Step | Hyp | Ref
| Expression |
1 | | psgnunilem2.w |
. . . . . . 7
⊢ (𝜑 → 𝑊 ∈ Word 𝑇) |
2 | | wrd0 13330 |
. . . . . . 7
⊢ ∅
∈ Word 𝑇 |
3 | | splcl 13503 |
. . . . . . 7
⊢ ((𝑊 ∈ Word 𝑇 ∧ ∅ ∈ Word 𝑇) → (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉) ∈ Word 𝑇) |
4 | 1, 2, 3 | sylancl 694 |
. . . . . 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 12488 |
. . . . . . . . . . 11
⊢
(0..^𝐿) ⊆
(0...𝐿) |
7 | | psgnunilem2.ix |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐼 ∈ (0..^𝐿)) |
8 | 6, 7 | sseldi 3601 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐼 ∈ (0...𝐿)) |
9 | | elfznn0 12433 |
. . . . . . . . . 10
⊢ (𝐼 ∈ (0...𝐿) → 𝐼 ∈
ℕ0) |
10 | 8, 9 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐼 ∈
ℕ0) |
11 | | 2nn0 11309 |
. . . . . . . . . 10
⊢ 2 ∈
ℕ0 |
12 | | nn0addcl 11328 |
. . . . . . . . . 10
⊢ ((𝐼 ∈ ℕ0
∧ 2 ∈ ℕ0) → (𝐼 + 2) ∈
ℕ0) |
13 | 10, 11, 12 | sylancl 694 |
. . . . . . . . 9
⊢ (𝜑 → (𝐼 + 2) ∈
ℕ0) |
14 | 10 | nn0red 11352 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐼 ∈ ℝ) |
15 | | nn0addge1 11339 |
. . . . . . . . . 10
⊢ ((𝐼 ∈ ℝ ∧ 2 ∈
ℕ0) → 𝐼 ≤ (𝐼 + 2)) |
16 | 14, 11, 15 | sylancl 694 |
. . . . . . . . 9
⊢ (𝜑 → 𝐼 ≤ (𝐼 + 2)) |
17 | | elfz2nn0 12431 |
. . . . . . . . 9
⊢ (𝐼 ∈ (0...(𝐼 + 2)) ↔ (𝐼 ∈ ℕ0 ∧ (𝐼 + 2) ∈ ℕ0
∧ 𝐼 ≤ (𝐼 + 2))) |
18 | 10, 13, 16, 17 | syl3anbrc 1246 |
. . . . . . . 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 17914 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐼 + 1) ∈ (0..^𝐿)) |
27 | | fzofzp1 12565 |
. . . . . . . . . 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 11353 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐼 ∈ ℂ) |
30 | | 1cnd 10056 |
. . . . . . . . . . 11
⊢ (𝜑 → 1 ∈
ℂ) |
31 | 29, 30, 30 | addassd 10062 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐼 + 1) + 1) = (𝐼 + (1 + 1))) |
32 | | df-2 11079 |
. . . . . . . . . . 11
⊢ 2 = (1 +
1) |
33 | 32 | oveq2i 6661 |
. . . . . . . . . 10
⊢ (𝐼 + 2) = (𝐼 + (1 + 1)) |
34 | 31, 33 | syl6reqr 2675 |
. . . . . . . . 9
⊢ (𝜑 → (𝐼 + 2) = ((𝐼 + 1) + 1)) |
35 | 23 | oveq2d 6666 |
. . . . . . . . 9
⊢ (𝜑 → (0...(#‘𝑊)) = (0...𝐿)) |
36 | 28, 34, 35 | 3eltr4d 2716 |
. . . . . . . 8
⊢ (𝜑 → (𝐼 + 2) ∈ (0...(#‘𝑊))) |
37 | 2 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → ∅ ∈ Word 𝑇) |
38 | 1, 18, 36, 37 | spllen 13505 |
. . . . . . 7
⊢ (𝜑 → (#‘(𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉)) = ((#‘𝑊) + ((#‘∅) −
((𝐼 + 2) − 𝐼)))) |
39 | | hash0 13158 |
. . . . . . . . . . 11
⊢
(#‘∅) = 0 |
40 | 39 | oveq1i 6660 |
. . . . . . . . . 10
⊢
((#‘∅) − ((𝐼 + 2) − 𝐼)) = (0 − ((𝐼 + 2) − 𝐼)) |
41 | | df-neg 10269 |
. . . . . . . . . 10
⊢ -((𝐼 + 2) − 𝐼) = (0 − ((𝐼 + 2) − 𝐼)) |
42 | 40, 41 | eqtr4i 2647 |
. . . . . . . . 9
⊢
((#‘∅) − ((𝐼 + 2) − 𝐼)) = -((𝐼 + 2) − 𝐼) |
43 | | 2cn 11091 |
. . . . . . . . . . 11
⊢ 2 ∈
ℂ |
44 | | pncan2 10288 |
. . . . . . . . . . 11
⊢ ((𝐼 ∈ ℂ ∧ 2 ∈
ℂ) → ((𝐼 + 2)
− 𝐼) =
2) |
45 | 29, 43, 44 | sylancl 694 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐼 + 2) − 𝐼) = 2) |
46 | 45 | negeqd 10275 |
. . . . . . . . 9
⊢ (𝜑 → -((𝐼 + 2) − 𝐼) = -2) |
47 | 42, 46 | syl5eq 2668 |
. . . . . . . 8
⊢ (𝜑 → ((#‘∅) −
((𝐼 + 2) − 𝐼)) = -2) |
48 | 23, 47 | oveq12d 6668 |
. . . . . . 7
