝜑 → ( I ↾ 𝐷) = (𝐺 Σg
∅)) |
104 | 103 | adantr 474 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ( I ↾ 𝐷) = (𝐺 Σg
∅)) |
105 | 97, 98, 104 | 3eqtrd 2818 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐺 Σg (𝑊 substr 〈𝐼, (𝐼 + 2)〉)) = (𝐺 Σg
∅)) |
106 | 60, 65, 70, 71, 72, 75, 77, 105 | gsumspl 17767 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), (𝑊 substr 〈𝐼, (𝐼 + 2)〉)〉)) = (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉))) |
107 | 22 | adantr 474 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐺 Σg 𝑊) = ( I ↾ 𝐷)) |
108 | 59, 106, 107 | 3eqtr3d 2822 |
. . . . 5
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉)) = ( I ↾ 𝐷)) |
109 | | fveqeq2 6455 |
. . . . . . 7
⊢ (𝑥 = (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉) →
((♯‘𝑥) = (𝐿 − 2) ↔
(♯‘(𝑊 splice
〈𝐼, (𝐼 + 2), ∅〉)) = (𝐿 − 2))) |
110 | | oveq2 6930 |
. . . . . . . 8
⊢ (𝑥 = (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉) → (𝐺 Σg
𝑥) = (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉))) |
111 | 110 | eqeq1d 2780 |
. . . . . . 7
⊢ (𝑥 = (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉) → ((𝐺 Σg
𝑥) = ( I ↾ 𝐷) ↔ (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉)) = ( I ↾ 𝐷))) |
112 | 109, 111 | anbi12d 624 |
. . . . . 6
⊢ (𝑥 = (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉) →
(((♯‘𝑥) =
(𝐿 − 2) ∧ (𝐺 Σg
𝑥) = ( I ↾ 𝐷)) ↔ ((♯‘(𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉)) = (𝐿 − 2) ∧ (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉)) = ( I ↾ 𝐷)))) |
113 | 112 | rspcev 3511 |
. . . . 5
⊢ (((𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉) ∈ Word 𝑇 ∧ ((♯‘(𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉)) = (𝐿 − 2) ∧ (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉)) = ( I ↾ 𝐷))) → ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) |
114 | 5, 55, 108, 113 | syl12anc 827 |
. . . 4
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) |
115 | | psgnunilem2.in |
. . . . 5
⊢ (𝜑 → ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) |
116 | 115 | adantr 474 |
. . . 4
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) |
117 | 114, 116 | pm2.21dd 187 |
. . 3
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ ((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom ((𝑤‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom ((𝑤‘𝑗) ∖ I )))) |
118 | 117 | ex 403 |
. 2
⊢ (𝜑 → (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ ((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom ((𝑤‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom ((𝑤‘𝑗) ∖ I ))))) |
119 | 1 | adantr 474 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 𝑊 ∈ Word 𝑇) |
120 | | simprl 761 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 𝑟 ∈ 𝑇) |
121 | | simprr 763 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 𝑠 ∈ 𝑇) |
122 | 120, 121 | s2cld 14022 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 〈“𝑟𝑠”〉 ∈ Word 𝑇) |
123 | | splcl 13892 |
. . . . . . 7
⊢ ((𝑊 ∈ Word 𝑇 ∧ 〈“𝑟𝑠”〉 ∈ Word 𝑇) → (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) ∈ Word 𝑇) |
124 | 119, 122,
123 | syl2anc 579 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) ∈ Word 𝑇) |
125 | 124 | adantrr 707 |
. . . . 5
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) ∈ Word 𝑇) |
126 | 64 | adantr 474 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → 𝐺 ∈ Mnd) |
127 | 69 | adantr 474 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → 𝑊 ∈ Word (Base‘𝐺)) |
128 | 18 | adantr 474 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → 𝐼 ∈ (0...(𝐼 + 2))) |
