Step | Hyp | Ref
| Expression |
1 | | psgnunilem2.w |
. . . . . . 7
⊢ (𝜑 → 𝑊 ∈ Word 𝑇) |
2 | | wrd0 14252 |
. . . . . . 7
⊢ ∅
∈ Word 𝑇 |
3 | | splcl 14475 |
. . . . . . 7
⊢ ((𝑊 ∈ Word 𝑇 ∧ ∅ ∈ Word 𝑇) → (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉) ∈ Word 𝑇) |
4 | 1, 2, 3 | sylancl 586 |
. . . . . 6
⊢ (𝜑 → (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉) ∈ Word 𝑇) |
5 | 4 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉) ∈ Word 𝑇) |
6 | | fzossfz 13416 |
. . . . . . . . . . 11
⊢
(0..^𝐿) ⊆
(0...𝐿) |
7 | | psgnunilem2.ix |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐼 ∈ (0..^𝐿)) |
8 | 6, 7 | sselid 3918 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐼 ∈ (0...𝐿)) |
9 | | elfznn0 13359 |
. . . . . . . . . 10
⊢ (𝐼 ∈ (0...𝐿) → 𝐼 ∈
ℕ0) |
10 | 8, 9 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐼 ∈
ℕ0) |
11 | | 2nn0 12260 |
. . . . . . . . . 10
⊢ 2 ∈
ℕ0 |
12 | | nn0addcl 12278 |
. . . . . . . . . 10
⊢ ((𝐼 ∈ ℕ0
∧ 2 ∈ ℕ0) → (𝐼 + 2) ∈
ℕ0) |
13 | 10, 11, 12 | sylancl 586 |
. . . . . . . . 9
⊢ (𝜑 → (𝐼 + 2) ∈
ℕ0) |
14 | 10 | nn0red 12304 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐼 ∈ ℝ) |
15 | | nn0addge1 12289 |
. . . . . . . . . 10
⊢ ((𝐼 ∈ ℝ ∧ 2 ∈
ℕ0) → 𝐼 ≤ (𝐼 + 2)) |
16 | 14, 11, 15 | sylancl 586 |
. . . . . . . . 9
⊢ (𝜑 → 𝐼 ≤ (𝐼 + 2)) |
17 | | elfz2nn0 13357 |
. . . . . . . . 9
⊢ (𝐼 ∈ (0...(𝐼 + 2)) ↔ (𝐼 ∈ ℕ0 ∧ (𝐼 + 2) ∈ ℕ0
∧ 𝐼 ≤ (𝐼 + 2))) |
18 | 10, 13, 16, 17 | syl3anbrc 1342 |
. . . . . . . 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 19112 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐼 + 1) ∈ (0..^𝐿)) |
27 | | fzofzp1 13494 |
. . . . . . . . . 10
⊢ ((𝐼 + 1) ∈ (0..^𝐿) → ((𝐼 + 1) + 1) ∈ (0...𝐿)) |
28 | 26, 27 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐼 + 1) + 1) ∈ (0...𝐿)) |
29 | | df-2 12046 |
. . . . . . . . . . 11
⊢ 2 = (1 +
1) |
30 | 29 | oveq2i 7278 |
. . . . . . . . . 10
⊢ (𝐼 + 2) = (𝐼 + (1 + 1)) |
31 | 10 | nn0cnd 12305 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐼 ∈ ℂ) |
32 | | 1cnd 10980 |
. . . . . . . . . . 11
⊢ (𝜑 → 1 ∈
ℂ) |
33 | 31, 32, 32 | addassd 11007 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐼 + 1) + 1) = (𝐼 + (1 + 1))) |
34 | 30, 33 | eqtr4id 2797 |
. . . . . . . . 9
⊢ (𝜑 → (𝐼 + 2) = ((𝐼 + 1) + 1)) |
35 | 23 | oveq2d 7283 |
. . . . . . . . 9
⊢ (𝜑 → (0...(♯‘𝑊)) = (0...𝐿)) |
36 | 28, 34, 35 | 3eltr4d 2854 |
. . . . . . . 8
⊢ (𝜑 → (𝐼 + 2) ∈ (0...(♯‘𝑊))) |
37 | 2 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → ∅ ∈ Word 𝑇) |
38 | 1, 18, 36, 37 | spllen 14477 |
. . . . . . 7
⊢ (𝜑 → (♯‘(𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉)) =
((♯‘𝑊) +
((♯‘∅) − ((𝐼 + 2) − 𝐼)))) |
39 | | hash0 14092 |
. . . . . . . . . . 11
⊢
(♯‘∅) = 0 |
40 | 39 | oveq1i 7277 |
. . . . . . . . . 10
⊢
((♯‘∅) − ((𝐼 + 2) − 𝐼)) = (0 − ((𝐼 + 2) − 𝐼)) |
41 | | df-neg 11218 |
. . . . . . . . . 10
⊢ -((𝐼 + 2) − 𝐼) = (0 − ((𝐼 + 2) − 𝐼)) |
42 | 40, 41 | eqtr4i 2769 |
. . . . . . . . 9
⊢
((♯‘∅) − ((𝐼 + 2) − 𝐼)) = -((𝐼 + 2) − 𝐼) |
43 | | 2cn 12058 |
. . . . . . . . . . 11
⊢ 2 ∈
ℂ |
44 | | pncan2 11238 |
. . . . . . . . . . 11
⊢ ((𝐼 ∈ ℂ ∧ 2 ∈
ℂ) → ((𝐼 + 2)
− 𝐼) =
2) |
45 | 31, 43, 44 | sylancl 586 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐼 + 2) − 𝐼) = 2) |
46 | 45 | negeqd 11225 |
. . . . . . . . 9
⊢ (𝜑 → -((𝐼 + 2) − 𝐼) = -2) |
