Theorem gsumpart 33383

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 2763	. . . 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 32995	. . 3 ⊢ (𝜑 → (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)–1-1-onto→𝐴)
12	1, 2, 3, 4, 5, 6, 11	gsumf1o 19981	. 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))))
13		vsnex 5406	. . . . . . 7 ⊢ {𝑥} ∈ V
14	13	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ V)
15	4	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑉)
16		ssidd 3960	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ 𝐴)
17	10, 16	eqsstrd 3971	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝑋 𝐶 ⊆ 𝐴)
18		iunss 5009	. . . . . . . . 9 ⊢ (∪ 𝑥 ∈ 𝑋 𝐶 ⊆ 𝐴 ↔ ∀𝑥 ∈ 𝑋 𝐶 ⊆ 𝐴)
19	17, 18	sylib 221	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐶 ⊆ 𝐴)
20	19	r19.21bi 3257	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ⊆ 𝐴)
21	15, 20	ssexd 5295	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ V)
22	14, 21	xpexd 7746	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ({𝑥} × 𝐶) ∈ V)
23	22	ralrimiva 3157	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ∈ V)
24		iunexg 7956	. . . 4 ⊢ ((𝑋 ∈ 𝑊 ∧ ∀𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ∈ V) → ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ∈ V)
25	8, 23, 24	syl2anc 595	. . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ∈ V)
26		relxp 5679	. . . . . 6 ⊢ Rel ({𝑥} × 𝐶)
27	26	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Rel ({𝑥} × 𝐶))
28	27	ralrimiva 3157	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 Rel ({𝑥} × 𝐶))
29		reliun 5803	. . . 4 ⊢ (Rel ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ↔ ∀𝑥 ∈ 𝑋 Rel ({𝑥} × 𝐶))
30	28, 29	sylibr 237	. . 3 ⊢ (𝜑 → Rel ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))
31		dmiun 5903	. . . . . 6 ⊢ dom ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) = ∪ 𝑥 ∈ 𝑋 dom ({𝑥} × 𝐶)
32		dmxpss 6169	. . . . . . . 8 ⊢ dom ({𝑥} × 𝐶) ⊆ {𝑥}
33	32	rgenw 3083	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝑋 dom ({𝑥} × 𝐶) ⊆ {𝑥}
34		ss2iun 4975	. . . . . . 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 3983	. . . . 5 ⊢ dom ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ ∪ 𝑥 ∈ 𝑋 {𝑥}
37		iunid 5025	. . . . 5 ⊢ ∪ 𝑥 ∈ 𝑋 {𝑥} = 𝑋
38	36, 37	sseqtri 3985	. . . 4 ⊢ dom ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ 𝑋
39	38	a1i 11	. . 3 ⊢ (𝜑 → dom ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ 𝑋)
40		fo2nd 8003	. . . . . . . 8 ⊢ 2nd :V–onto→V
41		fof 6792	. . . . . . . 8 ⊢ (2nd :V–onto→V → 2nd :V⟶V)
42	40, 41	ax-mp 5	. . . . . . 7 ⊢ 2nd :V⟶V
43		ssv 3961	. . . . . . 7 ⊢ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ V
44		fssres 6744	. . . . . . 7 ⊢ ((2nd :V⟶V ∧ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ V) → (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)⟶V)
45	42, 43, 44	mp2an 704	. . . . . 6 ⊢ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)⟶V
46		ffn 6705	. . . . . 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 14	. . . . 5 ⊢ (𝜑 → (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)) Fn ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))
48		djussxp2 32993	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)
49		imass2 6104	. . . . . . . 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 6079	. . . . . . . . . . 11 ⊢ (2nd “ ∅) = ∅
52		xpeq1 5675	. . . . . . . . . . . . 13 ⊢ (𝑋 = ∅ → (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶) = (∅ × ∪ 𝑥 ∈ 𝑋 𝐶))
53		0xp 5760	. . . . . . . . . . . . 13 ⊢ (∅ × ∪ 𝑥 ∈ 𝑋 𝐶) = ∅
54	52, 53	eqtrdi 2814	. . . . . . . . . . . 12 ⊢ (𝑋 = ∅ → (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶) = ∅)
55	54	imaeq2d 6062	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → (2nd “ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)) = (2nd “ ∅))
56		iuneq1 4973	. . . . . . . . . . . 12 ⊢ (𝑋 = ∅ → ∪ 𝑥 ∈ 𝑋 𝐶 = ∪ 𝑥 ∈ ∅ 𝐶)
57		0iun 5027	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ ∅ 𝐶 = ∅
58	56, 57	eqtrdi 2814	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → ∪ 𝑥 ∈ 𝑋 𝐶 = ∅)
59	51, 55, 58	3eqtr4a 2824	. . . . . . . . . 10 ⊢ (𝑋 = ∅ → (2nd “ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)) = ∪ 𝑥 ∈ 𝑋 𝐶)
60	59	adantl 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (2nd “ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)) = ∪ 𝑥 ∈ 𝑋 𝐶)