⊢ (𝜑 → ((#‘𝑊) + ((#‘∅) −
((𝐼 + 2) − 𝐼))) = (𝐿 + -2)) |
49 | | elfzel2 12340 |
. . . . . . . . . 10
⊢ (𝐼 ∈ (0...𝐿) → 𝐿 ∈ ℤ) |
50 | 8, 49 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐿 ∈ ℤ) |
51 | 50 | zcnd 11483 |
. . . . . . . 8
⊢ (𝜑 → 𝐿 ∈ ℂ) |
52 | | negsub 10329 |
. . . . . . . 8
⊢ ((𝐿 ∈ ℂ ∧ 2 ∈
ℂ) → (𝐿 + -2) =
(𝐿 −
2)) |
53 | 51, 43, 52 | sylancl 694 |
. . . . . . 7
⊢ (𝜑 → (𝐿 + -2) = (𝐿 − 2)) |
54 | 38, 48, 53 | 3eqtrd 2660 |
. . . . . 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 13504 |
. . . . . . . . 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 1324 |
. . . . . . . 8
⊢ (𝜑 → (𝑊 splice 〈𝐼, (𝐼 + 2), (𝑊 substr 〈𝐼, (𝐼 + 2)〉)〉) = 𝑊) |
58 | 57 | oveq2d 6666 |
. . . . . . 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 2622 |
. . . . . . 7
⊢
(Base‘𝐺) =
(Base‘𝐺) |
61 | 19 | symggrp 17820 |
. . . . . . . . . 10
⊢ (𝐷 ∈ 𝑉 → 𝐺 ∈ Grp) |
62 | 21, 61 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐺 ∈ Grp) |
63 | | grpmnd 17429 |
. . . . . . . . 9
⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) |
64 | 62, 63 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐺 ∈ Mnd) |
65 | 64 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → 𝐺 ∈ Mnd) |
66 | 20, 19, 60 | symgtrf 17889 |
. . . . . . . . . 10
⊢ 𝑇 ⊆ (Base‘𝐺) |
67 | | sswrd 13313 |
. . . . . . . . . 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 3601 |
. . . . . . . 8
⊢ (𝜑 → 𝑊 ∈ Word (Base‘𝐺)) |
70 | 69 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → 𝑊 ∈ Word (Base‘𝐺)) |
71 | 18 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → 𝐼 ∈ (0...(𝐼 + 2))) |
72 | 36 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐼 + 2) ∈ (0...(#‘𝑊))) |
73 | | swrdcl 13419 |
. . . . . . . . 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 481 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝑊 substr 〈𝐼, (𝐼 + 2)〉) ∈ Word (Base‘𝐺)) |
76 | | wrd0 13330 |
. . . . . . . 8
⊢ ∅
∈ Word (Base‘𝐺) |
77 | 76 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ∅ ∈ Word
(Base‘𝐺)) |
78 | 23 | oveq2d 6666 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (0..^(#‘𝑊)) = (0..^𝐿)) |
79 | 26, 78 | eleqtrrd 2704 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐼 + 1) ∈ (0..^(#‘𝑊))) |
80 | | swrds2 13685 |
. . . . . . . . . . . 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 1326 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑊 substr 〈𝐼, (𝐼 + 2)〉) = 〈“(𝑊‘𝐼)(𝑊‘(𝐼 + 1))”〉) |
82 | 81 | oveq2d 6666 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐺 Σg (𝑊 substr 〈𝐼, (𝐼 + 2)〉)) = (𝐺 Σg
〈“(𝑊‘𝐼)(𝑊‘(𝐼 + 1))”〉)) |
83 | | wrdf 13310 |
. . . . . . . . . . . . . . 15
⊢ (𝑊 ∈ Word 𝑇 → 𝑊:(0..^(#‘𝑊))⟶𝑇) |
84 | 1, 83 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑊:(0..^(#‘𝑊))⟶𝑇) |
85 | 78 | feq2d 6031 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑊:(0..^(#‘𝑊))⟶𝑇 ↔ 𝑊:(0..^𝐿)⟶𝑇)) |
86 | 84, 85 | mpbid 222 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑊:(0..^𝐿)⟶𝑇) |
87 | 86, 7 | ffvelrnd 6360 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑊‘𝐼) ∈ 𝑇) |
88 | 66, 87 | sseldi 3601 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑊‘𝐼) ∈ (Base‘𝐺)) |
89 | 86, 26 | ffvelrnd 6360 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑊‘(𝐼 + 1)) ∈ 𝑇) |