129 | 36 | adantr 474 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝐼 + 2) ∈ (0...(♯‘𝑊))) |
130 | 68, 122 | sseldi 3819 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 〈“𝑟𝑠”〉 ∈ Word (Base‘𝐺)) |
131 | 130 | adantrr 707 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → 〈“𝑟𝑠”〉 ∈ Word (Base‘𝐺)) |
132 | 74 | adantr 474 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝑊 substr 〈𝐼, (𝐼 + 2)〉) ∈ Word (Base‘𝐺)) |
133 | | simprr1 1244 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠)) |
134 | 96 | adantr 474 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝐺 Σg (𝑊 substr 〈𝐼, (𝐼 + 2)〉)) = ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1)))) |
135 | 64 | adantr 474 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 𝐺 ∈ Mnd) |
136 | 66 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑇 ⊆ (Base‘𝐺)) |
137 | 136 | sselda 3821 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑇) → 𝑟 ∈ (Base‘𝐺)) |
138 | 137 | adantrr 707 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 𝑟 ∈ (Base‘𝐺)) |
139 | 136 | sselda 3821 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑇) → 𝑠 ∈ (Base‘𝐺)) |
140 | 139 | adantrl 706 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 𝑠 ∈ (Base‘𝐺)) |
141 | 60, 91 | gsumws2 17765 |
. . . . . . . . . . . 12
⊢ ((𝐺 ∈ Mnd ∧ 𝑟 ∈ (Base‘𝐺) ∧ 𝑠 ∈ (Base‘𝐺)) → (𝐺 Σg
〈“𝑟𝑠”〉) = (𝑟(+g‘𝐺)𝑠)) |
142 | 135, 138,
140, 141 | syl3anc 1439 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝐺 Σg
〈“𝑟𝑠”〉) = (𝑟(+g‘𝐺)𝑠)) |
143 | 19, 60, 91 | symgov 18193 |
. . . . . . . . . . . 12
⊢ ((𝑟 ∈ (Base‘𝐺) ∧ 𝑠 ∈ (Base‘𝐺)) → (𝑟(+g‘𝐺)𝑠) = (𝑟 ∘ 𝑠)) |
144 | 138, 140,
143 | syl2anc 579 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝑟(+g‘𝐺)𝑠) = (𝑟 ∘ 𝑠)) |
145 | 142, 144 | eqtrd 2814 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝐺 Σg
〈“𝑟𝑠”〉) = (𝑟 ∘ 𝑠)) |
146 | 145 | adantrr 707 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝐺 Σg
〈“𝑟𝑠”〉) = (𝑟 ∘ 𝑠)) |
147 | 133, 134,
146 | 3eqtr4rd 2825 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝐺 Σg
〈“𝑟𝑠”〉) = (𝐺 Σg
(𝑊 substr 〈𝐼, (𝐼 + 2)〉))) |
148 | 60, 126, 127, 128, 129, 131, 132, 147 | gsumspl 17767 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)) = (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), (𝑊 substr 〈𝐼, (𝐼 + 2)〉)〉))) |
149 | 58 | adantr 474 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), (𝑊 substr 〈𝐼, (𝐼 + 2)〉)〉)) = (𝐺 Σg 𝑊)) |
150 | 22 | adantr 474 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝐺 Σg 𝑊) = ( I ↾ 𝐷)) |
151 | 148, 149,
150 | 3eqtrd 2818 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)) = ( I ↾ 𝐷)) |
152 | 18 | adantr 474 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 𝐼 ∈ (0...(𝐼 + 2))) |
153 | 36 | adantr 474 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝐼 + 2) ∈ (0...(♯‘𝑊))) |
154 | 119, 152,
153, 122 | spllen 13896 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (♯‘(𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)) = ((♯‘𝑊) +
((♯‘〈“𝑟𝑠”〉) − ((𝐼 + 2) − 𝐼)))) |
155 | | s2len 14040 |
. . . . . . . . . . . . 13
⊢
(♯‘〈“𝑟𝑠”〉) = 2 |
156 | 155 | oveq1i 6932 |
. . . . . . . . . . . 12
⊢
((♯‘〈“𝑟𝑠”〉) − ((𝐼 + 2) − 𝐼)) = (2 − ((𝐼 + 2) − 𝐼)) |
157 | 45 | oveq2d 6938 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (2 − ((𝐼 + 2) − 𝐼)) = (2 − 2)) |
158 | 43 | subidi 10694 |
. . . . . . . . . . . . 13
⊢ (2
− 2) = 0 |
159 | 157, 158 | syl6eq 2830 |
. . . . . . . . . . . 12
⊢ (𝜑 → (2 − ((𝐼 + 2) − 𝐼)) = 0) |
160 | 156, 159 | syl5eq 2826 |
. . . . . . . . . . 11
⊢ (𝜑 →
((♯‘〈“𝑟𝑠”〉) − ((𝐼 + 2) − 𝐼)) = 0) |
161 | 160 | oveq2d 6938 |
