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Theorem gsumwrd2dccat 33070
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 5324 . . . . . 6 (𝜑𝐴 ∈ V)
8 wrdexg 14562 . . . . . 6 (𝐴 ∈ V → Word 𝐴 ∈ V)
97, 8syl 17 . . . . 5 (𝜑 → Word 𝐴 ∈ V)
109, 9xpexd 7771 . . . 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 2737 . . . . . . . 8 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) = 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤)))
14 eqid 2737 . . . . . . . 8 (𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ ⟨((1st𝑎) ++ (2nd𝑎)), (♯‘(1st𝑎))⟩) = (𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ ⟨((1st𝑎) ++ (2nd𝑎)), (♯‘(1st𝑎))⟩)
15 eqid 2737 . . . . . . . 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 33069 . . . . . . 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 19934 . . 3 (𝜑 → (𝑀 Σg 𝐹) = (𝑀 Σg (𝐹 ∘ (𝑏 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦ ⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩))))
24 relxp 5703 . . . . . . . . . . . 12 Rel ({𝑥} × (0...(♯‘𝑥)))
2524a1i 11 . . . . . . . . . . 11 ((𝜑𝑥 ∈ Word 𝐴) → Rel ({𝑥} × (0...(♯‘𝑥))))
2625ralrimiva 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))))
2826, 27sylibr 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)))) → (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 14571 . . . . . . . . . . . . 13 (𝑥 ∈ Word 𝐴 → (♯‘𝑥) ∈ ℕ0)
3231adantl 481 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ Word 𝐴) → (♯‘𝑥) ∈ ℕ0)
33 nn0uz 12920 . . . . . . . . . . . 12 0 = (ℤ‘0)
3432, 33eleqtrdi 2851 . . . . . . . . . . 11 ((𝜑𝑥 ∈ Word 𝐴) → (♯‘𝑥) ∈ (ℤ‘0))
35 fzn0 13578 . . . . . . . . . . 11 ((0...(♯‘𝑥)) ≠ ∅ ↔ (♯‘𝑥) ∈ (ℤ‘0))
3634, 35sylibr 234 . . . . . . . . . 10 ((𝜑𝑥 ∈ Word 𝐴) → (0...(♯‘𝑥)) ≠ ∅)
3736dmdju 32657 . . . . . . . . 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 2843 . . . . . . 7 ((𝜑𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (1st𝑏) ∈ Word 𝐴)
40 pfxcl 14715 . . . . . . 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 14683 . . . . . . 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 5724 . . . . 5 ((𝜑𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → ⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩ ∈ (Word 𝐴 × Word 𝐴))
45 sneq 4636 . . . . . . . . 9 (𝑤 = 𝑥 → {𝑤} = {𝑥})
46 fveq2 6906 . . . . . . . . . 10 (𝑤 = 𝑥 → (♯‘𝑤) = (♯‘𝑥))
4746oveq2d 7447 . . . . . . . . 9 (𝑤 = 𝑥 → (0...(♯‘𝑤)) = (0...(♯‘𝑥)))
4845, 47xpeq12d 5716 . . . . . . . 8 (𝑤 = 𝑥 → ({𝑤} × (0...(♯‘𝑤))) = ({𝑥} × (0...(♯‘𝑥))))
4948cbviunv 5040 . . . . . . 7 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) = 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))
5049mpteq1i 5238 . . . . . 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 6977 . . . . 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 7149 . . . 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 7447 . . 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 1914 . . . 4 𝑤𝜑
5711, 44cofmpt 7152 . . . . 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 2778 . . . . . . . . 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 2746 . . . . . . . . . 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 2738 . . . . . . . . 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 7247 . . . . . 6 (𝜑𝐹 ∈ V)
6912, 65, 67, 68fsuppco 9442 . . . . 5 (𝜑 → (𝐹 ∘ (𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ ⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩)) finSupp 𝑍)
7057, 69eqbrtrrd 5167 . . . 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 7105 . . . . 5 ((𝜑𝑏 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (𝐹‘⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩) ∈ 𝐵)
