Theorem gsum2d2lem 20192

Description: Lemma for gsum2d2 20193: 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 2760	. . . 4 ⊢ (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) = (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)
2	1	mpofun 7544	. . 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 3205	. . . . 5 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐶 𝑋 ∈ 𝐵)
7	1	fmpox 8078	. . . . 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 1946	. . . . . 6 ⊢ Ⅎ𝑗𝜑
10		nfiu1 4985	. . . . . . . 8 ⊢ Ⅎ𝑗∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)
11		nfcv 2922	. . . . . . . 8 ⊢ Ⅎ𝑗𝑈
12	10, 11	nfdif 4066	. . . . . . 7 ⊢ Ⅎ𝑗(∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈)
13	12	nfcri 2914	. . . . . 6 ⊢ Ⅎ𝑗 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈)
14	9, 13	nfan 1931	. . . . 5 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈))
15		nfmpo1 7500	. . . . . . 7 ⊢ Ⅎ𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)
16		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑗𝑧
17	15, 16	nffv 6895	. . . . . 6 ⊢ Ⅎ𝑗((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑧)
18	17	nfeq1 2937	. . . . 5 ⊢ Ⅎ𝑗((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑧) = 0
19		relxp 5668	. . . . . . . 8 ⊢ Rel ({𝑗} × 𝐶)
20	19	rgenw 3080	. . . . . . 7 ⊢ ∀𝑗 ∈ 𝐴 Rel ({𝑗} × 𝐶)
21		reliun 5793	. . . . . . 7 ⊢ (Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ↔ ∀𝑗 ∈ 𝐴 Rel ({𝑗} × 𝐶))
22	20, 21	mpbir 234	. . . . . 6 ⊢ Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)
23		eldifi 4067	. . . . . . 7 ⊢ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) → 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶))
24	23	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶))
25		elrel 5773	. . . . . 6 ⊢ ((Rel ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶)) → ∃𝑗∃𝑘 𝑧 = ⟨𝑗, 𝑘⟩)
26	22, 24, 25	sylancr 599	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈)) → ∃𝑗∃𝑘 𝑧 = ⟨𝑗, 𝑘⟩)
27		nfv 1946	. . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈))
28		nfmpo2 7501	. . . . . . . 8 ⊢ Ⅎ𝑘(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)
29		nfcv 2922	. . . . . . . 8 ⊢ Ⅎ𝑘𝑧
30	28, 29	nffv 6895	. . . . . . 7 ⊢ Ⅎ𝑘((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑧)
31	30	nfeq1 2937	. . . . . 6 ⊢ Ⅎ𝑘((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑧) = 0
32		simprr 785	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑧 = ⟨𝑗, 𝑘⟩)
33	32	fveq2d 6889	. . . . . . . 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 7423	. . . . . . . . 9 ⊢ (𝑗(𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)𝑘) = ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘⟨𝑗, 𝑘⟩)
35		simprl 783	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈))
36	32, 35	eqeltrrd 2861	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ⟨𝑗, 𝑘⟩ ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈))
37	36	eldifad 4051	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ⟨𝑗, 𝑘⟩ ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶))
38		opeliunxp 5717	. . . . . . . . . . . 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 500	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑗 ∈ 𝐴)
41	39	simprd 501	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑘 ∈ 𝐶)
42	39, 5	syldan 603	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑋 ∈ 𝐵)
43	1	ovmpt4g 7567	. . . . . . . . . 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 2809	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘⟨𝑗, 𝑘⟩) = 𝑋)
46		eldifn 4068	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) → ¬ 𝑧 ∈ 𝑈)
47	46	ad2antrl 741	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ¬ 𝑧 ∈ 𝑈)
48	32	eleq1d 2845	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → (𝑧 ∈ 𝑈 ↔ ⟨𝑗, 𝑘⟩ ∈ 𝑈))
49		df-br 5103	. . . . . . . . . . . 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 521	. . . . . . . . 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 603	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → 𝑋 = 0 )
55	33, 45, 54	3eqtrd 2799	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ (∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐶) ∖ 𝑈) ∧ 𝑧 = ⟨𝑗, 𝑘⟩)) → ((𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋)‘𝑧) = 0 )
56	55	expr 462	. . . . . 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 8214	. . 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 3154	. . . 4 ⊢ (𝜑 → ∀𝑗 ∈ 𝐴 𝐶 ∈ 𝑊)
64	1	mpoexxg 8088	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑗 ∈ 𝐴 𝐶 ∈ 𝑊) → (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) ∈ V)
65	61, 63, 64	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) ∈ V)
66		gsum2d2.z	. . . . 5 ⊢ 0 = (0g‘𝐺)
67	66	fvexi 6899	. . . 4 ⊢ 0 ∈ V
68	67	a1i 11	. . 3 ⊢ (𝜑 → 0 ∈ V)
69		isfsupp 9360	. . 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 596	. 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 726	1 ⊢ (𝜑 → (𝑗 ∈ 𝐴, 𝑘 ∈ 𝐶 ↦ 𝑋) finSupp 0 )

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∀wral 3076  Vcvv 3450   ∖ cdif 4045  {csn 4584  ⟨cop 4834  ∪ ciun 4950   class class class wbr 5102   × cxp 5648  Rel wrel 5655  Fun wfun 6532  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422   supp csupp 8179  Fincfn 8976   finSupp cfsupp 9356  Basecbs 17392  0_gc0g 17615  CMndccmn 19999

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  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 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-supp 8180  df-1o 8479  df-en 8977  df-fin 8980  df-fsupp 9357

This theorem is used by:  gsum2d2  20193  gsumcom2  20194