47 | 42, 46 | eqtrid 2790 |
. . . . . . . 8
⊢ (𝜑 → ((♯‘∅)
− ((𝐼 + 2) −
𝐼)) = -2) |
48 | 23, 47 | oveq12d 7285 |
. . . . . . 7
⊢ (𝜑 → ((♯‘𝑊) + ((♯‘∅)
− ((𝐼 + 2) −
𝐼))) = (𝐿 + -2)) |
49 | | elfzel2 13264 |
. . . . . . . . . 10
⊢ (𝐼 ∈ (0...𝐿) → 𝐿 ∈ ℤ) |
50 | 8, 49 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐿 ∈ ℤ) |
51 | 50 | zcnd 12437 |
. . . . . . . 8
⊢ (𝜑 → 𝐿 ∈ ℂ) |
52 | | negsub 11279 |
. . . . . . . 8
⊢ ((𝐿 ∈ ℂ ∧ 2 ∈
ℂ) → (𝐿 + -2) =
(𝐿 −
2)) |
53 | 51, 43, 52 | sylancl 586 |
. . . . . . 7
⊢ (𝜑 → (𝐿 + -2) = (𝐿 − 2)) |
54 | 38, 48, 53 | 3eqtrd 2782 |
. . . . . 6
⊢ (𝜑 → (♯‘(𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉)) = (𝐿 − 2)) |
55 | 54 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (♯‘(𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉)) = (𝐿 − 2)) |
56 | | splid 14476 |
. . . . . . . . 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 834 |
. . . . . . . 8
⊢ (𝜑 → (𝑊 splice 〈𝐼, (𝐼 + 2), (𝑊 substr 〈𝐼, (𝐼 + 2)〉)〉) = 𝑊) |
58 | 57 | oveq2d 7283 |
. . . . . . 7
⊢ (𝜑 → (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), (𝑊 substr 〈𝐼, (𝐼 + 2)〉)〉)) = (𝐺 Σg 𝑊)) |
59 | 58 | adantr 481 |
. . . . . 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 2738 |
. . . . . . 7
⊢
(Base‘𝐺) =
(Base‘𝐺) |
61 | 19 | symggrp 19018 |
. . . . . . . . . 10
⊢ (𝐷 ∈ 𝑉 → 𝐺 ∈ Grp) |
62 | 21, 61 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐺 ∈ Grp) |
63 | 62 | grpmndd 18599 |
. . . . . . . 8
⊢ (𝜑 → 𝐺 ∈ Mnd) |
64 | 63 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → 𝐺 ∈ Mnd) |
65 | 20, 19, 60 | symgtrf 19087 |
. . . . . . . . . 10
⊢ 𝑇 ⊆ (Base‘𝐺) |
66 | | sswrd 14235 |
. . . . . . . . . 10
⊢ (𝑇 ⊆ (Base‘𝐺) → Word 𝑇 ⊆ Word (Base‘𝐺)) |
67 | 65, 66 | ax-mp 5 |
. . . . . . . . 9
⊢ Word
𝑇 ⊆ Word
(Base‘𝐺) |
68 | 67, 1 | sselid 3918 |
. . . . . . . 8
⊢ (𝜑 → 𝑊 ∈ Word (Base‘𝐺)) |
69 | 68 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → 𝑊 ∈ Word (Base‘𝐺)) |
70 | 18 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → 𝐼 ∈ (0...(𝐼 + 2))) |
71 | 36 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐼 + 2) ∈ (0...(♯‘𝑊))) |
72 | | swrdcl 14368 |
. . . . . . . . 9
⊢ (𝑊 ∈ Word (Base‘𝐺) → (𝑊 substr 〈𝐼, (𝐼 + 2)〉) ∈ Word (Base‘𝐺)) |
73 | 68, 72 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (𝑊 substr 〈𝐼, (𝐼 + 2)〉) ∈ Word (Base‘𝐺)) |
74 | 73 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝑊 substr 〈𝐼, (𝐼 + 2)〉) ∈ Word (Base‘𝐺)) |
75 | | wrd0 14252 |
. . . . . . . 8
⊢ ∅
∈ Word (Base‘𝐺) |
76 | 75 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ∅ ∈ Word
(Base‘𝐺)) |
77 | 23 | oveq2d 7283 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (0..^(♯‘𝑊)) = (0..^𝐿)) |
78 | 26, 77 | eleqtrrd 2842 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐼 + 1) ∈ (0..^(♯‘𝑊))) |
79 | | swrds2 14663 |
. . . . . . . . . . . 12
⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝐼 ∈ ℕ0 ∧ (𝐼 + 1) ∈
(0..^(♯‘𝑊)))
→ (𝑊 substr
〈𝐼, (𝐼 + 2)〉) = 〈“(𝑊‘𝐼)(𝑊‘(𝐼 + 1))”〉) |
80 | 1, 10, 78, 79 | syl3anc 1370 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑊 substr 〈𝐼, (𝐼 + 2)〉) = 〈“(𝑊‘𝐼)(𝑊‘(𝐼 + 1))”〉) |
81 | 80 | oveq2d 7283 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐺 Σg (𝑊 substr 〈𝐼, (𝐼 + 2)〉)) = (𝐺 Σg
〈“(𝑊‘𝐼)(𝑊‘(𝐼 + 1))”〉)) |
82 | | wrdf 14232 |
. . . . . . . . . . . . . . 15
⊢ (𝑊 ∈ Word 𝑇 → 𝑊:(0..^(♯‘𝑊))⟶𝑇) |
