Theorem gsumpart 32713

Description: Express a group sum as a double sum, grouping along a (possibly infinite) partition. (Contributed by Thierry Arnoux, 22-Jun-2024.)

Hypotheses

Ref	Expression
gsumpart.b	⊢ 𝐵 = (Base‘𝐺)
gsumpart.z	⊢ 0 = (0g‘𝐺)
gsumpart.g	⊢ (𝜑 → 𝐺 ∈ CMnd)
gsumpart.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
gsumpart.x	⊢ (𝜑 → 𝑋 ∈ 𝑊)
gsumpart.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
gsumpart.w	⊢ (𝜑 → 𝐹 finSupp 0 )
gsumpart.1	⊢ (𝜑 → Disj 𝑥 ∈ 𝑋 𝐶)
gsumpart.2	⊢ (𝜑 → ∪ 𝑥 ∈ 𝑋 𝐶 = 𝐴)

Assertion

Ref	Expression
gsumpart	⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥 ∈ 𝑋 ↦ (𝐺 Σg (𝐹 ↾ 𝐶)))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺   𝑥,𝑋   𝜑,𝑥

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)   𝑊(𝑥)   0 (𝑥)

Proof of Theorem gsumpart

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumpart.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		gsumpart.z	. . 3 ⊢ 0 = (0g‘𝐺)
3		gsumpart.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd)
4		gsumpart.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5		gsumpart.f	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
6		gsumpart.w	. . 3 ⊢ (𝜑 → 𝐹 finSupp 0 )
7		eqid 2726	. . . 4 ⊢ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) = ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)
8		gsumpart.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
9		gsumpart.1	. . . 4 ⊢ (𝜑 → Disj 𝑥 ∈ 𝑋 𝐶)
10		gsumpart.2	. . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝑋 𝐶 = 𝐴)
11	7, 4, 8, 9, 10	2ndresdjuf1o 32384	. . 3 ⊢ (𝜑 → (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)–1-1-onto→𝐴)
12	1, 2, 3, 4, 5, 6, 11	gsumf1o 19836	. 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))))
13		vsnex 5422	. . . . . . 7 ⊢ {𝑥} ∈ V
14	13	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ V)
15	4	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑉)
16		ssidd 4000	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ 𝐴)
17	10, 16	eqsstrd 4015	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝑋 𝐶 ⊆ 𝐴)
18		iunss 5041	. . . . . . . . 9 ⊢ (∪ 𝑥 ∈ 𝑋 𝐶 ⊆ 𝐴 ↔ ∀𝑥 ∈ 𝑋 𝐶 ⊆ 𝐴)
19	17, 18	sylib 217	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐶 ⊆ 𝐴)
20	19	r19.21bi 3242	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ⊆ 𝐴)
21	15, 20	ssexd 5317	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ V)
22	14, 21	xpexd 7735	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ({𝑥} × 𝐶) ∈ V)
23	22	ralrimiva 3140	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ∈ V)
24		iunexg 7949	. . . 4 ⊢ ((𝑋 ∈ 𝑊 ∧ ∀𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ∈ V) → ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ∈ V)
25	8, 23, 24	syl2anc 583	. . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ∈ V)
26		relxp 5687	. . . . . 6 ⊢ Rel ({𝑥} × 𝐶)
27	26	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Rel ({𝑥} × 𝐶))
28	27	ralrimiva 3140	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 Rel ({𝑥} × 𝐶))
29		reliun 5809	. . . 4 ⊢ (Rel ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ↔ ∀𝑥 ∈ 𝑋 Rel ({𝑥} × 𝐶))
30	28, 29	sylibr 233	. . 3 ⊢ (𝜑 → Rel ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))
31		dmiun 5907	. . . . . 6 ⊢ dom ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) = ∪ 𝑥 ∈ 𝑋 dom ({𝑥} × 𝐶)
32		dmxpss 6164	. . . . . . . 8 ⊢ dom ({𝑥} × 𝐶) ⊆ {𝑥}
33	32	rgenw 3059	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝑋 dom ({𝑥} × 𝐶) ⊆ {𝑥}
34		ss2iun 5008	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 dom ({𝑥} × 𝐶) ⊆ {𝑥} → ∪ 𝑥 ∈ 𝑋 dom ({𝑥} × 𝐶) ⊆ ∪ 𝑥 ∈ 𝑋 {𝑥})
35	33, 34	ax-mp 5	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝑋 dom ({𝑥} × 𝐶) ⊆ ∪ 𝑥 ∈ 𝑋 {𝑥}
36	31, 35	eqsstri 4011	. . . . 5 ⊢ dom ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ ∪ 𝑥 ∈ 𝑋 {𝑥}
37		iunid 5056	. . . . 5 ⊢ ∪ 𝑥 ∈ 𝑋 {𝑥} = 𝑋
38	36, 37	sseqtri 4013	. . . 4 ⊢ dom ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ 𝑋
39	38	a1i 11	. . 3 ⊢ (𝜑 → dom ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ 𝑋)
40		fo2nd 7995	. . . . . . . 8 ⊢ 2nd :V–onto→V
41		fof 6799	. . . . . . . 8 ⊢ (2nd :V–onto→V → 2nd :V⟶V)
42	40, 41	ax-mp 5	. . . . . . 7 ⊢ 2nd :V⟶V