. . . . . . . . . 10
⊢ (𝜑 → ((♯‘𝑊) +
((♯‘〈“𝑟𝑠”〉) − ((𝐼 + 2) − 𝐼))) = ((♯‘𝑊) + 0)) |
162 | 23, 51 | eqeltrd 2859 |
. . . . . . . . . . 11
⊢ (𝜑 → (♯‘𝑊) ∈
ℂ) |
163 | 162 | addid1d 10576 |
. . . . . . . . . 10
⊢ (𝜑 → ((♯‘𝑊) + 0) = (♯‘𝑊)) |
164 | 161, 163,
23 | 3eqtrd 2818 |
. . . . . . . . 9
⊢ (𝜑 → ((♯‘𝑊) +
((♯‘〈“𝑟𝑠”〉) − ((𝐼 + 2) − 𝐼))) = 𝐿) |
165 | 164 | adantr 474 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → ((♯‘𝑊) + ((♯‘〈“𝑟𝑠”〉) − ((𝐼 + 2) − 𝐼))) = 𝐿) |
166 | 154, 165 | eqtrd 2814 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (♯‘(𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)) = 𝐿) |
167 | 166 | adantrr 707 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (♯‘(𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)) = 𝐿) |
168 | 151, 167 | jca 507 |
. . . . 5
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → ((𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)) = ( I ↾ 𝐷) ∧ (♯‘(𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)) = 𝐿)) |
169 | 26 | adantr 474 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝐼 + 1) ∈ (0..^𝐿)) |
170 | | simprr2 1246 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → 𝐴 ∈ dom (𝑠 ∖ I )) |
171 | | 1nn0 11660 |
. . . . . . . . . . . . . . 15
⊢ 1 ∈
ℕ0 |
172 | | 2nn 11448 |
. . . . . . . . . . . . . . 15
⊢ 2 ∈
ℕ |
173 | | 1lt2 11553 |
. . . . . . . . . . . . . . 15
⊢ 1 <
2 |
174 | | elfzo0 12828 |
. . . . . . . . . . . . . . 15
⊢ (1 ∈
(0..^2) ↔ (1 ∈ ℕ0 ∧ 2 ∈ ℕ ∧ 1
< 2)) |
175 | 171, 172,
173, 174 | mpbir3an 1398 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
(0..^2) |
176 | 155 | oveq2i 6933 |
. . . . . . . . . . . . . 14
⊢
(0..^(♯‘〈“𝑟𝑠”〉)) = (0..^2) |
177 | 175, 176 | eleqtrri 2858 |
. . . . . . . . . . . . 13
⊢ 1 ∈
(0..^(♯‘〈“𝑟𝑠”〉)) |
178 | 177 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 1 ∈
(0..^(♯‘〈“𝑟𝑠”〉))) |
179 | 119, 152,
153, 122, 178 | splfv2a 13900 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 1)) = (〈“𝑟𝑠”〉‘1)) |
180 | | s2fv1 14039 |
. . . . . . . . . . . 12
⊢ (𝑠 ∈ 𝑇 → (〈“𝑟𝑠”〉‘1) = 𝑠) |
181 | 180 | ad2antll 719 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (〈“𝑟𝑠”〉‘1) = 𝑠) |
182 | 179, 181 | eqtrd 2814 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 1)) = 𝑠) |
183 | 182 | adantrr 707 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 1)) = 𝑠) |
184 | 183 | difeq1d 3950 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 1)) ∖ I ) = (𝑠 ∖ I )) |
185 | 184 | dmeqd 5571 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 1)) ∖ I ) = dom (𝑠 ∖ I )) |
186 | 170, 185 | eleqtrrd 2862 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 1)) ∖ I )) |
187 | | fzosplitsni 12898 |
. . . . . . . . . . 11
⊢ (𝐼 ∈
(ℤ≥‘0) → (𝑗 ∈ (0..^(𝐼 + 1)) ↔ (𝑗 ∈ (0..^𝐼) ∨ 𝑗 = 𝐼))) |
188 | | nn0uz 12028 |
. . . . . . . . . . 11
⊢
ℕ0 = (ℤ≥‘0) |
189 | 187, 188 | eleq2s 2877 |
. . . . . . . . . 10
⊢ (𝐼 ∈ ℕ0
→ (𝑗 ∈
(0..^(𝐼 + 1)) ↔ (𝑗 ∈ (0..^𝐼) ∨ 𝑗 = 𝐼))) |
190 | 10, 189 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (𝑗 ∈ (0..^(𝐼 + 1)) ↔ (𝑗 ∈ (0..^𝐼) ∨ 𝑗 = 𝐼))) |
191 | 190 | adantr 474 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝑗 ∈ (0..^(𝐼 + 1)) ↔ (𝑗 ∈ (0..^𝐼) ∨ 𝑗 = 𝐼))) |
192 | | fveq2 6446 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑗 → (𝑊‘𝑘) = (𝑊‘𝑗)) |
193 | 192 | difeq1d 3950 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑗 → ((𝑊‘𝑘) ∖ I ) = ((𝑊‘𝑗) ∖ I )) |
194 | 193 | dmeqd 5571 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑗 → dom ((𝑊‘𝑘) ∖ I ) = dom ((𝑊‘𝑗) ∖ I )) |
195 | 194 | eleq2d 2845 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑗 → (𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ) ↔ 𝐴 ∈ dom ((𝑊‘𝑗) ∖ I ))) |
