Theorem gsumpart 33042

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 2734	. . . 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 32666	. . 3 ⊢ (𝜑 → (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)–1-1-onto→𝐴)
12	1, 2, 3, 4, 5, 6, 11	gsumf1o 19948	. 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))))
13		vsnex 5439	. . . . . . 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 4018	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ 𝐴)
17	10, 16	eqsstrd 4033	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝑋 𝐶 ⊆ 𝐴)
18		iunss 5049	. . . . . . . . 9 ⊢ (∪ 𝑥 ∈ 𝑋 𝐶 ⊆ 𝐴 ↔ ∀𝑥 ∈ 𝑋 𝐶 ⊆ 𝐴)
19	17, 18	sylib 218	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐶 ⊆ 𝐴)
20	19	r19.21bi 3248	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ⊆ 𝐴)
21	15, 20	ssexd 5329	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ V)
22	14, 21	xpexd 7769	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ({𝑥} × 𝐶) ∈ V)
23	22	ralrimiva 3143	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ∈ V)
24		iunexg 7986	. . . 4 ⊢ ((𝑋 ∈ 𝑊 ∧ ∀𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ∈ V) → ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ∈ V)
25	8, 23, 24	syl2anc 584	. . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ∈ V)
26		relxp 5706	. . . . . 6 ⊢ Rel ({𝑥} × 𝐶)
27	26	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Rel ({𝑥} × 𝐶))
28	27	ralrimiva 3143	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 Rel ({𝑥} × 𝐶))
29		reliun 5828	. . . 4 ⊢ (Rel ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ↔ ∀𝑥 ∈ 𝑋 Rel ({𝑥} × 𝐶))
30	28, 29	sylibr 234	. . 3 ⊢ (𝜑 → Rel ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))
31		dmiun 5926	. . . . . 6 ⊢ dom ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) = ∪ 𝑥 ∈ 𝑋 dom ({𝑥} × 𝐶)
32		dmxpss 6192	. . . . . . . 8 ⊢ dom ({𝑥} × 𝐶) ⊆ {𝑥}
33	32	rgenw 3062	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝑋 dom ({𝑥} × 𝐶) ⊆ {𝑥}
34		ss2iun 5014	. . . . . . 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 4029	. . . . 5 ⊢ dom ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ ∪ 𝑥 ∈ 𝑋 {𝑥}
37		iunid 5064	. . . . 5 ⊢ ∪ 𝑥 ∈ 𝑋 {𝑥} = 𝑋
38	36, 37	sseqtri 4031	. . . 4 ⊢ dom ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ 𝑋
39	38	a1i 11	. . 3 ⊢ (𝜑 → dom ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ 𝑋)
40		fo2nd 8033	. . . . . . . 8 ⊢ 2nd :V–onto→V
41		fof 6820	. . . . . . . 8 ⊢ (2nd :V–onto→V → 2nd :V⟶V)
42	40, 41	ax-mp 5	. . . . . . 7 ⊢ 2nd :V⟶V
43		ssv 4019	. . . . . . 7 ⊢ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ V
44		fssres 6774	. . . . . . 7 ⊢ ((2nd :V⟶V ∧ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ V) → (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)⟶V)
45	42, 43, 44	mp2an 692	. . . . . 6 ⊢ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)⟶V
46		ffn 6736	. . . . . 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 32664	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)
49		imass2 6122	. . . . . . . 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 6096	. . . . . . . . . . 11 ⊢ (2nd “ ∅) = ∅
52		xpeq1 5702	. . . . . . . . . . . . 13 ⊢ (𝑋 = ∅ → (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶) = (∅ × ∪ 𝑥 ∈ 𝑋 𝐶))
53		0xp 5786	. . . . . . . . . . . . 13 ⊢ (∅ × ∪ 𝑥 ∈ 𝑋 𝐶) = ∅
54	52, 53	eqtrdi 2790	. . . . . . . . . . . 12 ⊢ (𝑋 = ∅ → (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶) = ∅)
55	54	imaeq2d 6079	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → (2nd “ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)) = (2nd “ ∅))
56		iuneq1 5012	. . . . . . . . . . . 12 ⊢ (𝑋 = ∅ → ∪ 𝑥 ∈ 𝑋 𝐶 = ∪ 𝑥 ∈ ∅ 𝐶)
57		0iun 5067	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ ∅ 𝐶 = ∅
58	56, 57	eqtrdi 2790	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → ∪ 𝑥 ∈ 𝑋 𝐶 = ∅)
