| Step | Hyp | Ref
| Expression |
| 1 | | psgnunilem2.w |
. . . . . . 7
⊢ (𝜑 → 𝑊 ∈ Word 𝑇) |
| 2 | | wrd0 14577 |
. . . . . . 7
⊢ ∅
∈ Word 𝑇 |
| 3 | | splcl 14790 |
. . . . . . 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 480 |
. . . . 5
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉) ∈ Word 𝑇) |
| 6 | | fzossfz 13718 |
. . . . . . . . . . 11
⊢
(0..^𝐿) ⊆
(0...𝐿) |
| 7 | | psgnunilem2.ix |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐼 ∈ (0..^𝐿)) |
| 8 | 6, 7 | sselid 3981 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐼 ∈ (0...𝐿)) |
| 9 | | elfznn0 13660 |
. . . . . . . . . 10
⊢ (𝐼 ∈ (0...𝐿) → 𝐼 ∈
ℕ0) |
| 10 | 8, 9 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐼 ∈
ℕ0) |
| 11 | | 2nn0 12543 |
. . . . . . . . . 10
⊢ 2 ∈
ℕ0 |
| 12 | | nn0addcl 12561 |
. . . . . . . . . 10
⊢ ((𝐼 ∈ ℕ0
∧ 2 ∈ ℕ0) → (𝐼 + 2) ∈
ℕ0) |
| 13 | 10, 11, 12 | sylancl 586 |
. . . . . . . . 9
⊢ (𝜑 → (𝐼 + 2) ∈
ℕ0) |
| 14 | 10 | nn0red 12588 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐼 ∈ ℝ) |
| 15 | | nn0addge1 12572 |
. . . . . . . . . 10
⊢ ((𝐼 ∈ ℝ ∧ 2 ∈
ℕ0) → 𝐼 ≤ (𝐼 + 2)) |
| 16 | 14, 11, 15 | sylancl 586 |
. . . . . . . . 9
⊢ (𝜑 → 𝐼 ≤ (𝐼 + 2)) |
| 17 | | elfz2nn0 13658 |
. . . . . . . . 9
⊢ (𝐼 ∈ (0...(𝐼 + 2)) ↔ (𝐼 ∈ ℕ0 ∧ (𝐼 + 2) ∈ ℕ0
∧ 𝐼 ≤ (𝐼 + 2))) |
| 18 | 10, 13, 16, 17 | syl3anbrc 1344 |
. . . . . . . 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 19512 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐼 + 1) ∈ (0..^𝐿)) |
| 27 | | fzofzp1 13803 |
. . . . . . . . . 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 12329 |
. . . . . . . . . . 11
⊢ 2 = (1 +
1) |
| 30 | 29 | oveq2i 7442 |
. . . . . . . . . 10
⊢ (𝐼 + 2) = (𝐼 + (1 + 1)) |
| 31 | 10 | nn0cnd 12589 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐼 ∈ ℂ) |
| 32 | | 1cnd 11256 |
. . . . . . . . . . 11
⊢ (𝜑 → 1 ∈
ℂ) |
| 33 | 31, 32, 32 | addassd 11283 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐼 + 1) + 1) = (𝐼 + (1 + 1))) |
| 34 | 30, 33 | eqtr4id 2796 |
. . . . . . . . 9
⊢ (𝜑 → (𝐼 + 2) = ((𝐼 + 1) + 1)) |
| 35 | 23 | oveq2d 7447 |
. . . . . . . . 9
⊢ (𝜑 → (0...(♯‘𝑊)) = (0...𝐿)) |
| 36 | 28, 34, 35 | 3eltr4d 2856 |
. . . . . . . 8
⊢ (𝜑 → (𝐼 + 2) ∈ (0...(♯‘𝑊))) |
| 37 | 2 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → ∅ ∈ Word 𝑇) |
| 38 | 1, 18, 36, 37 | spllen 14792 |
. . . . . . 7
⊢ (𝜑 → (♯‘(𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉)) =
((♯‘𝑊) +
((♯‘∅) − ((𝐼 + 2) − 𝐼)))) |
| 39 | | hash0 14406 |
. . . . . . . . . . 11
⊢
(♯‘∅) = 0 |
| 40 | 39 | oveq1i 7441 |
. . . . . . . . . 10
⊢
((♯‘∅) − ((𝐼 + 2) − 𝐼)) = (0 − ((𝐼 + 2) − 𝐼)) |
| 41 | | df-neg 11495 |
. . . . . . . . . 10
⊢ -((𝐼 + 2) − 𝐼) = (0 − ((𝐼 + 2) − 𝐼)) |
| 42 | 40, 41 | eqtr4i 2768 |
. . . . . . . . 9
⊢
((♯‘∅) − ((𝐼 + 2) − 𝐼)) = -((𝐼 + 2) − 𝐼) |
| 43 | | 2cn 12341 |
. . . . . . . . . . 11
⊢ 2 ∈
ℂ |
| 44 | | pncan2 11515 |
. . . . . . . . . . 11
⊢ ((𝐼 ∈ ℂ ∧ 2 ∈
ℂ) → ((𝐼 + 2)
− 𝐼) =
2) |
| 45 | 31, 43, 44 | sylancl 586 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐼 + 2) − 𝐼) = 2) |
| 46 | 45 | negeqd 11502 |
. . . . . . . . 9
⊢ (𝜑 → -((𝐼 + 2) − 𝐼) = -2) |
| 47 | 42, 46 | eqtrid 2789 |
. . . . . . . 8
⊢ (𝜑 → ((♯‘∅)
− ((𝐼 + 2) −
𝐼)) = -2) |
