Theorem gsumpart 30826

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 2759	. . . 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 30495	. . 3 ⊢ (𝜑 → (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)–1-1-onto→𝐴)
12	1, 2, 3, 4, 5, 6, 11	gsumf1o 19089	. 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))))
13		snex 5293	. . . . . . 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 3911	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ 𝐴)
17	10, 16	eqsstrd 3926	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝑋 𝐶 ⊆ 𝐴)
18		iunss 4927	. . . . . . . . 9 ⊢ (∪ 𝑥 ∈ 𝑋 𝐶 ⊆ 𝐴 ↔ ∀𝑥 ∈ 𝑋 𝐶 ⊆ 𝐴)
19	17, 18	sylib 221	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐶 ⊆ 𝐴)
20	19	r19.21bi 3135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ⊆ 𝐴)
21	15, 20	ssexd 5187	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ V)
22	14, 21	xpexd 7465	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ({𝑥} × 𝐶) ∈ V)
23	22	ralrimiva 3111	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ∈ V)
24		iunexg 7661	. . . 4 ⊢ ((𝑋 ∈ 𝑊 ∧ ∀𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ∈ V) → ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ∈ V)
25	8, 23, 24	syl2anc 588	. . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ∈ V)
26		relxp 5535	. . . . . 6 ⊢ Rel ({𝑥} × 𝐶)
27	26	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Rel ({𝑥} × 𝐶))
28	27	ralrimiva 3111	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 Rel ({𝑥} × 𝐶))
29		reliun 5651	. . . 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 5746	. . . . . 6 ⊢ dom ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) = ∪ 𝑥 ∈ 𝑋 dom ({𝑥} × 𝐶)
32		dmxpss 5993	. . . . . . . 8 ⊢ dom ({𝑥} × 𝐶) ⊆ {𝑥}
33	32	rgenw 3080	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝑋 dom ({𝑥} × 𝐶) ⊆ {𝑥}
34		ss2iun 4894	. . . . . . 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 3922	. . . . 5 ⊢ dom ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ ∪ 𝑥 ∈ 𝑋 {𝑥}
37		iunid 4942	. . . . 5 ⊢ ∪ 𝑥 ∈ 𝑋 {𝑥} = 𝑋
38	36, 37	sseqtri 3924	. . . 4 ⊢ dom ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ 𝑋
39	38	a1i 11	. . 3 ⊢ (𝜑 → dom ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ 𝑋)
40		fo2nd 7707	. . . . . . . 8 ⊢ 2nd :V–onto→V
41		fof 6569	. . . . . . . 8 ⊢ (2nd :V–onto→V → 2nd :V⟶V)
42	40, 41	ax-mp 5	. . . . . . 7 ⊢ 2nd :V⟶V
43		ssv 3912	. . . . . . 7 ⊢ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ V
44		fssres 6522	. . . . . . 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 6491	. . . . . 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 30493	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)
49		imass2 5930	. . . . . . . 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 5910	. . . . . . . . . . 11 ⊢ (2nd “ ∅) = ∅
52		xpeq1 5531	. . . . . . . . . . . . 13 ⊢ (𝑋 = ∅ → (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶) = (∅ × ∪ 𝑥 ∈ 𝑋 𝐶))
53		0xp 5611	. . . . . . . . . . . . 13 ⊢ (∅ × ∪ 𝑥 ∈ 𝑋 𝐶) = ∅
54	52, 53	eqtrdi 2810	. . . . . . . . . . . 12 ⊢ (𝑋 = ∅ → (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶) = ∅)
55	54	imaeq2d 5894	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → (2nd “ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)) = (2nd “ ∅))
56		iuneq1 4892	. . . . . . . . . . . 12 ⊢ (𝑋 = ∅ → ∪ 𝑥 ∈ 𝑋 𝐶 = ∪ 𝑥 ∈ ∅ 𝐶)
57		0iun 4944	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ ∅ 𝐶 = ∅
58	56, 57	eqtrdi 2810	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → ∪ 𝑥 ∈ 𝑋 𝐶 = ∅)
59	51, 55, 58	3eqtr4a 2820	. . . . . . . . . 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 30492	. . . . . . . . . 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 3036	. . . . . . . 8 ⊢ (𝜑 → (2nd “ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)) = ∪ 𝑥 ∈ 𝑋 𝐶)
64	63, 10	eqtrd 2794	. . . . . . 7 ⊢ (𝜑 → (2nd “ (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶)) = 𝐴)
65	50, 64	sseqtrid 3940	. . . . . 6 ⊢ (𝜑 → (2nd “ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)) ⊆ 𝐴)
66		resssxp 6092	. . . . . 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 6849	. . . . 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 587	. . . 4 ⊢ (𝜑 → (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)⟶𝐴)
