Theorem gsumpart 31294

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 2739	. . . 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 30966	. . 3 ⊢ (𝜑 → (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)–1-1-onto→𝐴)
12	1, 2, 3, 4, 5, 6, 11	gsumf1o 19498	. 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))))
13		snex 5357	. . . . . . 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 3948	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ 𝐴)
17	10, 16	eqsstrd 3963	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝑋 𝐶 ⊆ 𝐴)
18		iunss 4979	. . . . . . . . 9 ⊢ (∪ 𝑥 ∈ 𝑋 𝐶 ⊆ 𝐴 ↔ ∀𝑥 ∈ 𝑋 𝐶 ⊆ 𝐴)
19	17, 18	sylib 217	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐶 ⊆ 𝐴)
20	19	r19.21bi 3134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ⊆ 𝐴)
21	15, 20	ssexd 5251	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ V)
22	14, 21	xpexd 7592	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ({𝑥} × 𝐶) ∈ V)
23	22	ralrimiva 3109	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ∈ V)
24		iunexg 7792	. . . 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 5606	. . . . . 6 ⊢ Rel ({𝑥} × 𝐶)
27	26	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Rel ({𝑥} × 𝐶))
28	27	ralrimiva 3109	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 Rel ({𝑥} × 𝐶))
29		reliun 5723	. . . 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 5819	. . . . . 6 ⊢ dom ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) = ∪ 𝑥 ∈ 𝑋 dom ({𝑥} × 𝐶)
32		dmxpss 6071	. . . . . . . 8 ⊢ dom ({𝑥} × 𝐶) ⊆ {𝑥}
33	32	rgenw 3077	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝑋 dom ({𝑥} × 𝐶) ⊆ {𝑥}
34		ss2iun 4947	. . . . . . 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 3959	. . . . 5 ⊢ dom ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ ∪ 𝑥 ∈ 𝑋 {𝑥}
37		iunid 4994	. . . . 5 ⊢ ∪ 𝑥 ∈ 𝑋 {𝑥} = 𝑋
38	36, 37	sseqtri 3961	. . . 4 ⊢ dom ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ 𝑋
39	38	a1i 11	. . 3 ⊢ (𝜑 → dom ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ 𝑋)
40		fo2nd 7838	. . . . . . . 8 ⊢ 2nd :V–onto→V
41		fof 6684	. . . . . . . 8 ⊢ (2nd :V–onto→V → 2nd :V⟶V)
42	40, 41	ax-mp 5	. . . . . . 7 ⊢ 2nd :V⟶V
43		ssv 3949	. . . . . . 7 ⊢ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ V
44		fssres 6636	. . . . . . 7 ⊢ ((2nd :V⟶V ∧ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ V) → (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)⟶V)
45	42, 43, 44	mp2an 688	. . . . . 6 ⊢ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)⟶V
46		ffn 6596	. . . . . 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 30964	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)
49		imass2 6007	. . . . . . . 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 5982	. . . . . . . . . . 11 ⊢ (2nd “ ∅) = ∅
52		xpeq1 5602	. . . . . . . . . . . . 13 ⊢ (𝑋 = ∅ → (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶) = (∅ × ∪ 𝑥 ∈ 𝑋 𝐶))
53		0xp 5683	. . . . . . . . . . . . 13 ⊢ (∅ × ∪ 𝑥 ∈ 𝑋 𝐶) = ∅
54	52, 53	eqtrdi 2795	. . . . . . . . . . . 12 ⊢ (𝑋 = ∅ → (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶) = ∅)
55	54	imaeq2d 5966	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → (2nd “ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)) = (2nd “ ∅))
56		iuneq1 4945	. . . . . . . . . . . 12 ⊢ (𝑋 = ∅ → ∪ 𝑥 ∈ 𝑋 𝐶 = ∪ 𝑥 ∈ ∅ 𝐶)
57		0iun 4996	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ ∅ 𝐶 = ∅
58	56, 57	eqtrdi 2795	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → ∪ 𝑥 ∈ 𝑋 𝐶 = ∅)
59	51, 55, 58	3eqtr4a 2805	. . . . . . . . . 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 30963	. . . . . . . . . 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 3033	. . . . . . . 8 ⊢ (𝜑 → (2nd “ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)) = ∪ 𝑥 ∈ 𝑋 𝐶)
64	63, 10	eqtrd 2779	. . . . . . 7 ⊢ (𝜑 → (2nd “ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)) = 𝐴)
65	50, 64	sseqtrid 3977	. . . . . 6 ⊢ (𝜑 → (2nd “ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)) ⊆ 𝐴)