| 48 | 23, 47 | oveq12d 7449 |
. . . . . . 7
⊢ (𝜑 → ((♯‘𝑊) + ((♯‘∅)
− ((𝐼 + 2) −
𝐼))) = (𝐿 + -2)) |
| 49 | | elfzel2 13562 |
. . . . . . . . . 10
⊢ (𝐼 ∈ (0...𝐿) → 𝐿 ∈ ℤ) |
| 50 | 8, 49 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐿 ∈ ℤ) |
| 51 | 50 | zcnd 12723 |
. . . . . . . 8
⊢ (𝜑 → 𝐿 ∈ ℂ) |
| 52 | | negsub 11557 |
. . . . . . . 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 2781 |
. . . . . 6
⊢ (𝜑 → (♯‘(𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉)) = (𝐿 − 2)) |
| 55 | 54 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (♯‘(𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉)) = (𝐿 − 2)) |
| 56 | | splid 14791 |
. . . . . . . . 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 837 |
. . . . . . . 8
⊢ (𝜑 → (𝑊 splice 〈𝐼, (𝐼 + 2), (𝑊 substr 〈𝐼, (𝐼 + 2)〉)〉) = 𝑊) |
| 58 | 57 | oveq2d 7447 |
. . . . . . 7
⊢ (𝜑 → (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), (𝑊 substr 〈𝐼, (𝐼 + 2)〉)〉)) = (𝐺 Σg 𝑊)) |
| 59 | 58 | adantr 480 |
. . . . . 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 2737 |
. . . . . . 7
⊢
(Base‘𝐺) =
(Base‘𝐺) |
| 61 | 19 | symggrp 19418 |
. . . . . . . . . 10
⊢ (𝐷 ∈ 𝑉 → 𝐺 ∈ Grp) |
| 62 | 21, 61 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐺 ∈ Grp) |
| 63 | 62 | grpmndd 18964 |
. . . . . . . 8
⊢ (𝜑 → 𝐺 ∈ Mnd) |
| 64 | 63 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → 𝐺 ∈ Mnd) |
| 65 | 20, 19, 60 | symgtrf 19487 |
. . . . . . . . . 10
⊢ 𝑇 ⊆ (Base‘𝐺) |
| 66 | | sswrd 14560 |
. . . . . . . . . 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 3981 |
. . . . . . . 8
⊢ (𝜑 → 𝑊 ∈ Word (Base‘𝐺)) |
| 69 | 68 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → 𝑊 ∈ Word (Base‘𝐺)) |
| 70 | 18 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → 𝐼 ∈ (0...(𝐼 + 2))) |
| 71 | 36 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐼 + 2) ∈ (0...(♯‘𝑊))) |
| 72 | | swrdcl 14683 |
. . . . . . . . 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 480 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝑊 substr 〈𝐼, (𝐼 + 2)〉) ∈ Word (Base‘𝐺)) |
| 75 | | wrd0 14577 |
. . . . . . . 8
⊢ ∅
∈ Word (Base‘𝐺) |
| 76 | 75 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ∅ ∈ Word
(Base‘𝐺)) |
| 77 | 23 | oveq2d 7447 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (0..^(♯‘𝑊)) = (0..^𝐿)) |
| 78 | 26, 77 | eleqtrrd 2844 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐼 + 1) ∈ (0..^(♯‘𝑊))) |
| 79 | | swrds2 14979 |
. . . . . . . . . . . 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 1373 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑊 substr 〈𝐼, (𝐼 + 2)〉) = 〈“(𝑊‘𝐼)(𝑊‘(𝐼 + 1))”〉) |
| 81 | 80 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐺 Σg (𝑊 substr 〈𝐼, (𝐼 + 2)〉)) = (𝐺 Σg
〈“(𝑊‘𝐼)(𝑊‘(𝐼 + 1))”〉)) |
| 82 | | wrdf 14557 |
. . . . . . . . . . . . . . 15
⊢ (𝑊 ∈ Word 𝑇 → 𝑊:(0..^(♯‘𝑊))⟶𝑇) |
| 83 | 1, 82 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑊:(0..^(♯‘𝑊))⟶𝑇) |
| 84 | 77 | feq2d 6722 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑊:(0..^(♯‘𝑊))⟶𝑇 ↔ 𝑊:(0..^𝐿)⟶𝑇)) |
| 85 | 83, 84 | mpbid 232 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑊:(0..^𝐿)⟶𝑇) |
| 86 | 85, 7 | ffvelcdmd 7105 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑊‘𝐼) ∈ 𝑇) |