83 | 1, 82 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑊:(0..^(♯‘𝑊))⟶𝑇) |
84 | 77 | feq2d 6578 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑊:(0..^(♯‘𝑊))⟶𝑇 ↔ 𝑊:(0..^𝐿)⟶𝑇)) |
85 | 83, 84 | mpbid 231 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑊:(0..^𝐿)⟶𝑇) |
86 | 85, 7 | ffvelrnd 6954 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑊‘𝐼) ∈ 𝑇) |
87 | 65, 86 | sselid 3918 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑊‘𝐼) ∈ (Base‘𝐺)) |
88 | 85, 26 | ffvelrnd 6954 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑊‘(𝐼 + 1)) ∈ 𝑇) |
89 | 65, 88 | sselid 3918 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑊‘(𝐼 + 1)) ∈ (Base‘𝐺)) |
90 | | eqid 2738 |
. . . . . . . . . . . 12
⊢
(+g‘𝐺) = (+g‘𝐺) |
91 | 60, 90 | gsumws2 18491 |
. . . . . . . . . . 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)))) |
92 | 63, 87, 89, 91 | syl3anc 1370 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐺 Σg
〈“(𝑊‘𝐼)(𝑊‘(𝐼 + 1))”〉) = ((𝑊‘𝐼)(+g‘𝐺)(𝑊‘(𝐼 + 1)))) |
93 | 19, 60, 90 | symgov 19001 |
. . . . . . . . . . 11
⊢ (((𝑊‘𝐼) ∈ (Base‘𝐺) ∧ (𝑊‘(𝐼 + 1)) ∈ (Base‘𝐺)) → ((𝑊‘𝐼)(+g‘𝐺)(𝑊‘(𝐼 + 1))) = ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1)))) |
94 | 87, 89, 93 | syl2anc 584 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑊‘𝐼)(+g‘𝐺)(𝑊‘(𝐼 + 1))) = ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1)))) |
95 | 81, 92, 94 | 3eqtrd 2782 |
. . . . . . . . 9
⊢ (𝜑 → (𝐺 Σg (𝑊 substr 〈𝐼, (𝐼 + 2)〉)) = ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1)))) |
96 | 95 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐺 Σg (𝑊 substr 〈𝐼, (𝐼 + 2)〉)) = ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1)))) |
97 | | simpr 485 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) |
98 | 19 | symgid 19019 |
. . . . . . . . . . 11
⊢ (𝐷 ∈ 𝑉 → ( I ↾ 𝐷) = (0g‘𝐺)) |
99 | 21, 98 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → ( I ↾ 𝐷) = (0g‘𝐺)) |
100 | | eqid 2738 |
. . . . . . . . . . 11
⊢
(0g‘𝐺) = (0g‘𝐺) |
101 | 100 | gsum0 18378 |
. . . . . . . . . 10
⊢ (𝐺 Σg
∅) = (0g‘𝐺) |
102 | 99, 101 | eqtr4di 2796 |
. . . . . . . . 9
⊢ (𝜑 → ( I ↾ 𝐷) = (𝐺 Σg
∅)) |
103 | 102 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ( I ↾ 𝐷) = (𝐺 Σg
∅)) |
104 | 96, 97, 103 | 3eqtrd 2782 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐺 Σg (𝑊 substr 〈𝐼, (𝐼 + 2)〉)) = (𝐺 Σg
∅)) |
105 | 60, 64, 69, 70, 71, 74, 76, 104 | gsumspl 18493 |
. . . . . 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), ∅〉))) |
106 | 22 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐺 Σg 𝑊) = ( I ↾ 𝐷)) |
107 | 59, 105, 106 | 3eqtr3d 2786 |
. . . . 5
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉)) = ( I ↾ 𝐷)) |
108 | | fveqeq2 6775 |
. . . . . . 7
⊢ (𝑥 = (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉) →
((♯‘𝑥) = (𝐿 − 2) ↔
(♯‘(𝑊 splice
〈𝐼, (𝐼 + 2), ∅〉)) = (𝐿 − 2))) |
109 | | oveq2 7275 |
. . . . . . . 8
⊢ (𝑥 = (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉) → (𝐺 Σg
𝑥) = (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉))) |
110 | 109 | eqeq1d 2740 |
. . . . . . 7
⊢ (𝑥 = (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉) → ((𝐺 Σg
𝑥) = ( I ↾ 𝐷) ↔ (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉)) = ( I ↾ 𝐷))) |
111 | 108, 110 | anbi12d 631 |
. . . . . 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)))) |
