gsumwrd2dccat - Mathbox for Thierry Arnoux

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Theorem gsumwrd2dccat 33050

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 = (0_g‘𝑀)
gsumwrd2dccat.3	⊢ (𝜑 → 𝐹:(Word 𝐴 × Word 𝐴)⟶𝐵)
gsumwrd2dccat.4	⊢ (𝜑 → 𝐹 finSupp 𝑍)
gsumwrd2dccat.5	⊢ (𝜑 → 𝑀 ∈ CMnd)
gsumwrd2dccat.6	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

**Assertion**
Ref	Expression
gsumwrd2dccat	⊢ (𝜑 → (𝑀 Σ_g 𝐹) = (𝑀 Σ_g (𝑤 ∈ Word 𝐴 ↦ (𝑀 Σ_g (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.

Step	Hyp	Ref	Expression
1		gsumwrd2dccat.1	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
2		gsumwrd2dccat.2	. . . 4 ⊢ 𝑍 = (0_g‘𝑀)
3		gsumwrd2dccat.5	. . . 4 ⊢ (𝜑 → 𝑀 ∈ CMnd)
4	1	fvexi 6854	. . . . . . . 8 ⊢ 𝐵 ∈ V
5	4	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V)
6		gsumwrd2dccat.6	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
7	5, 6	ssexd 5274	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V)
8		wrdexg 14465	. . . . . 6 ⊢ (𝐴 ∈ V → Word 𝐴 ∈ V)
9	7, 8	syl 17	. . . . 5 ⊢ (𝜑 → Word 𝐴 ∈ V)
10	9, 9	xpexd 7707	. . . 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 2729	. . . . . . . 8 ⊢ ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) = ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤)))
14		eqid 2729	. . . . . . . 8 ⊢ (𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ ⟨((1^st ‘𝑎) ++ (2^nd ‘𝑎)), (♯‘(1^st ‘𝑎))⟩) = (𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ ⟨((1^st ‘𝑎) ++ (2^nd ‘𝑎)), (♯‘(1^st ‘𝑎))⟩)
15		eqid 2729	. . . . . . . 8 ⊢ (𝑏 ∈ ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦ ⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩) = (𝑏 ∈ ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦ ⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩)
16	13, 14, 15, 7	gsumwrd2dccatlem 33049	. . . . . . 7 ⊢ (𝜑 → ((𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ ⟨((1^st ‘𝑎) ++ (2^nd ‘𝑎)), (♯‘(1^st ‘𝑎))⟩):(Word 𝐴 × Word 𝐴)–1-1-onto→∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ∧ ^◡(𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ ⟨((1^st ‘𝑎) ++ (2^nd ‘𝑎)), (♯‘(1^st ‘𝑎))⟩) = (𝑏 ∈ ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦ ⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩)))
17	16	simpld 494	. . . . . 6 ⊢ (𝜑 → (𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ ⟨((1^st ‘𝑎) ++ (2^nd ‘𝑎)), (♯‘(1^st ‘𝑎))⟩):(Word 𝐴 × Word 𝐴)–1-1-onto→∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))))
18		f1ocnv 6794	. . . . . 6 ⊢ ((𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ ⟨((1^st ‘𝑎) ++ (2^nd ‘𝑎)), (♯‘(1^st ‘𝑎))⟩):(Word 𝐴 × Word 𝐴)–1-1-onto→∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) → ^◡(𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ ⟨((1^st ‘𝑎) ++ (2^nd ‘𝑎)), (♯‘(1^st ‘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) ↦ ⟨((1^st ‘𝑎) ++ (2^nd ‘𝑎)), (♯‘(1^st ‘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) ↦ ⟨((1^st ‘𝑎) ++ (2^nd ‘𝑎)), (♯‘(1^st ‘𝑎))⟩) = (𝑏 ∈ ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦ ⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩))
21	20	f1oeq1d 6777	. . . . 5 ⊢ (𝜑 → (^◡(𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ ⟨((1^st ‘𝑎) ++ (2^nd ‘𝑎)), (♯‘(1^st ‘𝑎))⟩):∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤)))–1-1-onto→(Word 𝐴 × Word 𝐴) ↔ (𝑏 ∈ ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦ ⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘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))) ↦ ⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩):∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤)))–1-1-onto→(Word 𝐴 × Word 𝐴))
23	1, 2, 3, 10, 11, 12, 22	gsumf1o 19830	. . 3 ⊢ (𝜑 → (𝑀 Σ_g 𝐹) = (𝑀 Σ_g (𝐹 ∘ (𝑏 ∈ ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦ ⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩))))
24		relxp 5649	. . . . . . . . . . . 12 ⊢ Rel ({𝑥} × (0...(♯‘𝑥)))
25	24	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → Rel ({𝑥} × (0...(♯‘𝑥))))
26	25	ralrimiva 3125	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ Word 𝐴Rel ({𝑥} × (0...(♯‘𝑥))))
