Theorem evlslem4 22011

Description: The support of a tensor product of ring element families is contained in the product of the supports. (Contributed by Stefan O'Rear, 8-Mar-2015.) (Revised by AV, 18-Jul-2019.)

Hypotheses

Ref	Expression
evlslem4.b	⊢ 𝐵 = (Base‘𝑅)
evlslem4.z	⊢ 0 = (0g‘𝑅)
evlslem4.t	⊢ · = (.r‘𝑅)
evlslem4.r	⊢ (𝜑 → 𝑅 ∈ Ring)
evlslem4.x	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑋 ∈ 𝐵)
evlslem4.y	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → 𝑌 ∈ 𝐵)
evlslem4.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
evlslem4.j	⊢ (𝜑 → 𝐽 ∈ 𝑊)

Assertion

Ref	Expression
evlslem4	⊢ (𝜑 → ((𝑥 ∈ 𝐼, 𝑦 ∈ 𝐽 ↦ (𝑋 · 𝑌)) supp 0 ) ⊆ (((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 ) × ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))

Distinct variable groups:   𝑥,𝑦,𝐼   𝑥,𝐽,𝑦   𝜑,𝑥,𝑦   𝑦,𝑋   𝑥,𝐵,𝑦   𝑥, · ,𝑦   𝑥,𝑌

Allowed substitution hints:   𝑅(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)   𝑋(𝑥)   𝑌(𝑦)   0 (𝑥,𝑦)