196 | 195 | notbid 310 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑗 → (¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ) ↔ ¬ 𝐴 ∈ dom ((𝑊‘𝑗) ∖ I ))) |
197 | 196 | rspccva 3510 |
. . . . . . . . . . . . . 14
⊢
((∀𝑘 ∈
(0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ) ∧ 𝑗 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom ((𝑊‘𝑗) ∖ I )) |
198 | 25, 197 | sylan 575 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom ((𝑊‘𝑗) ∖ I )) |
199 | 198 | adantlr 705 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom ((𝑊‘𝑗) ∖ I )) |
200 | 1 | ad2antrr 716 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → 𝑊 ∈ Word 𝑇) |
201 | 18 | ad2antrr 716 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → 𝐼 ∈ (0...(𝐼 + 2))) |
202 | 36 | ad2antrr 716 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → (𝐼 + 2) ∈ (0...(♯‘𝑊))) |
203 | 122 | adantr 474 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → 〈“𝑟𝑠”〉 ∈ Word 𝑇) |
204 | | simpr 479 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → 𝑗 ∈ (0..^𝐼)) |
205 | 200, 201,
202, 203, 204 | splfv1 13898 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) = (𝑊‘𝑗)) |
206 | 205 | difeq1d 3950 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ) = ((𝑊‘𝑗) ∖ I )) |
207 | 206 | dmeqd 5571 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ) = dom ((𝑊‘𝑗) ∖ I )) |
208 | 199, 207 | neleqtrrd 2881 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I )) |
209 | 208 | ex 403 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝑗 ∈ (0..^𝐼) → ¬ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ))) |
210 | 209 | adantrr 707 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝑗 ∈ (0..^𝐼) → ¬ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ))) |
211 | | simprr3 1248 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → ¬ 𝐴 ∈ dom (𝑟 ∖ I )) |
212 | | 0nn0 11659 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 0 ∈
ℕ0 |
213 | | 2pos 11485 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 0 <
2 |
214 | | elfzo0 12828 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (0 ∈
(0..^2) ↔ (0 ∈ ℕ0 ∧ 2 ∈ ℕ ∧ 0
< 2)) |
215 | 212, 172,
213, 214 | mpbir3an 1398 |
. . . . . . . . . . . . . . . . . . 19
⊢ 0 ∈
(0..^2) |
216 | 215, 176 | eleqtrri 2858 |
. . . . . . . . . . . . . . . . . 18
⊢ 0 ∈
(0..^(♯‘〈“𝑟𝑠”〉)) |
217 | 216 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 0 ∈
(0..^(♯‘〈“𝑟𝑠”〉))) |
218 | 119, 152,
153, 122, 217 | splfv2a 13900 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 0)) = (〈“𝑟𝑠”〉‘0)) |
219 | 29 | addid1d 10576 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝐼 + 0) = 𝐼) |
220 | 219 | adantr 474 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝐼 + 0) = 𝐼) |
221 | 220 | fveq2d 6450 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 0)) = ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼)) |
222 | | s2fv0 14038 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑟 ∈ 𝑇 → (〈“𝑟𝑠”〉‘0) = 𝑟) |
223 | 222 | ad2antrl 718 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (〈“𝑟𝑠”〉‘0) = 𝑟) |
224 | 218, 221,
223 | 3eqtr3d 2822 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) = 𝑟) |
225 | 224 | difeq1d 3950 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) ∖ I ) = (𝑟 ∖ I )) |
226 | 225 | dmeqd 5571 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) ∖ I ) = dom (𝑟 ∖ I )) |
227 | 226 | eleq2d 2845 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) ∖ I ) ↔ 𝐴 ∈ dom (𝑟 ∖ I ))) |
