Theorem evlslem4 19329

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		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑖(((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦))
2		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑗(((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦))
3		nffvmpt1 6111	. . . . . . 7 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖)
4		nfcv 2751	. . . . . . 7 ⊢ Ⅎ𝑥 ·
5		nfcv 2751	. . . . . . 7 ⊢ Ⅎ𝑥((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗)
6	3, 4, 5	nfov 6575	. . . . . 6 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗))
7		nfcv 2751	. . . . . . 7 ⊢ Ⅎ𝑦((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖)
8		nfcv 2751	. . . . . . 7 ⊢ Ⅎ𝑦 ·
9		nffvmpt1 6111	. . . . . . 7 ⊢ Ⅎ𝑦((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗)
10	7, 8, 9	nfov 6575	. . . . . 6 ⊢ Ⅎ𝑦(((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗))
11		fveq2 6103	. . . . . . 7 ⊢ (𝑥 = 𝑖 → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) = ((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖))
12		fveq2 6103	. . . . . . 7 ⊢ (𝑦 = 𝑗 → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦) = ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗))
13	11, 12	oveqan12d 6568	. . . . . 6 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦)) = (((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗)))
14	1, 2, 6, 10, 13	cbvmpt2 6632	. . . . 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)))
15		vex 3176	. . . . . . . 8 ⊢ 𝑖 ∈ V
16		vex 3176	. . . . . . . 8 ⊢ 𝑗 ∈ V
17	15, 16	eqop2 7100	. . . . . . 7 ⊢ (𝑧 = ⟨𝑖, 𝑗⟩ ↔ (𝑧 ∈ (V × V) ∧ ((1st ‘𝑧) = 𝑖 ∧ (2nd ‘𝑧) = 𝑗)))
18		fveq2 6103	. . . . . . . . 9 ⊢ ((1st ‘𝑧) = 𝑖 → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) = ((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖))
19		fveq2 6103	. . . . . . . . 9 ⊢ ((2nd ‘𝑧) = 𝑗 → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧)) = ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗))
20	18, 19	oveqan12d 6568	. . . . . . . 8 ⊢ (((1st ‘𝑧) = 𝑖 ∧ (2nd ‘𝑧) = 𝑗) → (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧))) = (((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗)))
21	20	adantl 481	. . . . . . 7 ⊢ ((𝑧 ∈ (V × V) ∧ ((1st ‘𝑧) = 𝑖 ∧ (2nd ‘𝑧) = 𝑗)) → (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧))) = (((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗)))
22	17, 21	sylbi 206	. . . . . 6 ⊢ (𝑧 = ⟨𝑖, 𝑗⟩ → (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧))) = (((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗)))
23	22	mpt2mpt 6650	. . . . 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)))
24	14, 23	eqtr4i 2635	. . . 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))))
25		simp2 1055	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽) → 𝑥 ∈ 𝐼)
26		evlslem4.x	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑋 ∈ 𝐵)
27	26	3adant3 1074	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽) → 𝑋 ∈ 𝐵)
28		eqid 2610	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐼 ↦ 𝑋) = (𝑥 ∈ 𝐼 ↦ 𝑋)
29	28	fvmpt2 6200	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵) → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) = 𝑋)
30	25, 27, 29	syl2anc 691	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽) → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) = 𝑋)
31		simp3 1056	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽) → 𝑦 ∈ 𝐽)
32		evlslem4.y	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → 𝑌 ∈ 𝐵)
33	32	3adant2 1073	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽) → 𝑌 ∈ 𝐵)
34		eqid 2610	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐽 ↦ 𝑌) = (𝑦 ∈ 𝐽 ↦ 𝑌)
35	34	fvmpt2 6200	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐽 ∧ 𝑌 ∈ 𝐵) → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦) = 𝑌)
36	31, 33, 35	syl2anc 691	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽) → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦) = 𝑌)
37	30, 36	oveq12d 6567	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽) → (((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦)) = (𝑋 · 𝑌))
38	37	mpt2eq3dva 6617	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐼, 𝑦 ∈ 𝐽 ↦ (((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦))) = (𝑥 ∈ 𝐼, 𝑦 ∈ 𝐽 ↦ (𝑋 · 𝑌)))
39	24, 38	syl5reqr 2659	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼, 𝑦 ∈ 𝐽 ↦ (𝑋 · 𝑌)) = (𝑧 ∈ (𝐼 × 𝐽) ↦ (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧)))))
40	39	oveq1d 6564	. 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 ))
41		difxp 5477	. . . . . 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 ))))
42	41	eleq2i 2680	. . . . 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 )))))
43		elun 3715	. . . . 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 )))))
44	42, 43	bitri 263	. . . 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 )))))
45		xp1st 7089	. . . . . . . 8 ⊢ (𝑧 ∈ ((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽) → (1st ‘𝑧) ∈ (𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )))
