Theorem gsummpt2d 33032

Description: Express a finite sum over a two-dimensional range as a double sum. See also gsum2d 20014. (Contributed by Thierry Arnoux, 27-Apr-2020.)

Hypotheses

Ref	Expression
gsummpt2d.c	⊢ Ⅎ𝑧𝐶
gsummpt2d.0	⊢ Ⅎ𝑦𝜑
gsummpt2d.b	⊢ 𝐵 = (Base‘𝑊)
gsummpt2d.1	⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → 𝐶 = 𝐷)
gsummpt2d.r	⊢ (𝜑 → Rel 𝐴)
gsummpt2d.2	⊢ (𝜑 → 𝐴 ∈ Fin)
gsummpt2d.m	⊢ (𝜑 → 𝑊 ∈ CMnd)
gsummpt2d.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)

Assertion

Ref	Expression
gsummpt2d	⊢ (𝜑 → (𝑊 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑊 Σg (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)))))

Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑦,𝐶   𝑥,𝐷   𝑥,𝑊,𝑦   𝜑,𝑥,𝑧

Allowed substitution hints:   𝜑(𝑦)   𝐶(𝑥,𝑧)   𝐷(𝑦,𝑧)   𝑊(𝑧)

Proof of Theorem gsummpt2d

Step	Hyp	Ref	Expression
1		gsummpt2d.b	. . 3 ⊢ 𝐵 = (Base‘𝑊)
2		eqid 2740	. . 3 ⊢ (0g‘𝑊) = (0g‘𝑊)
3		gsummpt2d.m	. . 3 ⊢ (𝜑 → 𝑊 ∈ CMnd)
4		gsummpt2d.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
5	4	dmexd 7943	. . 3 ⊢ (𝜑 → dom 𝐴 ∈ V)
6		gsummpt2d.3	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
7		gsummpt2d.r	. . . 4 ⊢ (𝜑 → Rel 𝐴)
8		1stdm 8081	. . . 4 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → (1st ‘𝑥) ∈ dom 𝐴)
9	7, 8	sylan 579	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (1st ‘𝑥) ∈ dom 𝐴)
10		fo1st 8050	. . . . . 6 ⊢ 1st :V–onto→V
11		fofn 6836	. . . . . 6 ⊢ (1st :V–onto→V → 1st Fn V)
12		dffn5 6980	. . . . . . 7 ⊢ (1st Fn V ↔ 1st = (𝑥 ∈ V ↦ (1st ‘𝑥)))
13	12	biimpi 216	. . . . . 6 ⊢ (1st Fn V → 1st = (𝑥 ∈ V ↦ (1st ‘𝑥)))
14	10, 11, 13	mp2b 10	. . . . 5 ⊢ 1st = (𝑥 ∈ V ↦ (1st ‘𝑥))
15	14	reseq1i 6005	. . . 4 ⊢ (1st ↾ 𝐴) = ((𝑥 ∈ V ↦ (1st ‘𝑥)) ↾ 𝐴)
16		ssv 4033	. . . . 5 ⊢ 𝐴 ⊆ V
17		resmpt 6066	. . . . 5 ⊢ (𝐴 ⊆ V → ((𝑥 ∈ V ↦ (1st ‘𝑥)) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥)))
18	16, 17	ax-mp 5	. . . 4 ⊢ ((𝑥 ∈ V ↦ (1st ‘𝑥)) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))
19	15, 18	eqtri 2768	. . 3 ⊢ (1st ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))
20	1, 2, 3, 4, 5, 6, 9, 19	gsummpt2co 33031	. 2 ⊢ (𝜑 → (𝑊 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑊 Σg (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦}) ↦ 𝐶)))))
21		gsummpt2d.0	. . . 4 ⊢ Ⅎ𝑦𝜑
22	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → 𝑊 ∈ CMnd)
23	4	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → 𝐴 ∈ Fin)
24		imaexg 7953	. . . . . . 7 ⊢ (𝐴 ∈ Fin → (𝐴 “ {𝑦}) ∈ V)
25	23, 24	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (𝐴 “ {𝑦}) ∈ V)
26		gsummpt2d.1	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → 𝐶 = 𝐷)
27	26	adantl 481	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝐶 = 𝐷)
28		simp-4l 782	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝜑)
29		simplr 768	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝑥 ∈ 𝐴)
30	28, 29, 6	syl2anc 583	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝐶 ∈ 𝐵)
