gsumwrd2dccat - Mathbox for Thierry Arnoux

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Theorem gsumwrd2dccat 33178

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 6858	. . . . . . . 8 ⊢ 𝐵 ∈ V
5	4	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V)
6		gsumwrd2dccat.6	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
7	5, 6	ssexd 5273	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V)
8		wrdexg 14461	. . . . . 6 ⊢ (𝐴 ∈ V → Word 𝐴 ∈ V)
9	7, 8	syl 17	. . . . 5 ⊢ (𝜑 → Word 𝐴 ∈ V)
10	9, 9	xpexd 7708	. . . 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) ↦ ⟨((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 2737	. . . . . . . 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 33177	. . . . . . 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 6796	. . . . . 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 6779	. . . . 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 19862	. . 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 5652	. . . . . . . . . . . 12 ⊢ Rel ({𝑥} × (0...(♯‘𝑥)))
25	24	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → Rel ({𝑥} × (0...(♯‘𝑥))))
26	25	ralrimiva 3130	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ Word 𝐴Rel ({𝑥} × (0...(♯‘𝑥))))
27		reliun 5775	. . . . . . . . . 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 7996	. . . . . . . . 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 581	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (1^st ‘𝑏) ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))))
31		lencl 14470	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Word 𝐴 → (♯‘𝑥) ∈ ℕ₀)
32	31	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → (♯‘𝑥) ∈ ℕ₀)
33		nn0uz 12803	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
34	32, 33	eleqtrdi 2847	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → (♯‘𝑥) ∈ (ℤ_≥‘0))
35		fzn0 13468	. . . . . . . . . . 11 ⊢ ((0...(♯‘𝑥)) ≠ ∅ ↔ (♯‘𝑥) ∈ (ℤ_≥‘0))
36	34, 35	sylibr 234	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → (0...(♯‘𝑥)) ≠ ∅)
37	36	dmdju 32743	. . . . . . . . 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 2839	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (1^st ‘𝑏) ∈ Word 𝐴)
40		pfxcl 14615	. . . . . . 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 14583	. . . . . . 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 5673	. . . . 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 4592	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → {𝑤} = {𝑥})
46		fveq2 6844	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (♯‘𝑤) = (♯‘𝑥))
47	46	oveq2d 7386	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (0...(♯‘𝑤)) = (0...(♯‘𝑥)))
48	45, 47	xpeq12d 5665	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ({𝑤} × (0...(♯‘𝑤))) = ({𝑥} × (0...(♯‘𝑥))))
49	48	cbviunv 4996	. . . . . . 7 ⊢ ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) = ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))
50	49	mpteq1i 5191	. . . . . 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 6912	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ (𝐹‘𝑎)))
53		fveq2 6844	. . . . 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 7086	. . . 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 7386	. . 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 1916	. . . 4 ⊢ Ⅎ𝑤𝜑
57	11, 44	cofmpt 7089	. . . . 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 2773	. . . . . . . . 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 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 6778	. . . . . . . 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 6783	. . . . . . 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 6858	. . . . . . 7 ⊢ 𝑍 ∈ V
67	66	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ V)
68	11, 10	fexd 7185	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V)
69	12, 65, 67, 68	fsuppco 9319	. . . . 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 5124	. . . 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 7041	. . . . 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 7071	. . . 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 5383	. . . . . . . 8 ⊢ {𝑥} ∈ V
75		ovex 7403	. . . . . . . 8 ⊢ (0...(♯‘𝑥)) ∈ V
76	74, 75	xpex 7710	. . . . . . 7 ⊢ ({𝑥} × (0...(♯‘𝑥))) ∈ V
77	76	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → ({𝑥} × (0...(♯‘𝑥))) ∈ V)
78	77	ralrimiva 3130	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ∈ V)
79		iunexg 7919	. . . . 5 ⊢ ((Word 𝐴 ∈ V ∧ ∀𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ∈ V) → ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ∈ V)
80	9, 78, 79	syl2anc 585	. . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ∈ V)
81	56, 1, 2, 28, 70, 3, 73, 80	gsumfs2d 33161	. . 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 2776	. 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 2737	. . . . . . 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 3446	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
85		vex 3446	. . . . . . . . . . 11 ⊢ 𝑗 ∈ V
86	84, 85	op1std 7955	. . . . . . . . . 10 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → (1^st ‘𝑏) = 𝑤)
87	84, 85	op2ndd 7956	. . . . . . . . . 10 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → (2^nd ‘𝑏) = 𝑗)
88	86, 87	oveq12d 7388	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → ((1^st ‘𝑏) prefix (2^nd ‘𝑏)) = (𝑤 prefix 𝑗))
89	86	fveq2d 6848	. . . . . . . . . . 11 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → (♯‘(1^st ‘𝑏)) = (♯‘𝑤))
90	87, 89	opeq12d 4839	. . . . . . . . . 10 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩ = ⟨𝑗, (♯‘𝑤)⟩)
91	86, 90	oveq12d 7388	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩) = (𝑤 substr ⟨𝑗, (♯‘𝑤)⟩))
92	88, 91	opeq12d 4839	. . . . . . . 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 6848	. . . . . . 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 2823	. . . . . . . . . 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 7405	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → (0...(♯‘𝑥)) ∈ V)
98		nfcv 2899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(0...(♯‘𝑤))
99		fveq2 6844	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (♯‘𝑥) = (♯‘𝑤))
100	99	oveq2d 7386	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (0...(♯‘𝑥)) = (0...(♯‘𝑤)))
101	9, 97, 98, 100	iunsnima2 32715	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ Word 𝐴) → (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) = (0...(♯‘𝑤)))
102	95, 101	syldan 592	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) = (0...(♯‘𝑤)))
103	102	eleq2d 2823	. . . . . . . . 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 5797	. . . . . . . 8 ⊢ (⟨𝑤, 𝑗⟩ ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↔ (𝑤 ∈ Word 𝐴 ∧ 𝑗 ∈ (0...(♯‘𝑤))))
106	96, 104, 105	sylanbrc 584	. . . . . . 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 6859	. . . . . . 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 6975	. . . . . 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 5193	. . . . 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 7386	. . . 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 5193	. . 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 7386	. 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 5190	. . . . 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 7386	. . . 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 5186	. . 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 7386	. 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 2776	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 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 Vcvv 3442 ⊆ wss 3903 ∅c0 4287 {csn 4582 ⟨cop 4588 ∪ ciun 4948 class class class wbr 5100 ↦ cmpt 5181 × cxp 5632 ^◡ccnv 5633 dom cdm 5634 “ cima 5637 ∘ ccom 5638 Rel wrel 5639 ⟶wf 6498 –1-1→wf1 6499 –1-1-onto→wf1o 6501 ‘cfv 6502 (class class class)co 7370 1^st c1st 7943 2^nd c2nd 7944 finSupp cfsupp 9278 0cc0 11040 ℕ₀cn0 12415 ℤ_≥cuz 12765 ...cfz 13437 ♯chash 14267 Word cword 14450 ++ cconcat 14507 substr csubstr 14578 prefix cpfx 14608 Basecbs 17150 0_gc0g 17373 Σ_g cgsu 17374 CMndccmn 19726