61		2ndimaxp 32991	. . . . . . . . . 10 ⊢ (𝑋 ≠ ∅ → (2nd “ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)) = ∪ 𝑥 ∈ 𝑋 𝐶)
62	61	adantl 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (2nd “ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)) = ∪ 𝑥 ∈ 𝑋 𝐶)
63	60, 62	pm2.61dane 3045	. . . . . . . 8 ⊢ (𝜑 → (2nd “ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)) = ∪ 𝑥 ∈ 𝑋 𝐶)
64	63, 10	eqtrd 2798	. . . . . . 7 ⊢ (𝜑 → (2nd “ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)) = 𝐴)
65	50, 64	sseqtrid 3979	. . . . . 6 ⊢ (𝜑 → (2nd “ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)) ⊆ 𝐴)
66		resssxp 6271	. . . . . 6 ⊢ ((2nd “ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)) ⊆ 𝐴 ↔ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)) ⊆ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) × 𝐴))
67	65, 66	sylib 221	. . . . 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 594	. . . 4 ⊢ (𝜑 → (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)⟶𝐴)
70	5, 69	fcod 6731	. . 3 ⊢ (𝜑 → (𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)⟶𝐵)
71	7, 4, 8, 9, 10	2ndresdju 32994	. . . 4 ⊢ (𝜑 → (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)–1-1→𝐴)
72	2	fvexi 6895	. . . . 5 ⊢ 0 ∈ V
73	72	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ V)
74	5, 4	fexd 7225	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
75	6, 71, 73, 74	fsuppco 9358	. . 3 ⊢ (𝜑 → (𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))) finSupp 0 )
76	1, 2, 3, 25, 30, 8, 39, 70, 75	gsum2d 20037	. 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 3877	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
78		csbeq1a 3867	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
79	8, 21, 77, 78	iunsnima2 32964	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦}) = ⦋𝑦 / 𝑥⦌𝐶)
80		df-ov 7413	. . . . . . . . 9 ⊢ (𝑦(𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))𝑧) = ((𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))‘⟨𝑦, 𝑧⟩)
81	69	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)⟶𝐴)
82		simplr 780	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → 𝑦 ∈ 𝑋)
83		vsnid 4629	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ {𝑦}
84	83	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → 𝑦 ∈ {𝑦})
85	79	eleq2d 2849	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↔ 𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐶))
86	85	biimpa 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → 𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐶)
87	84, 86	opelxpd 5700	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ⟨𝑦, 𝑧⟩ ∈ ({𝑦} × ⦋𝑦 / 𝑥⦌𝐶))
88		nfcv 2925	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥{𝑦}
89	88, 77	nfxp 5694	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥({𝑦} × ⦋𝑦 / 𝑥⦌𝐶)
90	89	nfel2 2943	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥⟨𝑦, 𝑧⟩ ∈ ({𝑦} × ⦋𝑦 / 𝑥⦌𝐶)
91		sneq 4599	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
92	91, 78	xpeq12d 5692	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → ({𝑥} × 𝐶) = ({𝑦} × ⦋𝑦 / 𝑥⦌𝐶))
93	92	eleq2d 2849	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (⟨𝑦, 𝑧⟩ ∈ ({𝑥} × 𝐶) ↔ ⟨𝑦, 𝑧⟩ ∈ ({𝑦} × ⦋𝑦 / 𝑥⦌𝐶)))
94	90, 93	rspce 3570	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝑋 ∧ ⟨𝑦, 𝑧⟩ ∈ ({𝑦} × ⦋𝑦 / 𝑥⦌𝐶)) → ∃𝑥 ∈ 𝑋 ⟨𝑦, 𝑧⟩ ∈ ({𝑥} × 𝐶))
95	82, 87, 94	syl2anc 595	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ∃𝑥 ∈ 𝑋 ⟨𝑦, 𝑧⟩ ∈ ({𝑥} × 𝐶))
96		eliun 4960	. . . . . . . . . . . 12 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ↔ ∃𝑥 ∈ 𝑋 ⟨𝑦, 𝑧⟩ ∈ ({𝑥} × 𝐶))
97	95, 96	sylibr 237	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))
98	81, 97	fvco3d 6982	. . . . . . . . . 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 6901	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ((2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))‘⟨𝑦, 𝑧⟩) = (2nd ‘⟨𝑦, 𝑧⟩))
100		vex 3459	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
101		vex 3459	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
102	100, 101	op2nd 7991	. . . . . . . . . . . 12 ⊢ (2nd ‘⟨𝑦, 𝑧⟩) = 𝑧
103	99, 102	eqtrdi 2814	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ((2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))‘⟨𝑦, 𝑧⟩) = 𝑧)
104	103	fveq2d 6885	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → (𝐹‘((2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))‘⟨𝑦, 𝑧⟩)) = (𝐹‘𝑧))