27		reliun 5770	. . . . . . . . . 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 7998	. . . . . . . . 9 ⊢ ((Rel ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ∧ 𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (1^st ‘𝑏) ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))))
30	28, 29	sylan 580	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (1^st ‘𝑏) ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))))
31		lencl 14474	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Word 𝐴 → (♯‘𝑥) ∈ ℕ₀)
32	31	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → (♯‘𝑥) ∈ ℕ₀)
33		nn0uz 12811	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
34	32, 33	eleqtrdi 2838	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → (♯‘𝑥) ∈ (ℤ_≥‘0))
35		fzn0 13475	. . . . . . . . . . 11 ⊢ ((0...(♯‘𝑥)) ≠ ∅ ↔ (♯‘𝑥) ∈ (ℤ_≥‘0))
36	34, 35	sylibr 234	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → (0...(♯‘𝑥)) ≠ ∅)
37	36	dmdju 32621	. . . . . . . . 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 2830	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (1^st ‘𝑏) ∈ Word 𝐴)
40		pfxcl 14618	. . . . . . 7 ⊢ ((1^st ‘𝑏) ∈ Word 𝐴 → ((1^st ‘𝑏) prefix (2^nd ‘𝑏)) ∈ Word 𝐴)
41	39, 40	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → ((1^st ‘𝑏) prefix (2^nd ‘𝑏)) ∈ Word 𝐴)
42		swrdcl 14586	. . . . . . 7 ⊢ ((1^st ‘𝑏) ∈ Word 𝐴 → ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩) ∈ Word 𝐴)
43	39, 42	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩) ∈ Word 𝐴)
44	41, 43	opelxpd 5670	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → ⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩ ∈ (Word 𝐴 × Word 𝐴))
45		sneq 4595	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → {𝑤} = {𝑥})
46		fveq2 6840	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (♯‘𝑤) = (♯‘𝑥))
47	46	oveq2d 7385	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (0...(♯‘𝑤)) = (0...(♯‘𝑥)))
48	45, 47	xpeq12d 5662	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ({𝑤} × (0...(♯‘𝑤))) = ({𝑥} × (0...(♯‘𝑥))))
49	48	cbviunv 4999	. . . . . . 7 ⊢ ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) = ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))
50	49	mpteq1i 5193	. . . . . 6 ⊢ (𝑏 ∈ ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦ ⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩) = (𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ ⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩)
51	50	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑏 ∈ ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦ ⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩) = (𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ ⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩))
52	11	feqmptd 6911	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ (𝐹‘𝑎)))
53		fveq2 6840	. . . . 5 ⊢ (𝑎 = ⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩ → (𝐹‘𝑎) = (𝐹‘⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩))
54	44, 51, 52, 53	fmptco 7083	. . . 4 ⊢ (𝜑 → (𝐹 ∘ (𝑏 ∈ ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦ ⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩)) = (𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩)))
55	54	oveq2d 7385	. . 3 ⊢ (𝜑 → (𝑀 Σ_g (𝐹 ∘ (𝑏 ∈ ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) ↦ ⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩))) = (𝑀 Σ_g (𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩))))
56		nfv 1914	. . . 4 ⊢ Ⅎ𝑤𝜑
57	11, 44	cofmpt 7086	. . . . 5 ⊢ (𝜑 → (𝐹 ∘ (𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ ⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩)) = (𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩)))
58	20, 51	eqtr2d 2765	. . . . . . . . 9 ⊢ (𝜑 → (𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ ⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩) = ^◡(𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ ⟨((1^st ‘𝑎) ++ (2^nd ‘𝑎)), (♯‘(1^st ‘𝑎))⟩))
59	49	eqcomi 2738	. . . . . . . . . 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 2730	. . . . . . . . 9 ⊢ (𝜑 → (Word 𝐴 × Word 𝐴) = (Word 𝐴 × Word 𝐴))