66		resssxp 6170	. . . . . 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 6969	. . . . 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 6622	. . 3 ⊢ (𝜑 → (𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)⟶𝐵)
71	7, 4, 8, 9, 10	2ndresdju 30965	. . . 4 ⊢ (𝜑 → (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)–1-1→𝐴)
72	2	fvexi 6782	. . . . 5 ⊢ 0 ∈ V
73	72	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ V)
74	5, 4	fexd 7097	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
75	6, 71, 73, 74	fsuppco 9122	. . 3 ⊢ (𝜑 → (𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))) finSupp 0 )
76	1, 2, 3, 25, 30, 8, 39, 70, 75	gsum2d 19554	. 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 3861	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
78		csbeq1a 3850	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
79	8, 21, 77, 78	iunsnima2 30938	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦}) = ⦋𝑦 / 𝑥⦌𝐶)
80		df-ov 7271	. . . . . . . . 9 ⊢ (𝑦(𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))𝑧) = ((𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))‘⟨𝑦, 𝑧⟩)
81	69	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)⟶𝐴)
82		simplr 765	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → 𝑦 ∈ 𝑋)
83		vsnid 4603	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ {𝑦}
84	83	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → 𝑦 ∈ {𝑦})
85	79	eleq2d 2825	. . . . . . . . . . . . . . 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 5626	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ⟨𝑦, 𝑧⟩ ∈ ({𝑦} × ⦋𝑦 / 𝑥⦌𝐶))
88		nfcv 2908	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥{𝑦}
89	88, 77	nfxp 5621	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥({𝑦} × ⦋𝑦 / 𝑥⦌𝐶)
90	89	nfel2 2926	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥⟨𝑦, 𝑧⟩ ∈ ({𝑦} × ⦋𝑦 / 𝑥⦌𝐶)
91		sneq 4576	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
92	91, 78	xpeq12d 5619	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → ({𝑥} × 𝐶) = ({𝑦} × ⦋𝑦 / 𝑥⦌𝐶))
93	92	eleq2d 2825	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (⟨𝑦, 𝑧⟩ ∈ ({𝑥} × 𝐶) ↔ ⟨𝑦, 𝑧⟩ ∈ ({𝑦} × ⦋𝑦 / 𝑥⦌𝐶)))
94	90, 93	rspce 3548	. . . . . . . . . . . . 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 4933	. . . . . . . . . . . 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 6862	. . . . . . . . . 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 6788	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ((2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))‘⟨𝑦, 𝑧⟩) = (2nd ‘⟨𝑦, 𝑧⟩))
100		vex 3434	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
101		vex 3434	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
102	100, 101	op2nd 7826	. . . . . . . . . . . 12 ⊢ (2nd ‘⟨𝑦, 𝑧⟩) = 𝑧
103	99, 102	eqtrdi 2795	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ((2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))‘⟨𝑦, 𝑧⟩) = 𝑧)
104	103	fveq2d 6772	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → (𝐹‘((2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))‘⟨𝑦, 𝑧⟩)) = (𝐹‘𝑧))
105	98, 104	eqtrd 2779	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ((𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))‘⟨𝑦, 𝑧⟩) = (𝐹‘𝑧))
106	80, 105	eqtrid 2791	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → (𝑦(𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))𝑧) = (𝐹‘𝑧))
107	79, 106	mpteq12dva 5167	. . . . . . 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 5977	. . . . . . . . . 10 ⊢ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦}) ⊆ ran ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)
110	10	xpeq2d 5618	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶) = (𝑋 × 𝐴))
111	48, 110	sseqtrid 3977	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ (𝑋 × 𝐴))
112		rnss 5845	. . . . . . . . . . . . 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 6072	. . . . . . . . . . 11 ⊢ ran (𝑋 × 𝐴) ⊆ 𝐴