70	5, 69	fcod 6510	. . 3 ⊢ (𝜑 → (𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)⟶𝐵)
71	7, 4, 8, 9, 10	2ndresdju 30494	. . . 4 ⊢ (𝜑 → (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)):∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)–1-1→𝐴)
72	2	fvexi 6665	. . . . 5 ⊢ 0 ∈ V
73	72	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ V)
74	5, 4	fexd 6974	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
75	6, 71, 73, 74	fsuppco 8884	. . 3 ⊢ (𝜑 → (𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))) finSupp 0 )
76	1, 2, 3, 25, 30, 8, 39, 70, 75	gsum2d 19145	. 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 3825	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
78		csbeq1a 3815	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
79	8, 21, 77, 78	iunsnima2 30466	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦}) = ⦋𝑦 / 𝑥⦌𝐶)
80		df-ov 7146	. . . . . . . . 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 4552	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ {𝑦}
84	83	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → 𝑦 ∈ {𝑦})
85	79	eleq2d 2836	. . . . . . . . . . . . . . 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 5555	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ⟨𝑦, 𝑧⟩ ∈ ({𝑦} × ⦋𝑦 / 𝑥⦌𝐶))
88		nfcv 2917	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥{𝑦}
89	88, 77	nfxp 5550	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥({𝑦} × ⦋𝑦 / 𝑥⦌𝐶)
90	89	nfel2 2935	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥⟨𝑦, 𝑧⟩ ∈ ({𝑦} × ⦋𝑦 / 𝑥⦌𝐶)
91		sneq 4525	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
92	91, 78	xpeq12d 5548	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → ({𝑥} × 𝐶) = ({𝑦} × ⦋𝑦 / 𝑥⦌𝐶))
93	92	eleq2d 2836	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (⟨𝑦, 𝑧⟩ ∈ ({𝑥} × 𝐶) ↔ ⟨𝑦, 𝑧⟩ ∈ ({𝑦} × ⦋𝑦 / 𝑥⦌𝐶)))
94	90, 93	rspce 3528	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝑋 ∧ ⟨𝑦, 𝑧⟩ ∈ ({𝑦} × ⦋𝑦 / 𝑥⦌𝐶)) → ∃𝑥 ∈ 𝑋 ⟨𝑦, 𝑧⟩ ∈ ({𝑥} × 𝐶))
95	82, 87, 94	syl2anc 588	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ∃𝑥 ∈ 𝑋 ⟨𝑦, 𝑧⟩ ∈ ({𝑥} × 𝐶))
96		eliun 4880	. . . . . . . . . . . 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 6745	. . . . . . . . . 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 6671	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ((2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))‘⟨𝑦, 𝑧⟩) = (2nd ‘⟨𝑦, 𝑧⟩))
100		vex 3411	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
101		vex 3411	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
102	100, 101	op2nd 7695	. . . . . . . . . . . 12 ⊢ (2nd ‘⟨𝑦, 𝑧⟩) = 𝑧
103	99, 102	eqtrdi 2810	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ((2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))‘⟨𝑦, 𝑧⟩) = 𝑧)
104	103	fveq2d 6655	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → (𝐹‘((2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶))‘⟨𝑦, 𝑧⟩)) = (𝐹‘𝑧))
105	98, 104	eqtrd 2794	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ((𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))‘⟨𝑦, 𝑧⟩) = (𝐹‘𝑧))
106	80, 105	syl5eq 2806	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦})) → (𝑦(𝐹 ∘ (2nd ↾ ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)))𝑧) = (𝐹‘𝑧))
107	79, 106	mpteq12dva 5109	. . . . . . 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 5905	. . . . . . . . . 10 ⊢ (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦}) ⊆ ran ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)
110	10	xpeq2d 5547	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋 × ∪ 𝑥 ∈ 𝑋 𝐶) = (𝑋 × 𝐴))
111	48, 110	sseqtrid 3940	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ (𝑋 × 𝐴))
112		rnss 5773	. . . . . . . . . . . . 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 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ran ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ ran (𝑋 × 𝐴))
115		rnxpss 5994	. . . . . . . . . . 11 ⊢ ran (𝑋 × 𝐴) ⊆ 𝐴
116	114, 115	sstrdi 3900	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ran ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) ⊆ 𝐴)