62	58, 60, 61	f1oeq123d 6776	. . . . . . . 8 ⊢ (𝜑 → ((𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ ⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩):∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))–1-1-onto→(Word 𝐴 × Word 𝐴) ↔ ^◡(𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ ⟨((1^st ‘𝑎) ++ (2^nd ‘𝑎)), (♯‘(1^st ‘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))) ↦ ⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩):∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))–1-1-onto→(Word 𝐴 × Word 𝐴))
64		f1of1 6781	. . . . . . 7 ⊢ ((𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ ⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩):∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))–1-1-onto→(Word 𝐴 × Word 𝐴) → (𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ ⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘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))) ↦ ⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩):∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))–1-1→(Word 𝐴 × Word 𝐴))
66	2	fvexi 6854	. . . . . . 7 ⊢ 𝑍 ∈ V
67	66	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ V)
68	11, 10	fexd 7183	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V)
69	12, 65, 67, 68	fsuppco 9329	. . . . 5 ⊢ (𝜑 → (𝐹 ∘ (𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ ⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩)) finSupp 𝑍)
70	57, 69	eqbrtrrd 5126	. . . 4 ⊢ (𝜑 → (𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘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 7039	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (𝐹‘⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩) ∈ 𝐵)
73	72	fmpttd 7069	. . . 4 ⊢ (𝜑 → (𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩)):∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))⟶𝐵)
74		vsnex 5384	. . . . . . . 8 ⊢ {𝑥} ∈ V
75		ovex 7402	. . . . . . . 8 ⊢ (0...(♯‘𝑥)) ∈ V
76	74, 75	xpex 7709	. . . . . . 7 ⊢ ({𝑥} × (0...(♯‘𝑥))) ∈ V
77	76	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → ({𝑥} × (0...(♯‘𝑥))) ∈ V)
78	77	ralrimiva 3125	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ∈ V)
79		iunexg 7921	. . . . 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 33038	. . 3 ⊢ (𝜑 → (𝑀 Σ_g (𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩))) = (𝑀 Σ_g (𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝑀 Σ_g (𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ ((𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩))‘⟨𝑤, 𝑗⟩))))))
82	23, 55, 81	3eqtrd 2768	. 2 ⊢ (𝜑 → (𝑀 Σ_g 𝐹) = (𝑀 Σ_g (𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝑀 Σ_g (𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ ((𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩))‘⟨𝑤, 𝑗⟩))))))
83		eqid 2729	. . . . . . 7 ⊢ (𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩)) = (𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩))
84		vex 3448	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
85		vex 3448	. . . . . . . . . . 11 ⊢ 𝑗 ∈ V
86	84, 85	op1std 7957	. . . . . . . . . 10 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → (1^st ‘𝑏) = 𝑤)
87	84, 85	op2ndd 7958	. . . . . . . . . 10 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → (2^nd ‘𝑏) = 𝑗)
88	86, 87	oveq12d 7387	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → ((1^st ‘𝑏) prefix (2^nd ‘𝑏)) = (𝑤 prefix 𝑗))
89	86	fveq2d 6844	. . . . . . . . . . 11 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → (♯‘(1^st ‘𝑏)) = (♯‘𝑤))
90	87, 89	opeq12d 4841	. . . . . . . . . 10 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩ = ⟨𝑗, (♯‘𝑤)⟩)
91	86, 90	oveq12d 7387	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩) = (𝑤 substr ⟨𝑗, (♯‘𝑤)⟩))
92	88, 91	opeq12d 4841	. . . . . . . 8 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → ⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩ = ⟨(𝑤 prefix 𝑗), (𝑤 substr ⟨𝑗, (♯‘𝑤)⟩)⟩)
93	92	fveq2d 6844	. . . . . . 7 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → (𝐹‘⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩) = (𝐹‘⟨(𝑤 prefix 𝑗), (𝑤 substr ⟨𝑗, (♯‘𝑤)⟩)⟩))
94	37	eleq2d 2814	. . . . . . . . . 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 7404	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → (0...(♯‘𝑥)) ∈ V)
98		nfcv 2891	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(0...(♯‘𝑤))