46	26, 28	fmptd 6292	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝑋):𝐼⟶𝐵)
47		ssid 3587	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 ) ⊆ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )
48	47	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 ) ⊆ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 ))
49		evlslem4.i	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
50		evlslem4.z	. . . . . . . . . . 11 ⊢ 0 = (0g‘𝑅)
51		fvex 6113	. . . . . . . . . . 11 ⊢ (0g‘𝑅) ∈ V
52	50, 51	eqeltri 2684	. . . . . . . . . 10 ⊢ 0 ∈ V
53	52	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ V)
54	46, 48, 49, 53	suppssr 7213	. . . . . . . 8 ⊢ ((𝜑 ∧ (1st ‘𝑧) ∈ (𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 ))) → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) = 0 )
55	45, 54	sylan2 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽)) → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) = 0 )
56	55	oveq1d 6564	. . . . . 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))))
57		evlslem4.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
58	57	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽)) → 𝑅 ∈ Ring)
59	32, 34	fmptd 6292	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ 𝑌):𝐽⟶𝐵)
60		xp2nd 7090	. . . . . . . 8 ⊢ (𝑧 ∈ ((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽) → (2nd ‘𝑧) ∈ 𝐽)
61		ffvelrn 6265	. . . . . . . 8 ⊢ (((𝑦 ∈ 𝐽 ↦ 𝑌):𝐽⟶𝐵 ∧ (2nd ‘𝑧) ∈ 𝐽) → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧)) ∈ 𝐵)
62	59, 60, 61	syl2an 493	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽)) → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧)) ∈ 𝐵)
63		evlslem4.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅)
64		evlslem4.t	. . . . . . . 8 ⊢ · = (.r‘𝑅)
65	63, 64, 50	ringlz 18410	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧)) ∈ 𝐵) → ( 0 · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧))) = 0 )
66	58, 62, 65	syl2anc 691	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽)) → ( 0 · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧))) = 0 )
67	56, 66	eqtrd 2644	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽)) → (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧))) = 0 )
68		xp2nd 7090	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 ))) → (2nd ‘𝑧) ∈ (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))
69		ssid 3587	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 ) ⊆ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )
70	69	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 ) ⊆ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 ))
71		evlslem4.j	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ 𝑊)
72	59, 70, 71, 53	suppssr 7213	. . . . . . . 8 ⊢ ((𝜑 ∧ (2nd ‘𝑧) ∈ (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 ))) → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧)) = 0 )
73	68, 72	sylan2 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))) → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧)) = 0 )
74	73	oveq2d 6565	. . . . . 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 ))
75	57	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))) → 𝑅 ∈ Ring)
76		xp1st 7089	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 ))) → (1st ‘𝑧) ∈ 𝐼)
77		ffvelrn 6265	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐼 ↦ 𝑋):𝐼⟶𝐵 ∧ (1st ‘𝑧) ∈ 𝐼) → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) ∈ 𝐵)
78	46, 76, 77	syl2an 493	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))) → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) ∈ 𝐵)
79	63, 64, 50	ringrz 18411	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ ((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) ∈ 𝐵) → (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · 0 ) = 0 )
80	75, 78, 79	syl2anc 691	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))) → (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · 0 ) = 0 )
81	74, 80	eqtrd 2644	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))) → (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧))) = 0 )
82	67, 81	jaodan 822	. . . 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 )
83	44, 82	sylan2b 491	. . 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 )
84		xpexg 6858	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ∈ 𝑊) → (𝐼 × 𝐽) ∈ V)
85	49, 71, 84	syl2anc 691	. . 3 ⊢ (𝜑 → (𝐼 × 𝐽) ∈ V)
86	83, 85	suppss2 7216	. 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 )))
87	40, 86	eqsstrd 3602	1 ⊢ (𝜑 → ((𝑥 ∈ 𝐼, 𝑦 ∈ 𝐽 ↦ (𝑋 · 𝑌)) supp 0 ) ⊆ (((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 ) × ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))

Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  ⟨cop 4131   ↦ cmpt 4643   × cxp 5036  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1^st c1st 7057  2^nd c2nd 7058   supp csupp 7182  Basecbs 15695  ._rcmulr 15769  0_gc0g 15923  Ringcrg 18370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-mgp 18313  df-ring 18372

This theorem is referenced by:  evlslem2  19333