Theorem gsummpt2d 33040

Description: Express a finite sum over a two-dimensional range as a double sum. See also gsum2d 19894. (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 2733	. . 3 ⊢ (0g‘𝑊) = (0g‘𝑊)
3		gsummpt2d.m	. . 3 ⊢ (𝜑 → 𝑊 ∈ CMnd)
4		gsummpt2d.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
5	4	dmexd 7842	. . 3 ⊢ (𝜑 → dom 𝐴 ∈ V)
6		gsummpt2d.3	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
7		gsummpt2d.r	. . . 4 ⊢ (𝜑 → Rel 𝐴)
8		1stdm 7981	. . . 4 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → (1st ‘𝑥) ∈ dom 𝐴)
9	7, 8	sylan 580	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (1st ‘𝑥) ∈ dom 𝐴)
10		fo1st 7950	. . . . . 6 ⊢ 1st :V–onto→V
11		fofn 6745	. . . . . 6 ⊢ (1st :V–onto→V → 1st Fn V)
12		dffn5 6889	. . . . . . 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 5931	. . . 4 ⊢ (1st ↾ 𝐴) = ((𝑥 ∈ V ↦ (1st ‘𝑥)) ↾ 𝐴)
16		ssv 3956	. . . . 5 ⊢ 𝐴 ⊆ V
17		resmpt 5993	. . . . 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 2756	. . 3 ⊢ (1st ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))
20	1, 2, 3, 4, 5, 6, 9, 19	gsummpt2co 33039	. 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 7852	. . . . . . 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 584	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝐶 ∈ 𝐵)
31	27, 30	eqeltrrd 2834	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝐷 ∈ 𝐵)
32		vex 3442	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
33		vex 3442	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
34	32, 33	elimasn 6046	. . . . . . . . . . 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 2735	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → (𝑥 = ⟨𝑦, 𝑧⟩ ↔ ⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑧⟩))
39		eqidd 2734	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → ⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑧⟩)
40	36, 38, 39	rspcedvd 3576	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → ∃𝑥 ∈ 𝐴 𝑥 = ⟨𝑦, 𝑧⟩)
41	31, 40	r19.29a 3142	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → 𝐷 ∈ 𝐵)
42	41	fmpttd 7057	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷):(𝐴 “ {𝑦})⟶𝐵)
43		eqid 2733	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) = (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)
44		imafi2 32704	. . . . . . . . 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 6844	. . . . . . . 8 ⊢ (0g‘𝑊) ∈ V
48	47	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (0g‘𝑊) ∈ V)
49	43, 46, 41, 48	fsuppmptdm 9270	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) finSupp (0g‘𝑊))
50		2ndconst 8040	. . . . . . . 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 32698	. . . . . . . . . 10 ⊢ ((Rel 𝐴 ∧ 𝑦 ∈ dom 𝐴) → (◡(1st ↾ 𝐴) “ {𝑦}) = ({𝑦} × (𝐴 “ {𝑦})))
53	7, 52	sylan 580	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (◡(1st ↾ 𝐴) “ {𝑦}) = ({𝑦} × (𝐴 “ {𝑦})))
54	53	reseq2d 5935	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (2nd ↾ (◡(1st ↾ 𝐴) “ {𝑦})) = (2nd ↾ ({𝑦} × (𝐴 “ {𝑦}))))
55	54	f1oeq1d 6766	. . . . . . 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 19838	. . . . 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 2835	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) → 𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})))
61		xp2nd 7963	. . . . . . . . . 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 3126	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → ∀𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})(2nd ‘𝑥) ∈ (𝐴 “ {𝑦}))
64		fo2nd 7951	. . . . . . . . . . . 12 ⊢ 2nd :V–onto→V
65		fofn 6745	. . . . . . . . . . . 12 ⊢ (2nd :V–onto→V → 2nd Fn V)
66		dffn5 6889	. . . . . . . . . . . . 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 5931	. . . . . . . . . 10 ⊢ (2nd ↾ (◡(1st ↾ 𝐴) “ {𝑦})) = ((𝑥 ∈ V ↦ (2nd ‘𝑥)) ↾ (◡(1st ↾ 𝐴) “ {𝑦}))
70		ssv 3956	. . . . . . . . . . 11 ⊢ (◡(1st ↾ 𝐴) “ {𝑦}) ⊆ V