| 87 | 65, 86 | sselid 3981 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑊‘𝐼) ∈ (Base‘𝐺)) |
| 88 | 85, 26 | ffvelcdmd 7105 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑊‘(𝐼 + 1)) ∈ 𝑇) |
| 89 | 65, 88 | sselid 3981 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑊‘(𝐼 + 1)) ∈ (Base‘𝐺)) |
| 90 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(+g‘𝐺) = (+g‘𝐺) |
| 91 | 60, 90 | gsumws2 18855 |
. . . . . . . . . . 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 1373 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐺 Σg
〈“(𝑊‘𝐼)(𝑊‘(𝐼 + 1))”〉) = ((𝑊‘𝐼)(+g‘𝐺)(𝑊‘(𝐼 + 1)))) |
| 93 | 19, 60, 90 | symgov 19401 |
. . . . . . . . . . 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 2781 |
. . . . . . . . 9
⊢ (𝜑 → (𝐺 Σg (𝑊 substr 〈𝐼, (𝐼 + 2)〉)) = ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1)))) |
| 96 | 95 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐺 Σg (𝑊 substr 〈𝐼, (𝐼 + 2)〉)) = ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1)))) |
| 97 | | simpr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) |
| 98 | 19 | symgid 19419 |
. . . . . . . . . . 11
⊢ (𝐷 ∈ 𝑉 → ( I ↾ 𝐷) = (0g‘𝐺)) |
| 99 | 21, 98 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → ( I ↾ 𝐷) = (0g‘𝐺)) |
| 100 | | eqid 2737 |
. . . . . . . . . . 11
⊢
(0g‘𝐺) = (0g‘𝐺) |
| 101 | 100 | gsum0 18697 |
. . . . . . . . . 10
⊢ (𝐺 Σg
∅) = (0g‘𝐺) |
| 102 | 99, 101 | eqtr4di 2795 |
. . . . . . . . 9
⊢ (𝜑 → ( I ↾ 𝐷) = (𝐺 Σg
∅)) |
| 103 | 102 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ( I ↾ 𝐷) = (𝐺 Σg
∅)) |
| 104 | 96, 97, 103 | 3eqtrd 2781 |
. . . . . . 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 18857 |
. . . . . 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 480 |
. . . . . 6
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐺 Σg 𝑊) = ( I ↾ 𝐷)) |
| 107 | 59, 105, 106 | 3eqtr3d 2785 |
. . . . 5
⊢ ((𝜑 ∧ ((𝑊‘𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉)) = ( I ↾ 𝐷)) |
| 108 | | fveqeq2 6915 |
. . . . . . 7
⊢ (𝑥 = (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉) →
((♯‘𝑥) = (𝐿 − 2) ↔
(♯‘(𝑊 splice
〈𝐼, (𝐼 + 2), ∅〉)) = (𝐿 − 2))) |
| 109 | | oveq2 7439 |
. . . . . . . 8
⊢ (𝑥 = (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉) → (𝐺 Σg
𝑥) = (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), ∅〉))) |
| 110 | 109 | eqeq1d 2739 |
. . . . . . 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 632 |
. . . . . 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 3622 |
. . . . 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 837 |
. . . 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 480 |
. . . 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 195 |
. . 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 412 |
. 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 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 𝑊 ∈ Word 𝑇) |
| 119 | | simprl 771 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 𝑟 ∈ 𝑇) |
| 120 | | simprr 773 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 𝑠 ∈ 𝑇) |
| 121 | 119, 120 | s2cld 14910 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 〈“𝑟𝑠”〉 ∈ Word 𝑇) |
| 122 | | splcl 14790 |