112 | 111 | rspcev 3559 |
. . . . 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))) |
113 | 5, 55, 107, 112 | syl12anc 834 |
. . . 4
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) |
114 | | psgnunilem2.in |
. . . . 5
⊢ (𝜑 → ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) |
115 | 114 | adantr 481 |
. . . 4
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) |
116 | 113, 115 | pm2.21dd 194 |
. . 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 )))) |
117 | 116 | ex 413 |
. 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 ))))) |
118 | 1 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 𝑊 ∈ Word 𝑇) |
119 | | simprl 768 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 𝑟 ∈ 𝑇) |
120 | | simprr 770 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 𝑠 ∈ 𝑇) |
121 | 119, 120 | s2cld 14594 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 〈“𝑟𝑠”〉 ∈ Word 𝑇) |
122 | | splcl 14475 |
. . . . . . 7
⊢ ((𝑊 ∈ Word 𝑇 ∧ 〈“𝑟𝑠”〉 ∈ Word 𝑇) → (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) ∈ Word 𝑇) |
123 | 118, 121,
122 | syl2anc 584 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) ∈ Word 𝑇) |
124 | 123 | adantrr 714 |
. . . . 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) |
125 | 63 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇) ∧ (((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟 ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → 𝐺 ∈ Mnd) |
126 | 68 | adantr 481 |
. . . . . . . 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)) |
127 | 18 | adantr 481 |
. . . . . . . 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))) |
128 | 36 | adantr 481 |
. . . . . . . 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))) |
129 | 67, 121 | sselid 3918 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 〈“𝑟𝑠”〉 ∈ Word (Base‘𝐺)) |
130 | 129 | adantrr 714 |
. . . . . . . 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)) |
131 | 73 | adantr 481 |
. . . . . . . 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)) |
132 | | simprr1 1220 |
. . . . . . . . 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)) |
133 | 95 | adantr 481 |
. . . . . . . . 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)))) |
134 | 63 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 𝐺 ∈ Mnd) |
135 | 65 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑇 ⊆ (Base‘𝐺)) |
136 | 135 | sselda 3920 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑇) → 𝑟 ∈ (Base‘𝐺)) |
137 | 136 | adantrr 714 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 𝑟 ∈ (Base‘𝐺)) |
138 | 135 | sselda 3920 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑇) → 𝑠 ∈ (Base‘𝐺)) |
139 | 138 | adantrl 713 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 𝑠 ∈ (Base‘𝐺)) |
140 | 60, 90 | gsumws2 18491 |
. . . . . . . . . . . 12
⊢ ((𝐺 ∈ Mnd ∧ 𝑟 ∈ (Base‘𝐺) ∧ 𝑠 ∈ (Base‘𝐺)) → (𝐺 Σg
〈“𝑟𝑠”〉) = (𝑟(+g‘𝐺)𝑠)) |
141 | 134, 137,
139, 140 | syl3anc 1370 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝐺 Σg
〈“𝑟𝑠”〉) = (𝑟(+g‘𝐺)𝑠)) |
142 | 19, 60, 90 | symgov 19001 |
. . . . . . . . . . . 12
⊢ ((𝑟 ∈ (Base‘𝐺) ∧ 𝑠 ∈ (Base‘𝐺)) → (𝑟(+g‘𝐺)𝑠) = (𝑟 ∘ 𝑠)) |
143 | 137, 139,
142 | syl2anc 584 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝑟(+g‘𝐺)𝑠) = (𝑟 ∘ 𝑠)) |
144 | 141, 143 | eqtrd 2778 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝐺 Σg
