Theorem gsum2d2lem 19833

Description: Lemma for gsum2d2 19834: 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 2733	. . . 4 ⊢ (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) = (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)
2	1	mpofun 7527	. . 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 3201	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐶 𝑋 ∈ 𝐵)
7	1	fmpox 8048	. . . . 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 1918	. . . . . 6 ⊢ Ⅎ𝑗𝜑
10		nfiu1 5030	. . . . . . . 8 ⊢ Ⅎ𝑗∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)
11		nfcv 2904	. . . . . . . 8 ⊢ Ⅎ𝑗𝑈
12	10, 11	nfdif 4124	. . . . . . 7 ⊢ Ⅎ𝑗(∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈)
13	12	nfcri 2891	. . . . . 6 ⊢ Ⅎ𝑗 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈)
14	9, 13	nfan 1903	. . . . 5 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈))
15		nfmpo1 7484	. . . . . . 7 ⊢ Ⅎ𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)
16		nfcv 2904	. . . . . . 7 ⊢ Ⅎ𝑗𝑧
17	15, 16	nffv 6898	. . . . . 6 ⊢ Ⅎ𝑗((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑧)
18	17	nfeq1 2919	. . . . 5 ⊢ Ⅎ𝑗((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑧) = 0
19		relxp 5693	. . . . . . . 8 ⊢ Rel ({𝑗} × 𝐶)
20	19	rgenw 3066	. . . . . . 7 ⊢ ∀𝑗 ∈ 𝐴 Rel ({𝑗} × 𝐶)
21		reliun 5814	. . . . . . 7 ⊢ (Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ∀𝑗 ∈ 𝐴 Rel ({𝑗} × 𝐶))
22	20, 21	mpbir 230	. . . . . 6 ⊢ Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)
23		eldifi 4125	. . . . . . 7 ⊢ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) → 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶))
24	23	adantl 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶))
25		elrel 5796	. . . . . 6 ⊢ ((Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)) → ∃𝑗∃𝑘 𝑧 = ⟨𝑗, 𝑘⟩)
26	22, 24, 25	sylancr 588	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → ∃𝑗∃𝑘 𝑧 = ⟨𝑗, 𝑘⟩)
27		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈))
28		nfmpo2 7485	. . . . . . . 8 ⊢ Ⅎ𝑘(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)
29		nfcv 2904	. . . . . . . 8 ⊢ Ⅎ𝑘𝑧
30	28, 29	nffv 6898	. . . . . . 7 ⊢ Ⅎ𝑘((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑧)
31	30	nfeq1 2919	. . . . . 6 ⊢ Ⅎ𝑘((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑧) = 0
32		simprr 772	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑧 = ⟨𝑗, 𝑘⟩)
33	32	fveq2d 6892	. . . . . . . 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 7407	. . . . . . . . 9 ⊢ (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘) = ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘⟨𝑗, 𝑘⟩)
35		simprl 770	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈))
36	32, 35	eqeltrrd 2835	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ⟨𝑗, 𝑘⟩ ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈))
37	36	eldifad 3959	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ⟨𝑗, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶))
38		opeliunxp 5741	. . . . . . . . . . . 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 496	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑗 ∈ 𝐴)
41	39	simprd 497	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑘 ∈ 𝐶)
42	39, 5	syldan 592	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑋 ∈ 𝐵)
43	1	ovmpt4g 7550	. . . . . . . . . 10 ⊢ ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘) = 𝑋)
44	40, 41, 42, 43	syl3anc 1372	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘) = 𝑋)
45	34, 44	eqtr3id 2787	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘⟨𝑗, 𝑘⟩) = 𝑋)
46		eldifn 4126	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) → ¬ 𝑧 ∈ 𝑈)
47	46	ad2antrl 727	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ¬ 𝑧 ∈ 𝑈)
48	32	eleq1d 2819	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → (𝑧 ∈ 𝑈 ↔ ⟨𝑗, 𝑘⟩ ∈ 𝑈))
49		df-br 5148	. . . . . . . . . . . 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 513	. . . . . . . . 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 592	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑋 = 0 )
55	33, 45, 54	3eqtrd 2777	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑧) = 0 )
56	55	expr 458	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → (𝑧 = ⟨𝑗, 𝑘⟩ → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑧) = 0 ))
57	27, 31, 56	exlimd 2212	. . . . 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 2213	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑧) = 0 )
59	8, 58	suppss 8174	. . 3 ⊢ (𝜑 → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) supp 0 ) ⊆ 𝑈)
60	4, 59	ssfid 9263	. 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 3147	. . . 4 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 𝐶 ∈ 𝑊)
64	1	mpoexxg 8057	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑗 ∈ 𝐴 𝐶 ∈ 𝑊) → (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) ∈ V)
65	61, 63, 64	syl2anc 585	. . 3 ⊢ (𝜑 → (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) ∈ V)
66		gsum2d2.z	. . . . 5 ⊢ 0 = (0g‘𝐺)
67	66	fvexi 6902	. . . 4 ⊢ 0 ∈ V
68	67	a1i 11	. . 3 ⊢ (𝜑 → 0 ∈ V)
69		isfsupp 9361	. . 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 585	. 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 712	1 ⊢ (𝜑 → (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) finSupp 0 )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  ∃wex 1782   ∈ wcel 2107  ∀wral 3062  Vcvv 3475   ∖ cdif 3944  {csn 4627  ⟨cop 4633  ∪ ciun 4996   class class class wbr 5147   × cxp 5673  Rel wrel 5680  Fun wfun 6534  ⟶wf 6536  ‘cfv 6540  (class class class)co 7404   ∈ cmpo 7406   supp csupp 8141  Fincfn 8935   finSupp cfsupp 9357  Basecbs 17140  0_gc0g 17381  CMndccmn 19641

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-1st 7970  df-2nd 7971  df-supp 8142  df-1o 8461  df-en 8936  df-fin 8939  df-fsupp 9358

This theorem is referenced by:  gsum2d2  19834  gsumcom2  19835