. . . . . . 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 717 |
. . . . 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 480 |
. . . . . . . 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 480 |
. . . . . . . 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 480 |
. . . . . . . 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 480 |
. . . . . . . 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 3981 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 〈“𝑟𝑠”〉 ∈ Word (Base‘𝐺)) |
| 130 | 129 | adantrr 717 |
. . . . . . . 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 480 |
. . . . . . . 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 1222 |
. . . . . . . . 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 480 |
. . . . . . . . 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 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 𝐺 ∈ Mnd) |
| 135 | 65 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑇 ⊆ (Base‘𝐺)) |
| 136 | 135 | sselda 3983 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑟 ∈ 𝑇) → 𝑟 ∈ (Base‘𝐺)) |
| 137 | 136 | adantrr 717 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 𝑟 ∈ (Base‘𝐺)) |
| 138 | 135 | sselda 3983 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑇) → 𝑠 ∈ (Base‘𝐺)) |
| 139 | 138 | adantrl 716 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 𝑠 ∈ (Base‘𝐺)) |
| 140 | 60, 90 | gsumws2 18855 |
. . . . . . . . . . . 12
⊢ ((𝐺 ∈ Mnd ∧ 𝑟 ∈ (Base‘𝐺) ∧ 𝑠 ∈ (Base‘𝐺)) → (𝐺 Σg
〈“𝑟𝑠”〉) = (𝑟(+g‘𝐺)𝑠)) |
| 141 | 134, 137,
139, 140 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝐺 Σg
〈“𝑟𝑠”〉) = (𝑟(+g‘𝐺)𝑠)) |
| 142 | 19, 60, 90 | symgov 19401 |
. . . . . . . . . . . 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 2777 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝐺 Σg
〈“𝑟𝑠”〉) = (𝑟 ∘ 𝑠)) |
| 145 | 144 | adantrr 717 |
. . . . . . . . 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 2788 |
. . . . . . . 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 18857 |
. . . . . . 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 480 |
. . . . . . 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 480 |
. . . . . . 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 2781 |
. . . . . 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 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 𝐼 ∈ (0...(𝐼 + 2))) |
| 152 | 36 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝐼 + 2) ∈ (0...(♯‘𝑊))) |
| 153 | 118, 151,
152, 121 | spllen 14792 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (♯‘(𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)) = ((♯‘𝑊) +
((♯‘〈“𝑟𝑠”〉) − ((𝐼 + 2) − 𝐼)))) |
| 154 | | s2len 14928 |
. . . . . . . . . . . . 13
⊢
(♯‘〈“𝑟𝑠”〉) = 2 |
| 155 | 154 | oveq1i 7441 |
. . . . . . . . . . . 12
⊢
((♯‘〈“𝑟𝑠”〉) − ((𝐼 + 2) − 𝐼)) = (2 − ((𝐼 + 2) − 𝐼)) |
| 156 | 45 | oveq2d 7447 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (2 − ((𝐼 + 2) − 𝐼)) = (2 − 2)) |
| 157 | 43 | subidi 11580 |
. . . . . . . . . . . . 13
⊢ (2
− 2) = 0 |
| 158 | 156, 157 | eqtrdi 2793 |
. . . . . . . . . . . 12
⊢ (𝜑 → (2 − ((𝐼 + 2) − 𝐼)) = 0) |
| 159 | 155, 158 | eqtrid 2789 |
. . . . . . . . . . 11
⊢ (𝜑 →
((♯‘〈“𝑟𝑠”〉) − ((𝐼 + 2) − 𝐼)) = 0) |
