Theorem gsummpt2d 33282

Description: Express a finite sum over a two-dimensional range as a double sum. See also gsum2d 20033. (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 2765	. . 3 ⊢ (0g‘𝑊) = (0g‘𝑊)
3		gsummpt2d.m	. . 3 ⊢ (𝜑 → 𝑊 ∈ CMnd)
4		gsummpt2d.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
5	4	dmexd 7888	. . 3 ⊢ (𝜑 → dom 𝐴 ∈ V)
6		gsummpt2d.3	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
7		gsummpt2d.r	. . . 4 ⊢ (𝜑 → Rel 𝐴)
8		1stdm 8025	. . . 4 ⊢ ((Rel 𝐴 ∧ 𝑥 ∈ 𝐴) → (1st ‘𝑥) ∈ dom 𝐴)
9	7, 8	sylan 591	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (1st ‘𝑥) ∈ dom 𝐴)
10		fo1st 7994	. . . . . 6 ⊢ 1st :V–onto→V
11		fofn 6784	. . . . . 6 ⊢ (1st :V–onto→V → 1st Fn V)
12		dffn5 6929	. . . . . . 7 ⊢ (1st Fn V ↔ 1st = (𝑥 ∈ V ↦ (1st ‘𝑥)))
13	12	biimpi 219	. . . . . 6 ⊢ (1st Fn V → 1st = (𝑥 ∈ V ↦ (1st ‘𝑥)))
14	10, 11, 13	mp2b 10	. . . . 5 ⊢ 1st = (𝑥 ∈ V ↦ (1st ‘𝑥))
15	14	reseq1i 5965	. . . 4 ⊢ (1st ↾ 𝐴) = ((𝑥 ∈ V ↦ (1st ‘𝑥)) ↾ 𝐴)
16		ssv 3963	. . . . 5 ⊢ 𝐴 ⊆ V
17		resmpt 6030	. . . . 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 2788	. . 3 ⊢ (1st ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))
20	1, 2, 3, 4, 5, 6, 9, 19	gsummpt2co 33281	. 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 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → 𝑊 ∈ CMnd)
23	4	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → 𝐴 ∈ Fin)
24		imaexg 7898	. . . . . . 7 ⊢ (𝐴 ∈ Fin → (𝐴 “ {𝑦}) ∈ V)
25	23, 24	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (𝐴 “ {𝑦}) ∈ V)
26		gsummpt2d.1	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → 𝐶 = 𝐷)
27	26	adantl 486	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝐶 = 𝐷)
28		simp-4l 794	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝜑)
29		simplr 780	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝑥 ∈ 𝐴)
30	28, 29, 6	syl2anc 595	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝐶 ∈ 𝐵)
31	27, 30	eqeltrrd 2866	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝐷 ∈ 𝐵)
32		vex 3461	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
33		vex 3461	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
34	32, 33	elimasn 6083	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝐴 “ {𝑦}) ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴)
35	34	biimpi 219	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝐴 “ {𝑦}) → ⟨𝑦, 𝑧⟩ ∈ 𝐴)
36	35	adantl 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → ⟨𝑦, 𝑧⟩ ∈ 𝐴)
37		simpr 489	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → 𝑥 = ⟨𝑦, 𝑧⟩)
38	37	eqeq1d 2767	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) → (𝑥 = ⟨𝑦, 𝑧⟩ ↔ ⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑧⟩))
39		eqidd 2766	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → ⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑧⟩)
40	36, 38, 39	rspcedvd 3586	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → ∃𝑥 ∈ 𝐴 𝑥 = ⟨𝑦, 𝑧⟩)
41	31, 40	r19.29a 3173	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑧 ∈ (𝐴 “ {𝑦})) → 𝐷 ∈ 𝐵)
42	41	fmpttd 7100	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷):(𝐴 “ {𝑦})⟶𝐵)
43		eqid 2765	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) = (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)
44		imafi2 9306	. . . . . . . . 9 ⊢ (𝐴 ∈ Fin → (𝐴 “ {𝑦}) ∈ Fin)
45	4, 44	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝐴 “ {𝑦}) ∈ Fin)