43		ssv 4001	. . . . . . 7 ⊢ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ V
44		fssres 6751	. . . . . . 7 ⊢ ((2nd :V⟶V ∧ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ V) → (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)⟶V)
45	42, 43, 44	mp2an 689	. . . . . 6 ⊢ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)⟶V
46		ffn 6711	. . . . . 6 ⊢ ((2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)⟶V → (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)) Fn ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))
47	45, 46	mp1i 13	. . . . 5 ⊢ (𝜑 → (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)) Fn ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))
48		djussxp2 32382	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)
49		imass2 6095	. . . . . . . 8 ⊢ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶) → (2nd “ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)) ⊆ (2nd “ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)))
50	48, 49	ax-mp 5	. . . . . . 7 ⊢ (2nd “ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)) ⊆ (2nd “ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶))
51		ima0 6070	. . . . . . . . . . 11 ⊢ (2nd “ ∅) = ∅
52		xpeq1 5683	. . . . . . . . . . . . 13 ⊢ (𝑋 = ∅ → (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶) = (∅ × ∪ 𝑥 ∈ 𝑋 𝐶))
53		0xp 5767	. . . . . . . . . . . . 13 ⊢ (∅ × ∪ 𝑥 ∈ 𝑋 𝐶) = ∅
54	52, 53	eqtrdi 2782	. . . . . . . . . . . 12 ⊢ (𝑋 = ∅ → (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶) = ∅)
55	54	imaeq2d 6053	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → (2nd “ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)) = (2nd “ ∅))
56		iuneq1 5006	. . . . . . . . . . . 12 ⊢ (𝑋 = ∅ → ∪ 𝑥 ∈ 𝑋 𝐶 = ∪ 𝑥 ∈ ∅ 𝐶)
57		0iun 5059	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ ∅ 𝐶 = ∅
58	56, 57	eqtrdi 2782	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → ∪ 𝑥 ∈ 𝑋 𝐶 = ∅)
59	51, 55, 58	3eqtr4a 2792	. . . . . . . . . 10 ⊢ (𝑋 = ∅ → (2nd “ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)) = ∪ 𝑥 ∈ 𝑋 𝐶)
60	59	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (2nd “ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)) = ∪ 𝑥 ∈ 𝑋 𝐶)
61		2ndimaxp 32381	. . . . . . . . . 10 ⊢ (𝑋 ≠ ∅ → (2nd “ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)) = ∪ 𝑥 ∈ 𝑋 𝐶)
62	61	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (2nd “ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)) = ∪ 𝑥 ∈ 𝑋 𝐶)
63	60, 62	pm2.61dane 3023	. . . . . . . 8 ⊢ (𝜑 → (2nd “ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)) = ∪ 𝑥 ∈ 𝑋 𝐶)
64	63, 10	eqtrd 2766	. . . . . . 7 ⊢ (𝜑 → (2nd “ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)) = 𝐴)
65	50, 64	sseqtrid 4029	. . . . . 6 ⊢ (𝜑 → (2nd “ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)) ⊆ 𝐴)
66		resssxp 6263	. . . . . 6 ⊢ ((2nd “ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)) ⊆ 𝐴 ↔ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)) ⊆ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) × 𝐴))
67	65, 66	sylib 217	. . . . 5 ⊢ (𝜑 → (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)) ⊆ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) × 𝐴))
68		dff2 7094	. . . . 5 ⊢ ((2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)⟶𝐴 ↔ ((2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)) Fn ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ∧ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)) ⊆ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) × 𝐴)))
69	47, 67, 68	sylanbrc 582	. . . 4 ⊢ (𝜑 → (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)⟶𝐴)
70	5, 69	fcod 6737	. . 3 ⊢ (𝜑 → (𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)⟶𝐵)
71	7, 4, 8, 9, 10	2ndresdju 32383	. . . 4 ⊢ (𝜑 → (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)–1-1→𝐴)
72	2	fvexi 6899	. . . . 5 ⊢ 0 ∈ V
73	72	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ V)
74	5, 4	fexd 7224	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
75	6, 71, 73, 74	fsuppco 9399	. . 3 ⊢ (𝜑 → (𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))) finSupp 0 )