116	114, 115	sstrdi 3937	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ran ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ 𝐴)
117	109, 116	sstrid 3936	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦}) ⊆ 𝐴)
118	79, 117	eqsstrrd 3964	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐶 ⊆ 𝐴)
119	108, 118	feqresmpt 6832	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶) = (𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐶 ↦ (𝐹‘𝑧)))
120	107, 119	eqtr4d 2782	. . . . . 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 7284	. . . . 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 5178	. . . 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 2908	. . . . 5 ⊢ Ⅎ𝑦(𝐺 Σg (𝐹 ↾ 𝐶))
124		nfcv 2908	. . . . . 6 ⊢ Ⅎ𝑥𝐺
125		nfcv 2908	. . . . . 6 ⊢ Ⅎ𝑥 Σg
126		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑥𝐹
127	126, 77	nfres 5890	. . . . . 6 ⊢ Ⅎ𝑥(𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶)
128	124, 125, 127	nfov 7298	. . . . 5 ⊢ Ⅎ𝑥(𝐺 Σg (𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶))
129	78	reseq2d 5888	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐹 ↾ 𝐶) = (𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶))
130	129	oveq2d 7284	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐺 Σg (𝐹 ↾ 𝐶)) = (𝐺 Σg (𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶)))
131	123, 128, 130	cbvmpt 5189	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ (𝐺 Σg (𝐹 ↾ 𝐶))) = (𝑦 ∈ 𝑋 ↦ (𝐺 Σg (𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶)))
132	122, 131	eqtr4di 2797	. . 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 7284	. 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 2783	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥 ∈ 𝑋 ↦ (𝐺 Σg (𝐹 ↾ 𝐶)))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2109   ≠ wne 2944  ∀wral 3065  ∃wrex 3066  Vcvv 3430  ⦋csb 3836   ⊆ wss 3891  ∅c0 4261  {csn 4566  ⟨cop 4572  ∪ ciun 4929  Disj wdisj 5043   class class class wbr 5078   ↦ cmpt 5161   × cxp 5586  dom cdm 5588  ran crn 5589   ↾ cres 5590   “ cima 5591   ∘ ccom 5592  Rel wrel 5593   Fn wfn 6425  ⟶wf 6426  –onto→wfo 6428  ‘cfv 6430  (class class class)co 7268  2^nd c2nd 7816   finSupp cfsupp 9089  Basecbs 16893  0_gc0g 17131   Σ_g cgsu 17132  CMndccmn 19367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579  ax-cnex 10911  ax-resscn 10912  ax-1cn 10913  ax-icn 10914  ax-addcl 10915  ax-addrcl 10916  ax-mulcl 10917  ax-mulrcl 10918  ax-mulcom 10919  ax-addass 10920  ax-mulass 10921  ax-distr 10922  ax-i2m1 10923  ax-1ne0 10924  ax-1rid 10925  ax-rnegex 10926  ax-rrecex 10927  ax-cnre 10928  ax-pre-lttri 10929  ax-pre-lttrn 10930  ax-pre-ltadd 10931  ax-pre-mulgt0 10932

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-reu 3072  df-rmo 3073  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-int 4885  df-iun 4931  df-iin 4932  df-disj 5044  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-se 5544  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-pred 6199  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-isom 6439  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-of 7524  df-om 7701  df-1st 7817  df-2nd 7818  df-supp 7962  df-frecs 8081  df-wrecs 8112  df-recs 8186  df-rdg 8225  df-1o 8281  df-er 8472  df-en 8708  df-dom 8709  df-sdom 8710  df-fin 8711  df-fsupp 9090  df-oi 9230  df-card 9681  df-pnf 10995  df-mnf 10996  df-xr 10997  df-ltxr 10998  df-le 10999  df-sub 11190  df-neg 11191  df-nn 11957  df-2 12019  df-n0 12217  df-z 12303  df-uz 12565  df-fz 13222  df-fzo 13365  df-seq 13703  df-hash 14026  df-sets 16846  df-slot 16864  df-ndx 16876  df-base 16894  df-ress 16923  df-plusg 16956  df-0g 17133  df-gsum 17134  df-mre 17276  df-mrc 17277  df-acs 17279  df-mgm 18307  df-sgrp 18356  df-mnd 18367  df-submnd 18412  df-mulg 18682  df-cntz 18904  df-cmn 19369

This theorem is referenced by:  elrspunidl  31585