31	27, 30	eqeltrrd 2845	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝐷 ∈ 𝐵)
32		vex 3492	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
33		vex 3492	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
34	32, 33	elimasn 6119	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝐴 “ {𝑦}) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴)
35	34	biimpi 216	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝐴 “ {𝑦}) → ⟨𝑦, 𝑧⟩ ∈ 𝐴)
36	35	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → ⟨𝑦, 𝑧⟩ ∈ 𝐴)
37		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝑥 = ⟨𝑦, 𝑧⟩)
38	37	eqeq1d 2742	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → (𝑥 = ⟨𝑦, 𝑧⟩ ↔ ⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑧⟩))
39		eqidd 2741	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → ⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑧⟩)
40	36, 38, 39	rspcedvd 3637	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → ∃𝑥 ∈ 𝐴 𝑥 = ⟨𝑦, 𝑧⟩)
41	31, 40	r19.29a 3168	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → 𝐷 ∈ 𝐵)
42	41	fmpttd 7149	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷):(𝐴 “ {𝑦})⟶𝐵)
43		eqid 2740	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) = (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)
44		imafi2 32725	. . . . . . . . 9 ⊢ (𝐴 ∈ Fin → (𝐴 “ {𝑦}) ∈ Fin)
45	4, 44	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐴 “ {𝑦}) ∈ Fin)
46	45	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (𝐴 “ {𝑦}) ∈ Fin)
47		fvex 6933	. . . . . . . 8 ⊢ (0g‘𝑊) ∈ V
48	47	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (0g‘𝑊) ∈ V)
49	43, 46, 41, 48	fsuppmptdm 9445	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) finSupp (0g‘𝑊))
50		2ndconst 8142	. . . . . . . 8 ⊢ (𝑦 ∈ dom 𝐴 → (2nd ↾ ({𝑦} × (𝐴 “ {𝑦}))):({𝑦} × (𝐴 “ {𝑦}))–1-1-onto→(𝐴 “ {𝑦}))
51	50	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (2nd ↾ ({𝑦} × (𝐴 “ {𝑦}))):({𝑦} × (𝐴 “ {𝑦}))–1-1-onto→(𝐴 “ {𝑦}))
52		1stpreimas 32717	. . . . . . . . . 10 ⊢ ((Rel 𝐴 ∧ 𝑦 ∈ dom 𝐴) → (◡(1st ↾ 𝐴) “ {𝑦}) = ({𝑦} × (𝐴 “ {𝑦})))
53	7, 52	sylan 579	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (◡(1st ↾ 𝐴) “ {𝑦}) = ({𝑦} × (𝐴 “ {𝑦})))
54	53	reseq2d 6009	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (2nd ↾ (◡(1st ↾ 𝐴) “ {𝑦})) = (2nd ↾ ({𝑦} × (𝐴 “ {𝑦}))))
55	54	f1oeq1d 6857	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → ((2nd ↾ (◡(1st ↾ 𝐴) “ {𝑦})):({𝑦} × (𝐴 “ {𝑦}))–1-1-onto→(𝐴 “ {𝑦}) ↔ (2nd ↾ ({𝑦} × (𝐴 “ {𝑦}))):({𝑦} × (𝐴 “ {𝑦}))–1-1-onto→(𝐴 “ {𝑦})))
56	51, 55	mpbird 257	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (2nd ↾ (◡(1st ↾ 𝐴) “ {𝑦})):({𝑦} × (𝐴 “ {𝑦}))–1-1-onto→(𝐴 “ {𝑦}))
57	1, 2, 22, 25, 42, 49, 56	gsumf1o 19958	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)) = (𝑊 Σg ((𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) ∘ (2nd ↾ (◡(1st ↾ 𝐴) “ {𝑦})))))
58		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) → 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦}))
