gsumwrd2dccat - Mathbox for Thierry Arnoux

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Theorem gsumwrd2dccat 33160

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 6842	. . . . . . . 8 ⊢ 𝐵 ∈ V
5	4	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V)
6		gsumwrd2dccat.6	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
7	5, 6	ssexd 5253	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V)
8		wrdexg 14478	. . . . . 6 ⊢ (𝐴 ∈ V → Word 𝐴 ∈ V)
9	7, 8	syl 17	. . . . 5 ⊢ (𝜑 → Word 𝐴 ∈ V)
10	9, 9	xpexd 7695	. . . 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 2739	. . . . . . . 8 ⊢ ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) = ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤)))
14		eqid 2739	. . . . . . . 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 2739	. . . . . . . 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 33159	. . . . . . 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 495	. . . . . 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 6780	. . . . . 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 496	. . . . . 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 6763	. . . . 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 233	. . . 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 19883	. . 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 5637	. . . . . . . . . . . 12 ⊢ Rel ({𝑥} × (0...(♯‘𝑥)))
25	24	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → Rel ({𝑥} × (0...(♯‘𝑥))))
26	25	ralrimiva 3131	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ Word 𝐴Rel ({𝑥} × (0...(♯‘𝑥))))
27		reliun 5760	. . . . . . . . . 10 ⊢ (Rel ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↔ ∀𝑥 ∈ Word 𝐴Rel ({𝑥} × (0...(♯‘𝑥))))
28	26, 27	sylibr 235	. . . . . . . . 9 ⊢ (𝜑 → Rel ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))))
29		1stdm 7983	. . . . . . . . 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 586	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (1^st ‘𝑏) ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))))
31		lencl 14487	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Word 𝐴 → (♯‘𝑥) ∈ ℕ₀)
32	31	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → (♯‘𝑥) ∈ ℕ₀)
33		nn0uz 12818	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
34	32, 33	eleqtrdi 2849	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → (♯‘𝑥) ∈ (ℤ_≥‘0))
35		fzn0 13484	. . . . . . . . . . 11 ⊢ ((0...(♯‘𝑥)) ≠ ∅ ↔ (♯‘𝑥) ∈ (ℤ_≥‘0))
36	34, 35	sylibr 235	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → (0...(♯‘𝑥)) ≠ ∅)
37	36	dmdju 32740	. . . . . . . . 9 ⊢ (𝜑 → dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) = Word 𝐴)
38	37	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) = Word 𝐴)
39	30, 38	eleqtrd 2841	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (1^st ‘𝑏) ∈ Word 𝐴)
40		pfxcl 14632	. . . . . . 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 14600	. . . . . . 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 5658	. . . . 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 4566	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → {𝑤} = {𝑥})
46		fveq2 6828	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (♯‘𝑤) = (♯‘𝑥))
47	46	oveq2d 7373	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (0...(♯‘𝑤)) = (0...(♯‘𝑥)))
48	45, 47	xpeq12d 5650	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ({𝑤} × (0...(♯‘𝑤))) = ({𝑥} × (0...(♯‘𝑥))))
49	48	cbviunv 4969	. . . . . . 7 ⊢ ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) = ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))
50	49	mpteq1i 5164	. . . . . 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 6896	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ (𝐹‘𝑎)))
53		fveq2 6828	. . . . 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 7072	. . . 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 7373	. . 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 1921	. . . 4 ⊢ Ⅎ𝑤𝜑
57	11, 44	cofmpt 7075	. . . . 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 2775	. . . . . . . . 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 2748	. . . . . . . . . 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 2740	. . . . . . . . 9 ⊢ (𝜑 → (Word 𝐴 × Word 𝐴) = (Word 𝐴 × Word 𝐴))
62	58, 60, 61	f1oeq123d 6762	. . . . . . . 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 258	. . . . . . 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 6767	. . . . . . 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 6842	. . . . . . 7 ⊢ 𝑍 ∈ V
67	66	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ V)
68	11, 10	fexd 7172	. . . . . 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 5097	. . . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → 𝐹:(Word 𝐴 × Word 𝐴)⟶𝐵)
72	71, 44	ffvelcdmd 7027	. . . . 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 7057	. . . 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 5365	. . . . . . . 8 ⊢ {𝑥} ∈ V
75		ovex 7390	. . . . . . . 8 ⊢ (0...(♯‘𝑥)) ∈ V
76	74, 75	xpex 7697	. . . . . . 7 ⊢ ({𝑥} × (0...(♯‘𝑥))) ∈ V
77	76	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → ({𝑥} × (0...(♯‘𝑥))) ∈ V)