71		resmpt 5993	. . . . . . . . . . 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 2756	. . . . . . . . 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 2734	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) = (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷))
76	63, 74, 75	fmptcos 7073	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → ((𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) ∘ (2nd ↾ (◡(1st ↾ 𝐴) “ {𝑦}))) = (𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦}) ↦ ⦋(2nd ‘𝑥) / 𝑧⦌𝐷))
77		nfv 1915	. . . . . . . . 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 7962	. . . . . . . . . . . . . 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 6844	. . . . . . . . . . . . . 14 ⊢ (1st ‘𝑥) ∈ V
84	83	elsn 4592	. . . . . . . . . . . . 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 2739	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) ∧ 𝑧 = (2nd ‘𝑥)) → (2nd ‘𝑥) = 𝑧)
88		eqopi 7966	. . . . . . . . . . . 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 2739	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) ∧ 𝑧 = (2nd ‘𝑥)) → 𝐷 = 𝐶)
92	77, 79, 62, 91	csbiedf 3877	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) → ⦋(2nd ‘𝑥) / 𝑧⦌𝐷 = 𝐶)
93	92	mpteq2dva 5188	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦}) ↦ ⦋(2nd ‘𝑥) / 𝑧⦌𝐷) = (𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦}) ↦ 𝐶))
94	76, 93	eqtrd 2768	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → ((𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) ∘ (2nd ↾ (◡(1st ↾ 𝐴) “ {𝑦}))) = (𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦}) ↦ 𝐶))
95	94	oveq2d 7371	. . . . 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 2769	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (𝑊 Σg (𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦}) ↦ 𝐶)) = (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)))
97	21, 96	mpteq2da 5187	. . 3 ⊢ (𝜑 → (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦}) ↦ 𝐶))) = (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷))))
98	97	oveq2d 7371	. 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 2768	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 1541  Ⅎwnf 1784   ∈ wcel 2113  Ⅎwnfc 2881  Vcvv 3438  ⦋csb 3847   ⊆ wss 3899  {csn 4577  ⟨cop 4583   ↦ cmpt 5176   × cxp 5619  ^◡ccnv 5620  dom cdm 5621   ↾ cres 5623   “ cima 5624   ∘ ccom 5625  Rel wrel 5626   Fn wfn 6484  –onto→wfo 6487  –1-1-onto→wf1o 6488  ‘cfv 6489  (class class class)co 7355  1^st c1st 7928  2^nd c2nd 7929  Fincfn 8878  Basecbs 17130  0_gc0g 17353   Σ_g cgsu 17354  CMndccmn 19702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11072  ax-resscn 11073  ax-1cn 11074  ax-icn 11075  ax-addcl 11076  ax-addrcl 11077  ax-mulcl 11078  ax-mulrcl 11079  ax-mulcom 11080  ax-addass 11081  ax-mulass 11082  ax-distr 11083  ax-i2m1 11084  ax-1ne0 11085  ax-1rid 11086  ax-rnegex 11087  ax-rrecex 11088  ax-cnre 11089  ax-pre-lttri 11090  ax-pre-lttrn 11091  ax-pre-ltadd 11092  ax-pre-mulgt0 11093

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-iin 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-isom 6498  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-of 7619  df-om 7806  df-1st 7930  df-2nd 7931  df-supp 8100  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-2o 8395  df-er 8631  df-en 8879  df-dom 8880  df-sdom 8881  df-fin 8882  df-fsupp 9256  df-oi 9406  df-card 9842  df-pnf 11158  df-mnf 11159  df-xr 11160  df-ltxr 11161  df-le 11162  df-sub 11356  df-neg 11357  df-nn 12136  df-2 12198  df-n0 12392  df-z 12479  df-uz 12743  df-fz 13418  df-fzo 13565  df-seq 13919  df-hash 14248  df-sets 17085  df-slot 17103  df-ndx 17115  df-base 17131  df-ress 17152  df-plusg 17184  df-0g 17355  df-gsum 17356  df-mre 17498  df-mrc 17499  df-acs 17501  df-mgm 18558  df-sgrp 18637  df-mnd 18653  df-submnd 18702  df-mulg 18991  df-cntz 19239  df-cmn 19704

This theorem is referenced by:  gsumfs2d  33046  gsumhashmul  33052  esum2d  34117