59	51, 55, 58	3eqtr4a 2800	. . . . . . . . . 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 32662	. . . . . . . . . 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 3026	. . . . . . . 8 ⊢ (𝜑 → (2nd “ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)) = ∪ 𝑥 ∈ 𝑋 𝐶)
64	63, 10	eqtrd 2774	. . . . . . 7 ⊢ (𝜑 → (2nd “ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)) = 𝐴)
65	50, 64	sseqtrid 4047	. . . . . 6 ⊢ (𝜑 → (2nd “ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)) ⊆ 𝐴)
66		resssxp 6291	. . . . . 6 ⊢ ((2nd “ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)) ⊆ 𝐴 ↔ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)) ⊆ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) × 𝐴))
67	65, 66	sylib 218	. . . . 5 ⊢ (𝜑 → (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)) ⊆ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) × 𝐴))
68		dff2 7118	. . . . 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 583	. . . 4 ⊢ (𝜑 → (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)⟶𝐴)
70	5, 69	fcod 6761	. . 3 ⊢ (𝜑 → (𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)⟶𝐵)
71	7, 4, 8, 9, 10	2ndresdju 32665	. . . 4 ⊢ (𝜑 → (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)–1-1→𝐴)
72	2	fvexi 6920	. . . . 5 ⊢ 0 ∈ V
73	72	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ V)
74	5, 4	fexd 7246	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
75	6, 71, 73, 74	fsuppco 9439	. . 3 ⊢ (𝜑 → (𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))) finSupp 0 )
76	1, 2, 3, 25, 30, 8, 39, 70, 75	gsum2d 20004	. 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 3932	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
78		csbeq1a 3921	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
79	8, 21, 77, 78	iunsnima2 32638	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦}) = ⦋𝑦 / 𝑥⦌𝐶)
80		df-ov 7433	. . . . . . . . 9 ⊢ (𝑦(𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))𝑧) = ((𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))‘⟨𝑦, 𝑧⟩)
81	69	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)⟶𝐴)
82		simplr 769	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → 𝑦 ∈ 𝑋)
83		vsnid 4667	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ {𝑦}
84	83	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → 𝑦 ∈ {𝑦})
85	79	eleq2d 2824	. . . . . . . . . . . . . . 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 5727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ⟨𝑦, 𝑧⟩ ∈ ({𝑦} × ⦋𝑦 / 𝑥⦌𝐶))
88		nfcv 2902	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥{𝑦}
89	88, 77	nfxp 5721	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥({𝑦} × ⦋𝑦 / 𝑥⦌𝐶)
90	89	nfel2 2921	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥⟨𝑦, 𝑧⟩ ∈ ({𝑦} × ⦋𝑦 / 𝑥⦌𝐶)
91		sneq 4640	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
92	91, 78	xpeq12d 5719	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → ({𝑥} × 𝐶) = ({𝑦} × ⦋𝑦 / 𝑥⦌𝐶))
93	92	eleq2d 2824	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (⟨𝑦, 𝑧⟩ ∈ ({𝑥} × 𝐶) ↔ ⟨𝑦, 𝑧⟩ ∈ ({𝑦} × ⦋𝑦 / 𝑥⦌𝐶)))
94	90, 93	rspce 3610	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝑋 ∧ ⟨𝑦, 𝑧⟩ ∈ ({𝑦} × ⦋𝑦 / 𝑥⦌𝐶)) → ∃𝑥 ∈ 𝑋 ⟨𝑦, 𝑧⟩ ∈ ({𝑥} × 𝐶))
95	82, 87, 94	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ∃𝑥 ∈ 𝑋 ⟨𝑦, 𝑧⟩ ∈ ({𝑥} × 𝐶))
96		eliun 4999	. . . . . . . . . . . 12 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ↔ ∃𝑥 ∈ 𝑋 ⟨𝑦, 𝑧⟩ ∈ ({𝑥} × 𝐶))
97	95, 96	sylibr 234	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))
98	81, 97	fvco3d 7008	. . . . . . . . . 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 6926	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ((2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))‘⟨𝑦, 𝑧⟩) = (2nd ‘⟨𝑦, 𝑧⟩))
100		vex 3481	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
101		vex 3481	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
102	100, 101	op2nd 8021	. . . . . . . . . . . 12 ⊢ (2nd ‘⟨𝑦, 𝑧⟩) = 𝑧
