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Theorem gsumwrd2dccat 33052
Description: Rewrite a sum ranging over pairs of words as a sum of sums over concatenated subwords. (Contributed by Thierry Arnoux, 5-Oct-2025.)
Hypotheses
Ref Expression
gsumwrd2dccat.1 𝐵 = (Base‘𝑀)
gsumwrd2dccat.2 𝑍 = (0g𝑀)
gsumwrd2dccat.3 (𝜑𝐹:(Word 𝐴 × Word 𝐴)⟶𝐵)
gsumwrd2dccat.4 (𝜑𝐹 finSupp 𝑍)
gsumwrd2dccat.5 (𝜑𝑀 ∈ CMnd)
gsumwrd2dccat.6 (𝜑𝐴𝐵)
Assertion
Ref Expression
gsumwrd2dccat (𝜑 → (𝑀 Σg 𝐹) = (𝑀 Σg (𝑤 ∈ Word 𝐴 ↦ (𝑀 Σg (𝑗 ∈ (0...(♯‘𝑤)) ↦ (𝐹‘⟨(𝑤 prefix 𝑗), (𝑤 substr ⟨𝑗, (♯‘𝑤)⟩)⟩))))))
Distinct variable groups:   𝑤,𝐴,𝑗   𝐵,𝑗,𝑤   𝑗,𝐹,𝑤   𝑗,𝑀,𝑤   𝑗,𝑍,𝑤   𝜑,𝑗,𝑤

Proof of Theorem gsumwrd2dccat
Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumwrd2dccat.1 . . . 4 𝐵 = (Base‘𝑀)
2 gsumwrd2dccat.2 . . . 4 𝑍 = (0g𝑀)
3 gsumwrd2dccat.5 . . . 4 (𝜑𝑀 ∈ CMnd)
41fvexi 6920 . . . . . . . 8 𝐵 ∈ V
54a1i 11 . . . . . . 7 (𝜑𝐵 ∈ V)
6 gsumwrd2dccat.6 . . . . . . 7 (𝜑𝐴𝐵)
75, 6ssexd 5329 . . . . . 6 (𝜑𝐴 ∈ V)
8 wrdexg 14558 . . . . . 6 (𝐴 ∈ V → Word 𝐴 ∈ V)
97, 8syl 17 . . . . 5 (𝜑 → Word 𝐴 ∈ V)
109, 9xpexd 7769 . . . 4 (𝜑 → (Word 𝐴 × Word 𝐴) ∈ V)
11 gsumwrd2dccat.3 . . . 4 (𝜑𝐹:(Word 𝐴 × Word 𝐴)⟶𝐵)
12 gsumwrd2dccat.4 . . . 4 (𝜑𝐹 finSupp 𝑍)
13 eqid 2734 . . . . . . . 8 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) = 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤)))
14 eqid 2734 . . . . . . . 8 (𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ ⟨((1st𝑎) ++ (2nd𝑎)), (♯‘(1st𝑎))⟩) = (𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ ⟨((1st𝑎) ++ (2nd𝑎)), (♯‘(1st𝑎))⟩)
15 eqid 2734 . . . . . . . 8 (𝑏 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦ ⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩) = (𝑏 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦ ⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩)
1613, 14, 15, 7gsumwrd2dccatlem 33051 . . . . . . 7 (𝜑 → ((𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ ⟨((1st𝑎) ++ (2nd𝑎)), (♯‘(1st𝑎))⟩):(Word 𝐴 × Word 𝐴)–1-1-onto 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ∧ (𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ ⟨((1st𝑎) ++ (2nd𝑎)), (♯‘(1st𝑎))⟩) = (𝑏 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦ ⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩)))
1716simpld 494 . . . . . 6 (𝜑 → (𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ ⟨((1st𝑎) ++ (2nd𝑎)), (♯‘(1st𝑎))⟩):(Word 𝐴 × Word 𝐴)–1-1-onto 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))))
18 f1ocnv 6860 . . . . . 6 ((𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ ⟨((1st𝑎) ++ (2nd𝑎)), (♯‘(1st𝑎))⟩):(Word 𝐴 × Word 𝐴)–1-1-onto 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) → (𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ ⟨((1st𝑎) ++ (2nd𝑎)), (♯‘(1st𝑎))⟩): 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤)))–1-1-onto→(Word 𝐴 × Word 𝐴))
1917, 18syl 17 . . . . 5 (𝜑(𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ ⟨((1st𝑎) ++ (2nd𝑎)), (♯‘(1st𝑎))⟩): 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤)))–1-1-onto→(Word 𝐴 × Word 𝐴))
2016simprd 495 . . . . . 6 (𝜑(𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ ⟨((1st𝑎) ++ (2nd𝑎)), (♯‘(1st𝑎))⟩) = (𝑏 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦ ⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩))
2120f1oeq1d 6843 . . . . 5 (𝜑 → ((𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ ⟨((1st𝑎) ++ (2nd𝑎)), (♯‘(1st𝑎))⟩): 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤)))–1-1-onto→(Word 𝐴 × Word 𝐴) ↔ (𝑏 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦ ⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩): 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤)))–1-1-onto→(Word 𝐴 × Word 𝐴)))
