Theorem evlslem4 20881

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 2899	. . . . . 6 ⊢ Ⅎ𝑖(((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦))
2		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑗(((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦))
3		nffvmpt1 6679	. . . . . . 7 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖)
4		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑥 ·
5		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑥((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗)
6	3, 4, 5	nfov 7194	. . . . . 6 ⊢ Ⅎ𝑥(((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗))
7		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑦((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖)
8		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑦 ·
9		nffvmpt1 6679	. . . . . . 7 ⊢ Ⅎ𝑦((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗)
10	7, 8, 9	nfov 7194	. . . . . 6 ⊢ Ⅎ𝑦(((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗))
11		fveq2 6668	. . . . . . 7 ⊢ (𝑥 = 𝑖 → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) = ((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖))
12		fveq2 6668	. . . . . . 7 ⊢ (𝑦 = 𝑗 → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦) = ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗))
13	11, 12	oveqan12d 7183	. . . . . 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	cbvmpo 7256	. . . . 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 3401	. . . . . . . 8 ⊢ 𝑖 ∈ V
16		vex 3401	. . . . . . . 8 ⊢ 𝑗 ∈ V
17	15, 16	eqop2 7750	. . . . . . 7 ⊢ (𝑧 = ⟨𝑖, 𝑗⟩ ↔ (𝑧 ∈ (V × V) ∧ ((1st ‘𝑧) = 𝑖 ∧ (2nd ‘𝑧) = 𝑗)))
18		fveq2 6668	. . . . . . . 8 ⊢ ((1st ‘𝑧) = 𝑖 → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) = ((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖))
19		fveq2 6668	. . . . . . . 8 ⊢ ((2nd ‘𝑧) = 𝑗 → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧)) = ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗))
20	18, 19	oveqan12d 7183	. . . . . . 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)))
21	17, 20	simplbiim 508	. . . . . 6 ⊢ (𝑧 = ⟨𝑖, 𝑗⟩ → (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧))) = (((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑖) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑗)))
22	21	mpompt 7274	. . . . 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)))
23	14, 22	eqtr4i 2764	. . . 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))))
24		simp2 1138	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽) → 𝑥 ∈ 𝐼)
25		evlslem4.x	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑋 ∈ 𝐵)
26	25	3adant3 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽) → 𝑋 ∈ 𝐵)
27		eqid 2738	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐼 ↦ 𝑋) = (𝑥 ∈ 𝐼 ↦ 𝑋)
28	27	fvmpt2 6780	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝑋 ∈ 𝐵) → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) = 𝑋)
29	24, 26, 28	syl2anc 587	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽) → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) = 𝑋)
30		simp3 1139	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽) → 𝑦 ∈ 𝐽)
31		evlslem4.y	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → 𝑌 ∈ 𝐵)
32		eqid 2738	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐽 ↦ 𝑌) = (𝑦 ∈ 𝐽 ↦ 𝑌)
33	32	fvmpt2 6780	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐽 ∧ 𝑌 ∈ 𝐵) → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦) = 𝑌)
34	30, 31, 33	3imp3i2an 1346	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽) → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦) = 𝑌)
35	29, 34	oveq12d 7182	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽) → (((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦)) = (𝑋 · 𝑌))
36	35	mpoeq3dva 7239	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐼, 𝑦 ∈ 𝐽 ↦ (((𝑥 ∈ 𝐼 ↦ 𝑋)‘𝑥) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘𝑦))) = (𝑥 ∈ 𝐼, 𝑦 ∈ 𝐽 ↦ (𝑋 · 𝑌)))
37	23, 36	syl5reqr 2788	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼, 𝑦 ∈ 𝐽 ↦ (𝑋 · 𝑌)) = (𝑧 ∈ (𝐼 × 𝐽) ↦ (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧)))))