Proof of Theorem evlslem4

Dummy variables 𝑖 𝑗 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1137	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽) → 𝑥 ∈ 𝐼)
2		evlslem4.x	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑋 ∈ 𝐵)
3	2	3adant3 1132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽) → 𝑋 ∈ 𝐵)
4		eqid 2731	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐼 ↦ 𝑋) = (𝑥 ∈ 𝐼 ↦ 𝑋)
5	4	fvmpt2 6940	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵) → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) = 𝑋)
6	1, 3, 5	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽) → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) = 𝑋)
7		simp3 1138	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽) → 𝑦 ∈ 𝐽)
8		evlslem4.y	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → 𝑌 ∈ 𝐵)
9		eqid 2731	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐽 ↦ 𝑌) = (𝑦 ∈ 𝐽 ↦ 𝑌)
10	9	fvmpt2 6940	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐽 ∧ 𝑌 ∈ 𝐵) → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦) = 𝑌)
11	7, 8, 10	3imp3i2an 1346	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽) → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦) = 𝑌)
12	6, 11	oveq12d 7364	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽) → (((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦)) = (𝑋 · 𝑌))
13	12	mpoeq3dva 7423	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐼, 𝑦 ∈ 𝐽 ↦ (((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦))) = (𝑥 ∈ 𝐼, 𝑦 ∈ 𝐽 ↦ (𝑋 · 𝑌)))
14		nfcv 2894	. . . . . 6 ⊢ Ⅎ𝑖(((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦))
15		nfcv 2894	. . . . . 6 ⊢ Ⅎ𝑗(((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦))
16		nffvmpt1 6833	. . . . . . 7 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖)
17		nfcv 2894	. . . . . . 7 ⊢ Ⅎ𝑥 ·
18		nfcv 2894	. . . . . . 7 ⊢ Ⅎ𝑥((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗)
19	16, 17, 18	nfov 7376	. . . . . 6 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗))
20		nfcv 2894	. . . . . . 7 ⊢ Ⅎ𝑦((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖)
21		nfcv 2894	. . . . . . 7 ⊢ Ⅎ𝑦 ·
22		nffvmpt1 6833	. . . . . . 7 ⊢ Ⅎ𝑦((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗)
23	20, 21, 22	nfov 7376	. . . . . 6 ⊢ Ⅎ𝑦(((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗))
24		fveq2 6822	. . . . . . 7 ⊢ (𝑥 = 𝑖 → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) = ((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖))
25		fveq2 6822	. . . . . . 7 ⊢ (𝑦 = 𝑗 → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦) = ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗))
26	24, 25	oveqan12d 7365	. . . . . 6 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦)) = (((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗)))
27	14, 15, 19, 23, 26	cbvmpo 7440	. . . . 5 ⊢ (𝑥 ∈ 𝐼, 𝑦 ∈ 𝐽 ↦ (((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦))) = (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ (((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗)))
28		vex 3440	. . . . . . . 8 ⊢ 𝑖 ∈ V
29		vex 3440	. . . . . . . 8 ⊢ 𝑗 ∈ V
30	28, 29	eqop2 7964	. . . . . . 7 ⊢ (𝑧 = ⟨𝑖, 𝑗⟩ ↔ (𝑧 ∈ (V × V) ∧ ((1st ‘𝑧) = 𝑖 ∧ (2nd ‘𝑧) = 𝑗)))
31		fveq2 6822	. . . . . . . 8 ⊢ ((1st ‘𝑧) = 𝑖 → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) = ((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖))
32		fveq2 6822	. . . . . . . 8 ⊢ ((2nd ‘𝑧) = 𝑗 → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧)) = ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗))
33	31, 32	oveqan12d 7365	. . . . . . 7 ⊢ (((1st ‘𝑧) = 𝑖 ∧ (2nd ‘𝑧) = 𝑗) → (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧))) = (((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗)))
34	30, 33	simplbiim 504	. . . . . 6 ⊢ (𝑧 = ⟨𝑖, 𝑗⟩ → (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧))) = (((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗)))
35	34	mpompt 7460	. . . . 5 ⊢ (𝑧 ∈ (𝐼 × 𝐽) ↦ (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧)))) = (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 ↦ (((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗)))
36	27, 35	eqtr4i 2757	. . . 4 ⊢ (𝑥 ∈ 𝐼, 𝑦 ∈ 𝐽 ↦ (((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦))) = (𝑧 ∈ (𝐼 × 𝐽) ↦ (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧))))
37	13, 36	eqtr3di 2781	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼, 𝑦 ∈ 𝐽 ↦ (𝑋 · 𝑌)) = (𝑧 ∈ (𝐼 × 𝐽) ↦ (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧)))))
38	37	oveq1d 7361	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐼, 𝑦 ∈ 𝐽 ↦ (𝑋 · 𝑌)) supp 0 ) = ((𝑧 ∈ (𝐼 × 𝐽) ↦ (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧)))) supp 0 ))
39		difxp 6111	. . . . . 6 ⊢ ((𝐼 × 𝐽) ∖ (((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 ) × ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 ))) = (((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽) ∪ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 ))))
40	39	eleq2i 2823	. . . . 5 ⊢ (𝑧 ∈ ((𝐼 × 𝐽) ∖ (((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 ) × ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 ))) ↔ 𝑧 ∈ (((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽) ∪ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))))
41		elun 4100	. . . . 5 ⊢ (𝑧 ∈ (((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽) ∪ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))) ↔ (𝑧 ∈ ((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽) ∨ 𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))))
42	40, 41	bitri 275	. . . 4 ⊢ (𝑧 ∈ ((𝐼 × 𝐽) ∖ (((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 ) × ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 ))) ↔ (𝑧 ∈ ((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽) ∨ 𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))))