〈“𝑟𝑠”〉) = (𝑟 ∘ 𝑠)) |
145 | 144 | adantrr 714 |
. . . . . . . . 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)) |
146 | 132, 133,
145 | 3eqtr4rd 2789 |
. . . . . . . 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)〉))) |
147 | 60, 125, 126, 127, 128, 130, 131, 146 | gsumspl 18493 |
. . . . . . 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)〉)〉))) |
148 | 58 | adantr 481 |
. . . . . . 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)) |
149 | 22 | adantr 481 |
. . . . . . 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)) |
150 | 147, 148,
149 | 3eqtrd 2782 |
. . . . . 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)) |
151 | 18 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 𝐼 ∈ (0...(𝐼 + 2))) |
152 | 36 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝐼 + 2) ∈ (0...(♯‘𝑊))) |
153 | 118, 151,
152, 121 | spllen 14477 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (♯‘(𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)) = ((♯‘𝑊) +
((♯‘〈“𝑟𝑠”〉) − ((𝐼 + 2) − 𝐼)))) |
154 | | s2len 14612 |
. . . . . . . . . . . . 13
⊢
(♯‘〈“𝑟𝑠”〉) = 2 |
155 | 154 | oveq1i 7277 |
. . . . . . . . . . . 12
⊢
((♯‘〈“𝑟𝑠”〉) − ((𝐼 + 2) − 𝐼)) = (2 − ((𝐼 + 2) − 𝐼)) |
156 | 45 | oveq2d 7283 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (2 − ((𝐼 + 2) − 𝐼)) = (2 − 2)) |
157 | 43 | subidi 11302 |
. . . . . . . . . . . . 13
⊢ (2
− 2) = 0 |
158 | 156, 157 | eqtrdi 2794 |
. . . . . . . . . . . 12
⊢ (𝜑 → (2 − ((𝐼 + 2) − 𝐼)) = 0) |
159 | 155, 158 | eqtrid 2790 |
. . . . . . . . . . 11
⊢ (𝜑 →
((♯‘〈“𝑟𝑠”〉) − ((𝐼 + 2) − 𝐼)) = 0) |
160 | 159 | oveq2d 7283 |
. . . . . . . . . 10
⊢ (𝜑 → ((♯‘𝑊) +
((♯‘〈“𝑟𝑠”〉) − ((𝐼 + 2) − 𝐼))) = ((♯‘𝑊) + 0)) |
161 | 23, 51 | eqeltrd 2839 |
. . . . . . . . . . 11
⊢ (𝜑 → (♯‘𝑊) ∈
ℂ) |
162 | 161 | addid1d 11185 |
. . . . . . . . . 10
⊢ (𝜑 → ((♯‘𝑊) + 0) = (♯‘𝑊)) |
163 | 160, 162,
23 | 3eqtrd 2782 |
. . . . . . . . 9
⊢ (𝜑 → ((♯‘𝑊) +
((♯‘〈“𝑟𝑠”〉) − ((𝐼 + 2) − 𝐼))) = 𝐿) |
164 | 163 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → ((♯‘𝑊) + ((♯‘〈“𝑟𝑠”〉) − ((𝐼 + 2) − 𝐼))) = 𝐿) |
165 | 153, 164 | eqtrd 2778 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (♯‘(𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)) = 𝐿) |
166 | 165 | adantrr 714 |
. . . . . 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) |
167 | 150, 166 | jca 512 |
. . . . 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)) |
168 | 26 | adantr 481 |
. . . . . 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)) |
169 | | simprr2 1221 |
. . . . . . 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 )) |
170 | | 1nn0 12259 |
. . . . . . . . . . . . . . 15
⊢ 1 ∈
ℕ0 |
171 | | 2nn 12056 |
. . . . . . . . . . . . . . 15
⊢ 2 ∈
ℕ |
172 | | 1lt2 12154 |
. . . . . . . . . . . . . . 15
⊢ 1 <
2 |
173 | | elfzo0 13438 |
. . . . . . . . . . . . . . 15
⊢ (1 ∈
(0..^2) ↔ (1 ∈ ℕ0 ∧ 2 ∈ ℕ ∧ 1
< 2)) |
174 | 170, 171,
172, 173 | mpbir3an 1340 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
(0..^2) |
175 | 154 | oveq2i 7278 |
. . . . . . . . . . . . . 14
⊢
(0..^(♯‘〈“𝑟𝑠”〉)) = (0..^2) |
176 | 174, 175 | eleqtrri 2838 |
. . . . . . . . . . . . 13
⊢ 1 ∈
(0..^(♯‘〈“𝑟𝑠”〉)) |
177 | 176 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 1 ∈
(0..^(♯‘〈“𝑟𝑠”〉))) |
178 | 118, 151,
