| Step | Hyp | Ref
| Expression |
| 1 | | gsumwrd2dccat.1 |
. . . 4
⊢ 𝐵 = (Base‘𝑀) |
| 2 | | gsumwrd2dccat.2 |
. . . 4
⊢ 𝑍 = (0g‘𝑀) |
| 3 | | gsumwrd2dccat.5 |
. . . 4
⊢ (𝜑 → 𝑀 ∈ CMnd) |
| 4 | 1 | fvexi 6920 |
. . . . . . . 8
⊢ 𝐵 ∈ V |
| 5 | 4 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ V) |
| 6 | | gsumwrd2dccat.6 |
. . . . . . 7
⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
| 7 | 5, 6 | ssexd 5324 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ V) |
| 8 | | wrdexg 14562 |
. . . . . 6
⊢ (𝐴 ∈ V → Word 𝐴 ∈ V) |
| 9 | 7, 8 | syl 17 |
. . . . 5
⊢ (𝜑 → Word 𝐴 ∈ V) |
| 10 | 9, 9 | xpexd 7771 |
. . . 4
⊢ (𝜑 → (Word 𝐴 × Word 𝐴) ∈ V) |
| 11 | | gsumwrd2dccat.3 |
. . . 4
⊢ (𝜑 → 𝐹:(Word 𝐴 × Word 𝐴)⟶𝐵) |
| 12 | | gsumwrd2dccat.4 |
. . . 4
⊢ (𝜑 → 𝐹 finSupp 𝑍) |
| 13 | | eqid 2737 |
. . . . . . . 8
⊢ ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) = ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) |
| 14 | | eqid 2737 |
. . . . . . . 8
⊢ (𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ 〈((1st
‘𝑎) ++
(2nd ‘𝑎)),
(♯‘(1st ‘𝑎))〉) = (𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ 〈((1st
‘𝑎) ++
(2nd ‘𝑎)),
(♯‘(1st ‘𝑎))〉) |
| 15 | | eqid 2737 |
. . . . . . . 8
⊢ (𝑏 ∈ ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦
〈((1st ‘𝑏) prefix (2nd ‘𝑏)), ((1st
‘𝑏) substr
〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉) =
(𝑏 ∈ ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦
〈((1st ‘𝑏) prefix (2nd ‘𝑏)), ((1st
‘𝑏) substr
〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉) |
| 16 | 13, 14, 15, 7 | gsumwrd2dccatlem 33069 |
. . . . . . 7
⊢ (𝜑 → ((𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ 〈((1st
‘𝑎) ++
(2nd ‘𝑎)),
(♯‘(1st ‘𝑎))〉):(Word 𝐴 × Word 𝐴)–1-1-onto→∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ∧ ◡(𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ 〈((1st
‘𝑎) ++
(2nd ‘𝑎)),
(♯‘(1st ‘𝑎))〉) = (𝑏 ∈ ∪
𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦
〈((1st ‘𝑏) prefix (2nd ‘𝑏)), ((1st
‘𝑏) substr
〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉))) |
| 17 | 16 | simpld 494 |
. . . . . 6
⊢ (𝜑 → (𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ 〈((1st
‘𝑎) ++
(2nd ‘𝑎)),
(♯‘(1st ‘𝑎))〉):(Word 𝐴 × Word 𝐴)–1-1-onto→∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤)))) |
| 18 | | f1ocnv 6860 |
. . . . . 6
⊢ ((𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ 〈((1st
‘𝑎) ++
(2nd ‘𝑎)),
(♯‘(1st ‘𝑎))〉):(Word 𝐴 × Word 𝐴)–1-1-onto→∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) → ◡(𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ 〈((1st
‘𝑎) ++
(2nd ‘𝑎)),
(♯‘(1st ‘𝑎))〉):∪
𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤)))–1-1-onto→(Word
𝐴 × Word 𝐴)) |
| 19 | 17, 18 | syl 17 |
. . . . 5
⊢ (𝜑 → ◡(𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ 〈((1st
‘𝑎) ++
(2nd ‘𝑎)),
(♯‘(1st ‘𝑎))〉):∪
𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤)))–1-1-onto→(Word
𝐴 × Word 𝐴)) |
| 20 | 16 | simprd 495 |
. . . . . 6
⊢ (𝜑 → ◡(𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ 〈((1st
‘𝑎) ++
(2nd ‘𝑎)),
(♯‘(1st ‘𝑎))〉) = (𝑏 ∈ ∪
𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦
〈((1st ‘𝑏) prefix (2nd ‘𝑏)), ((1st
‘𝑏) substr
〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉)) |
| 21 | 20 | f1oeq1d 6843 |
. . . . 5
⊢ (𝜑 → (◡(𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ 〈((1st
‘𝑎) ++
(2nd ‘𝑎)),
(♯‘(1st ‘𝑎))〉):∪
𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤)))–1-1-onto→(Word
𝐴 × Word 𝐴) ↔ (𝑏 ∈ ∪
𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦
〈((1st ‘𝑏) prefix (2nd ‘𝑏)), ((1st
‘𝑏) substr
〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉):∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤)))–1-1-onto→(Word
𝐴 × Word 𝐴))) |
| 22 | 19, 21 | mpbid 232 |
. . . 4
⊢ (𝜑 → (𝑏 ∈ ∪
𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦
〈((1st ‘𝑏) prefix (2nd ‘𝑏)), ((1st
‘𝑏) substr
〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉):∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤)))–1-1-onto→(Word
𝐴 × Word 𝐴)) |
| 23 | 1, 2, 3, 10, 11, 12, 22 | gsumf1o 19934 |
. . 3
⊢ (𝜑 → (𝑀 Σg 𝐹) = (𝑀 Σg (𝐹 ∘ (𝑏 ∈ ∪
𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦
〈((1st ‘𝑏) prefix (2nd ‘𝑏)), ((1st
‘𝑏) substr
〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉)))) |
| 24 | | relxp 5703 |
. . . . . . . . . . . 12
⊢ Rel
({𝑥} ×
(0...(♯‘𝑥))) |
| 25 | 24 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → Rel ({𝑥} × (0...(♯‘𝑥)))) |
| 26 | 25 | ralrimiva 3146 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑥 ∈ Word 𝐴Rel ({𝑥} × (0...(♯‘𝑥)))) |
| 27 | | reliun 5826 |
. . . . . . . . . 10
⊢ (Rel
∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↔ ∀𝑥 ∈ Word 𝐴Rel ({𝑥} × (0...(♯‘𝑥)))) |
| 28 | 26, 27 | sylibr 234 |
. . . . . . . . 9
⊢ (𝜑 → Rel ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) |
| 29 | | 1stdm 8065 |
. . . . . . . . 9
⊢ ((Rel
∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ∧ 𝑏 ∈ ∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (1st
‘𝑏) ∈ dom
∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) |
| 30 | 28, 29 | sylan 580 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ ∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (1st
‘𝑏) ∈ dom
∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) |
| 31 | | lencl 14571 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ Word 𝐴 → (♯‘𝑥) ∈
ℕ0) |
| 32 | 31 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → (♯‘𝑥) ∈
ℕ0) |
| 33 | | nn0uz 12920 |
. . . . . . . . . . . 12
⊢
ℕ0 = (ℤ≥‘0) |
| 34 | 32, 33 | eleqtrdi 2851 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → (♯‘𝑥) ∈
(ℤ≥‘0)) |
| 35 | | fzn0 13578 |
. . . . . . . . . . 11
⊢
((0...(♯‘𝑥)) ≠ ∅ ↔ (♯‘𝑥) ∈
(ℤ≥‘0)) |
| 36 | 34, 35 | sylibr 234 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → (0...(♯‘𝑥)) ≠
∅) |
| 37 | 36 | dmdju 32657 |
. . . . . . . . 9
⊢ (𝜑 → dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) = Word 𝐴) |
| 38 | 37 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ ∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) = Word 𝐴) |
| 39 | 30, 38 | eleqtrd 2843 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑏 ∈ ∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (1st
‘𝑏) ∈ Word 𝐴) |
| 40 | | pfxcl 14715 |
. . . . . . 7
⊢