103	99, 102	eqtrdi 2790	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ((2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))‘⟨𝑦, 𝑧⟩) = 𝑧)
104	103	fveq2d 6910	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → (𝐹‘((2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))‘⟨𝑦, 𝑧⟩)) = (𝐹‘𝑧))
105	98, 104	eqtrd 2774	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ((𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))‘⟨𝑦, 𝑧⟩) = (𝐹‘𝑧))
106	80, 105	eqtrid 2786	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → (𝑦(𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))𝑧) = (𝐹‘𝑧))
107	79, 106	mpteq12dva 5236	. . . . . . 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 6090	. . . . . . . . . 10 ⊢ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦}) ⊆ ran ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)
110	10	xpeq2d 5718	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶) = (𝑋 × 𝐴))
111	48, 110	sseqtrid 4047	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ (𝑋 × 𝐴))
112		rnss 5952	. . . . . . . . . . . . 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 6193	. . . . . . . . . . 11 ⊢ ran (𝑋 × 𝐴) ⊆ 𝐴
116	114, 115	sstrdi 4007	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ran ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ 𝐴)
117	109, 116	sstrid 4006	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦}) ⊆ 𝐴)
118	79, 117	eqsstrrd 4034	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐶 ⊆ 𝐴)
119	108, 118	feqresmpt 6977	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶) = (𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐶 ↦ (𝐹‘𝑧)))
120	107, 119	eqtr4d 2777	. . . . . 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 7446	. . . . 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 5247	. . . 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 2902	. . . . 5 ⊢ Ⅎ𝑦(𝐺 Σg (𝐹 ↾ 𝐶))
124		nfcv 2902	. . . . . 6 ⊢ Ⅎ𝑥𝐺
125		nfcv 2902	. . . . . 6 ⊢ Ⅎ𝑥 Σg
126		nfcv 2902	. . . . . . 7 ⊢ Ⅎ𝑥𝐹
127	126, 77	nfres 6001	. . . . . 6 ⊢ Ⅎ𝑥(𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶)
128	124, 125, 127	nfov 7460	. . . . 5 ⊢ Ⅎ𝑥(𝐺 Σg (𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶))
129	78	reseq2d 5999	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐹 ↾ 𝐶) = (𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶))
130	129	oveq2d 7446	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐺 Σg (𝐹 ↾ 𝐶)) = (𝐺 Σg (𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶)))
131	123, 128, 130	cbvmpt 5258	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ (𝐺 Σg (𝐹 ↾ 𝐶))) = (𝑦 ∈ 𝑋 ↦ (𝐺 Σg (𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶)))
132	122, 131	eqtr4di 2792	. . 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 7446	. 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 2778	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥 ∈ 𝑋 ↦ (𝐺 Σg (𝐹 ↾ 𝐶)))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1536   ∈ wcel 2105   ≠ wne 2937  ∀wral 3058  ∃wrex 3067  Vcvv 3477  ⦋csb 3907   ⊆ wss 3962  ∅c0 4338  {csn 4630  ⟨cop 4636  ∪ ciun 4995  Disj wdisj 5114   class class class wbr 5147   ↦ cmpt 5230   × cxp 5686  dom cdm 5688  ran crn 5689   ↾ cres 5690   “ cima 5691   ∘ ccom 5692  Rel wrel 5693   Fn wfn 6557  ⟶wf 6558  –onto→wfo 6560  ‘cfv 6562  (class class class)co 7430  2^nd c2nd 8011   finSupp cfsupp 9398  Basecbs 17244  0_gc0g 17485   Σ_g cgsu 17486  CMndccmn 19812

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-disj 5115  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-fzo 13691  df-seq 14039  df-hash 14366  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-0g 17487  df-gsum 17488  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-submnd 18809  df-mulg 19098  df-cntz 19347  df-cmn 19814

This theorem is referenced by:  elrspunidl  33435