59	53	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) → (◡(1st ↾ 𝐴) “ {𝑦}) = ({𝑦} × (𝐴 “ {𝑦})))
60	58, 59	eleqtrd 2846	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) → 𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})))
61		xp2nd 8063	. . . . . . . . . 10 ⊢ (𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})) → (2nd ‘𝑥) ∈ (𝐴 “ {𝑦}))
62	60, 61	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) → (2nd ‘𝑥) ∈ (𝐴 “ {𝑦}))
63	62	ralrimiva 3152	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → ∀𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})(2nd ‘𝑥) ∈ (𝐴 “ {𝑦}))
64		fo2nd 8051	. . . . . . . . . . . 12 ⊢ 2nd :V–onto→V
65		fofn 6836	. . . . . . . . . . . 12 ⊢ (2nd :V–onto→V → 2nd Fn V)
66		dffn5 6980	. . . . . . . . . . . . 13 ⊢ (2nd Fn V ↔ 2nd = (𝑥 ∈ V ↦ (2nd ‘𝑥)))
67	66	biimpi 216	. . . . . . . . . . . 12 ⊢ (2nd Fn V → 2nd = (𝑥 ∈ V ↦ (2nd ‘𝑥)))
68	64, 65, 67	mp2b 10	. . . . . . . . . . 11 ⊢ 2nd = (𝑥 ∈ V ↦ (2nd ‘𝑥))
69	68	reseq1i 6005	. . . . . . . . . 10 ⊢ (2nd ↾ (◡(1st ↾ 𝐴) “ {𝑦})) = ((𝑥 ∈ V ↦ (2nd ‘𝑥)) ↾ (◡(1st ↾ 𝐴) “ {𝑦}))
70		ssv 4033	. . . . . . . . . . 11 ⊢ (◡(1st ↾ 𝐴) “ {𝑦}) ⊆ V
71		resmpt 6066	. . . . . . . . . . 11 ⊢ ((◡(1st ↾ 𝐴) “ {𝑦}) ⊆ V → ((𝑥 ∈ V ↦ (2nd ‘𝑥)) ↾ (◡(1st ↾ 𝐴) “ {𝑦})) = (𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦}) ↦ (2nd ‘𝑥)))
72	70, 71	ax-mp 5	. . . . . . . . . 10 ⊢ ((𝑥 ∈ V ↦ (2nd ‘𝑥)) ↾ (◡(1st ↾ 𝐴) “ {𝑦})) = (𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦}) ↦ (2nd ‘𝑥))
73	69, 72	eqtri 2768	. . . . . . . . 9 ⊢ (2nd ↾ (◡(1st ↾ 𝐴) “ {𝑦})) = (𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦}) ↦ (2nd ‘𝑥))
74	73	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (2nd ↾ (◡(1st ↾ 𝐴) “ {𝑦})) = (𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦}) ↦ (2nd ‘𝑥)))
75		eqidd 2741	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) = (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷))
76	63, 74, 75	fmptcos 7165	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → ((𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) ∘ (2nd ↾ (◡(1st ↾ 𝐴) “ {𝑦}))) = (𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦}) ↦ ⦋(2nd ‘𝑥) / 𝑧⦌𝐷))
77		nfv 1913	. . . . . . . . 9 ⊢ Ⅎ𝑧((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦}))
78		gsummpt2d.c	. . . . . . . . . 10 ⊢ Ⅎ𝑧𝐶
79	78	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) → Ⅎ𝑧𝐶)
80	60	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) ∧ 𝑧 = (2nd ‘𝑥)) → 𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})))
81		xp1st 8062	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})) → (1st ‘𝑥) ∈ {𝑦})
82	80, 81	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) ∧ 𝑧 = (2nd ‘𝑥)) → (1st ‘𝑥) ∈ {𝑦})
83		fvex 6933	. . . . . . . . . . . . . 14 ⊢ (1st ‘𝑥) ∈ V
84	83	elsn 4663	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑥) ∈ {𝑦} ↔ (1st ‘𝑥) = 𝑦)
85	82, 84	sylib 218	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) ∧ 𝑧 = (2nd ‘𝑥)) → (1st ‘𝑥) = 𝑦)
86		simpr 484	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) ∧ 𝑧 = (2nd ‘𝑥)) → 𝑧 = (2nd ‘𝑥))
