Theorem gsum2d2lem 20044

Description: Lemma for gsum2d2 20045: 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 2763	. . . 4 ⊢ (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) = (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)
2	1	mpofun 7536	. . 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 3208	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐶 𝑋 ∈ 𝐵)
7	1	fmpox 8065	. . . . 5 ⊢ (∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐶 𝑋 ∈ 𝐵 ↔ (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋):∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)⟶𝐵)
8	6, 7	sylib 221	. . . 4 ⊢ (𝜑 → (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋):∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)⟶𝐵)
9		nfv 1944	. . . . . 6 ⊢ Ⅎ𝑗𝜑
10		nfiu1 4993	. . . . . . . 8 ⊢ Ⅎ𝑗∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)
11		nfcv 2925	. . . . . . . 8 ⊢ Ⅎ𝑗𝑈
12	10, 11	nfdif 4085	. . . . . . 7 ⊢ Ⅎ𝑗(∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈)
13	12	nfcri 2917	. . . . . 6 ⊢ Ⅎ𝑗 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈)
14	9, 13	nfan 1929	. . . . 5 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈))
15		nfmpo1 7492	. . . . . . 7 ⊢ Ⅎ𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)
16		nfcv 2925	. . . . . . 7 ⊢ Ⅎ𝑗𝑧
17	15, 16	nffv 6893	. . . . . 6 ⊢ Ⅎ𝑗((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑧)
18	17	nfeq1 2940	. . . . 5 ⊢ Ⅎ𝑗((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑧) = 0
19		relxp 5681	. . . . . . . 8 ⊢ Rel ({𝑗} × 𝐶)
20	19	rgenw 3083	. . . . . . 7 ⊢ ∀𝑗 ∈ 𝐴 Rel ({𝑗} × 𝐶)
21		reliun 5805	. . . . . . 7 ⊢ (Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ∀𝑗 ∈ 𝐴 Rel ({𝑗} × 𝐶))
22	20, 21	mpbir 234	. . . . . 6 ⊢ Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)
23		eldifi 4086	. . . . . . 7 ⊢ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) → 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶))
24	23	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶))
25		elrel 5786	. . . . . 6 ⊢ ((Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)) → ∃𝑗∃𝑘 𝑧 = ⟨𝑗, 𝑘⟩)
26	22, 24, 25	sylancr 598	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → ∃𝑗∃𝑘 𝑧 = ⟨𝑗, 𝑘⟩)
27		nfv 1944	. . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈))
28		nfmpo2 7493	. . . . . . . 8 ⊢ Ⅎ𝑘(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)
29		nfcv 2925	. . . . . . . 8 ⊢ Ⅎ𝑘𝑧
30	28, 29	nffv 6893	. . . . . . 7 ⊢ Ⅎ𝑘((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑧)
31	30	nfeq1 2940	. . . . . 6 ⊢ Ⅎ𝑘((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑧) = 0
32		simprr 784	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑧 = ⟨𝑗, 𝑘⟩)
33	32	fveq2d 6887	. . . . . . . 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 7415	. . . . . . . . 9 ⊢ (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘) = ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘⟨𝑗, 𝑘⟩)
35		simprl 782	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈))
36	32, 35	eqeltrrd 2864	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ⟨𝑗, 𝑘⟩ ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈))
37	36	eldifad 3918	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ⟨𝑗, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶))
38		opeliunxp 5730	. . . . . . . . . . . 12 ⊢ (⟨𝑗, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶))
39	37, 38	sylib 221	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶))
40	39	simpld 499	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑗 ∈ 𝐴)
41	39	simprd 500	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑘 ∈ 𝐶)
42	39, 5	syldan 602	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑋 ∈ 𝐵)
43	1	ovmpt4g 7559	. . . . . . . . . 10 ⊢ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘) = 𝑋)
44	40, 41, 42, 43	syl3anc 1398	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘) = 𝑋)
45	34, 44	eqtr3id 2812	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘⟨𝑗, 𝑘⟩) = 𝑋)
46		eldifn 4087	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) → ¬ 𝑧 ∈ 𝑈)
47	46	ad2antrl 740	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ¬ 𝑧 ∈ 𝑈)
48	32	eleq1d 2848	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → (𝑧 ∈ 𝑈 ↔ ⟨𝑗, 𝑘⟩ ∈ 𝑈))
49		df-br 5111	. . . . . . . . . . . 12 ⊢ (𝑗𝑈𝑘 ↔ ⟨𝑗, 𝑘⟩ ∈ 𝑈)
50	48, 49	bitr4di 292	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → (𝑧 ∈ 𝑈 ↔ 𝑗𝑈𝑘))
51	47, 50	mtbid 327	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ¬ 𝑗𝑈𝑘)
52	39, 51	jca 520	. . . . . . . . 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 602	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑋 = 0 )
55	33, 45, 54	3eqtrd 2802	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑧) = 0 )
56	55	expr 461	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → (𝑧 = ⟨𝑗, 𝑘⟩ → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑧) = 0 ))
57	27, 31, 56	exlimd 2254	. . . . 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 2255	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑧) = 0 )
59	8, 58	suppss 8191	. . 3 ⊢ (𝜑 → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) supp 0 ) ⊆ 𝑈)
60	4, 59	ssfid 9230	. 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 3157	. . . 4 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 𝐶 ∈ 𝑊)
64	1	mpoexxg 8073	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑗 ∈ 𝐴 𝐶 ∈ 𝑊) → (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) ∈ V)
65	61, 63, 64	syl2anc 595	. . 3 ⊢ (𝜑 → (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) ∈ V)
66		gsum2d2.z	. . . . 5 ⊢ 0 = (0g‘𝐺)
67	66	fvexi 6897	. . . 4 ⊢ 0 ∈ V
68	67	a1i 11	. . 3 ⊢ (𝜑 → 0 ∈ V)
69		isfsupp 9326	. . 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 595	. 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 725	1 ⊢ (𝜑 → (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) finSupp 0 )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   ∖ cdif 3903  {csn 4590  ⟨cop 4596  ∪ ciun 4957   class class class wbr 5110   × cxp 5661  Rel wrel 5668  Fun wfun 6532  ⟶wf 6534  ‘cfv 6538  (class class class)co 7412   ∈ cmpo 7414   supp csupp 8157  Fincfn 8944   finSupp cfsupp 9322  Basecbs 17270  0_gc0g 17493  CMndccmn 19851

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 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

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-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-supp 8158  df-1o 8454  df-en 8945  df-fin 8948  df-fsupp 9323

This theorem is referenced by:  gsum2d2  20045  gsumcom2  20046