gsumwrd2dccat - Mathbox for Thierry Arnoux

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Theorem gsumwrd2dccat 33144

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 6846	. . . . . . . 8 ⊢ 𝐵 ∈ V
5	4	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V)
6		gsumwrd2dccat.6	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
7	5, 6	ssexd 5259	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V)
8		wrdexg 14448	. . . . . 6 ⊢ (𝐴 ∈ V → Word 𝐴 ∈ V)
9	7, 8	syl 17	. . . . 5 ⊢ (𝜑 → Word 𝐴 ∈ V)
10	9, 9	xpexd 7696	. . . 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 33143	. . . . . . 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 6784	. . . . . 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 6767	. . . . 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 19849	. . 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 5640	. . . . . . . . . . . 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 5763	. . . . . . . . . 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 7984	. . . . . . . . 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 14457	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Word 𝐴 → (♯‘𝑥) ∈ ℕ₀)
32	31	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → (♯‘𝑥) ∈ ℕ₀)
33		nn0uz 12790	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
34	32, 33	eleqtrdi 2847	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → (♯‘𝑥) ∈ (ℤ_≥‘0))
35		fzn0 13455	. . . . . . . . . . 11 ⊢ ((0...(♯‘𝑥)) ≠ ∅ ↔ (♯‘𝑥) ∈ (ℤ_≥‘0))
36	34, 35	sylibr 234	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → (0...(♯‘𝑥)) ≠ ∅)
37	36	dmdju 32709	. . . . . . . . 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 14602	. . . . . . 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 14570	. . . . . . 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 5661	. . . . 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 4578	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → {𝑤} = {𝑥})
46		fveq2 6832	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (♯‘𝑤) = (♯‘𝑥))
47	46	oveq2d 7374	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (0...(♯‘𝑤)) = (0...(♯‘𝑥)))
48	45, 47	xpeq12d 5653	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ({𝑤} × (0...(♯‘𝑤))) = ({𝑥} × (0...(♯‘𝑥))))
49	48	cbviunv 4982	. . . . . . 7 ⊢ ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) = ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))
50	49	mpteq1i 5177	. . . . . 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 6900	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ (𝐹‘𝑎)))
53		fveq2 6832	. . . . 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 7074	. . . 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 7374	. . 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 7077	. . . . 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 6766	. . . . . . . 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 6771	. . . . . . 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 6846	. . . . . . 7 ⊢ 𝑍 ∈ V
67	66	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ V)
68	11, 10	fexd 7173	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V)
69	12, 65, 67, 68	fsuppco 9306	. . . . 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 5110	. . . 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 7029	. . . . 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 7059	. . . 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 5370	. . . . . . . 8 ⊢ {𝑥} ∈ V
75		ovex 7391	. . . . . . . 8 ⊢ (0...(♯‘𝑥)) ∈ V
76	74, 75	xpex 7698	. . . . . . 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 7907	. . . . 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 33127	. . 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 3434	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
85		vex 3434	. . . . . . . . . . 11 ⊢ 𝑗 ∈ V
86	84, 85	op1std 7943	. . . . . . . . . 10 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → (1^st ‘𝑏) = 𝑤)
87	84, 85	op2ndd 7944	. . . . . . . . . 10 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → (2^nd ‘𝑏) = 𝑗)
88	86, 87	oveq12d 7376	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → ((1^st ‘𝑏) prefix (2^nd ‘𝑏)) = (𝑤 prefix 𝑗))
89	86	fveq2d 6836	. . . . . . . . . . 11 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → (♯‘(1^st ‘𝑏)) = (♯‘𝑤))
90	87, 89	opeq12d 4825	. . . . . . . . . 10 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩ = ⟨𝑗, (♯‘𝑤)⟩)
91	86, 90	oveq12d 7376	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩) = (𝑤 substr ⟨𝑗, (♯‘𝑤)⟩))
92	88, 91	opeq12d 4825	. . . . . . . 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 6836	. . . . . . 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 7393	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → (0...(♯‘𝑥)) ∈ V)
98		nfcv 2899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(0...(♯‘𝑤))
99		fveq2 6832	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (♯‘𝑥) = (♯‘𝑤))
100	99	oveq2d 7374	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (0...(♯‘𝑥)) = (0...(♯‘𝑤)))
101	9, 97, 98, 100	iunsnima2 32681	. . . . . . . . . . 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 5785	. . . . . . . 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 6847	. . . . . . 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 6963	. . . . . 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 5179	. . . . 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 7374	. . . 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 5179	. . 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 7374	. 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 5176	. . . . 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 7374	. . . 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 5172	. . 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 7374	. 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 3430 ⊆ wss 3890 ∅c0 4274 {csn 4568 ⟨cop 4574 ∪ ciun 4934 class class class wbr 5086 ↦ cmpt 5167 × cxp 5620 ^◡ccnv 5621 dom cdm 5622 “ cima 5625 ∘ ccom 5626 Rel wrel 5627 ⟶wf 6486 –1-1→wf1 6487 –1-1-onto→wf1o 6489 ‘cfv 6490 (class class class)co 7358 1^st c1st 7931 2^nd c2nd 7932 finSupp cfsupp 9265 0cc0 11027 ℕ₀cn0 12402 ℤ_≥cuz 12752 ...cfz 13424 ♯chash 14254 Word cword 14437 ++ cconcat 14494 substr csubstr 14565 prefix cpfx 14595 Basecbs 17137 0_gc0g 17360 Σ_g cgsu 17361 CMndccmn 19713

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 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104

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 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-er 8634 df-map 8766 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-oi 9416 df-card 9852 df-pnf 11169 df-mnf 11170 df-xr 11171 df-ltxr 11172 df-le 11173 df-sub 11367 df-neg 11368 df-nn 12147 df-2 12209 df-n0 12403 df-z 12490 df-uz 12753 df-fz 13425 df-fzo 13572 df-seq 13926 df-hash 14255 df-word 14438 df-concat 14495 df-substr 14566 df-pfx 14596 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17138 df-ress 17159 df-plusg 17191 df-0g 17362 df-gsum 17363 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18710 df-mulg 19002 df-cntz 19250 df-cmn 19715

This theorem is referenced by: elrgspnlem2 33309

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