38	37	oveq1d 7179	. 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 5990	. . . . . 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 2824	. . . . 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 4037	. . . . 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 278	. . . 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 7739	. . . . . . . 8 ⊢ (𝑧 ∈ ((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽) → (1st ‘𝑧) ∈ (𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )))
44	25	fmpttd 6883	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝑋):𝐼⟶𝐵)
45		ssidd 3898	. . . . . . . . 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 6682	. . . . . . . . . 10 ⊢ 0 ∈ V
49	48	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ V)
50	44, 45, 46, 49	suppssr 7884	. . . . . . . 8 ⊢ ((𝜑 ∧ (1st ‘𝑧) ∈ (𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 ))) → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) = 0 )
51	43, 50	sylan2 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽)) → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) = 0 )
52	51	oveq1d 7179	. . . . . 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	31	fmpttd 6883	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ 𝑌):𝐽⟶𝐵)
55		xp2nd 7740	. . . . . . . 8 ⊢ (𝑧 ∈ ((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽) → (2nd ‘𝑧) ∈ 𝐽)
56		ffvelrn 6853	. . . . . . . 8 ⊢ (((𝑦 ∈ 𝐽 ↦ 𝑌):𝐽⟶𝐵 ∧ (2nd ‘𝑧) ∈ 𝐽) → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧)) ∈ 𝐵)
57	54, 55, 56	syl2an 599	. . . . . . 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 19452	. . . . . . 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 2773	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ 𝑋) supp 0 )) × 𝐽)) → (((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) · ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧))) = 0 )
63		xp2nd 7740	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 ))) → (2nd ‘𝑧) ∈ (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))
64		ssidd 3898	. . . . . . . . 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 7884	. . . . . . . 8 ⊢ ((𝜑 ∧ (2nd ‘𝑧) ∈ (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 ))) → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧)) = 0 )
67	63, 66	sylan2 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))) → ((𝑦 ∈ 𝐽 ↦ 𝑌)‘(2nd ‘𝑧)) = 0 )
68	67	oveq2d 7180	. . . . . 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 7739	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 ))) → (1st ‘𝑧) ∈ 𝐼)
70		ffvelrn 6853	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐼 ↦ 𝑋):𝐼⟶𝐵 ∧ (1st ‘𝑧) ∈ 𝐼) → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) ∈ 𝐵)
71	44, 69, 70	syl2an 599	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦 ∈ 𝐽 ↦ 𝑌) supp 0 )))) → ((𝑥 ∈ 𝐼 ↦ 𝑋)‘(1st ‘𝑧)) ∈ 𝐵)
72	58, 59, 47	ringrz 19453	. . . . . . 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 2773	. . . . 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 957	. . . 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 597	. . 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 7486	. . 3 ⊢ (𝜑 → (𝐼 × 𝐽) ∈ V)
78	76, 77	suppss2 7888	. 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 3913	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 399   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2113  Vcvv 3397   ∖ cdif 3838   ∪ cun 3839   ⊆ wss 3841  ⟨cop 4519   ↦ cmpt 5107   × cxp 5517  ⟶wf 6329  ‘cfv 6333  (class class class)co 7164   ∈ cmpo 7166  1^st c1st 7705  2^nd c2nd 7706   supp csupp 7849  Basecbs 16579  ._rcmulr 16662  0_gc0g 16809  Ringcrg 19409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473  ax-cnex 10664  ax-resscn 10665  ax-1cn 10666  ax-icn 10667  ax-addcl 10668  ax-addrcl 10669  ax-mulcl 10670  ax-mulrcl 10671  ax-mulcom 10672  ax-addass 10673  ax-mulass 10674  ax-distr 10675  ax-i2m1 10676  ax-1ne0 10677  ax-1rid 10678  ax-rnegex 10679  ax-rrecex 10680  ax-cnre 10681  ax-pre-lttri 10682  ax-pre-lttrn 10683  ax-pre-ltadd 10684  ax-pre-mulgt0 10685

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7121  df-ov 7167  df-oprab 7168  df-mpo 7169  df-om 7594  df-1st 7707  df-2nd 7708  df-supp 7850  df-wrecs 7969  df-recs 8030  df-rdg 8068  df-er 8313  df-en 8549  df-dom 8550  df-sdom 8551  df-pnf 10748  df-mnf 10749  df-xr 10750  df-ltxr 10751  df-le 10752  df-sub 10943  df-neg 10944  df-nn 11710  df-2 11772  df-ndx 16582  df-slot 16583  df-base 16585  df-sets 16586  df-plusg 16674  df-0g 16811  df-mgm 17961  df-sgrp 18010  df-mnd 18021  df-grp 18215  df-minusg 18216  df-mgp 19352  df-ring 19411

This theorem is referenced by:  evlslem2  20886