90 | 66, 89 | sseldi 3601 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑊‘(𝐼 + 1)) ∈ (Base‘𝐺)) |
91 | | eqid 2622 |
. . . . . . . . . . . 12
⊢
(+g‘𝐺) = (+g‘𝐺) |
92 | 60, 91 | gsumws2 17379 |
. . . . . . . . . . 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 1326 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐺 Σg
〈“(𝑊‘𝐼)(𝑊‘(𝐼 + 1))”〉) = ((𝑊‘𝐼)(+g‘𝐺)(𝑊‘(𝐼 + 1)))) |
94 | 19, 60, 91 | symgov 17810 |
. . . . . . . . . . 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 693 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑊‘𝐼)(+g‘𝐺)(𝑊‘(𝐼 + 1))) = ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1)))) |
96 | 82, 93, 95 | 3eqtrd 2660 |
. . . . . . . . 9
⊢ (𝜑 → (𝐺 Σg (𝑊 substr 〈𝐼, (𝐼 + 2)〉)) = ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1)))) |
97 | 96 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐺 Σg (𝑊 substr 〈𝐼, (𝐼 + 2)〉)) = ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1)))) |
98 | | simpr 477 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) |
99 | 19 | symgid 17821 |
. . . . . . . . . . 11
⊢ (𝐷 ∈ 𝑉 → ( I ↾ 𝐷) = (0g‘𝐺)) |
100 | 21, 99 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → ( I ↾ 𝐷) = (0g‘𝐺)) |
101 | | eqid 2622 |
. . . . . . . . . . 11
⊢
(0g‘𝐺) = (0g‘𝐺) |
102 | 101 | gsum0 17278 |
. . . . . . . . . 10
⊢ (𝐺 Σg
∅) = (0g‘𝐺) |
103 | 100, 102 | syl6eqr 2674 |
. . . . . . . . 9
⊢ (𝜑 → ( I ↾ 𝐷) = (𝐺 Σg
∅)) |
104 | 103 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ( I ↾ 𝐷) = (𝐺 Σg
∅)) |
105 | 97, 98, 104 | 3eqtrd 2660 |
. . . . . . 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 17381 |
. . . . . 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 481 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐺 Σg 𝑊) = ( I ↾ 𝐷)) |
108 | 59, 106, 107 | 3eqtr3d 2664 |
. . . . 5
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉)) = ( I ↾ 𝐷)) |
109 | | fveq2 6191 |
. . . . . . . 8
⊢ (𝑥 = (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉) → (#‘𝑥) = (#‘(𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉))) |
110 | 109 | eqeq1d 2624 |
. . . . . . 7
⊢ (𝑥 = (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉) → ((#‘𝑥) = (𝐿 − 2) ↔ (#‘(𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉)) = (𝐿 − 2))) |
111 | | oveq2 6658 |
. . . . . . . 8
⊢ (𝑥 = (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉) → (𝐺 Σg
𝑥) = (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉))) |
112 | 111 | eqeq1d 2624 |
. . . . . . 7
⊢ (𝑥 = (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉) → ((𝐺 Σg
𝑥) = ( I ↾ 𝐷) ↔ (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉)) = ( I ↾ 𝐷))) |
113 | 110, 112 | anbi12d 747 |
. . . . . 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)))) |
114 | 113 | rspcev 3309 |
. . . . 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))) |
115 | 5, 55, 108, 114 | syl12anc 1324 |
. . . 4
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ∃𝑥 ∈ Word 𝑇((#‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) |
116 | | psgnunilem2.in |
. . . . 5
⊢ (𝜑 → ¬ ∃𝑥 ∈ Word 𝑇((#‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) |
117 | 116 | adantr 481 |
. . . 4