228 | 227 | adantrr 707 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) ∖ I ) ↔ 𝐴 ∈ dom (𝑟 ∖ I ))) |
229 | 211, 228 | mtbird 317 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → ¬ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) ∖ I )) |
230 | | fveq2 6446 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝐼 → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) = ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼)) |
231 | 230 | difeq1d 3950 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝐼 → (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ) = (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) ∖ I )) |
232 | 231 | dmeqd 5571 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝐼 → dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ) = dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) ∖ I )) |
233 | 232 | eleq2d 2845 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝐼 → (𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ) ↔ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) ∖ I ))) |
234 | 233 | notbid 310 |
. . . . . . . . . 10
⊢ (𝑗 = 𝐼 → (¬ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ) ↔ ¬ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) ∖ I ))) |
235 | 229, 234 | syl5ibrcom 239 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝑗 = 𝐼 → ¬ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ))) |
236 | 210, 235 | jaod 848 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → ((𝑗 ∈ (0..^𝐼) ∨ 𝑗 = 𝐼) → ¬ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ))) |
237 | 191, 236 | sylbid 232 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝑗 ∈ (0..^(𝐼 + 1)) → ¬ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ))) |
238 | 237 | ralrimiv 3147 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I )) |
239 | 169, 186,
238 | 3jca 1119 |
. . . . 5
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → ((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ))) |
240 | | oveq2 6930 |
. . . . . . . . 9
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → (𝐺 Σg
𝑤) = (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉))) |
241 | 240 | eqeq1d 2780 |
. . . . . . . 8
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → ((𝐺 Σg
𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)) = ( I ↾ 𝐷))) |
242 | | fveqeq2 6455 |
. . . . . . . 8
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) →
((♯‘𝑤) = 𝐿 ↔ (♯‘(𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)) = 𝐿)) |
243 | 241, 242 | anbi12d 624 |
. . . . . . 7
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → (((𝐺 Σg
𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ↔ ((𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)) = ( I ↾ 𝐷) ∧ (♯‘(𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)) = 𝐿))) |
244 | | fveq1 6445 |
. . . . . . . . . . 11
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → (𝑤‘(𝐼 + 1)) = ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 1))) |
245 | 244 | difeq1d 3950 |
. . . . . . . . . 10
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → ((𝑤‘(𝐼 + 1)) ∖ I ) = (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 1)) ∖ I )) |
246 | 245 | dmeqd 5571 |
. . . . . . . . 9
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → dom ((𝑤‘(𝐼 + 1)) ∖ I ) = dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 1)) ∖ I )) |
247 | 246 | eleq2d 2845 |
. . . . . . . 8
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → (𝐴 ∈ dom ((𝑤‘(𝐼 + 1)) ∖ I ) ↔ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 1)) ∖ I ))) |
248 | | fveq1 6445 |
. . . . . . . . . . . . 13