99		fveq2 6840	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (♯‘𝑥) = (♯‘𝑤))
100	99	oveq2d 7385	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (0...(♯‘𝑥)) = (0...(♯‘𝑤)))
101	9, 97, 98, 100	iunsnima2 32597	. . . . . . . . . . 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 2814	. . . . . . . . 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 5792	. . . . . . . 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 6855	. . . . . . 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 6973	. . . . . 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‘⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩))‘⟨𝑤, 𝑗⟩) = (𝐹‘⟨(𝑤 prefix 𝑗), (𝑤 substr ⟨𝑗, (♯‘𝑤)⟩)⟩))
109	108	mpteq2dva 5195	. . . . 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‘⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩))‘⟨𝑤, 𝑗⟩)) = (𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ (𝐹‘⟨(𝑤 prefix 𝑗), (𝑤 substr ⟨𝑗, (♯‘𝑤)⟩)⟩)))
110	109	oveq2d 7385	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (𝑀 Σ_g (𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ ((𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩))‘⟨𝑤, 𝑗⟩))) = (𝑀 Σ_g (𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ (𝐹‘⟨(𝑤 prefix 𝑗), (𝑤 substr ⟨𝑗, (♯‘𝑤)⟩)⟩))))
111	110	mpteq2dva 5195	. . 3 ⊢ (𝜑 → (𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝑀 Σ_g (𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ ((𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩))‘⟨𝑤, 𝑗⟩)))) = (𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝑀 Σ_g (𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ (𝐹‘⟨(𝑤 prefix 𝑗), (𝑤 substr ⟨𝑗, (♯‘𝑤)⟩)⟩)))))
112	111	oveq2d 7385	. 2 ⊢ (𝜑 → (𝑀 Σ_g (𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝑀 Σ_g (𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ ((𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝐹‘⟨((1^st ‘𝑏) prefix (2^nd ‘𝑏)), ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩)⟩))‘⟨𝑤, 𝑗⟩))))) = (𝑀 Σ_g (𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝑀 Σ_g (𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ (𝐹‘⟨(𝑤 prefix 𝑗), (𝑤 substr ⟨𝑗, (♯‘𝑤)⟩)⟩))))))
113	102	mpteq1d 5192	. . . . 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 7385	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (𝑀 Σ_g (𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ (𝐹‘⟨(𝑤 prefix 𝑗), (𝑤 substr ⟨𝑗, (♯‘𝑤)⟩)⟩))) = (𝑀 Σ_g (𝑗 ∈ (0...(♯‘𝑤)) ↦ (𝐹‘⟨(𝑤 prefix 𝑗), (𝑤 substr ⟨𝑗, (♯‘𝑤)⟩)⟩))))
115	37, 114	mpteq12dva 5188	. . 3 ⊢ (𝜑 → (𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝑀 Σ_g (𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ (𝐹‘⟨(𝑤 prefix 𝑗), (𝑤 substr ⟨𝑗, (♯‘𝑤)⟩)⟩)))) = (𝑤 ∈ Word 𝐴 ↦ (𝑀 Σ_g (𝑗 ∈ (0...(♯‘𝑤)) ↦ (𝐹‘⟨(𝑤 prefix 𝑗), (𝑤 substr ⟨𝑗, (♯‘𝑤)⟩)⟩)))))
116	115	oveq2d 7385	. 2 ⊢ (𝜑 → (𝑀 Σ_g (𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↦ (𝑀 Σ_g (𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↦ (𝐹‘⟨(𝑤 prefix 𝑗), (𝑤 substr ⟨𝑗, (♯‘𝑤)⟩)⟩))))) = (𝑀 Σ_g (𝑤 ∈ Word 𝐴 ↦ (𝑀 Σ_g (𝑗 ∈ (0...(♯‘𝑤)) ↦ (𝐹‘⟨(𝑤 prefix 𝑗), (𝑤 substr ⟨𝑗, (♯‘𝑤)⟩)⟩))))))
117	82, 112, 116	3eqtrd 2768	1 ⊢ (𝜑 → (𝑀 Σ_g 𝐹) = (𝑀 Σ_g (𝑤 ∈ Word 𝐴 ↦ (𝑀 Σ_g (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 2109 ≠ wne 2925 ∀wral 3044 Vcvv 3444 ⊆ wss 3911 ∅c0 4292 {csn 4585 ⟨cop 4591 ∪ ciun 4951 class class class wbr 5102 ↦ cmpt 5183 × cxp 5629 ^◡ccnv 5630 dom cdm 5631 “ cima 5634 ∘ ccom 5635 Rel wrel 5636 ⟶wf 6495 –1-1→wf1 6496 –1-1-onto→wf1o 6498 ‘cfv 6499 (class class class)co 7369 1^st c1st 7945 2^nd c2nd 7946 finSupp cfsupp 9288 0cc0 11044 ℕ₀cn0 12418 ℤ_≥cuz 12769 ...cfz 13444 ♯chash 14271 Word cword 14454 ++ cconcat 14511 substr csubstr 14581 prefix cpfx 14611 Basecbs 17155 0_gc0g 17378 Σ_g cgsu 17379 CMndccmn 19694

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-2 12225 df-n0 12419 df-z 12506 df-uz 12770 df-fz 13445 df-fzo 13592 df-seq 13943 df-hash 14272 df-word 14455 df-concat 14512 df-substr 14582 df-pfx 14612 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-0g 17380 df-gsum 17381 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-submnd 18693 df-mulg 18982 df-cntz 19231 df-cmn 19696

This theorem is referenced by: elrgspnlem2 33210

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