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ¬ ∃𝑥 ∈ Word 𝑇((#‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) |
118 | 115, 117 | pm2.21dd 186 |
. . 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 )))) |
119 | 118 | ex 450 |
. 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 ))))) |
120 | 1 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 𝑊 ∈ Word 𝑇) |
121 | | simprl 794 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 𝑟 ∈ 𝑇) |
122 | | simprr 796 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 𝑠 ∈ 𝑇) |
123 | 121, 122 | s2cld 13616 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 〈“𝑟𝑠”〉 ∈ Word 𝑇) |
124 | | splcl 13503 |
. . . . . . 7
⊢ ((𝑊 ∈ Word 𝑇 ∧ 〈“𝑟𝑠”〉 ∈ Word 𝑇) → (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) ∈ Word 𝑇) |
125 | 120, 123,
124 | syl2anc 693 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) ∈ Word 𝑇) |
126 | 125 | adantrr 753 |
. . . . 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) |
127 | 64 | 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) |
128 | 69 | 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)) |
129 | 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))) |
130 | 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))) |
131 | 68, 123 | sseldi 3601 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 〈“𝑟𝑠”〉 ∈ Word (Base‘𝐺)) |
132 | 131 | adantrr 753 |
. . . . . . . 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)) |
133 | 74 | 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)) |
134 | | simprr1 1109 |
. . . . . . . . 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)) |
135 | 96 | 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)))) |
136 | 64 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 𝐺 ∈ Mnd) |
137 | 66 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑇 ⊆ (Base‘𝐺)) |
138 | 137 | sselda 3603 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑇) → 𝑟 ∈ (Base‘𝐺)) |
139 | 138 | adantrr 753 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 𝑟 ∈ (Base‘𝐺)) |
140 | 137 | sselda 3603 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑇) → 𝑠 ∈ (Base‘𝐺)) |
141 | 140 | adantrl 752 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 𝑠 ∈ (Base‘𝐺)) |
142 | 60, 91 | gsumws2 17379 |
. . . . . . . . . . . 12
⊢ ((𝐺 ∈ Mnd ∧ 𝑟 ∈ (Base‘𝐺) ∧ 𝑠 ∈ (Base‘𝐺)) → (𝐺 Σg
〈“𝑟𝑠”〉) = (𝑟(+g‘𝐺)𝑠)) |
143 | 136, 139,
141, 142 | syl3anc 1326 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝐺 Σg
〈“𝑟𝑠”〉) = (𝑟(+g‘𝐺)𝑠)) |
144 | 19, 60, 91 | symgov 17810 |
. . . . . . . . . . . 12
⊢ ((𝑟 ∈ (Base‘𝐺) ∧ 𝑠 ∈ (Base‘𝐺)) → (𝑟(+g‘𝐺)𝑠) = (𝑟 ∘ 𝑠)) |
145 | 139, 141,
144 | syl2anc 693 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝑟(+g‘𝐺)𝑠) = (𝑟 ∘ 𝑠)) |
146 | 143, 145 | eqtrd 2656 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝐺 Σg
〈“𝑟𝑠”〉) = (𝑟 ∘ 𝑠)) |
147 | 146 | adantrr 753 |
. . . . . . . . 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)) |
148 | 134, 135,
147 | 3eqtr4rd 2667 |
. . . . . . . 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)〉))) |
149 | 60, 127, 128, 129, 130, 132, 133, 148 | gsumspl 17381 |
. . . . . . 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)〉)〉))) |
150 | 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)) |