78	77	ralrimiva 3131	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ∈ V)
79		iunexg 7906	. . . . 5 ⊢ ((Word 𝐴 ∈ V ∧ ∀𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ∈ V) → ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ∈ V)
80	9, 78, 79	syl2anc 590	. . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ∈ V)
81	56, 1, 2, 28, 70, 3, 73, 80	gsumfs2d 33143	. . 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 2778	. 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 2739	. . . . . . 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 3435	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
85		vex 3435	. . . . . . . . . . 11 ⊢ 𝑗 ∈ V
86	84, 85	op1std 7942	. . . . . . . . . 10 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → (1^st ‘𝑏) = 𝑤)
87	84, 85	op2ndd 7943	. . . . . . . . . 10 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → (2^nd ‘𝑏) = 𝑗)
88	86, 87	oveq12d 7375	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → ((1^st ‘𝑏) prefix (2^nd ‘𝑏)) = (𝑤 prefix 𝑗))
89	86	fveq2d 6832	. . . . . . . . . . 11 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → (♯‘(1^st ‘𝑏)) = (♯‘𝑤))
90	87, 89	opeq12d 4813	. . . . . . . . . 10 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩ = ⟨𝑗, (♯‘𝑤)⟩)
91	86, 90	oveq12d 7375	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩) = (𝑤 substr ⟨𝑗, (♯‘𝑤)⟩))
92	88, 91	opeq12d 4813	. . . . . . . 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 6832	. . . . . . 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 2825	. . . . . . . . . 10 ⊢ (𝜑 → (𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↔ 𝑤 ∈ Word 𝐴))
95	94	biimpa 477	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → 𝑤 ∈ Word 𝐴)
96	95	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) ∧ 𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤})) → 𝑤 ∈ Word 𝐴)
97		ovexd 7392	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → (0...(♯‘𝑥)) ∈ V)
98		nfcv 2901	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(0...(♯‘𝑤))
99		fveq2 6828	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (♯‘𝑥) = (♯‘𝑤))
100	99	oveq2d 7373	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (0...(♯‘𝑥)) = (0...(♯‘𝑤)))
101	9, 97, 98, 100	iunsnima2 32712	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ Word 𝐴) → (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) = (0...(♯‘𝑤)))
102	95, 101	syldan 597	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) = (0...(♯‘𝑤)))
103	102	eleq2d 2825	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↔ 𝑗 ∈ (0...(♯‘𝑤))))
104	103	biimpa 477	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) ∧ 𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤})) → 𝑗 ∈ (0...(♯‘𝑤)))
105	100	opeliunxp2 5781	. . . . . . . 8 ⊢ (⟨𝑤, 𝑗⟩ ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↔ (𝑤 ∈ Word 𝐴 ∧ 𝑗 ∈ (0...(♯‘𝑤))))
106	96, 104, 105	sylanbrc 589	. . . . . . 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 6843	. . . . . . 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 6960	. . . . . 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 5166	. . . . 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 7373	. . . 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 5166	. . 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 7373	. 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 5163	. . . . 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 7373	. . . 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 5159	. . 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 7373	. 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 2778	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 396 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ∀wral 3053 Vcvv 3431 ⊆ wss 3883 ∅c0 4262 {csn 4556 ⟨cop 4562 ∪ ciun 4922 class class class wbr 5073 ↦ cmpt 5154 × cxp 5617 ^◡ccnv 5618 dom cdm 5619 “ cima 5622 ∘ ccom 5623 Rel wrel 5624 ⟶wf 6482 –1-1→wf1 6483 –1-1-onto→wf1o 6485 ‘cfv 6486 (class class class)co 7357 1^st c1st 7930 2^nd c2nd 7931 finSupp cfsupp 9265 0cc0 11030 ℕ₀cn0 12429 ℤ_≥cuz 12780 ...cfz 13453 ♯chash 14284 Word cword 14467 ++ cconcat 14524 substr csubstr 14595 prefix cpfx 14625 Basecbs 17171 0_gc0g 17394 Σ_g cgsu 17395 CMndccmn 19747

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5200 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4840 df-int 4879 df-iun 4924 df-iin 4925 df-br 5074 df-opab 5136 df-mpt 5155 df-tr 5181 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7314 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7621 df-om 7808 df-1st 7932 df-2nd 7933 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 9855 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-nn 12167 df-2 12236 df-n0 12430 df-z 12517 df-uz 12781 df-fz 13454 df-fzo 13601 df-seq 13956 df-hash 14285 df-word 14468 df-concat 14525 df-substr 14596 df-pfx 14626 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17172 df-ress 17193 df-plusg 17225 df-0g 17396 df-gsum 17397 df-mre 17540 df-mrc 17541 df-acs 17543 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18744 df-mulg 19036 df-cntz 19284 df-cmn 19749

This theorem is referenced by: elrgspnlem2 33325

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