7372fmpttd 7135 . . . 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 5434 . . . . . . . 8 {𝑥} ∈ V
75 ovex 7464 . . . . . . . 8 (0...(♯‘𝑥)) ∈ V
7674, 75xpex 7773 . . . . . . 7 ({𝑥} × (0...(♯‘𝑥))) ∈ V
7776a1i 11 . . . . . 6 ((𝜑𝑥 ∈ Word 𝐴) → ({𝑥} × (0...(♯‘𝑥))) ∈ V)
7877ralrimiva 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)
809, 78, 79syl2anc 584 . . . 4 (𝜑 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ∈ V)
8156, 1, 2, 28, 70, 3, 73, 80gsumfs2d 33058 . . 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 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‘⟨((1st𝑏) prefix (2nd𝑏)), ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩)⟩))‘⟨𝑤, 𝑗⟩))))))
83 eqid 2737 . . . . . . 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 3484 . . . . . . . . . . 11 𝑤 ∈ V
85 vex 3484 . . . . . . . . . . 11 𝑗 ∈ V
8684, 85op1std 8024 . . . . . . . . . 10 (𝑏 = ⟨𝑤, 𝑗⟩ → (1st𝑏) = 𝑤)
8784, 85op2ndd 8025 . . . . . . . . . 10 (𝑏 = ⟨𝑤, 𝑗⟩ → (2nd𝑏) = 𝑗)
8886, 87oveq12d 7449 . . . . . . . . 9 (𝑏 = ⟨𝑤, 𝑗⟩ → ((1st𝑏) prefix (2nd𝑏)) = (𝑤 prefix 𝑗))
8986fveq2d 6910 . . . . . . . . . . 11 (𝑏 = ⟨𝑤, 𝑗⟩ → (♯‘(1st𝑏)) = (♯‘𝑤))
9087, 89opeq12d 4881 . . . . . . . . . 10 (𝑏 = ⟨𝑤, 𝑗⟩ → ⟨(2nd𝑏), (♯‘(1st𝑏))⟩ = ⟨𝑗, (♯‘𝑤)⟩)
9186, 90oveq12d 7449 . . . . . . . . 9 (𝑏 = ⟨𝑤, 𝑗⟩ → ((1st𝑏) substr ⟨(2nd𝑏), (♯‘(1st𝑏))⟩) = (𝑤 substr ⟨𝑗, (♯‘𝑤)⟩))
9288, 91opeq12d 4881 . . . . . . . 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 2827 . . . . . . . . . 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 7466 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ Word 𝐴) → (0...(♯‘𝑥)) ∈ V)
98 nfcv 2905 . . . . . . . . . . . 12 𝑥(0...(♯‘𝑤))
99 fveq2 6906 . . . . . . . . . . . . 13 (𝑥 = 𝑤 → (♯‘𝑥) = (♯‘𝑤))
10099oveq2d 7447 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (0...(♯‘𝑥)) = (0...(♯‘𝑤)))
1019, 97, 98, 100iunsnima2 32631 . . . . . . . . . . 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 2827 . . . . . . . . 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 5849 . . . . . . . 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 7039 . . . . . 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 5242 . . . . 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 7447 . . . 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 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‘⟨((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 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‘⟨((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 5237 . . . . 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 7447 . . . 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 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)⟩)⟩)))))
116115oveq2d 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)⟩)⟩))))))
11782, 112, 1163eqtrd 2781 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 1540  wcel 2108  wne 2940  wral 3061  Vcvv 3480  wss 3951  c0 4333  {csn 4626  cop 4632   ciun 4991   class class class wbr 5143  cmpt 5225   × cxp 5683  ccnv 5684  dom cdm 5685  cima 5688  ccom 5689  Rel wrel 5690  wf 6557  1-1wf1 6558  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013   finSupp cfsupp 9401  0cc0 11155  0cn0 12526  cuz 12878  ...cfz 13547  chash 14369  Word cword 14552   ++ cconcat 14608   substr csubstr 14678   prefix cpfx 14708  Basecbs 17247  0gc0g 17484   Σg cgsu 17485  CMndccmn 19798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-word 14553  df-concat 14609  df-substr 14679  df-pfx 14709  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-gsum 17487  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-mulg 19086  df-cntz 19335  df-cmn 19800
This theorem is referenced by:  elrgspnlem2  33247
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