151 | 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)) |
152 | 149, 150,
151 | 3eqtrd 2660 |
. . . . . 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)) |
153 | 18 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 𝐼 ∈ (0...(𝐼 + 2))) |
154 | 36 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝐼 + 2) ∈ (0...(#‘𝑊))) |
155 | 120, 153,
154, 123 | spllen 13505 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (#‘(𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)) = ((#‘𝑊) + ((#‘〈“𝑟𝑠”〉) − ((𝐼 + 2) − 𝐼)))) |
156 | | s2len 13634 |
. . . . . . . . . . . . 13
⊢
(#‘〈“𝑟𝑠”〉) = 2 |
157 | 156 | oveq1i 6660 |
. . . . . . . . . . . 12
⊢
((#‘〈“𝑟𝑠”〉) − ((𝐼 + 2) − 𝐼)) = (2 − ((𝐼 + 2) − 𝐼)) |
158 | 45 | oveq2d 6666 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (2 − ((𝐼 + 2) − 𝐼)) = (2 − 2)) |
159 | 43 | subidi 10352 |
. . . . . . . . . . . . 13
⊢ (2
− 2) = 0 |
160 | 158, 159 | syl6eq 2672 |
. . . . . . . . . . . 12
⊢ (𝜑 → (2 − ((𝐼 + 2) − 𝐼)) = 0) |
161 | 157, 160 | syl5eq 2668 |
. . . . . . . . . . 11
⊢ (𝜑 →
((#‘〈“𝑟𝑠”〉) − ((𝐼 + 2) − 𝐼)) = 0) |
162 | 161 | oveq2d 6666 |
. . . . . . . . . 10
⊢ (𝜑 → ((#‘𝑊) + ((#‘〈“𝑟𝑠”〉) − ((𝐼 + 2) − 𝐼))) = ((#‘𝑊) + 0)) |
163 | 23, 51 | eqeltrd 2701 |
. . . . . . . . . . 11
⊢ (𝜑 → (#‘𝑊) ∈ ℂ) |
164 | 163 | addid1d 10236 |
. . . . . . . . . 10
⊢ (𝜑 → ((#‘𝑊) + 0) = (#‘𝑊)) |
165 | 162, 164,
23 | 3eqtrd 2660 |
. . . . . . . . 9
⊢ (𝜑 → ((#‘𝑊) + ((#‘〈“𝑟𝑠”〉) − ((𝐼 + 2) − 𝐼))) = 𝐿) |
166 | 165 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → ((#‘𝑊) + ((#‘〈“𝑟𝑠”〉) − ((𝐼 + 2) − 𝐼))) = 𝐿) |
167 | 155, 166 | eqtrd 2656 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (#‘(𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)) = 𝐿) |
168 | 167 | adantrr 753 |
. . . . . 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) |
169 | 152, 168 | jca 554 |
. . . . 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)) |
170 | 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)) |
171 | | simprr2 1110 |
. . . . . . 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 )) |
172 | | 1nn0 11308 |
. . . . . . . . . . . . . . 15
⊢ 1 ∈
ℕ0 |
173 | | 2nn 11185 |
. . . . . . . . . . . . . . 15
⊢ 2 ∈
ℕ |
174 | | 1lt2 11194 |
. . . . . . . . . . . . . . 15
⊢ 1 <
2 |
175 | | elfzo0 12508 |
. . . . . . . . . . . . . . 15
⊢ (1 ∈
(0..^2) ↔ (1 ∈ ℕ0 ∧ 2 ∈ ℕ ∧ 1
< 2)) |
176 | 172, 173,
174, 175 | mpbir3an 1244 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
(0..^2) |
177 | 156 | oveq2i 6661 |
. . . . . . . . . . . . . 14
⊢
(0..^(#‘〈“𝑟𝑠”〉)) = (0..^2) |
178 | 176, 177 | eleqtrri 2700 |
. . . . . . . . . . . . 13
⊢ 1 ∈
(0..^(#‘〈“𝑟𝑠”〉)) |
179 | 178 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 1 ∈
(0..^(#‘〈“𝑟𝑠”〉))) |
180 | 120, 153,
154, 123, 179 | splfv2a 13507 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 1)) = (〈“𝑟𝑠”〉‘1)) |
181 | | s2fv1 13633 |
. . . . . . . . . . . 12
⊢ (𝑠 ∈ 𝑇 → (〈“𝑟𝑠”〉‘1) = 𝑠) |
182 | 181 | ad2antll 765 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (〈“𝑟𝑠”〉‘1) = 𝑠) |
183 | 180, 182 | eqtrd 2656 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 1)) = 𝑠) |
184 | 183 | adantrr 753 |
. . . . . . . . 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) |
185 | 184 | difeq1d 3727 |
