gsumwrd2dccat - Mathbox for Thierry Arnoux

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Theorem gsumwrd2dccat 33339

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 6896	. . . . . . . 8 ⊢ 𝐵 ∈ V
5	4	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V)
6		gsumwrd2dccat.6	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
7	5, 6	ssexd 5295	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V)
8		wrdexg 14561	. . . . . 6 ⊢ (𝐴 ∈ V → Word 𝐴 ∈ V)
9	7, 8	syl 18	. . . . 5 ⊢ (𝜑 → Word 𝐴 ∈ V)
10	9, 9	xpexd 7750	. . . 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 2769	. . . . . . . 8 ⊢ ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) = ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤)))
14		eqid 2769	. . . . . . . 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 2769	. . . . . . . 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 33338	. . . . . . 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 499	. . . . . 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 6834	. . . . . 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 18	. . . . 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 500	. . . . . 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 6816	. . . . 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 235	. . . 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 19986	. . 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 5680	. . . . . . . . . . . 12 ⊢ Rel ({𝑥} × (0...(♯‘𝑥)))
25	24	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → Rel ({𝑥} × (0...(♯‘𝑥))))
26	25	ralrimiva 3163	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ Word 𝐴Rel ({𝑥} × (0...(♯‘𝑥))))
27		reliun 5804	. . . . . . . . . 10 ⊢ (Rel ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↔ ∀𝑥 ∈ Word 𝐴Rel ({𝑥} × (0...(♯‘𝑥))))
28	26, 27	sylibr 237	. . . . . . . . 9 ⊢ (𝜑 → Rel ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))))
29		1stdm 8037	. . . . . . . . 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 591	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (1^st ‘𝑏) ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))))
31		lencl 14570	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Word 𝐴 → (♯‘𝑥) ∈ ℕ₀)
32	31	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → (♯‘𝑥) ∈ ℕ₀)
33		nn0uz 12900	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
34	32, 33	eleqtrdi 2879	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → (♯‘𝑥) ∈ (ℤ_≥‘0))
35		fzn0 13566	. . . . . . . . . . 11 ⊢ ((0...(♯‘𝑥)) ≠ ∅ ↔ (♯‘𝑥) ∈ (ℤ_≥‘0))
36	34, 35	sylibr 237	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → (0...(♯‘𝑥)) ≠ ∅)
37	36	dmdju 32933	. . . . . . . . 9 ⊢ (𝜑 → dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) = Word 𝐴)
38	37	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) = Word 𝐴)
39	30, 38	eleqtrd 2871	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (1^st ‘𝑏) ∈ Word 𝐴)
40		pfxcl 14715	. . . . . . 7 ⊢ ((1^st ‘𝑏) ∈ Word 𝐴 → ((1^st ‘𝑏) prefix (2^nd ‘𝑏)) ∈ Word 𝐴)
41	39, 40	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → ((1^st ‘𝑏) prefix (2^nd ‘𝑏)) ∈ Word 𝐴)
42		swrdcl 14683	. . . . . . 7 ⊢ ((1^st ‘𝑏) ∈ Word 𝐴 → ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩) ∈ Word 𝐴)
43	39, 42	syl 18	. . . . . 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 5701	. . . . 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 4604	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → {𝑤} = {𝑥})
46		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (♯‘𝑤) = (♯‘𝑥))
47	46	oveq2d 7427	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (0...(♯‘𝑤)) = (0...(♯‘𝑥)))
48	45, 47	xpeq12d 5693	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ({𝑤} × (0...(♯‘𝑤))) = ({𝑥} × (0...(♯‘𝑥))))
49	48	cbviunv 5007	. . . . . . 7 ⊢ ∪ 𝑤 ∈ Word 𝐴({𝑤} × (0...(♯‘𝑤))) = ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))
50	49	mpteq1i 5206	. . . . . 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 6950	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑎 ∈ (Word 𝐴 × Word 𝐴) ↦ (𝐹‘𝑎)))
53		fveq2 6882	. . . . 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 7126	. . . 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 7427	. . 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 1941	. . . 4 ⊢ Ⅎ𝑤𝜑
57	11, 44	cofmpt 7129	. . . . 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 2805	. . . . . . . . 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 2778	. . . . . . . . . 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 2770	. . . . . . . . 9 ⊢ (𝜑 → (Word 𝐴 × Word 𝐴) = (Word 𝐴 × Word 𝐴))
62	58, 60, 61	f1oeq123d 6815	. . . . . . . 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 260	. . . . . . 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 6820	. . . . . . 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 18	. . . . . 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 6896	. . . . . . 7 ⊢ 𝑍 ∈ V
67	66	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ V)
68	11, 10	fexd 7226	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V)
69	12, 65, 67, 68	fsuppco 9362	. . . . 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 5139	. . . 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 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → 𝐹:(Word 𝐴 × Word 𝐴)⟶𝐵)
72	71, 44	ffvelcdmd 7081	. . . . 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 7111	. . . 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 5407	. . . . . . . 8 ⊢ {𝑥} ∈ V