2219, 21mpbid 232 . . . 4 (𝜑 → (𝑏 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦ ⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩): 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤)))–1-1-onto→(Word 𝐴 × Word 𝐴))
231, 2, 3, 10, 11, 12, 22gsumf1o 19948 . . 3 (𝜑 → (𝑀 Σg 𝐹) = (𝑀 Σg (𝐹 ∘ (𝑏 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦ ⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩))))
24 relxp 5706 . . . . . . . . . . . 12 Rel ({𝑥} × (0...(♯‘𝑥)))
2524a1i 11 . . . . . . . . . . 11 ((𝜑𝑥 ∈ Word 𝐴) → Rel ({𝑥} × (0...(♯‘𝑥))))
2625ralrimiva 3143 . . . . . . . . . 10 (𝜑 → ∀𝑥 ∈ Word 𝐴Rel ({𝑥} × (0...(♯‘𝑥))))
27 reliun 5828 . . . . . . . . . 10 (Rel 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↔ ∀𝑥 ∈ Word 𝐴Rel ({𝑥} × (0...(♯‘𝑥))))
2826, 27sylibr 234 . . . . . . . . 9 (𝜑 → Rel 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))))
29 1stdm 8063 . . . . . . . . 9 ((Rel 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ∧ 𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (1st𝑏) ∈ dom 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))))
3028, 29sylan 580 . . . . . . . 8 ((𝜑𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (1st𝑏) ∈ dom 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))))
31 lencl 14567 . . . . . . . . . . . . 13 (𝑥 ∈ Word 𝐴 → (♯‘𝑥) ∈ ℕ0)
3231adantl 481 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ Word 𝐴) → (♯‘𝑥) ∈ ℕ0)
33 nn0uz 12917 . . . . . . . . . . . 12 0 = (ℤ‘0)
3432, 33eleqtrdi 2848 . . . . . . . . . . 11 ((𝜑𝑥 ∈ Word 𝐴) → (♯‘𝑥) ∈ (ℤ‘0))
35 fzn0 13574 . . . . . . . . . . 11 ((0...(♯‘𝑥)) ≠ ∅ ↔ (♯‘𝑥) ∈ (ℤ‘0))
3634, 35sylibr 234 . . . . . . . . . 10 ((𝜑𝑥 ∈ Word 𝐴) → (0...(♯‘𝑥)) ≠ ∅)
3736dmdju 32663 . . . . . . . . 9 (𝜑 → dom 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) = Word 𝐴)
3837adantr 480 . . . . . . . 8 ((𝜑𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → dom 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) = Word 𝐴)
3930, 38eleqtrd 2840 . . . . . . 7 ((𝜑𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (1st𝑏) ∈ Word 𝐴)
40 pfxcl 14711 . . . . . . 7 ((1st𝑏) ∈ Word 𝐴 → ((1st𝑏) prefix (2nd𝑏)) ∈ Word 𝐴)
4139, 40syl 17 . . . . . 6 ((𝜑𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → ((1st𝑏) prefix (2nd𝑏)) ∈ Word 𝐴)
42 swrdcl 14679 . . . . . . 7 ((1st𝑏) ∈ Word 𝐴 → ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩) ∈ Word 𝐴)
4339, 42syl 17 . . . . . 6 ((𝜑𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩) ∈ Word 𝐴)
4441, 43opelxpd 5727 . . . . 5 ((𝜑𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → ⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩ ∈ (Word 𝐴 × Word 𝐴))
45 sneq 4640 . . . . . . . . 9 (𝑤 = 𝑥 → {𝑤} = {𝑥})
46 fveq2 6906 . . . . . . . . . 10 (𝑤 = 𝑥 → (♯‘𝑤) = (♯‘𝑥))
4746oveq2d 7446 . . . . . . . . 9 (𝑤 = 𝑥 → (0...(♯‘𝑤)) = (0...(♯‘𝑥)))
4845, 47xpeq12d 5719 . . . . . . . 8 (𝑤 = 𝑥 → ({𝑤} × (0...(♯‘𝑤))) = ({𝑥} × (0...(♯‘𝑥))))
4948cbviunv 5044 . . . . . . 7 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) = 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))
5049mpteq1i 5243 . . . . . 6 (𝑏 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦ ⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩) = (𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ ⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩)
