Theorem evlslem4 21856

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 1135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽) → 𝑥 ∈ 𝐼)
2		evlslem4.x	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑋 ∈ 𝐵)
3	2	3adant3 1130	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽) → 𝑋 ∈ 𝐵)
4		eqid 2730	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐼 ↦ 𝑋) = (𝑥 ∈ 𝐼 ↦ 𝑋)
5	4	fvmpt2 7008	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵) → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) = 𝑋)
6	1, 3, 5	syl2anc 582	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽) → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) = 𝑋)
7		simp3 1136	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽) → 𝑦 ∈ 𝐽)
8		evlslem4.y	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → 𝑌 ∈ 𝐵)
9		eqid 2730	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐽 ↦ 𝑌) = (𝑦 ∈ 𝐽 ↦ 𝑌)
10	9	fvmpt2 7008	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐽 ∧ 𝑌 ∈ 𝐵) → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦) = 𝑌)
11	7, 8, 10	3imp3i2an 1343	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽) → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦) = 𝑌)
12	6, 11	oveq12d 7429	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽) → (((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦)) = (𝑋 · 𝑌))
13	12	mpoeq3dva 7488	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐼, 𝑦 ∈ 𝐽 ↦ (((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦))) = (𝑥 ∈ 𝐼, 𝑦 ∈ 𝐽 ↦ (𝑋 · 𝑌)))
14		nfcv 2901	. . . . . 6 ⊢ Ⅎ𝑖(((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦))
15		nfcv 2901	. . . . . 6 ⊢ Ⅎ𝑗(((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦))
16		nffvmpt1 6901	. . . . . . 7 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖)
17		nfcv 2901	. . . . . . 7 ⊢ Ⅎ𝑥 ·
18		nfcv 2901	. . . . . . 7 ⊢ Ⅎ𝑥((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗)
19	16, 17, 18	nfov 7441	. . . . . 6 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗))
20		nfcv 2901	. . . . . . 7 ⊢ Ⅎ𝑦((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖)
21		nfcv 2901	. . . . . . 7 ⊢ Ⅎ𝑦 ·
22		nffvmpt1 6901	. . . . . . 7 ⊢ Ⅎ𝑦((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗)
23	20, 21, 22	nfov 7441	. . . . . 6 ⊢ Ⅎ𝑦(((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗))
24		fveq2 6890	. . . . . . 7 ⊢ (𝑥 = 𝑖 → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) = ((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖))
25		fveq2 6890	. . . . . . 7 ⊢ (𝑦 = 𝑗 → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦) = ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗))
26	24, 25	oveqan12d 7430	. . . . . 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 7505	. . . . 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 3476	. . . . . . . 8 ⊢ 𝑖 ∈ V
29		vex 3476	. . . . . . . 8 ⊢ 𝑗 ∈ V
30	28, 29	eqop2 8020	. . . . . . 7 ⊢ (𝑧 = ⟨𝑖, 𝑗⟩ ↔ (𝑧 ∈ (V × V) ∧ ((1st ‘𝑧) = 𝑖 ∧ (2nd ‘𝑧) = 𝑗)))
31		fveq2 6890	. . . . . . . 8 ⊢ ((1st ‘𝑧) = 𝑖 → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) = ((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖))
32		fveq2 6890	. . . . . . . 8 ⊢ ((2nd ‘𝑧) = 𝑗 → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧)) = ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗))
33	31, 32	oveqan12d 7430	. . . . . . 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 503	. . . . . 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 7524	. . . . 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 2761	. . . 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 2785	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼, 𝑦 ∈ 𝐽 ↦ (𝑋 · 𝑌)) = (𝑧 ∈ (𝐼 × 𝐽) ↦ (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧)))))
38	37	oveq1d 7426	. 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 6162	. . . . . 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 4147	. . . . 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 274	. . . 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 8009	. . . . . . . 8 ⊢ (𝑧 ∈ ((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽) → (1st ‘𝑧) ∈ (𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )))
44	2	fmpttd 7115	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝑋):𝐼⟶𝐵)
45		ssidd 4004	. . . . . . . . 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 6904	. . . . . . . . . 10 ⊢ 0 ∈ V
49	48	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ V)
50	44, 45, 46, 49	suppssr 8183	. . . . . . . 8 ⊢ ((𝜑 ∧ (1st ‘𝑧) ∈ (𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 ))) → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) = 0 )
51	43, 50	sylan2 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽)) → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) = 0 )
52	51	oveq1d 7426	. . . . . 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 7115	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ 𝑌):𝐽⟶𝐵)
55		xp2nd 8010	. . . . . . . 8 ⊢ (𝑧 ∈ ((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽) → (2nd ‘𝑧) ∈ 𝐽)
56		ffvelcdm 7082	. . . . . . . 8 ⊢ (((𝑦 ∈ 𝐽 ↦ 𝑌):𝐽⟶𝐵 ∧ (2nd ‘𝑧) ∈ 𝐽) → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧)) ∈ 𝐵)
57	54, 55, 56	syl2an 594	. . . . . . 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 20181	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧)) ∈ 𝐵) → ( 0 · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧))) = 0 )
61	53, 57, 60	syl2an2r 681	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽)) → ( 0 · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧))) = 0 )
62	52, 61	eqtrd 2770	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽)) → (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧))) = 0 )
63		xp2nd 8010	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 ))) → (2nd ‘𝑧) ∈ (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))
64		ssidd 4004	. . . . . . . . 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 8183	. . . . . . . 8 ⊢ ((𝜑 ∧ (2nd ‘𝑧) ∈ (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 ))) → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧)) = 0 )
67	63, 66	sylan2 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))) → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧)) = 0 )
68	67	oveq2d 7427	. . . . . 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 8009	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 ))) → (1st ‘𝑧) ∈ 𝐼)
70		ffvelcdm 7082	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐼 ↦ 𝑋):𝐼⟶𝐵 ∧ (1st ‘𝑧) ∈ 𝐼) → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) ∈ 𝐵)
71	44, 69, 70	syl2an 594	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))) → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) ∈ 𝐵)
72	58, 59, 47	ringrz 20182	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ ((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) ∈ 𝐵) → (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · 0 ) = 0 )
73	53, 71, 72	syl2an2r 681	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))) → (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · 0 ) = 0 )
74	68, 73	eqtrd 2770	. . . . 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 954	. . . 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 592	. . 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 7740	. . 3 ⊢ (𝜑 → (𝐼 × 𝐽) ∈ V)
78	76, 77	suppss2 8187	. 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 4019	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 394   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  Vcvv 3472   ∖ cdif 3944   ∪ cun 3945   ⊆ wss 3947  ⟨cop 4633   ↦ cmpt 5230   × cxp 5673  ⟶wf 6538  ‘cfv 6542  (class class class)co 7411   ∈ cmpo 7413  1^st c1st 7975  2^nd c2nd 7976   supp csupp 8148  Basecbs 17148  ._rcmulr 17202  0_gc0g 17389  Ringcrg 20127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  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-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-supp 8149  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-plusg 17214  df-0g 17391  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858  df-minusg 18859  df-cmn 19691  df-abl 19692  df-mgp 20029  df-rng 20047  df-ur 20076  df-ring 20129

This theorem is referenced by:  evlslem2  21861