((1st ‘𝑏) ∈ Word 𝐴 → ((1st ‘𝑏) prefix (2nd
‘𝑏)) ∈ Word
𝐴) |
| 41 | 39, 40 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑏 ∈ ∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → ((1st
‘𝑏) prefix
(2nd ‘𝑏))
∈ Word 𝐴) |
| 42 | | swrdcl 14683 |
. . . . . . 7
⊢
((1st ‘𝑏) ∈ Word 𝐴 → ((1st ‘𝑏) substr 〈(2nd
‘𝑏),
(♯‘(1st ‘𝑏))〉) ∈ Word 𝐴) |
| 43 | 39, 42 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑏 ∈ ∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → ((1st
‘𝑏) substr
〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉) ∈
Word 𝐴) |
| 44 | 41, 43 | opelxpd 5724 |
. . . . 5
⊢ ((𝜑 ∧ 𝑏 ∈ ∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) →
〈((1st ‘𝑏) prefix (2nd ‘𝑏)), ((1st
‘𝑏) substr
〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉
∈ (Word 𝐴 × Word
𝐴)) |
| 45 | | sneq 4636 |
. . . . . . . . 9
⊢ (𝑤 = 𝑥 → {𝑤} = {𝑥}) |
| 46 | | fveq2 6906 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑥 → (♯‘𝑤) = (♯‘𝑥)) |
| 47 | 46 | oveq2d 7447 |
. . . . . . . . 9
⊢ (𝑤 = 𝑥 → (0...(♯‘𝑤)) = (0...(♯‘𝑥))) |
| 48 | 45, 47 | xpeq12d 5716 |
. . . . . . . 8
⊢ (𝑤 = 𝑥 → ({𝑤} × (0...(♯‘𝑤))) = ({𝑥} × (0...(♯‘𝑥)))) |
| 49 | 48 | cbviunv 5040 |
. . . . . . 7
⊢ ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) = ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) |
| 50 | 49 | mpteq1i 5238 |
. . . . . 6
⊢ (𝑏 ∈ ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦
〈((1st ‘𝑏) prefix (2nd ‘𝑏)), ((1st
‘𝑏) substr
〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉) =
(𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦
〈((1st ‘𝑏) prefix (2nd ‘𝑏)), ((1st
‘𝑏) substr
〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉) |
| 51 | 50 | a1i 11 |
. . . . 5
⊢ (𝜑 → (𝑏 ∈ ∪
𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦
〈((1st ‘𝑏) prefix (2nd ‘𝑏)), ((1st
‘𝑏) substr
〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉) =
(𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦
〈((1st ‘𝑏) prefix (2nd ‘𝑏)), ((1st
‘𝑏) substr
〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉)) |
| 52 | 11 | feqmptd 6977 |
. . . . 5
⊢ (𝜑 → 𝐹 = (𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ (𝐹‘𝑎))) |
| 53 | | fveq2 6906 |
. . . . 5
⊢ (𝑎 = 〈((1st
‘𝑏) prefix
(2nd ‘𝑏)),
((1st ‘𝑏)
substr 〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉
→ (𝐹‘𝑎) = (𝐹‘〈((1st ‘𝑏) prefix (2nd
‘𝑏)),
((1st ‘𝑏)
substr 〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉)) |
| 54 | 44, 51, 52, 53 | fmptco 7149 |
. . . 4
⊢ (𝜑 → (𝐹 ∘ (𝑏 ∈ ∪
𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦
〈((1st ‘𝑏) prefix (2nd ‘𝑏)), ((1st
‘𝑏) substr
〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉)) =
(𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘〈((1st ‘𝑏) prefix (2nd
‘𝑏)),
((1st ‘𝑏)
substr 〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉))) |
| 55 | 54 | oveq2d 7447 |
. . 3
⊢ (𝜑 → (𝑀 Σg (𝐹 ∘ (𝑏 ∈ ∪
𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦
〈((1st ‘𝑏) prefix (2nd ‘𝑏)), ((1st
‘𝑏) substr
〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉)))
= (𝑀
Σg (𝑏 ∈ ∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘〈((1st ‘𝑏) prefix (2nd
‘𝑏)),
((1st ‘𝑏)