5150a1i 11 . . . . 5 (𝜑 → (𝑏 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦ ⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩) = (𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ ⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩))
5211feqmptd 6976 . . . . 5 (𝜑𝐹 = (𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ (𝐹𝑎)))
53 fveq2 6906 . . . . 5 (𝑎 = ⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩ → (𝐹𝑎) = (𝐹‘⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩))
5444, 51, 52, 53fmptco 7148 . . . 4 (𝜑 → (𝐹 ∘ (𝑏 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦ ⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩)) = (𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩)))
5554oveq2d 7446 . . 3 (𝜑 → (𝑀 Σg (𝐹 ∘ (𝑏 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦ ⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩))) = (𝑀 Σg (𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩))))
56 nfv 1911 . . . 4 𝑤𝜑
5711, 44cofmpt 7151 . . . . 5 (𝜑 → (𝐹 ∘ (𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ ⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩)) = (𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩)))
5820, 51eqtr2d 2775 . . . . . . . . 9 (𝜑 → (𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ ⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩) = (𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ ⟨((1st𝑎) ++ (2nd𝑎)), (♯‘(1st𝑎))⟩))
5949eqcomi 2743 . . . . . . . . . 10 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) = 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤)))
6059a1i 11 . . . . . . . . 9 (𝜑 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) = 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))))
61 eqidd 2735 . . . . . . . . 9 (𝜑 → (Word 𝐴 × Word 𝐴) = (Word 𝐴 × Word 𝐴))
6258, 60, 61f1oeq123d 6842 . . . . . . . 8 (𝜑 → ((𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ ⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩): 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))–1-1-onto→(Word 𝐴 × Word 𝐴) ↔ (𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ ⟨((1st𝑎) ++ (2nd𝑎)), (♯‘(1st𝑎))⟩): 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤)))–1-1-onto→(Word 𝐴 × Word 𝐴)))
6319, 62mpbird 257 . . . . . . 7 (𝜑 → (𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ ⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩): 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))–1-1-onto→(Word 𝐴 × Word 𝐴))
64 f1of1 6847 . . . . . . 7 ((𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ ⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩): 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))–1-1-onto→(Word 𝐴 × Word 𝐴) → (𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ ⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩): 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))–1-1→(Word 𝐴 × Word 𝐴))
6563, 64syl 17 . . . . . 6 (𝜑 → (𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ ⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩): 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))–1-1→(Word 𝐴 × Word 𝐴))
662fvexi 6920 . . . . . . 7 𝑍 ∈ V
6766a1i 11 . . . . . 6 (𝜑𝑍 ∈ V)
6811, 10fexd 7246 . . . . . 6 (𝜑𝐹 ∈ V)
6912, 65, 67, 68fsuppco 9439 . . . . 5 (𝜑 → (𝐹 ∘ (𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ ⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩)) finSupp 𝑍)
7057, 69eqbrtrrd 5171 . . . 4 (𝜑 → (𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩)) finSupp 𝑍)