152, 121, 177 | splfv2a 14479 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 1)) = (〈“𝑟𝑠”〉‘1)) |
179 | | s2fv1 14611 |
. . . . . . . . . . . 12
⊢ (𝑠 ∈ 𝑇 → (〈“𝑟𝑠”〉‘1) = 𝑠) |
180 | 179 | ad2antll 726 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (〈“𝑟𝑠”〉‘1) = 𝑠) |
181 | 178, 180 | eqtrd 2778 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 1)) = 𝑠) |
182 | 181 | adantrr 714 |
. . . . . . . . 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) |
183 | 182 | difeq1d 4055 |
. . . . . . . 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 )) |
184 | 183 | dmeqd 5807 |
. . . . . . 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 )) |
185 | 169, 184 | eleqtrrd 2842 |
. . . . . 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 )) |
186 | | fzosplitsni 13508 |
. . . . . . . . . . 11
⊢ (𝐼 ∈
(ℤ≥‘0) → (𝑗 ∈ (0..^(𝐼 + 1)) ↔ (𝑗 ∈ (0..^𝐼) ∨ 𝑗 = 𝐼))) |
187 | | nn0uz 12630 |
. . . . . . . . . . 11
⊢
ℕ0 = (ℤ≥‘0) |
188 | 186, 187 | eleq2s 2857 |
. . . . . . . . . 10
⊢ (𝐼 ∈ ℕ0
→ (𝑗 ∈
(0..^(𝐼 + 1)) ↔ (𝑗 ∈ (0..^𝐼) ∨ 𝑗 = 𝐼))) |
189 | 10, 188 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (𝑗 ∈ (0..^(𝐼 + 1)) ↔ (𝑗 ∈ (0..^𝐼) ∨ 𝑗 = 𝐼))) |
190 | 189 | adantr 481 |
. . . . . . . 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))) |
191 | | fveq2 6766 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑗 → (𝑊‘𝑘) = (𝑊‘𝑗)) |
192 | 191 | difeq1d 4055 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑗 → ((𝑊‘𝑘) ∖ I ) = ((𝑊‘𝑗) ∖ I )) |
193 | 192 | dmeqd 5807 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑗 → dom ((𝑊‘𝑘) ∖ I ) = dom ((𝑊‘𝑗) ∖ I )) |
194 | 193 | eleq2d 2824 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑗 → (𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ) ↔ 𝐴 ∈ dom ((𝑊‘𝑗) ∖ I ))) |
195 | 194 | notbid 318 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑗 → (¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ) ↔ ¬ 𝐴 ∈ dom ((𝑊‘𝑗) ∖ I ))) |
196 | 195 | rspccva 3558 |
. . . . . . . . . . . . . 14
⊢
((∀𝑘 ∈
(0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ) ∧ 𝑗 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom ((𝑊‘𝑗) ∖ I )) |
197 | 25, 196 | sylan 580 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom ((𝑊‘𝑗) ∖ I )) |
198 | 197 | adantlr 712 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom ((𝑊‘𝑗) ∖ I )) |
199 | 1 | ad2antrr 723 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → 𝑊 ∈ Word 𝑇) |
200 | 18 | ad2antrr 723 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → 𝐼 ∈ (0...(𝐼 + 2))) |
201 | 36 | ad2antrr 723 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → (𝐼 + 2) ∈ (0...(♯‘𝑊))) |
202 | 121 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → 〈“𝑟𝑠”〉 ∈ Word 𝑇) |
203 | | simpr 485 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → 𝑗 ∈ (0..^𝐼)) |
204 | 199, 200,
201, 202, 203 | splfv1 14478 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) = (𝑊‘𝑗)) |
205 | 204 | difeq1d 4055 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ) = ((𝑊‘𝑗) ∖ I )) |
206 | 205 | dmeqd 5807 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ) = dom ((𝑊‘𝑗) ∖ I )) |
207 | 198, 206 | neleqtrrd 2861 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I )) |
208 | 207 | ex 413 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝑗 ∈ (0..^𝐼) → ¬ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ))) |