76	1, 2, 3, 25, 30, 8, 39, 70, 75	gsum2d 19892	. 2 ⊢ (𝜑 → (𝐺 Σg (𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))) = (𝐺 Σg (𝑦 ∈ 𝑋 ↦ (𝐺 Σg (𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↦ (𝑦(𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))𝑧))))))
77		nfcsb1v 3913	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
78		csbeq1a 3902	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
79	8, 21, 77, 78	iunsnima2 32357	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦}) = ⦋𝑦 / 𝑥⦌𝐶)
80		df-ov 7408	. . . . . . . . 9 ⊢ (𝑦(𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))𝑧) = ((𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))‘⟨𝑦, 𝑧⟩)
81	69	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)⟶𝐴)
82		simplr 766	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → 𝑦 ∈ 𝑋)
83		vsnid 4660	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ {𝑦}
84	83	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → 𝑦 ∈ {𝑦})
85	79	eleq2d 2813	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↔ 𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐶))
86	85	biimpa 476	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → 𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐶)
87	84, 86	opelxpd 5708	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ⟨𝑦, 𝑧⟩ ∈ ({𝑦} × ⦋𝑦 / 𝑥⦌𝐶))
88		nfcv 2897	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥{𝑦}
89	88, 77	nfxp 5702	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥({𝑦} × ⦋𝑦 / 𝑥⦌𝐶)
90	89	nfel2 2915	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥⟨𝑦, 𝑧⟩ ∈ ({𝑦} × ⦋𝑦 / 𝑥⦌𝐶)
91		sneq 4633	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
92	91, 78	xpeq12d 5700	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → ({𝑥} × 𝐶) = ({𝑦} × ⦋𝑦 / 𝑥⦌𝐶))
93	92	eleq2d 2813	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (⟨𝑦, 𝑧⟩ ∈ ({𝑥} × 𝐶) ↔ ⟨𝑦, 𝑧⟩ ∈ ({𝑦} × ⦋𝑦 / 𝑥⦌𝐶)))
94	90, 93	rspce 3595	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝑋 ∧ ⟨𝑦, 𝑧⟩ ∈ ({𝑦} × ⦋𝑦 / 𝑥⦌𝐶)) → ∃𝑥 ∈ 𝑋 ⟨𝑦, 𝑧⟩ ∈ ({𝑥} × 𝐶))
95	82, 87, 94	syl2anc 583	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ∃𝑥 ∈ 𝑋 ⟨𝑦, 𝑧⟩ ∈ ({𝑥} × 𝐶))
96		eliun 4994	. . . . . . . . . . . 12 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ↔ ∃𝑥 ∈ 𝑋 ⟨𝑦, 𝑧⟩ ∈ ({𝑥} × 𝐶))
97	95, 96	sylibr 233	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))
98	81, 97	fvco3d 6985	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ((𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))‘⟨𝑦, 𝑧⟩) = (𝐹‘((2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))‘⟨𝑦, 𝑧⟩)))
99	97	fvresd 6905	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ((2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))‘⟨𝑦, 𝑧⟩) = (2nd ‘⟨𝑦, 𝑧⟩))
100		vex 3472	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
101		vex 3472	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
102	100, 101	op2nd 7983	. . . . . . . . . . . 12 ⊢ (2nd ‘⟨𝑦, 𝑧⟩) = 𝑧
103	99, 102	eqtrdi 2782	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ((2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))‘⟨𝑦, 𝑧⟩) = 𝑧)
104	103	fveq2d 6889	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → (𝐹‘((2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))‘⟨𝑦, 𝑧⟩)) = (𝐹‘𝑧))
105	98, 104	eqtrd 2766	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ((𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))‘⟨𝑦, 𝑧⟩) = (𝐹‘𝑧))
106	80, 105	eqtrid 2778	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → (𝑦(𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))𝑧) = (𝐹‘𝑧))
107	79, 106	mpteq12dva 5230	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↦ (𝑦(𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))𝑧)) = (𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐶 ↦ (𝐹‘𝑧)))
108	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝐹:𝐴⟶𝐵)
109		imassrn 6064	. . . . . . . . . 10 ⊢ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦}) ⊆ ran ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)
110	10	xpeq2d 5699	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶) = (𝑋 × 𝐴))
111	48, 110	sseqtrid 4029	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ (𝑋 × 𝐴))
112		rnss 5932	. . . . . . . . . . . . 13 ⊢ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ (𝑋 × 𝐴) → ran ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ ran (𝑋 × 𝐴))