7111adantr 480 . . . . . 6 ((𝜑𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → 𝐹:(Word 𝐴 × Word 𝐴)⟶𝐵)
7271, 44ffvelcdmd 7104 . . . . 5 ((𝜑𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (𝐹‘⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩) ∈ 𝐵)
7372fmpttd 7134 . . . 4 (𝜑 → (𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩)): 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))⟶𝐵)
74 vsnex 5439 . . . . . . . 8 {𝑥} ∈ V
75 ovex 7463 . . . . . . . 8 (0...(♯‘𝑥)) ∈ V
7674, 75xpex 7771 . . . . . . 7 ({𝑥} × (0...(♯‘𝑥))) ∈ V
7776a1i 11 . . . . . 6 ((𝜑𝑥 ∈ Word 𝐴) → ({𝑥} × (0...(♯‘𝑥))) ∈ V)
7877ralrimiva 3143 . . . . 5 (𝜑 → ∀𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ∈ V)
79 iunexg 7986 . . . . 5 ((Word 𝐴 ∈ V ∧ ∀𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ∈ V) → 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ∈ V)
809, 78, 79syl2anc 584 . . . 4 (𝜑 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ∈ V)
8156, 1, 2, 28, 70, 3, 73, 80gsumfs2d 33040 . . 3 (𝜑 → (𝑀 Σg (𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩))) = (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‘⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩))‘⟨𝑤, 𝑗⟩))))))
8223, 55, 813eqtrd 2778 . 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‘⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩))‘⟨𝑤, 𝑗⟩))))))
83 eqid 2734 . . . . . . 7 (𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩)) = (𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩))
84 vex 3481 . . . . . . . . . . 11 𝑤 ∈ V
85 vex 3481 . . . . . . . . . . 11 𝑗 ∈ V
8684, 85op1std 8022 . . . . . . . . . 10 (𝑏 = ⟨𝑤, 𝑗⟩ → (1st𝑏) = 𝑤)
8784, 85op2ndd 8023 . . . . . . . . . 10 (𝑏 = ⟨𝑤, 𝑗⟩ → (2nd𝑏) = 𝑗)
8886, 87oveq12d 7448 . . . . . . . . 9 (𝑏 = ⟨𝑤, 𝑗⟩ → ((1st𝑏) prefix (2nd𝑏)) = (𝑤 prefix 𝑗))
8986fveq2d 6910 . . . . . . . . . . 11 (𝑏 = ⟨𝑤, 𝑗⟩ → (♯‘(1st𝑏)) = (♯‘𝑤))
9087, 89opeq12d 4885 . . . . . . . . . 10 (𝑏 = ⟨𝑤, 𝑗⟩ → ⟨(2nd𝑏), (♯‘(1st𝑏))⟩ = ⟨𝑗, (♯‘𝑤)⟩)
9186, 90oveq12d 7448 . . . . . . . . 9 (𝑏 = ⟨𝑤, 𝑗⟩ → ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩) = (𝑤 substr ⟨𝑗, (♯‘𝑤)⟩))
9288, 91opeq12d 4885 . . . . . . . 8 (𝑏 = ⟨𝑤, 𝑗⟩ → ⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩ = ⟨(𝑤 prefix 𝑗), (𝑤 substr ⟨𝑗, (♯‘𝑤)⟩)⟩)
9392fveq2d 6910 . . . . . . 7 (𝑏 = ⟨𝑤, 𝑗⟩ → (𝐹‘⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩) = (𝐹‘⟨(𝑤 prefix 𝑗), (𝑤 substr ⟨𝑗, (♯‘𝑤)⟩)⟩))
9437eleq2d 2824 . . . . . . . . . 10 (𝜑 → (𝑤 ∈ dom 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↔ 𝑤 ∈ Word 𝐴))
9594biimpa 476 . . . . . . . . 9 ((𝜑𝑤 ∈ dom 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → 𝑤 ∈ Word 𝐴)
9695adantr 480 . . . . . . . 8 (((𝜑𝑤 ∈ dom 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) ∧ 𝑗 ∈ ( 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤})) → 𝑤 ∈ Word 𝐴)
97 ovexd 7465 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ Word 𝐴) → (0...(♯‘𝑥)) ∈ V)
98 nfcv 2902 . . . . . . . . . . . 12 𝑥(0...(♯‘𝑤))
99 fveq2 6906 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (♯‘𝑥) = (♯‘𝑤))
10099oveq2d 7446 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (0...(♯‘𝑥)) = (0...(♯‘𝑤)))
1019, 97, 98, 100iunsnima2 32638 . . . . . . . . . . 11 ((𝜑𝑤 ∈ Word 𝐴) → ( 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) = (0...(♯‘𝑤)))
10295, 101syldan 591 . . . . . . . . . 10 ((𝜑𝑤 ∈ dom 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → ( 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) = (0...(♯‘𝑤)))