46	45	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (𝐴 “ {𝑦}) ∈ Fin)
47		fvex 6884	. . . . . . . 8 ⊢ (0g‘𝑊) ∈ V
48	47	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (0g‘𝑊) ∈ V)
49	43, 46, 41, 48	fsuppmptdm 9324	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) finSupp (0g‘𝑊))
50		2ndconst 8084	. . . . . . . 8 ⊢ (𝑦 ∈ dom 𝐴 → (2nd ↾ ({𝑦} × (𝐴 “ {𝑦}))):({𝑦} × (𝐴 “ {𝑦}))–1-1-onto→(𝐴 “ {𝑦}))
51	50	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (2nd ↾ ({𝑦} × (𝐴 “ {𝑦}))):({𝑦} × (𝐴 “ {𝑦}))–1-1-onto→(𝐴 “ {𝑦}))
52		1stpreimas 32963	. . . . . . . . . 10 ⊢ ((Rel 𝐴 ∧ 𝑦 ∈ dom 𝐴) → (◡(1st ↾ 𝐴) “ {𝑦}) = ({𝑦} × (𝐴 “ {𝑦})))
53	7, 52	sylan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (◡(1st ↾ 𝐴) “ {𝑦}) = ({𝑦} × (𝐴 “ {𝑦})))
54	53	reseq2d 5969	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (2nd ↾ (◡(1st ↾ 𝐴) “ {𝑦})) = (2nd ↾ ({𝑦} × (𝐴 “ {𝑦}))))
55	54	f1oeq1d 6805	. . . . . . 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 260	. . . . . 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 19977	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)) = (𝑊 Σg ((𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) ∘ (2nd ↾ (◡(1st ↾ 𝐴) “ {𝑦})))))
58		simpr 489	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) → 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦}))
59	53	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) → (◡(1st ↾ 𝐴) “ {𝑦}) = ({𝑦} × (𝐴 “ {𝑦})))
60	58, 59	eleqtrd 2867	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) → 𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})))
61		xp2nd 8007	. . . . . . . . . 10 ⊢ (𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})) → (2nd ‘𝑥) ∈ (𝐴 “ {𝑦}))
62	60, 61	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) → (2nd ‘𝑥) ∈ (𝐴 “ {𝑦}))
63	62	ralrimiva 3157	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → ∀𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})(2nd ‘𝑥) ∈ (𝐴 “ {𝑦}))
64		fo2nd 7995	. . . . . . . . . . . 12 ⊢ 2nd :V–onto→V
65		fofn 6784	. . . . . . . . . . . 12 ⊢ (2nd :V–onto→V → 2nd Fn V)
66		dffn5 6929	. . . . . . . . . . . . 13 ⊢ (2nd Fn V ↔ 2nd = (𝑥 ∈ V ↦ (2nd ‘𝑥)))
67	66	biimpi 219	. . . . . . . . . . . 12 ⊢ (2nd Fn V → 2nd = (𝑥 ∈ V ↦ (2nd ‘𝑥)))
68	64, 65, 67	mp2b 10	. . . . . . . . . . 11 ⊢ 2nd = (𝑥 ∈ V ↦ (2nd ‘𝑥))
69	68	reseq1i 5965	. . . . . . . . . 10 ⊢ (2nd ↾ (◡(1st ↾ 𝐴) “ {𝑦})) = ((𝑥 ∈ V ↦ (2nd ‘𝑥)) ↾ (◡(1st ↾ 𝐴) “ {𝑦}))
70		ssv 3963	. . . . . . . . . . 11 ⊢ (◡(1st ↾ 𝐴) “ {𝑦}) ⊆ V
71		resmpt 6030	. . . . . . . . . . 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 2788	. . . . . . . . 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 2766	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) = (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷))
76	63, 74, 75	fmptcos 7117	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → ((𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) ∘ (2nd ↾ (◡(1st ↾ 𝐴) “ {𝑦}))) = (𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦}) ↦ ⦋(2nd ‘𝑥) / 𝑧⦌𝐷))
77		nfv 1937	. . . . . . . . 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 485	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) ∧ 𝑧 = (2nd ‘𝑥)) → 𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})))
81		xp1st 8006	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})) → (1st ‘𝑥) ∈ {𝑦})
82	80, 81	syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) ∧ 𝑧 = (2nd ‘𝑥)) → (1st ‘𝑥) ∈ {𝑦})
83		fvex 6884	. . . . . . . . . . . . . 14 ⊢ (1st ‘𝑥) ∈ V