. . . . . . . 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 )) |
186 | 185 | dmeqd 5326 |
. . . . . . 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 )) |
187 | 171, 186 | eleqtrrd 2704 |
. . . . . 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 )) |
188 | | fzosplitsni 12579 |
. . . . . . . . . . 11
⊢ (𝐼 ∈
(ℤ≥‘0) → (𝑗 ∈ (0..^(𝐼 + 1)) ↔ (𝑗 ∈ (0..^𝐼) ∨ 𝑗 = 𝐼))) |
189 | | nn0uz 11722 |
. . . . . . . . . . 11
⊢
ℕ0 = (ℤ≥‘0) |
190 | 188, 189 | eleq2s 2719 |
. . . . . . . . . 10
⊢ (𝐼 ∈ ℕ0
→ (𝑗 ∈
(0..^(𝐼 + 1)) ↔ (𝑗 ∈ (0..^𝐼) ∨ 𝑗 = 𝐼))) |
191 | 10, 190 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (𝑗 ∈ (0..^(𝐼 + 1)) ↔ (𝑗 ∈ (0..^𝐼) ∨ 𝑗 = 𝐼))) |
192 | 191 | 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))) |
193 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑗 → (𝑊‘𝑘) = (𝑊‘𝑗)) |
194 | 193 | difeq1d 3727 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑗 → ((𝑊‘𝑘) ∖ I ) = ((𝑊‘𝑗) ∖ I )) |
195 | 194 | dmeqd 5326 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑗 → dom ((𝑊‘𝑘) ∖ I ) = dom ((𝑊‘𝑗) ∖ I )) |
196 | 195 | eleq2d 2687 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑗 → (𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ) ↔ 𝐴 ∈ dom ((𝑊‘𝑗) ∖ I ))) |
197 | 196 | notbid 308 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑗 → (¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ) ↔ ¬ 𝐴 ∈ dom ((𝑊‘𝑗) ∖ I ))) |
198 | 197 | rspccva 3308 |
. . . . . . . . . . . . . 14
⊢
((∀𝑘 ∈
(0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ) ∧ 𝑗 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom ((𝑊‘𝑗) ∖ I )) |
199 | 25, 198 | sylan 488 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom ((𝑊‘𝑗) ∖ I )) |
200 | 199 | adantlr 751 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom ((𝑊‘𝑗) ∖ I )) |
201 | 1 | ad2antrr 762 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → 𝑊 ∈ Word 𝑇) |
202 | 18 | ad2antrr 762 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → 𝐼 ∈ (0...(𝐼 + 2))) |
203 | 36 | ad2antrr 762 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → (𝐼 + 2) ∈ (0...(#‘𝑊))) |
204 | 123 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → 〈“𝑟𝑠”〉 ∈ Word 𝑇) |
205 | | simpr 477 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → 𝑗 ∈ (0..^𝐼)) |
206 | 201, 202,
203, 204, 205 | splfv1 13506 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) = (𝑊‘𝑗)) |
207 | 206 | difeq1d 3727 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ) = ((𝑊‘𝑗) ∖ I )) |
208 | 207 | dmeqd 5326 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ) = dom ((𝑊‘𝑗) ∖ I )) |
209 | 200, 208 | neleqtrrd 2723 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I )) |
210 | 209 | ex 450 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝑗 ∈ (0..^𝐼) → ¬ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ))) |
211 | 210 | adantrr 753 |
. . . . . . . . 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 ))) |
212 | | simprr3 1111 |
. . . . . . . . . . 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 )) |
213 | | 0nn0 11307 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 0 ∈
ℕ0 |
214 | | 2pos 11112 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 0 <
2 |