43		xp1st 7953	. . . . . . . 8 ⊢ (𝑧 ∈ ((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽) → (1st ‘𝑧) ∈ (𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )))
44	2	fmpttd 7048	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝑋):𝐼⟶𝐵)
45		ssidd 3953	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 ) ⊆ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 ))
46		evlslem4.i	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
47		evlslem4.z	. . . . . . . . . . 11 ⊢ 0 = (0g‘𝑅)
48	47	fvexi 6836	. . . . . . . . . 10 ⊢ 0 ∈ V
49	48	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ V)
50	44, 45, 46, 49	suppssr 8125	. . . . . . . 8 ⊢ ((𝜑 ∧ (1st ‘𝑧) ∈ (𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 ))) → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) = 0 )
51	43, 50	sylan2 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽)) → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) = 0 )
52	51	oveq1d 7361	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽)) → (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧))) = ( 0 · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧))))
53		evlslem4.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring)
54	8	fmpttd 7048	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ 𝑌):𝐽⟶𝐵)
55		xp2nd 7954	. . . . . . . 8 ⊢ (𝑧 ∈ ((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽) → (2nd ‘𝑧) ∈ 𝐽)
56		ffvelcdm 7014	. . . . . . . 8 ⊢ (((𝑦 ∈ 𝐽 ↦ 𝑌):𝐽⟶𝐵 ∧ (2nd ‘𝑧) ∈ 𝐽) → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧)) ∈ 𝐵)
57	54, 55, 56	syl2an 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽)) → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧)) ∈ 𝐵)
58		evlslem4.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅)
59		evlslem4.t	. . . . . . . 8 ⊢ · = (.r‘𝑅)
60	58, 59, 47	ringlz 20211	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧)) ∈ 𝐵) → ( 0 · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧))) = 0 )
61	53, 57, 60	syl2an2r 685	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽)) → ( 0 · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧))) = 0 )
62	52, 61	eqtrd 2766	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽)) → (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧))) = 0 )
63		xp2nd 7954	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 ))) → (2nd ‘𝑧) ∈ (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))
64		ssidd 3953	. . . . . . . . 9 ⊢ (𝜑 → ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 ) ⊆ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 ))
65		evlslem4.j	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ 𝑊)
66	54, 64, 65, 49	suppssr 8125	. . . . . . . 8 ⊢ ((𝜑 ∧ (2nd ‘𝑧) ∈ (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 ))) → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧)) = 0 )
67	63, 66	sylan2 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))) → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧)) = 0 )
68	67	oveq2d 7362	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))) → (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧))) = (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · 0 ))
69		xp1st 7953	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 ))) → (1st ‘𝑧) ∈ 𝐼)
70		ffvelcdm 7014	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐼 ↦ 𝑋):𝐼⟶𝐵 ∧ (1st ‘𝑧) ∈ 𝐼) → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) ∈ 𝐵)
71	44, 69, 70	syl2an 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))) → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) ∈ 𝐵)
72	58, 59, 47	ringrz 20212	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ ((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) ∈ 𝐵) → (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · 0 ) = 0 )
73	53, 71, 72	syl2an2r 685	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))) → (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · 0 ) = 0 )
74	68, 73	eqtrd 2766	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))) → (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧))) = 0 )
75	62, 74	jaodan 959	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ ((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽) ∨ 𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 ))))) → (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧))) = 0 )
76	42, 75	sylan2b 594	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐼 × 𝐽) ∖ (((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 ) × ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))) → (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧))) = 0 )
77	46, 65	xpexd 7684	. . 3 ⊢ (𝜑 → (𝐼 × 𝐽) ∈ V)
78	76, 77	suppss2 8130	. 2 ⊢ (𝜑 → ((𝑧 ∈ (𝐼 × 𝐽) ↦ (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧)))) supp 0 ) ⊆ (((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 ) × ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))
79	38, 78	eqsstrd 3964	1 ⊢ (𝜑 → ((𝑥 ∈ 𝐼, 𝑦 ∈ 𝐽 ↦ (𝑋 · 𝑌)) supp 0 ) ⊆ (((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 ) × ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ∖ cdif 3894   ∪ cun 3895   ⊆ wss 3897  ⟨cop 4579   ↦ cmpt 5170   × cxp 5612  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  1^st c1st 7919  2^nd c2nd 7920   supp csupp 8090  Basecbs 17120  ._rcmulr 17162  0_gc0g 17343  Ringcrg 20151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-supp 8091  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-plusg 17174  df-0g 17345  df-mgm 18548  df-sgrp 18627  df-mnd 18643  df-grp 18849  df-minusg 18850  df-cmn 19694  df-abl 19695  df-mgp 20059  df-rng 20071  df-ur 20100  df-ring 20153

This theorem is referenced by:  evlslem2  22014