substr 〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉)))) |
| 56 | | nfv 1914 |
. . . 4
⊢
Ⅎ𝑤𝜑 |
| 57 | 11, 44 | cofmpt 7152 |
. . . . 5
⊢ (𝜑 → (𝐹 ∘ (𝑏 ∈ ∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦
〈((1st ‘𝑏) prefix (2nd ‘𝑏)), ((1st
‘𝑏) substr
〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉)) =
(𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘〈((1st ‘𝑏) prefix (2nd
‘𝑏)),
((1st ‘𝑏)
substr 〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉))) |
| 58 | 20, 51 | eqtr2d 2778 |
. . . . . . . . 9
⊢ (𝜑 → (𝑏 ∈ ∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦
〈((1st ‘𝑏) prefix (2nd ‘𝑏)), ((1st
‘𝑏) substr
〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉) =
◡(𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ 〈((1st
‘𝑎) ++
(2nd ‘𝑎)),
(♯‘(1st ‘𝑎))〉)) |
| 59 | 49 | eqcomi 2746 |
. . . . . . . . . 10
⊢ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) = ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) |
| 60 | 59 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) = ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤)))) |
| 61 | | eqidd 2738 |
. . . . . . . . 9
⊢ (𝜑 → (Word 𝐴 × Word 𝐴) = (Word 𝐴 × Word 𝐴)) |
| 62 | 58, 60, 61 | f1oeq123d 6842 |
. . . . . . . 8
⊢ (𝜑 → ((𝑏 ∈ ∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦
〈((1st ‘𝑏) prefix (2nd ‘𝑏)), ((1st
‘𝑏) substr
〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉):∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))–1-1-onto→(Word
𝐴 × Word 𝐴) ↔ ◡(𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ 〈((1st
‘𝑎) ++
(2nd ‘𝑎)),
(♯‘(1st ‘𝑎))〉):∪
𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤)))–1-1-onto→(Word
𝐴 × Word 𝐴))) |
| 63 | 19, 62 | mpbird 257 |
. . . . . . 7
⊢ (𝜑 → (𝑏 ∈ ∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦
〈((1st ‘𝑏) prefix (2nd ‘𝑏)), ((1st
‘𝑏) substr
〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉):∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))–1-1-onto→(Word
𝐴 × Word 𝐴)) |
| 64 | | f1of1 6847 |
. . . . . . 7
⊢ ((𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦
〈((1st ‘𝑏) prefix (2nd ‘𝑏)), ((1st
‘𝑏) substr
〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉):∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))–1-1-onto→(Word
𝐴 × Word 𝐴) → (𝑏 ∈ ∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦
〈((1st ‘𝑏) prefix (2nd ‘𝑏)), ((1st
‘𝑏) substr
〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉):∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))–1-1→(Word 𝐴 × Word 𝐴)) |
| 65 | 63, 64 | syl 17 |
. . . . . 6
⊢ (𝜑 → (𝑏 ∈ ∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦
〈((1st ‘𝑏) prefix (2nd ‘𝑏)), ((1st
‘𝑏) substr
〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉):∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))–1-1→(Word 𝐴 × Word 𝐴)) |
| 66 | 2 | fvexi 6920 |
. . . . . . 7
⊢ 𝑍 ∈ V |
| 67 | 66 | a1i 11 |
. . . . . 6
⊢ (𝜑 → 𝑍 ∈ V) |
| 68 | 11, 10 | fexd 7247 |
. . . . . 6
⊢ (𝜑 → 𝐹 ∈ V) |
| 69 | 12, 65, 67, 68 | fsuppco 9442 |
. . . . 5
⊢ (𝜑 → (𝐹 ∘ (𝑏 ∈ ∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦
〈((1st ‘𝑏) prefix (2nd ‘𝑏)), ((1st
‘𝑏) substr
〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉))
finSupp 𝑍) |
| 70 | 57, 69 | eqbrtrrd 5167 |
. . . 4
⊢ (𝜑 → (𝑏 ∈ ∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘〈((1st ‘𝑏) prefix (2nd
‘𝑏)),
((1st ‘𝑏)
substr 〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉))
finSupp 𝑍) |
| 71 | 11 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑏 ∈ ∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → 𝐹:(Word 𝐴 × Word 𝐴)⟶𝐵) |
| 72 | 71, 44 | ffvelcdmd 7105 |
. . . . 5
⊢ ((𝜑 ∧ 𝑏 ∈ ∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (𝐹‘〈((1st ‘𝑏) prefix (2nd
‘𝑏)),
((1st ‘𝑏)
substr 〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉)
∈ 𝐵) |
| 73 | 72 | fmpttd 7135 |
. . . 4
⊢ (𝜑 → (𝑏 ∈ ∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘〈((1st ‘𝑏) prefix (2nd
‘𝑏)),
((1st ‘𝑏)
substr 〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉)):∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))⟶𝐵) |
| 74 | | vsnex 5434 |
. . . . . . . 8
⊢ {𝑥} ∈ V |
| 75 | | ovex 7464 |
. . . . . . . 8
⊢
(0...(♯‘𝑥)) ∈ V |
| 76 | 74, 75 | xpex 7773 |
. . . . . . 7
⊢ ({𝑥} ×
(0...(♯‘𝑥)))
∈ V |
| 77 | 76 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → ({𝑥} × (0...(♯‘𝑥))) ∈ V) |
| 78 | 77 | ralrimiva 3146 |
. . . . 5
⊢ (𝜑 → ∀𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ∈ V) |
| 79 | | iunexg 7988 |
. . . . 5
⊢ ((Word
𝐴 ∈ V ∧
∀𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ∈ V) → ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ∈ V) |
| 80 | 9, 78, 79 | syl2anc 584 |
. . . 4
⊢ (𝜑 → ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ∈ V) |
| 81 | 56, 1, 2, 28, 70, 3, 73, 80 | gsumfs2d 33058 |
. . 3
⊢ (𝜑 → (𝑀 Σg (𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘〈((1st ‘𝑏) prefix (2nd
‘𝑏)),
((1st ‘𝑏)
substr 〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉)))
= (𝑀
Σg (𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝑀 Σg (𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ ((𝑏 ∈ ∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘〈((1st ‘𝑏) prefix (2nd
‘𝑏)),
((1st ‘𝑏)
substr 〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉))‘〈𝑤, 𝑗〉)))))) |
| 82 | 23, 55, 81 | 3eqtrd 2781 |
. 2
⊢ (𝜑 → (𝑀 Σg 𝐹) = (𝑀 Σg (𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝑀 Σg (𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ ((𝑏 ∈ ∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘〈((1st ‘𝑏) prefix (2nd
‘𝑏)),
((1st ‘𝑏)
substr 〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉))‘〈𝑤, 𝑗〉)))))) |
| 83 | | eqid 2737 |
. . . . . . 7
⊢ (𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘〈((1st ‘𝑏) prefix (2nd
‘𝑏)),
((1st ‘𝑏)
substr 〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉)) =
(𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘〈((1st ‘𝑏) prefix (2nd
‘𝑏)),
((1st ‘𝑏)
substr 〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉)) |
| 84 | | vex 3484 |
. . . . . . . . . . 11
⊢ 𝑤 ∈ V |
| 85 | | vex 3484 |
. . . . . . . . . . 11
⊢ 𝑗 ∈ V |
| 86 | 84, 85 | op1std 8024 |
. . . . . . . . . 10
⊢ (𝑏 = 〈𝑤, 𝑗〉 → (1st ‘𝑏) = 𝑤) |
| 87 | 84, 85 | op2ndd 8025 |
. . . . . . . . . 10
⊢ (𝑏 = 〈𝑤, 𝑗〉 → (2nd ‘𝑏) = 𝑗) |
| 88 | 86, 87 | oveq12d 7449 |