75		ovex 7444	. . . . . . . 8 ⊢ (0...(♯‘𝑥)) ∈ V
76	74, 75	xpex 7752	. . . . . . 7 ⊢ ({𝑥} × (0...(♯‘𝑥))) ∈ V
77	76	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → ({𝑥} × (0...(♯‘𝑥))) ∈ V)
78	77	ralrimiva 3163	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ∈ V)
79		iunexg 7960	. . . . 5 ⊢ ((Word 𝐴 ∈ V ∧ ∀𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ∈ V) → ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ∈ V)
80	9, 78, 79	syl2anc 595	. . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ∈ V)
81	56, 1, 2, 28, 70, 3, 73, 80	gsumfs2d 33322	. . 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 2808	. 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 2769	. . . . . . 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 3467	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
85		vex 3467	. . . . . . . . . . 11 ⊢ 𝑗 ∈ V
86	84, 85	op1std 7996	. . . . . . . . . 10 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → (1^st ‘𝑏) = 𝑤)
87	84, 85	op2ndd 7997	. . . . . . . . . 10 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → (2^nd ‘𝑏) = 𝑗)
88	86, 87	oveq12d 7429	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → ((1^st ‘𝑏) prefix (2^nd ‘𝑏)) = (𝑤 prefix 𝑗))
89	86	fveq2d 6886	. . . . . . . . . . 11 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → (♯‘(1^st ‘𝑏)) = (♯‘𝑤))
90	87, 89	opeq12d 4850	. . . . . . . . . 10 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩ = ⟨𝑗, (♯‘𝑤)⟩)
91	86, 90	oveq12d 7429	. . . . . . . . 9 ⊢ (𝑏 = ⟨𝑤, 𝑗⟩ → ((1^st ‘𝑏) substr ⟨(2^nd ‘𝑏), (♯‘(1^st ‘𝑏))⟩) = (𝑤 substr ⟨𝑗, (♯‘𝑤)⟩))
92	88, 91	opeq12d 4850	. . . . . . . 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 6886	. . . . . . 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 2855	. . . . . . . . . 10 ⊢ (𝜑 → (𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↔ 𝑤 ∈ Word 𝐴))
95	94	biimpa 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → 𝑤 ∈ Word 𝐴)
96	95	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) ∧ 𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤})) → 𝑤 ∈ Word 𝐴)
97		ovexd 7446	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐴) → (0...(♯‘𝑥)) ∈ V)
98		nfcv 2931	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(0...(♯‘𝑤))
99		fveq2 6882	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (♯‘𝑥) = (♯‘𝑤))
100	99	oveq2d 7427	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (0...(♯‘𝑥)) = (0...(♯‘𝑤)))
101	9, 97, 98, 100	iunsnima2 32905	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ Word 𝐴) → (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) = (0...(♯‘𝑤)))
102	95, 101	syldan 602	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) = (0...(♯‘𝑤)))
103	102	eleq2d 2855	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) → (𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤}) ↔ 𝑗 ∈ (0...(♯‘𝑤))))
104	103	biimpa 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ dom ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥)))) ∧ 𝑗 ∈ (∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) “ {𝑤})) → 𝑗 ∈ (0...(♯‘𝑤)))
105	100	opeliunxp2 5825	. . . . . . . 8 ⊢ (⟨𝑤, 𝑗⟩ ∈ ∪ 𝑥 ∈ Word 𝐴({𝑥} × (0...(♯‘𝑥))) ↔ (𝑤 ∈ Word 𝐴 ∧ 𝑗 ∈ (0...(♯‘𝑤))))
106	96, 104, 105	sylanbrc 594	. . . . . . 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 6897	. . . . . . 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 7014	. . . . . 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 5208	. . . . 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 7427	. . . 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 5208	. . 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 7427	. 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 5205	. . . . 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 7427	. . . 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 5201	. . 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 7427	. 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 2808	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 400 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∀wral 3085 Vcvv 3463 ⊆ wss 3913 ∅c0 4294 {csn 4594 ⟨cop 4600 ∪ ciun 4960 class class class wbr 5113 ↦ cmpt 5196 × cxp 5660 ^◡ccnv 5661 dom cdm 5662 “ cima 5665 ∘ ccom 5666 Rel wrel 5667 ⟶wf 6533 –1-1→wf1 6534 –1-1-onto→wf1o 6536 ‘cfv 6537 (class class class)co 7411 1^st c1st 7984 2^nd c2nd 7985 finSupp cfsupp 9321 0cc0 11100 ℕ₀cn0 12504 ℤ_≥cuz 12862 ...cfz 13535 ♯chash 14366 Word cword 14550 ++ cconcat 14607 substr csubstr 14678 prefix cpfx 14708 Basecbs 17269 0_gc0g 17492 Σ_g cgsu 17493 CMndccmn 19850

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8157 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-er 8694 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9322 df-oi 9472 df-card 9925 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-nn 12234 df-2 12303 df-n0 12505 df-z 12592 df-uz 12863 df-fz 13536 df-fzo 13683 df-seq 14038 df-hash 14367 df-word 14551 df-concat 14608 df-substr 14679 df-pfx 14709 df-sets 17224 df-slot 17242 df-ndx 17254 df-base 17270 df-ress 17291 df-plusg 17323 df-0g 17494 df-gsum 17495 df-mre 17638 df-mrc 17639 df-acs 17641 df-mgm 18698 df-sgrp 18777 df-mnd 18793 df-submnd 18842 df-mulg 19134 df-cntz 19387 df-cmn 19852

This theorem is referenced by: elrgspnlem2 33504

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