215 | | elfzo0 12508 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (0 ∈
(0..^2) ↔ (0 ∈ ℕ0 ∧ 2 ∈ ℕ ∧ 0
< 2)) |
216 | 213, 173,
214, 215 | mpbir3an 1244 |
. . . . . . . . . . . . . . . . . . 19
⊢ 0 ∈
(0..^2) |
217 | 216, 177 | eleqtrri 2700 |
. . . . . . . . . . . . . . . . . 18
⊢ 0 ∈
(0..^(#‘〈“𝑟𝑠”〉)) |
218 | 217 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 0 ∈
(0..^(#‘〈“𝑟𝑠”〉))) |
219 | 120, 153,
154, 123, 218 | splfv2a 13507 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 0)) = (〈“𝑟𝑠”〉‘0)) |
220 | 29 | addid1d 10236 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝐼 + 0) = 𝐼) |
221 | 220 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝐼 + 0) = 𝐼) |
222 | 221 | fveq2d 6195 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 0)) = ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼)) |
223 | | s2fv0 13632 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑟 ∈ 𝑇 → (〈“𝑟𝑠”〉‘0) = 𝑟) |
224 | 223 | ad2antrl 764 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (〈“𝑟𝑠”〉‘0) = 𝑟) |
225 | 219, 222,
224 | 3eqtr3d 2664 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) = 𝑟) |
226 | 225 | difeq1d 3727 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) ∖ I ) = (𝑟 ∖ I )) |
227 | 226 | dmeqd 5326 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) ∖ I ) = dom (𝑟 ∖ I )) |
228 | 227 | eleq2d 2687 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) ∖ I ) ↔ 𝐴 ∈ dom (𝑟 ∖ I ))) |
229 | 228 | adantrr 753 |
. . . . . . . . . . 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 ))) |
230 | 212, 229 | mtbird 315 |
. . . . . . . . . 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 )) |
231 | | fveq2 6191 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝐼 → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) = ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼)) |
232 | 231 | difeq1d 3727 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝐼 → (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ) = (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) ∖ I )) |
233 | 232 | dmeqd 5326 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝐼 → dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ) = dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) ∖ I )) |
234 | 233 | eleq2d 2687 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝐼 → (𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ) ↔ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) ∖ I ))) |
235 | 234 | notbid 308 |
. . . . . . . . . 10
⊢ (𝑗 = 𝐼 → (¬ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ) ↔ ¬ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) ∖ I ))) |
236 | 230, 235 | syl5ibrcom 237 |
. . . . . . . . 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 ))) |
237 | 211, 236 | jaod 395 |
. . . . . . . 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 ))) |
238 | 192, 237 | sylbid 230 |
. . . . . . 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 ))) |
239 | 238 | ralrimiv 2965 |
. . . . . 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 )) |
240 | 170, 187,
239 | 3jca 1242 |
. . . . 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 ))) |
241 | | oveq2 6658 |
. . . . . . . . 9