. . . . . . . . 9
⊢ (𝑏 = 〈𝑤, 𝑗〉 → ((1st ‘𝑏) prefix (2nd
‘𝑏)) = (𝑤 prefix 𝑗)) |
| 89 | 86 | fveq2d 6910 |
. . . . . . . . . . 11
⊢ (𝑏 = 〈𝑤, 𝑗〉 → (♯‘(1st
‘𝑏)) =
(♯‘𝑤)) |
| 90 | 87, 89 | opeq12d 4881 |
. . . . . . . . . 10
⊢ (𝑏 = 〈𝑤, 𝑗〉 → 〈(2nd
‘𝑏),
(♯‘(1st ‘𝑏))〉 = 〈𝑗, (♯‘𝑤)〉) |
| 91 | 86, 90 | oveq12d 7449 |
. . . . . . . . 9
⊢ (𝑏 = 〈𝑤, 𝑗〉 → ((1st ‘𝑏) substr 〈(2nd
‘𝑏),
(♯‘(1st ‘𝑏))〉) = (𝑤 substr 〈𝑗, (♯‘𝑤)〉)) |
| 92 | 88, 91 | opeq12d 4881 |
. . . . . . . 8
⊢ (𝑏 = 〈𝑤, 𝑗〉 → 〈((1st
‘𝑏) prefix
(2nd ‘𝑏)),
((1st ‘𝑏)
substr 〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉 =
〈(𝑤 prefix 𝑗), (𝑤 substr 〈𝑗, (♯‘𝑤)〉)〉) |
| 93 | 92 | fveq2d 6910 |
. . . . . . 7
⊢ (𝑏 = 〈𝑤, 𝑗〉 → (𝐹‘〈((1st ‘𝑏) prefix (2nd
‘𝑏)),
((1st ‘𝑏)
substr 〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉) =
(𝐹‘〈(𝑤 prefix 𝑗), (𝑤 substr 〈𝑗, (♯‘𝑤)〉)〉)) |
| 94 | 37 | eleq2d 2827 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↔ 𝑤 ∈ Word 𝐴)) |
| 95 | 94 | biimpa 476 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → 𝑤 ∈ Word 𝐴) |
| 96 | 95 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) ∧ 𝑗 ∈ (∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤})) → 𝑤 ∈ Word 𝐴) |
| 97 | | ovexd 7466 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → (0...(♯‘𝑥)) ∈ V) |
| 98 | | nfcv 2905 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥(0...(♯‘𝑤)) |
| 99 | | fveq2 6906 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑤 → (♯‘𝑥) = (♯‘𝑤)) |
| 100 | 99 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑤 → (0...(♯‘𝑥)) = (0...(♯‘𝑤))) |
| 101 | 9, 97, 98, 100 | iunsnima2 32631 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑤 ∈ Word 𝐴) → (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) = (0...(♯‘𝑤))) |
| 102 | 95, 101 | syldan 591 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) = (0...(♯‘𝑤))) |
| 103 | 102 | eleq2d 2827 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (𝑗 ∈ (∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↔ 𝑗 ∈ (0...(♯‘𝑤)))) |
| 104 | 103 | biimpa 476 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) ∧ 𝑗 ∈ (∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤})) → 𝑗 ∈ (0...(♯‘𝑤))) |
| 105 | 100 | opeliunxp2 5849 |
. . . . . . . 8
⊢
(〈𝑤, 𝑗〉 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↔ (𝑤 ∈ Word 𝐴 ∧ 𝑗 ∈ (0...(♯‘𝑤)))) |
| 106 | 96, 104, 105 | sylanbrc 583 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) ∧ 𝑗 ∈ (∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤})) → 〈𝑤, 𝑗〉 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) |
| 107 | | fvexd 6921 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) ∧ 𝑗 ∈ (∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤})) → (𝐹‘〈(𝑤 prefix 𝑗), (𝑤 substr 〈𝑗, (♯‘𝑤)〉)〉) ∈ V) |
| 108 | 83, 93, 106, 107 | fvmptd3 7039 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) ∧ 𝑗 ∈ (∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤})) → ((𝑏 ∈ ∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘〈((1st ‘𝑏) prefix (2nd
‘𝑏)),
((1st ‘𝑏)
substr 〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉))‘〈𝑤, 𝑗〉) = (𝐹‘〈(𝑤 prefix 𝑗), (𝑤 substr 〈𝑗, (♯‘𝑤)〉)〉)) |
| 109 | 108 | mpteq2dva 5242 |
. . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (𝑗 ∈ (∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ ((𝑏 ∈ ∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘〈((1st ‘𝑏) prefix (2nd
‘𝑏)),
((1st ‘𝑏)
substr 〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉))‘〈𝑤, 𝑗〉)) = (𝑗 ∈ (∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ (𝐹‘〈(𝑤 prefix 𝑗), (𝑤 substr 〈𝑗, (♯‘𝑤)〉)〉))) |
| 110 | 109 | oveq2d 7447 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (𝑀 Σg (𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ ((𝑏 ∈ ∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘〈((1st ‘𝑏) prefix (2nd
‘𝑏)),
((1st ‘𝑏)
substr 〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉))‘〈𝑤, 𝑗〉))) = (𝑀 Σg (𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ (𝐹‘〈(𝑤 prefix 𝑗), (𝑤 substr 〈𝑗, (♯‘𝑤)〉)〉)))) |
| 111 | 110 | mpteq2dva 5242 |
. . 3
⊢ (𝜑 → (𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝑀 Σg (𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ ((𝑏 ∈ ∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘〈((1st ‘𝑏) prefix (2nd
‘𝑏)),
((1st ‘𝑏)
substr 〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉))‘〈𝑤, 𝑗〉)))) = (𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝑀 Σg (𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ (𝐹‘〈(𝑤 prefix 𝑗), (𝑤 substr 〈𝑗, (♯‘𝑤)〉)〉))))) |
| 112 | 111 | oveq2d 7447 |
. 2
⊢ (𝜑 → (𝑀 Σg (𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝑀 Σg (𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ ((𝑏 ∈ ∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘〈((1st ‘𝑏) prefix (2nd
‘𝑏)),
((1st ‘𝑏)
substr 〈(2nd ‘𝑏), (♯‘(1st
‘𝑏))〉)〉))‘〈𝑤, 𝑗〉))))) = (𝑀 Σg (𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝑀 Σg (𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ (𝐹‘〈(𝑤 prefix 𝑗), (𝑤 substr 〈𝑗, (♯‘𝑤)〉)〉)))))) |
| 113 | 102 | mpteq1d 5237 |
. . . . 5
⊢ ((𝜑 ∧ 𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (𝑗 ∈ (∪
𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ (𝐹‘〈(𝑤 prefix 𝑗), (𝑤 substr 〈𝑗, (♯‘𝑤)〉)〉)) = (𝑗 ∈ (0...(♯‘𝑤)) ↦ (𝐹‘〈(𝑤 prefix 𝑗), (𝑤 substr 〈𝑗, (♯‘𝑤)〉)〉))) |
| 114 | 113 | oveq2d 7447 |
. . . 4
⊢ ((𝜑 ∧ 𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (𝑀 Σg (𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ (𝐹‘〈(𝑤 prefix 𝑗), (𝑤 substr 〈𝑗, (♯‘𝑤)〉)〉))) = (𝑀 Σg (𝑗 ∈
(0...(♯‘𝑤))
↦ (𝐹‘〈(𝑤 prefix 𝑗), (𝑤 substr 〈𝑗, (♯‘𝑤)〉)〉)))) |
| 115 | 37, 114 | mpteq12dva 5231 |
. . 3
⊢ (𝜑 → (𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝑀 Σg (𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ (𝐹‘〈(𝑤 prefix 𝑗), (𝑤 substr 〈𝑗, (♯‘𝑤)〉)〉)))) = (𝑤 ∈ Word 𝐴 ↦ (𝑀 Σg (𝑗 ∈
(0...(♯‘𝑤))
↦ (𝐹‘〈(𝑤 prefix 𝑗), (𝑤 substr 〈𝑗, (♯‘𝑤)〉)〉))))) |
| 116 | 115 | oveq2d 7447 |
. 2
⊢ (𝜑 → (𝑀 Σg (𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝑀 Σg (𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ (𝐹‘〈(𝑤 prefix 𝑗), (𝑤 substr 〈𝑗, (♯‘𝑤)〉)〉))))) = (𝑀 Σg (𝑤 ∈ Word 𝐴 ↦ (𝑀 Σg (𝑗 ∈
(0...(♯‘𝑤))
↦ (𝐹‘〈(𝑤 prefix 𝑗), (𝑤 substr 〈𝑗, (♯‘𝑤)〉)〉)))))) |
| 117 | 82, 112, 116 | 3eqtrd 2781 |
1
⊢ (𝜑 → (𝑀 Σg 𝐹) = (𝑀 Σg (𝑤 ∈ Word 𝐴 ↦ (𝑀 Σg (𝑗 ∈
(0...(♯‘𝑤))
↦ (𝐹‘〈(𝑤 prefix 𝑗), (𝑤 substr 〈𝑗, (♯‘𝑤)〉)〉)))))) |