103102eleq2d 2824 . . . . . . . . 9 ((𝜑𝑤 ∈ dom 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (𝑗 ∈ ( 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↔ 𝑗 ∈ (0...(♯‘𝑤))))
104103biimpa 476 . . . . . . . 8 (((𝜑𝑤 ∈ dom 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) ∧ 𝑗 ∈ ( 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤})) → 𝑗 ∈ (0...(♯‘𝑤)))
105100opeliunxp2 5851 . . . . . . . 8 (⟨𝑤, 𝑗⟩ ∈ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↔ (𝑤 ∈ Word 𝐴𝑗 ∈ (0...(♯‘𝑤))))
10696, 104, 105sylanbrc 583 . . . . . . 7 (((𝜑𝑤 ∈ 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 (((𝜑𝑤 ∈ dom 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) ∧ 𝑗 ∈ ( 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤})) → (𝐹‘⟨(𝑤 prefix 𝑗), (𝑤 substr ⟨𝑗, (♯‘𝑤)⟩)⟩) ∈ V)
10883, 93, 106, 107fvmptd3 7038 . . . . . 6 (((𝜑𝑤 ∈ dom 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) ∧ 𝑗 ∈ ( 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤})) → ((𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩))‘⟨𝑤, 𝑗⟩) = (𝐹‘⟨(𝑤 prefix 𝑗), (𝑤 substr ⟨𝑗, (♯‘𝑤)⟩)⟩))
109108mpteq2dva 5247 . . . . 5 ((𝜑𝑤 ∈ dom 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (𝑗 ∈ ( 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ ((𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩))‘⟨𝑤, 𝑗⟩)) = (𝑗 ∈ ( 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ (𝐹‘⟨(𝑤 prefix 𝑗), (𝑤 substr ⟨𝑗, (♯‘𝑤)⟩)⟩)))
110109oveq2d 7446 . . . 4 ((𝜑𝑤 ∈ 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‘⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩))‘⟨𝑤, 𝑗⟩))) = (𝑀 Σg (𝑗 ∈ ( 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ (𝐹‘⟨(𝑤 prefix 𝑗), (𝑤 substr ⟨𝑗, (♯‘𝑤)⟩)⟩))))
111110mpteq2dva 5247 . . 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‘⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩))‘⟨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)⟩)⟩)))))
112111oveq2d 7446 . 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‘⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩))‘⟨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)⟩)⟩))))))
113102mpteq1d 5242 . . . . 5 ((𝜑𝑤 ∈ 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)⟩)⟩)))
114113oveq2d 7446 . . . 4 ((𝜑𝑤 ∈ 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)⟩)⟩))))
11537, 114mpteq12dva 5236 . . 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)⟩)⟩)))))
116115oveq2d 7446 . 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)⟩)⟩))))))
11782, 112, 1163eqtrd 2778 1 (𝜑 → (𝑀 Σg 𝐹) = (𝑀 Σg (𝑤 ∈ Word 𝐴 ↦ (𝑀 Σg (𝑗 ∈ (0...(♯‘𝑤)) ↦ (𝐹‘⟨(𝑤 prefix 𝑗), (𝑤 substr ⟨𝑗, (♯‘𝑤)⟩)⟩))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  wne 2937  wral 3058  Vcvv 3477  wss 3962  c0 4338  {csn 4630  cop 4636   ciun 4995   class class class wbr 5147  cmpt 5230   × cxp 5686  ccnv 5687  dom cdm 5688  cima 5691  ccom 5692  Rel wrel 5693  wf 6558  1-1wf1 6559  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  1st c1st 8010  2nd c2nd 8011   finSupp cfsupp 9398  0cc0 11152  0cn0 12523  cuz 12875  ...cfz 13543  chash 14365  Word cword 14548   ++ cconcat 14604   substr csubstr 14674   prefix cpfx 14704  Basecbs 17244  0gc0g 17485   Σg cgsu 17486  CMndccmn 19812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-fzo 13691  df-seq 14039  df-hash 14366  df-word 14549  df-concat 14605  df-substr 14675  df-pfx 14705  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-0g 17487  df-gsum 17488  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-submnd 18809  df-mulg 19098  df-cntz 19347  df-cmn 19814
This theorem is referenced by:  elrgspnlem2  33232
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