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → (𝐺 Σg
𝑤) = (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉))) |
242 | 241 | eqeq1d 2624 |
. . . . . . . 8
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → ((𝐺 Σg
𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)) = ( I ↾ 𝐷))) |
243 | | fveq2 6191 |
. . . . . . . . 9
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → (#‘𝑤) = (#‘(𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉))) |
244 | 243 | eqeq1d 2624 |
. . . . . . . 8
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → ((#‘𝑤) = 𝐿 ↔ (#‘(𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)) = 𝐿)) |
245 | 242, 244 | anbi12d 747 |
. . . . . . 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))) |
246 | | fveq1 6190 |
. . . . . . . . . . 11
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → (𝑤‘(𝐼 + 1)) = ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 1))) |
247 | 246 | difeq1d 3727 |
. . . . . . . . . 10
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → ((𝑤‘(𝐼 + 1)) ∖ I ) = (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 1)) ∖ I )) |
248 | 247 | dmeqd 5326 |
. . . . . . . . 9
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → dom ((𝑤‘(𝐼 + 1)) ∖ I ) = dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 1)) ∖ I )) |
249 | 248 | eleq2d 2687 |
. . . . . . . 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 ))) |
250 | | fveq1 6190 |
. . . . . . . . . . . . 13
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → (𝑤‘𝑗) = ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗)) |
251 | 250 | difeq1d 3727 |
. . . . . . . . . . . 12
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → ((𝑤‘𝑗) ∖ I ) = (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I )) |
252 | 251 | dmeqd 5326 |
. . . . . . . . . . 11
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → dom ((𝑤‘𝑗) ∖ I ) = dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I )) |
253 | 252 | eleq2d 2687 |
. . . . . . . . . 10
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → (𝐴 ∈ dom ((𝑤‘𝑗) ∖ I ) ↔ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ))) |
254 | 253 | notbid 308 |
. . . . . . . . 9
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → (¬ 𝐴 ∈ dom ((𝑤‘𝑗) ∖ I ) ↔ ¬ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ))) |
255 | 254 | ralbidv 2986 |
. . . . . . . 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 ))) |
256 | 249, 255 | 3anbi23d 1402 |
. . . . . . 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 )))) |
257 | 245, 256 | anbi12d 747 |
. . . . . 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 ))))) |
258 | 257 | rspcev 3309 |
. . . . 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 )))) |
259 | 126, 169,
240, 258 | syl12anc 1324 |
. . . 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 )))) |
260 | 259 | expr 643 |
. . 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 ))))) |
261 | 260 | rexlimdvva 3038 |
. 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 ))))) |
262 | 20, 21, 87, 89, 24 | psgnunilem1 17913 |
. 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 )))) |
263 | 119, 261,
262 | mpjaod 396 |
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 )))) |