Theorem gsum2d2lem 19574

Description: Lemma for gsum2d2 19575: show the function is finitely supported. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 9-Jun-2019.)

Hypotheses

Ref	Expression
gsum2d2.b	⊢ 𝐵 = (Base‘𝐺)
gsum2d2.z	⊢ 0 = (0g‘𝐺)
gsum2d2.g	⊢ (𝜑 → 𝐺 ∈ CMnd)
gsum2d2.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
gsum2d2.r	⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ 𝑊)
gsum2d2.f	⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → 𝑋 ∈ 𝐵)
gsum2d2.u	⊢ (𝜑 → 𝑈 ∈ Fin)
gsum2d2.n	⊢ ((𝜑 ∧ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 )

Assertion

Ref	Expression
gsum2d2lem	⊢ (𝜑 → (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) finSupp 0 )

Distinct variable groups:   𝑗,𝑘,𝐵   𝜑,𝑗,𝑘   𝐴,𝑗,𝑘   𝑗,𝐺,𝑘   𝑈,𝑗,𝑘   𝐶,𝑘   𝑗,𝑉   0 ,𝑗,𝑘

Allowed substitution hints:   𝐶(𝑗)   𝑉(𝑘)   𝑊(𝑗,𝑘)   𝑋(𝑗,𝑘)