117	109, 116	sstrid 3899	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶) “ {𝑦}) ⊆ 𝐴)
118	79, 117	eqsstrrd 3927	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → ⦋𝑦 / 𝑥⦌𝐶 ⊆ 𝐴)
119	108, 118	feqresmpt 6715	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶) = (𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐶 ↦ (𝐹‘𝑧)))
120	107, 119	eqtr4d 2797	. . . . . 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 7159	. . . . 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 5120	. . . 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 2917	. . . . 5 ⊢ Ⅎ𝑦(𝐺 Σg (𝐹 ↾ 𝐶))
124		nfcv 2917	. . . . . 6 ⊢ Ⅎ𝑥𝐺
125		nfcv 2917	. . . . . 6 ⊢ Ⅎ𝑥 Σg
126		nfcv 2917	. . . . . . 7 ⊢ Ⅎ𝑥𝐹
127	126, 77	nfres 5818	. . . . . 6 ⊢ Ⅎ𝑥(𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶)
128	124, 125, 127	nfov 7173	. . . . 5 ⊢ Ⅎ𝑥(𝐺 Σg (𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶))
129	78	reseq2d 5816	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐹 ↾ 𝐶) = (𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶))
130	129	oveq2d 7159	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐺 Σg (𝐹 ↾ 𝐶)) = (𝐺 Σg (𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶)))
131	123, 128, 130	cbvmpt 5126	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ (𝐺 Σg (𝐹 ↾ 𝐶))) = (𝑦 ∈ 𝑋 ↦ (𝐺 Σg (𝐹 ↾ ⦋𝑦 / 𝑥⦌𝐶)))
132	122, 131	eqtr4di 2812	. . 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 7159	. 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 2798	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥 ∈ 𝑋 ↦ (𝐺 Σg (𝐹 ↾ 𝐶)))))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112   ≠ wne 2949  ∀wral 3068  ∃wrex 3069  Vcvv 3407  ⦋csb 3801   ⊆ wss 3854  ∅c0 4221  {csn 4515  ⟨cop 4521  ∪ ciun 4876  Disj wdisj 4990   class class class wbr 5025   ↦ cmpt 5105   × cxp 5515  dom cdm 5517  ran crn 5518   ↾ cres 5519   “ cima 5520   ∘ ccom 5521  Rel wrel 5522   Fn wfn 6323  ⟶wf 6324  –onto→wfo 6326  ‘cfv 6328  (class class class)co 7143  2^nd c2nd 7685   finSupp cfsupp 8851  Basecbs 16526  0_gc0g 16756   Σ_g cgsu 16757  CMndccmn 18958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5149  ax-sep 5162  ax-nul 5169  ax-pow 5227  ax-pr 5291  ax-un 7452  ax-cnex 10616  ax-resscn 10617  ax-1cn 10618  ax-icn 10619  ax-addcl 10620  ax-addrcl 10621  ax-mulcl 10622  ax-mulrcl 10623  ax-mulcom 10624  ax-addass 10625  ax-mulass 10626  ax-distr 10627  ax-i2m1 10628  ax-1ne0 10629  ax-1rid 10630  ax-rnegex 10631  ax-rrecex 10632  ax-cnre 10633  ax-pre-lttri 10634  ax-pre-lttrn 10635  ax-pre-ltadd 10636  ax-pre-mulgt0 10637

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ne 2950  df-nel 3054  df-ral 3073  df-rex 3074  df-reu 3075  df-rmo 3076  df-rab 3077  df-v 3409  df-sbc 3694  df-csb 3802  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-pss 3873  df-nul 4222  df-if 4414  df-pw 4489  df-sn 4516  df-pr 4518  df-tp 4520  df-op 4522  df-uni 4792  df-int 4832  df-iun 4878  df-iin 4879  df-disj 4991  df-br 5026  df-opab 5088  df-mpt 5106  df-tr 5132  df-id 5423  df-eprel 5428  df-po 5436  df-so 5437  df-fr 5476  df-se 5477  df-we 5478  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-rn 5528  df-res 5529  df-ima 5530  df-pred 6119  df-ord 6165  df-on 6166  df-lim 6167  df-suc 6168  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7101  df-ov 7146  df-oprab 7147  df-mpo 7148  df-of 7398  df-om 7573  df-1st 7686  df-2nd 7687  df-supp 7829  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-oadd 8109  df-er 8292  df-en 8521  df-dom 8522  df-sdom 8523  df-fin 8524  df-fsupp 8852  df-oi 8992  df-card 9386  df-pnf 10700  df-mnf 10701  df-xr 10702  df-ltxr 10703  df-le 10704  df-sub 10895  df-neg 10896  df-nn 11660  df-2 11722  df-n0 11920  df-z 12006  df-uz 12268  df-fz 12925  df-fzo 13068  df-seq 13404  df-hash 13726  df-ndx 16529  df-slot 16530  df-base 16532  df-sets 16533  df-ress 16534  df-plusg 16621  df-0g 16758  df-gsum 16759  df-mre 16900  df-mrc 16901  df-acs 16903  df-mgm 17903  df-sgrp 17952  df-mnd 17963  df-submnd 18008  df-mulg 18277  df-cntz 18499  df-cmn 18960

This theorem is referenced by:  elrspunidl  31112