87	86	eqcomd 2746	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) ∧ 𝑧 = (2nd ‘𝑥)) → (2nd ‘𝑥) = 𝑧)
88		eqopi 8066	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})) ∧ ((1st ‘𝑥) = 𝑦 ∧ (2nd ‘𝑥) = 𝑧)) → 𝑥 = ⟨𝑦, 𝑧⟩)
89	80, 85, 87, 88	syl12anc 836	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) ∧ 𝑧 = (2nd ‘𝑥)) → 𝑥 = ⟨𝑦, 𝑧⟩)
90	89, 26	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) ∧ 𝑧 = (2nd ‘𝑥)) → 𝐶 = 𝐷)
91	90	eqcomd 2746	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) ∧ 𝑧 = (2nd ‘𝑥)) → 𝐷 = 𝐶)
92	77, 79, 62, 91	csbiedf 3952	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) → ⦋(2nd ‘𝑥) / 𝑧⦌𝐷 = 𝐶)
93	92	mpteq2dva 5266	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦}) ↦ ⦋(2nd ‘𝑥) / 𝑧⦌𝐷) = (𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦}) ↦ 𝐶))
94	76, 93	eqtrd 2780	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → ((𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) ∘ (2nd ↾ (◡(1st ↾ 𝐴) “ {𝑦}))) = (𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦}) ↦ 𝐶))
95	94	oveq2d 7464	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (𝑊 Σg ((𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) ∘ (2nd ↾ (◡(1st ↾ 𝐴) “ {𝑦})))) = (𝑊 Σg (𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦}) ↦ 𝐶)))
96	57, 95	eqtr2d 2781	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (𝑊 Σg (𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦}) ↦ 𝐶)) = (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)))
97	21, 96	mpteq2da 5264	. . 3 ⊢ (𝜑 → (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦}) ↦ 𝐶))) = (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷))))
98	97	oveq2d 7464	. 2 ⊢ (𝜑 → (𝑊 Σg (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦}) ↦ 𝐶)))) = (𝑊 Σg (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)))))
99	20, 98	eqtrd 2780	1 ⊢ (𝜑 → (𝑊 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑊 Σg (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537  Ⅎwnf 1781   ∈ wcel 2108  Ⅎwnfc 2893  Vcvv 3488  ⦋csb 3921   ⊆ wss 3976  {csn 4648  ⟨cop 4654   ↦ cmpt 5249   × cxp 5698  ^◡ccnv 5699  dom cdm 5700   ↾ cres 5702   “ cima 5703   ∘ ccom 5704  Rel wrel 5705   Fn wfn 6568  –onto→wfo 6571  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448  1^st c1st 8028  2^nd c2nd 8029  Fincfn 9003  Basecbs 17258  0_gc0g 17499   Σ_g cgsu 17500  CMndccmn 19822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-om 7904  df-1st 8030  df-2nd 8031  df-supp 8202  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fsupp 9432  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-fzo 13712  df-seq 14053  df-hash 14380  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-0g 17501  df-gsum 17502  df-mre 17644  df-mrc 17645  df-acs 17647  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-submnd 18819  df-mulg 19108  df-cntz 19357  df-cmn 19824

This theorem is referenced by:  gsumhashmul  33040  esum2d  34057