209 | 208 | adantrr 714 |
. . . . . . . . 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 ))) |
210 | | simprr3 1222 |
. . . . . . . . . . 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 )) |
211 | | 0nn0 12258 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 0 ∈
ℕ0 |
212 | | 2pos 12086 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 0 <
2 |
213 | | elfzo0 13438 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (0 ∈
(0..^2) ↔ (0 ∈ ℕ0 ∧ 2 ∈ ℕ ∧ 0
< 2)) |
214 | 211, 171,
212, 213 | mpbir3an 1340 |
. . . . . . . . . . . . . . . . . . 19
⊢ 0 ∈
(0..^2) |
215 | 214, 175 | eleqtrri 2838 |
. . . . . . . . . . . . . . . . . 18
⊢ 0 ∈
(0..^(♯‘〈“𝑟𝑠”〉)) |
216 | 215 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 0 ∈
(0..^(♯‘〈“𝑟𝑠”〉))) |
217 | 118, 151,
152, 121, 216 | splfv2a 14479 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 0)) = (〈“𝑟𝑠”〉‘0)) |
218 | 31 | addid1d 11185 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝐼 + 0) = 𝐼) |
219 | 218 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝐼 + 0) = 𝐼) |
220 | 219 | fveq2d 6770 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 0)) = ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼)) |
221 | | s2fv0 14610 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑟 ∈ 𝑇 → (〈“𝑟𝑠”〉‘0) = 𝑟) |
222 | 221 | ad2antrl 725 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (〈“𝑟𝑠”〉‘0) = 𝑟) |
223 | 217, 220,
222 | 3eqtr3d 2786 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) = 𝑟) |
224 | 223 | difeq1d 4055 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) ∖ I ) = (𝑟 ∖ I )) |
225 | 224 | dmeqd 5807 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) ∖ I ) = dom (𝑟 ∖ I )) |
226 | 225 | eleq2d 2824 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) ∖ I ) ↔ 𝐴 ∈ dom (𝑟 ∖ I ))) |
227 | 226 | adantrr 714 |
. . . . . . . . . . 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 ))) |
228 | 210, 227 | mtbird 325 |
. . . . . . . . . 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 )) |
229 | | fveq2 6766 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝐼 → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) = ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼)) |
230 | 229 | difeq1d 4055 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝐼 → (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ) = (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) ∖ I )) |
231 | 230 | dmeqd 5807 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝐼 → dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ) = dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) ∖ I )) |
232 | 231 | eleq2d 2824 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝐼 → (𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ) ↔ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) ∖ I ))) |
233 | 232 | notbid 318 |
. . . . . . . . . 10
⊢ (𝑗 = 𝐼 → (¬ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ) ↔ ¬ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) ∖ I ))) |
234 | 228, 233 | syl5ibrcom 246 |
. . . . . . . . 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 ))) |
235 | 209, 234 | jaod 856 |
. . . . . . . 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 ))) |
236 | 190, 235 | sylbid 239 |
. . . . . . 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 ))) |