84	83	elsn 4600	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑥) ∈ {𝑦} ↔ (1st ‘𝑥) = 𝑦)
85	82, 84	sylib 221	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) ∧ 𝑧 = (2nd ‘𝑥)) → (1st ‘𝑥) = 𝑦)
86		simpr 489	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) ∧ 𝑧 = (2nd ‘𝑥)) → 𝑧 = (2nd ‘𝑥))
87	86	eqcomd 2771	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) ∧ 𝑧 = (2nd ‘𝑥)) → (2nd ‘𝑥) = 𝑧)
88		eqopi 8010	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ({𝑦} × (𝐴 “ {𝑦})) ∧ ((1st ‘𝑥) = 𝑦 ∧ (2nd ‘𝑥) = 𝑧)) → 𝑥 = ⟨𝑦, 𝑧⟩)
89	80, 85, 87, 88	syl12anc 849	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) ∧ 𝑧 = (2nd ‘𝑥)) → 𝑥 = ⟨𝑦, 𝑧⟩)
90	89, 26	syl 18	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) ∧ 𝑧 = (2nd ‘𝑥)) → 𝐶 = 𝐷)
91	90	eqcomd 2771	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) ∧ 𝑧 = (2nd ‘𝑥)) → 𝐷 = 𝐶)
92	77, 79, 62, 91	csbiedf 3885	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ dom 𝐴) ∧ 𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦})) → ⦋(2nd ‘𝑥) / 𝑧⦌𝐷 = 𝐶)
93	92	mpteq2dva 5198	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦}) ↦ ⦋(2nd ‘𝑥) / 𝑧⦌𝐷) = (𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦}) ↦ 𝐶))
94	76, 93	eqtrd 2800	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → ((𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷) ∘ (2nd ↾ (◡(1st ↾ 𝐴) “ {𝑦}))) = (𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦}) ↦ 𝐶))
95	94	oveq2d 7416	. . . . 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 2801	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ dom 𝐴) → (𝑊 Σg (𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦}) ↦ 𝐶)) = (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)))
97	21, 96	mpteq2da 5197	. . 3 ⊢ (𝜑 → (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑥 ∈ (◡(1st ↾ 𝐴) “ {𝑦}) ↦ 𝐶))) = (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷))))
98	97	oveq2d 7416	. 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 2800	1 ⊢ (𝜑 → (𝑊 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑊 Σg (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)))))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563  Ⅎwnf 1806   ∈ wcel 2145  Ⅎwnfc 2912  Vcvv 3457  ⦋csb 3855   ⊆ wss 3907  {csn 4585  ⟨cop 4591   ↦ cmpt 5186   × cxp 5650  ^◡ccnv 5651  dom cdm 5652   ↾ cres 5654   “ cima 5655   ∘ ccom 5656  Rel wrel 5657   Fn wfn 6520  –onto→wfo 6523  –1-1-onto→wf1o 6524  ‘cfv 6525  (class class class)co 7400  1^st c1st 7972  2^nd c2nd 7973  Fincfn 8931  Basecbs 17259  0_gc0g 17482   Σ_g cgsu 17483  CMndccmn 19841

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-iin 4955  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-of 7664  df-om 7851  df-1st 7974  df-2nd 7975  df-supp 8145  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-er 8682  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-fsupp 9310  df-oi 9460  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12225  df-2 12294  df-n0 12496  df-z 12583  df-uz 12854  df-fz 13527  df-fzo 13674  df-seq 14029  df-hash 14358  df-sets 17214  df-slot 17232  df-ndx 17244  df-base 17260  df-ress 17281  df-plusg 17313  df-0g 17484  df-gsum 17485  df-mre 17628  df-mrc 17629  df-acs 17631  df-mgm 18688  df-sgrp 18767  df-mnd 18783  df-submnd 18832  df-mulg 19125  df-cntz 19378  df-cmn 19843

This theorem is referenced by:  gsumfs2d  33294  gsumhashmul  33300  esum2d  34400