| 160 | 159 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (𝜑 → ((♯‘𝑊) +
((♯‘〈“𝑟𝑠”〉) − ((𝐼 + 2) − 𝐼))) = ((♯‘𝑊) + 0)) |
| 161 | 23, 51 | eqeltrd 2841 |
. . . . . . . . . . 11
⊢ (𝜑 → (♯‘𝑊) ∈
ℂ) |
| 162 | 161 | addridd 11461 |
. . . . . . . . . 10
⊢ (𝜑 → ((♯‘𝑊) + 0) = (♯‘𝑊)) |
| 163 | 160, 162,
23 | 3eqtrd 2781 |
. . . . . . . . 9
⊢ (𝜑 → ((♯‘𝑊) +
((♯‘〈“𝑟𝑠”〉) − ((𝐼 + 2) − 𝐼))) = 𝐿) |
| 164 | 163 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → ((♯‘𝑊) + ((♯‘〈“𝑟𝑠”〉) − ((𝐼 + 2) − 𝐼))) = 𝐿) |
| 165 | 153, 164 | eqtrd 2777 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (♯‘(𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)) = 𝐿) |
| 166 | 165 | adantrr 717 |
. . . . . 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 511 |
. . . . 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 480 |
. . . . . 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 1223 |
. . . . . . 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 12542 |
. . . . . . . . . . . . . . 15
⊢ 1 ∈
ℕ0 |
| 171 | | 2nn 12339 |
. . . . . . . . . . . . . . 15
⊢ 2 ∈
ℕ |
| 172 | | 1lt2 12437 |
. . . . . . . . . . . . . . 15
⊢ 1 <
2 |
| 173 | | elfzo0 13740 |
. . . . . . . . . . . . . . 15
⊢ (1 ∈
(0..^2) ↔ (1 ∈ ℕ0 ∧ 2 ∈ ℕ ∧ 1
< 2)) |
| 174 | 170, 171,
172, 173 | mpbir3an 1342 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
(0..^2) |
| 175 | 154 | oveq2i 7442 |
. . . . . . . . . . . . . 14
⊢
(0..^(♯‘〈“𝑟𝑠”〉)) = (0..^2) |
| 176 | 174, 175 | eleqtrri 2840 |
. . . . . . . . . . . . 13
⊢ 1 ∈
(0..^(♯‘〈“𝑟𝑠”〉)) |
| 177 | 176 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 1 ∈
(0..^(♯‘〈“𝑟𝑠”〉))) |
| 178 | 118, 151,
152, 121, 177 | splfv2a 14794 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 1)) = (〈“𝑟𝑠”〉‘1)) |
| 179 | | s2fv1 14927 |
. . . . . . . . . . . 12
⊢ (𝑠 ∈ 𝑇 → (〈“𝑟𝑠”〉‘1) = 𝑠) |
| 180 | 179 | ad2antll 729 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (〈“𝑟𝑠”〉‘1) = 𝑠) |
| 181 | 178, 180 | eqtrd 2777 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 1)) = 𝑠) |
| 182 | 181 | adantrr 717 |
. . . . . . . . 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 4125 |
. . . . . . . 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 5916 |
. . . . . . 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 2844 |
. . . . . 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 13817 |
. . . . . . . . . . 11
⊢ (𝐼 ∈
(ℤ≥‘0) → (𝑗 ∈ (0..^(𝐼 + 1)) ↔ (𝑗 ∈ (0..^𝐼) ∨ 𝑗 = 𝐼))) |
| 187 | | nn0uz 12920 |
. . . . . . . . . . 11
⊢
ℕ0 = (ℤ≥‘0) |
| 188 | 186, 187 | eleq2s 2859 |
. . . . . . . . . 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 480 |
. . . . . . . 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 6906 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑗 → (𝑊‘𝑘) = (𝑊‘𝑗)) |
| 192 | 191 | difeq1d 4125 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑗 → ((𝑊‘𝑘) ∖ I ) = ((𝑊‘𝑗) ∖ I )) |
| 193 | 192 | dmeqd 5916 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑗 → dom ((𝑊‘𝑘) ∖ I ) = dom ((𝑊‘𝑗) ∖ I )) |
| 194 | 193 | eleq2d 2827 |
. . . . . . . . . . . . . . . 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 3621 |
. . . . . . . . . . . . . 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 715 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom ((𝑊‘𝑗) ∖ I )) |