105	98, 104	eqtrd 2798	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ((𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))‘⟨𝑦, 𝑧⟩) = (𝐹‘𝑧))
106	80, 105	eqtrid 2810	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → (𝑦(𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))𝑧) = (𝐹‘𝑧))
107	79, 106	mpteq12dva 5197	. . . . . . 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 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝐹:𝐴⟶𝐵)
109		imassrn 6073	. . . . . . . . . 10 ⊢ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦}) ⊆ ran ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)
110	10	xpeq2d 5691	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶) = (𝑋 × 𝐴))
111	48, 110	sseqtrid 3979	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ (𝑋 × 𝐴))
112		rnss 5929	. . . . . . . . . . . . 13 ⊢ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ (𝑋 × 𝐴) → ran ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ ran (𝑋 × 𝐴))
113	111, 112	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ran ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ ran (𝑋 × 𝐴))
114	113	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ran ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ ran (𝑋 × 𝐴))
115		rnxpss 6170	. . . . . . . . . . 11 ⊢ ran (𝑋 × 𝐴) ⊆ 𝐴
116	114, 115	sstrdi 3949	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ran ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ 𝐴)
117	109, 116	sstrid 3948	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦}) ⊆ 𝐴)
118	79, 117	eqsstrrd 3972	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐶 ⊆ 𝐴)
119	108, 118	feqresmpt 6950	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶) = (𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐶 ↦ (𝐹‘𝑧)))
120	107, 119	eqtr4d 2801	. . . . . 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 7426	. . . . 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 5204	. . . 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 2925	. . . . 5 ⊢ Ⅎ𝑦(𝐺 Σg (𝐹 ↾ 𝐶))
124		nfcv 2925	. . . . . 6 ⊢ Ⅎ𝑥𝐺
125		nfcv 2925	. . . . . 6 ⊢ Ⅎ𝑥 Σg
126		nfcv 2925	. . . . . . 7 ⊢ Ⅎ𝑥𝐹
127	126, 77	nfres 5980	. . . . . 6 ⊢ Ⅎ𝑥(𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶)
128	124, 125, 127	nfov 7440	. . . . 5 ⊢ Ⅎ𝑥(𝐺 Σg (𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶))
129	78	reseq2d 5978	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐹 ↾ 𝐶) = (𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶))
130	129	oveq2d 7426	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐺 Σg (𝐹 ↾ 𝐶)) = (𝐺 Σg (𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶)))
131	123, 128, 130	cbvmpt 5213	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ (𝐺 Σg (𝐹 ↾ 𝐶))) = (𝑦 ∈ 𝑋 ↦ (𝐺 Σg (𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶)))
132	122, 131	eqtr4di 2816	. . 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 7426	. 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 2802	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥 ∈ 𝑋 ↦ (𝐺 Σg (𝐹 ↾ 𝐶)))))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  Vcvv 3455  ⦋csb 3853   ⊆ wss 3905  ∅c0 4286  {csn 4589  ⟨cop 4595  ∪ ciun 4956  Disj wdisj 5076   class class class wbr 5109   ↦ cmpt 5192   × cxp 5659  dom cdm 5661  ran crn 5662   ↾ cres 5663   “ cima 5664   ∘ ccom 5665  Rel wrel 5666   Fn wfn 6531  ⟶wf 6532  –onto→wfo 6534  ‘cfv 6536  (class class class)co 7410  2^nd c2nd 7981   finSupp cfsupp 9317  Basecbs 17264  0_gc0g 17487   Σ_g cgsu 17488  CMndccmn 19845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-disj 5077  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  df-om 7859  df-1st 7982  df-2nd 7983  df-supp 8153  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9318  df-oi 9468  df-card 9921  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-2 12298  df-n0 12500  df-z 12587  df-uz 12858  df-fz 13531  df-fzo 13679  df-seq 14034  df-hash 14363  df-sets 17219  df-slot 17237  df-ndx 17249  df-base 17265  df-ress 17286  df-plusg 17318  df-0g 17489  df-gsum 17490  df-mre 17633  df-mrc 17634  df-acs 17636  df-mgm 18693  df-sgrp 18772  df-mnd 18788  df-submnd 18837  df-mulg 19129  df-cntz 19382  df-cmn 19847

This theorem is referenced by:  elrspunidl  33736