Proof of Theorem gsum2d2lem

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2738	. . . 4 ⊢ (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) = (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)
2	1	mpofun 7398	. . 3 ⊢ Fun (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)
3	2	a1i 11	. 2 ⊢ (𝜑 → Fun (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋))
4		gsum2d2.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ Fin)
5		gsum2d2.f	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶)) → 𝑋 ∈ 𝐵)
6	5	ralrimivva 3123	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐶 𝑋 ∈ 𝐵)
7	1	fmpox 7907	. . . . 5 ⊢ (∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐶 𝑋 ∈ 𝐵 ↔ (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋):∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)⟶𝐵)
8	6, 7	sylib 217	. . . 4 ⊢ (𝜑 → (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋):∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)⟶𝐵)
9		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑗𝜑
10		nfiu1 4958	. . . . . . . 8 ⊢ Ⅎ𝑗∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)
11		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑗𝑈
12	10, 11	nfdif 4060	. . . . . . 7 ⊢ Ⅎ𝑗(∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈)
13	12	nfcri 2894	. . . . . 6 ⊢ Ⅎ𝑗 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈)
14	9, 13	nfan 1902	. . . . 5 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈))
15		nfmpo1 7355	. . . . . . 7 ⊢ Ⅎ𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)
16		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑗𝑧
17	15, 16	nffv 6784	. . . . . 6 ⊢ Ⅎ𝑗((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑧)
18	17	nfeq1 2922	. . . . 5 ⊢ Ⅎ𝑗((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑧) = 0
19		relxp 5607	. . . . . . . 8 ⊢ Rel ({𝑗} × 𝐶)
20	19	rgenw 3076	. . . . . . 7 ⊢ ∀𝑗 ∈ 𝐴 Rel ({𝑗} × 𝐶)
21		reliun 5726	. . . . . . 7 ⊢ (Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ∀𝑗 ∈ 𝐴 Rel ({𝑗} × 𝐶))
22	20, 21	mpbir 230	. . . . . 6 ⊢ Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)
23		eldifi 4061	. . . . . . 7 ⊢ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) → 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶))
24	23	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶))
25		elrel 5708	. . . . . 6 ⊢ ((Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)) → ∃𝑗∃𝑘 𝑧 = ⟨𝑗, 𝑘⟩)
26	22, 24, 25	sylancr 587	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → ∃𝑗∃𝑘 𝑧 = ⟨𝑗, 𝑘⟩)
27		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈))
28		nfmpo2 7356	. . . . . . . 8 ⊢ Ⅎ𝑘(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)
29		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑘𝑧
30	28, 29	nffv 6784	. . . . . . 7 ⊢ Ⅎ𝑘((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑧)
31	30	nfeq1 2922	. . . . . 6 ⊢ Ⅎ𝑘((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑧) = 0
32		simprr 770	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑧 = ⟨𝑗, 𝑘⟩)
33	32	fveq2d 6778	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑧) = ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘⟨𝑗, 𝑘⟩))
34		df-ov 7278	. . . . . . . . 9 ⊢ (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘) = ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘⟨𝑗, 𝑘⟩)
35		simprl 768	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈))
36	32, 35	eqeltrrd 2840	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ⟨𝑗, 𝑘⟩ ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈))
37	36	eldifad 3899	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ⟨𝑗, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶))
38		opeliunxp 5654	. . . . . . . . . . . 12 ⊢ (⟨𝑗, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶))
39	37, 38	sylib 217	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶))
40	39	simpld 495	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑗 ∈ 𝐴)
41	39	simprd 496	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑘 ∈ 𝐶)
42	39, 5	syldan 591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑋 ∈ 𝐵)
43	1	ovmpt4g 7420	. . . . . . . . . 10 ⊢ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘) = 𝑋)
44	40, 41, 42, 43	syl3anc 1370	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘) = 𝑋)
45	34, 44	eqtr3id 2792	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘⟨𝑗, 𝑘⟩) = 𝑋)
46		eldifn 4062	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) → ¬ 𝑧 ∈ 𝑈)
47	46	ad2antrl 725	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ¬ 𝑧 ∈ 𝑈)
48	32	eleq1d 2823	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → (𝑧 ∈ 𝑈 ↔ ⟨𝑗, 𝑘⟩ ∈ 𝑈))
49		df-br 5075	. . . . . . . . . . . 12 ⊢ (𝑗𝑈𝑘 ↔ ⟨𝑗, 𝑘⟩ ∈ 𝑈)
50	48, 49	bitr4di 289	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → (𝑧 ∈ 𝑈 ↔ 𝑗𝑈𝑘))
51	47, 50	mtbid 324	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ¬ 𝑗𝑈𝑘)
52	39, 51	jca 512	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) ∧ ¬ 𝑗𝑈𝑘))
53		gsum2d2.n	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 )
54	52, 53	syldan 591	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑋 = 0 )
55	33, 45, 54	3eqtrd 2782	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑧) = 0 )
56	55	expr 457	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → (𝑧 = ⟨𝑗, 𝑘⟩ → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑧) = 0 ))
57	27, 31, 56	exlimd 2211	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → (∃𝑘 𝑧 = ⟨𝑗, 𝑘⟩ → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑧) = 0 ))
58	14, 18, 26, 57	exlimimdd 2212	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑧) = 0 )
59	8, 58	suppss 8010	. . 3 ⊢ (𝜑 → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) supp 0 ) ⊆ 𝑈)
60	4, 59	ssfid 9042	. 2 ⊢ (𝜑 → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) supp 0 ) ∈ Fin)
61		gsum2d2.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
62		gsum2d2.r	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ 𝑊)
63	62	ralrimiva 3103	. . . 4 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 𝐶 ∈ 𝑊)
64	1	mpoexxg 7916	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑗 ∈ 𝐴 𝐶 ∈ 𝑊) → (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) ∈ V)
65	61, 63, 64	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) ∈ V)
66		gsum2d2.z	. . . . 5 ⊢ 0 = (0g‘𝐺)
67	66	fvexi 6788	. . . 4 ⊢ 0 ∈ V
68	67	a1i 11	. . 3 ⊢ (𝜑 → 0 ∈ V)
69		isfsupp 9132	. . 3 ⊢ (((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) ∈ V ∧ 0 ∈ V) → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) finSupp 0 ↔ (Fun (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) ∧ ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) supp 0 ) ∈ Fin)))
70	65, 68, 69	syl2anc 584	. 2 ⊢ (𝜑 → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) finSupp 0 ↔ (Fun (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) ∧ ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) supp 0 ) ∈ Fin)))
71	3, 60, 70	mpbir2and 710	1 ⊢ (𝜑 → (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) finSupp 0 )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∀wral 3064  Vcvv 3432   ∖ cdif 3884  {csn 4561  ⟨cop 4567  ∪ ciun 4924   class class class wbr 5074   × cxp 5587  Rel wrel 5594  Fun wfun 6427  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277   supp csupp 7977  Fincfn 8733   finSupp cfsupp 9128  Basecbs 16912  0_gc0g 17150  CMndccmn 19386

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-1o 8297  df-en 8734  df-fin 8737  df-fsupp 9129

This theorem is referenced by:  gsum2d2  19575  gsumcom2  19576