| 199 | 1 | ad2antrr 726 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → 𝑊 ∈ Word 𝑇) |
| 200 | 18 | ad2antrr 726 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → 𝐼 ∈ (0...(𝐼 + 2))) |
| 201 | 36 | ad2antrr 726 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → (𝐼 + 2) ∈ (0...(♯‘𝑊))) |
| 202 | 121 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → 〈“𝑟𝑠”〉 ∈ Word 𝑇) |
| 203 | | simpr 484 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → 𝑗 ∈ (0..^𝐼)) |
| 204 | 199, 200,
201, 202, 203 | splfv1 14793 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) = (𝑊‘𝑗)) |
| 205 | 204 | difeq1d 4125 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ) = ((𝑊‘𝑗) ∖ I )) |
| 206 | 205 | dmeqd 5916 |
. . . . . . . . . . . 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 2864 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I )) |
| 208 | 207 | ex 412 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝑗 ∈ (0..^𝐼) → ¬ 𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ))) |
| 209 | 208 | adantrr 717 |
. . . . . . . . 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 1224 |
. . . . . . . . . . 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 12541 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 0 ∈
ℕ0 |
| 212 | | 2pos 12369 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 0 <
2 |
| 213 | | elfzo0 13740 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (0 ∈
(0..^2) ↔ (0 ∈ ℕ0 ∧ 2 ∈ ℕ ∧ 0
< 2)) |
| 214 | 211, 171,
212, 213 | mpbir3an 1342 |
. . . . . . . . . . . . . . . . . . 19
⊢ 0 ∈
(0..^2) |
| 215 | 214, 175 | eleqtrri 2840 |
. . . . . . . . . . . . . . . . . 18
⊢ 0 ∈
(0..^(♯‘〈“𝑟𝑠”〉)) |
| 216 | 215 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → 0 ∈
(0..^(♯‘〈“𝑟𝑠”〉))) |
| 217 | 118, 151,
152, 121, 216 | splfv2a 14794 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 0)) = (〈“𝑟𝑠”〉‘0)) |
| 218 | 31 | addridd 11461 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝐼 + 0) = 𝐼) |
| 219 | 218 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝐼 + 0) = 𝐼) |
| 220 | 219 | fveq2d 6910 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 0)) = ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼)) |
| 221 | | s2fv0 14926 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑟 ∈ 𝑇 → (〈“𝑟𝑠”〉‘0) = 𝑟) |
| 222 | 221 | ad2antrl 728 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (〈“𝑟𝑠”〉‘0) = 𝑟) |
| 223 | 217, 220,
222 | 3eqtr3d 2785 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) = 𝑟) |
| 224 | 223 | difeq1d 4125 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) ∖ I ) = (𝑟 ∖ I )) |
| 225 | 224 | dmeqd 5916 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) ∖ I ) = dom (𝑟 ∖ I )) |
| 226 | 225 | eleq2d 2827 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑠 ∈ 𝑇)) → (𝐴 ∈ dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) ∖ I ) ↔ 𝐴 ∈ dom (𝑟 ∖ I ))) |
| 227 | 226 | adantrr 717 |
. . . . . . . . . . 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 6906 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝐼 → ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) = ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼)) |