113	111, 112	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ran ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ ran (𝑋 × 𝐴))
114	113	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ran ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ ran (𝑋 × 𝐴))
115		rnxpss 6165	. . . . . . . . . . 11 ⊢ ran (𝑋 × 𝐴) ⊆ 𝐴
116	114, 115	sstrdi 3989	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ran ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ 𝐴)
117	109, 116	sstrid 3988	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦}) ⊆ 𝐴)
118	79, 117	eqsstrrd 4016	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐶 ⊆ 𝐴)
119	108, 118	feqresmpt 6955	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶) = (𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐶 ↦ (𝐹‘𝑧)))
120	107, 119	eqtr4d 2769	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↦ (𝑦(𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))𝑧)) = (𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶))
121	120	oveq2d 7421	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐺 Σg (𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↦ (𝑦(𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))𝑧))) = (𝐺 Σg (𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶)))
122	121	mpteq2dva 5241	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑋 ↦ (𝐺 Σg (𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↦ (𝑦(𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))𝑧)))) = (𝑦 ∈ 𝑋 ↦ (𝐺 Σg (𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶))))
123		nfcv 2897	. . . . 5 ⊢ Ⅎ𝑦(𝐺 Σg (𝐹 ↾ 𝐶))
124		nfcv 2897	. . . . . 6 ⊢ Ⅎ𝑥𝐺
125		nfcv 2897	. . . . . 6 ⊢ Ⅎ𝑥 Σg
126		nfcv 2897	. . . . . . 7 ⊢ Ⅎ𝑥𝐹
127	126, 77	nfres 5977	. . . . . 6 ⊢ Ⅎ𝑥(𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶)
128	124, 125, 127	nfov 7435	. . . . 5 ⊢ Ⅎ𝑥(𝐺 Σg (𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶))
129	78	reseq2d 5975	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐹 ↾ 𝐶) = (𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶))
130	129	oveq2d 7421	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐺 Σg (𝐹 ↾ 𝐶)) = (𝐺 Σg (𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶)))
131	123, 128, 130	cbvmpt 5252	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ (𝐺 Σg (𝐹 ↾ 𝐶))) = (𝑦 ∈ 𝑋 ↦ (𝐺 Σg (𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶)))
132	122, 131	eqtr4di 2784	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝑋 ↦ (𝐺 Σg (𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↦ (𝑦(𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))𝑧)))) = (𝑥 ∈ 𝑋 ↦ (𝐺 Σg (𝐹 ↾ 𝐶))))
133	132	oveq2d 7421	. 2 ⊢ (𝜑 → (𝐺 Σg (𝑦 ∈ 𝑋 ↦ (𝐺 Σg (𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↦ (𝑦(𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))𝑧))))) = (𝐺 Σg (𝑥 ∈ 𝑋 ↦ (𝐺 Σg (𝐹 ↾ 𝐶)))))
134	12, 76, 133	3eqtrd 2770	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥 ∈ 𝑋 ↦ (𝐺 Σg (𝐹 ↾ 𝐶)))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   ≠ wne 2934  ∀wral 3055  ∃wrex 3064  Vcvv 3468  ⦋csb 3888   ⊆ wss 3943  ∅c0 4317  {csn 4623  ⟨cop 4629  ∪ ciun 4990  Disj wdisj 5106   class class class wbr 5141   ↦ cmpt 5224   × cxp 5667  dom cdm 5669  ran crn 5670   ↾ cres 5671   “ cima 5672   ∘ ccom 5673  Rel wrel 5674   Fn wfn 6532  ⟶wf 6533  –onto→wfo 6535  ‘cfv 6537  (class class class)co 7405  2^nd c2nd 7973   finSupp cfsupp 9363  Basecbs 17153  0_gc0g 17394   Σ_g cgsu 17395  CMndccmn 19700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-disj 5107  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  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 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7667  df-om 7853  df-1st 7974  df-2nd 7975  df-supp 8147  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13491  df-fzo 13634  df-seq 13973  df-hash 14296  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-0g 17396  df-gsum 17397  df-mre 17539  df-mrc 17540  df-acs 17542  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-submnd 18714  df-mulg 18996  df-cntz 19233  df-cmn 19702

This theorem is referenced by:  elrspunidl  33052