237 | 236 | ralrimiv 3107 |
. . . . . 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 )) |
238 | 168, 185,
237 | 3jca 1127 |
. . . . 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 ))) |
239 | | oveq2 7275 |
. . . . . . . . 9
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → (𝐺 Σg
𝑤) = (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉))) |
240 | 239 | eqeq1d 2740 |
. . . . . . . 8
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → ((𝐺 Σg
𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)) = ( I ↾ 𝐷))) |
241 | | fveqeq2 6775 |
. . . . . . . 8
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) →
((♯‘𝑤) = 𝐿 ↔ (♯‘(𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)) = 𝐿)) |
242 | 240, 241 | anbi12d 631 |
. . . . . . 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))) |
243 | | fveq1 6765 |
. . . . . . . . . . 11
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → (𝑤‘(𝐼 + 1)) = ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 1))) |
244 | 243 | difeq1d 4055 |
. . . . . . . . . 10
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → ((𝑤‘(𝐼 + 1)) ∖ I ) = (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 1)) ∖ I )) |
245 | 244 | dmeqd 5807 |
. . . . . . . . 9
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → dom ((𝑤‘(𝐼 + 1)) ∖ I ) = dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 1)) ∖ I )) |
246 | 245 | eleq2d 2824 |
. . . . . . . 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 ))) |
247 | | fveq1 6765 |
. . . . . . . . . . . . 13
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → (𝑤‘𝑗) = ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗)) |
248 | 247 | difeq1d 4055 |
. . . . . . . . . . . 12
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → ((𝑤‘𝑗) ∖ I ) = (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I )) |
249 | 248 | dmeqd 5807 |
. . . . . . . . . . 11
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → dom ((𝑤‘𝑗) ∖ I ) = dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I )) |
250 | 249 | eleq2d 2824 |
. . . . . . . . . 10
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → (𝐴 ∈ dom ((𝑤‘𝑗) ∖ I ) ↔ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ))) |
251 | 250 | notbid 318 |
. . . . . . . . 9
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → (¬ 𝐴 ∈ dom ((𝑤‘𝑗) ∖ I ) ↔ ¬ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ))) |
252 | 251 | ralbidv 3121 |
. . . . . . . 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 ))) |
253 | 246, 252 | 3anbi23d 1438 |
. . . . . . 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 )))) |
254 | 242, 253 | anbi12d 631 |
. . . . . 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 ))))) |
255 | 254 | rspcev 3559 |
. . . . 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 )))) |
256 | 124, 167,
238, 255 | syl12anc 834 |
. . . 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 )))) |
257 | 256 | expr 457 |
. . 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 ))))) |
258 | 257 | 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 ))))) |
259 | 20, 21, 86, 88, 24 | psgnunilem1 19111 |
. 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 )))) |
260 | 117, 258,
259 | mpjaod 857 |
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 )))) |