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → (𝑤‘𝑗) = ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗)) |
249 | 248 | difeq1d 3950 |
. . . . . . . . . . . 12
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → ((𝑤‘𝑗) ∖ I ) = (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I )) |
250 | 249 | dmeqd 5571 |
. . . . . . . . . . 11
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → dom ((𝑤‘𝑗) ∖ I ) = dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I )) |
251 | 250 | eleq2d 2845 |
. . . . . . . . . 10
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → (𝐴 ∈ dom ((𝑤‘𝑗) ∖ I ) ↔ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ))) |
252 | 251 | notbid 310 |
. . . . . . . . 9
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → (¬ 𝐴 ∈ dom ((𝑤‘𝑗) ∖ I ) ↔ ¬ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ))) |
253 | 252 | ralbidv 3168 |
. . . . . . . 8
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → (∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom ((𝑤‘𝑗) ∖ I ) ↔ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ))) |
254 | 247, 253 | 3anbi23d 1512 |
. . . . . . 7
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → (((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom ((𝑤‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom ((𝑤‘𝑗) ∖ I )) ↔ ((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I )))) |
255 | 243, 254 | anbi12d 624 |
. . . . . 6
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → ((((𝐺 Σg
𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ ((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom ((𝑤‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom ((𝑤‘𝑗) ∖ I ))) ↔ (((𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)) = ( I ↾ 𝐷) ∧ (♯‘(𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)) = 𝐿) ∧ ((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ))))) |
256 | 255 | rspcev 3511 |
. . . . 5
⊢ (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) ∈ Word 𝑇 ∧ (((𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)) = ( I ↾ 𝐷) ∧ (♯‘(𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)) = 𝐿) ∧ ((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I )))) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ ((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom ((𝑤‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom ((𝑤‘𝑗) ∖ I )))) |
257 | 125, 168,
239, 256 | syl12anc 827 |
. . . 4
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ ((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom ((𝑤‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom ((𝑤‘𝑗) ∖ I )))) |
258 | 257 | expr 450 |
. . 3
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → ((((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ ((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom ((𝑤‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom ((𝑤‘𝑗) ∖ I ))))) |
259 | 258 | rexlimdvva 3221 |
. 2
⊢ (𝜑 → (∃𝑟 ∈ 𝑇 ∃𝑠 ∈ 𝑇 (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ ((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom ((𝑤‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom ((𝑤‘𝑗) ∖ I ))))) |
260 | 20, 21, 87, 89, 24 | psgnunilem1 18296 |
. 2
⊢ (𝜑 → (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷) ∨ ∃𝑟 ∈ 𝑇 ∃𝑠 ∈ 𝑇 (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) |
261 | 118, 259,
260 | mpjaod 849 |
1
⊢ (𝜑 → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ ((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom ((𝑤‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom ((𝑤‘𝑗) ∖ I )))) |