| 230 | 229 | difeq1d 4125 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝐼 → (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ) = (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) ∖ I )) |
| 231 | 230 | dmeqd 5916 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝐼 → dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I ) = dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝐼) ∖ I )) |
| 232 | 231 | eleq2d 2827 |
. . . . . . . . . . 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 247 |
. . . . . . . . 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 860 |
. . . . . . . 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 240 |
. . . . . . 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 3145 |
. . . . . 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 1129 |
. . . . 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 7439 |
. . . . . . . . 9
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → (𝐺 Σg
𝑤) = (𝐺 Σg (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉))) |
| 240 | 239 | eqeq1d 2739 |
. . . . . . . 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 6915 |
. . . . . . . 8
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) →
((♯‘𝑤) = 𝐿 ↔ (♯‘(𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)) = 𝐿)) |
| 242 | 240, 241 | anbi12d 632 |
. . . . . . 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 6905 |
. . . . . . . . . . 11
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → (𝑤‘(𝐼 + 1)) = ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 1))) |
| 244 | 243 | difeq1d 4125 |
. . . . . . . . . 10
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → ((𝑤‘(𝐼 + 1)) ∖ I ) = (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘(𝐼 + 1)) ∖ I )) |
| 245 | 244 | dmeqd 5916 |
. . . . . . . . 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 2827 |
. . . . . . . 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 6905 |
. . . . . . . . . . . . 13
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → (𝑤‘𝑗) = ((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗)) |
| 248 | 247 | difeq1d 4125 |
. . . . . . . . . . . 12
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → ((𝑤‘𝑗) ∖ I ) = (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I )) |
| 249 | 248 | dmeqd 5916 |
. . . . . . . . . . 11
⊢ (𝑤 = (𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉) → dom ((𝑤‘𝑗) ∖ I ) = dom (((𝑊 splice 〈𝐼, (𝐼 + 2), 〈“𝑟𝑠”〉〉)‘𝑗) ∖ I )) |
| 250 | 249 | eleq2d 2827 |
. . . . . . . . . 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 3178 |
. . . . . . . 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 1441 |
. . . . . . 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 632 |
. . . . . 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 3622 |
. . . . 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 837 |
. . . 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 456 |
. . 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 3213 |
. 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 19511 |
. 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 861 |
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 )))) |