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5529 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5587 df-se 5588 df-we 5589 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6269 df-ord 6330 df-on 6331 df-lim 6332 df-suc 6333 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-isom 6511 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-of 7634 df-om 7821 df-1st 7945 df-2nd 7946 df-supp 8115 df-frecs 8235 df-wrecs 8266 df-recs 8315 df-rdg 8353 df-1o 8409 df-2o 8410 df-er 8647 df-map 8779 df-en 8898 df-dom 8899 df-sdom 8900 df-fin 8901 df-fsupp 9279 df-oi 9429 df-card 9865 df-pnf 11182 df-mnf 11183 df-xr 11184 df-ltxr 11185 df-le 11186 df-sub 11380 df-neg 11381 df-nn 12160 df-2 12222 df-n0 12416 df-z 12503 df-uz 12766 df-fz 13438 df-fzo 13585 df-seq 13939 df-hash 14268 df-word 14451 df-concat 14508 df-substr 14579 df-pfx 14609 df-sets 17105 df-slot 17123 df-ndx 17135 df-base 17151 df-ress 17172 df-plusg 17204 df-0g 17375 df-gsum 17376 df-mre 17519 df-mrc 17520 df-acs 17522 df-mgm 18579 df-sgrp 18658 df-mnd 18674 df-submnd 18723 df-mulg 19015 df-cntz 19263 df-cmn 19728

This theorem is referenced by: elrgspnlem2 33343

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