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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 = (0g𝑅)
evlslem4.t · = (.r𝑅)
evlslem4.r (𝜑𝑅 ∈ Ring)
evlslem4.x ((𝜑𝑥𝐼) → 𝑋𝐵)
evlslem4.y ((𝜑𝑦𝐽) → 𝑌𝐵)
evlslem4.i (𝜑𝐼𝑉)
evlslem4.j (𝜑𝐽𝑊)
Assertion
Ref Expression
evlslem4 (𝜑 → ((𝑥𝐼, 𝑦𝐽 ↦ (𝑋 · 𝑌)) supp 0 ) ⊆ (((𝑥𝐼𝑋) supp 0 ) × ((𝑦𝐽𝑌) 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.
StepHypRef Expression
1 nfcv 2899 . . . . . 6 𝑖(((𝑥𝐼𝑋)‘𝑥) · ((𝑦𝐽𝑌)‘𝑦))
2 nfcv 2899 . . . . . 6 𝑗(((𝑥𝐼𝑋)‘𝑥) · ((𝑦𝐽𝑌)‘𝑦))
3 nffvmpt1 6679 . . . . . . 7 𝑥((𝑥𝐼𝑋)‘𝑖)
4 nfcv 2899 . . . . . . 7 𝑥 ·
5 nfcv 2899 . . . . . . 7 𝑥((𝑦𝐽𝑌)‘𝑗)
63, 4, 5nfov 7194 . . . . . 6 𝑥(((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗))
7 nfcv 2899 . . . . . . 7 𝑦((𝑥𝐼𝑋)‘𝑖)
8 nfcv 2899 . . . . . . 7 𝑦 ·
9 nffvmpt1 6679 . . . . . . 7 𝑦((𝑦𝐽𝑌)‘𝑗)
107, 8, 9nfov 7194 . . . . . 6 𝑦(((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗))
11 fveq2 6668 . . . . . . 7 (𝑥 = 𝑖 → ((𝑥𝐼𝑋)‘𝑥) = ((𝑥𝐼𝑋)‘𝑖))
12 fveq2 6668 . . . . . . 7 (𝑦 = 𝑗 → ((𝑦𝐽𝑌)‘𝑦) = ((𝑦𝐽𝑌)‘𝑗))
1311, 12oveqan12d 7183 . . . . . 6 ((𝑥 = 𝑖𝑦 = 𝑗) → (((𝑥𝐼𝑋)‘𝑥) · ((𝑦𝐽𝑌)‘𝑦)) = (((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗)))
141, 2, 6, 10, 13cbvmpo 7256 . . . . 5 (𝑥𝐼, 𝑦𝐽 ↦ (((𝑥𝐼𝑋)‘𝑥) · ((𝑦𝐽𝑌)‘𝑦))) = (𝑖𝐼, 𝑗𝐽 ↦ (((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗)))
15 vex 3401 . . . . . . . 8 𝑖 ∈ V
16 vex 3401 . . . . . . . 8 𝑗 ∈ V
1715, 16eqop2 7750 . . . . . . 7 (𝑧 = ⟨𝑖, 𝑗⟩ ↔ (𝑧 ∈ (V × V) ∧ ((1st𝑧) = 𝑖 ∧ (2nd𝑧) = 𝑗)))
18 fveq2 6668 . . . . . . . 8 ((1st𝑧) = 𝑖 → ((𝑥𝐼𝑋)‘(1st𝑧)) = ((𝑥𝐼𝑋)‘𝑖))
19 fveq2 6668 . . . . . . . 8 ((2nd𝑧) = 𝑗 → ((𝑦𝐽𝑌)‘(2nd𝑧)) = ((𝑦𝐽𝑌)‘𝑗))
2018, 19oveqan12d 7183 . . . . . . 7 (((1st𝑧) = 𝑖 ∧ (2nd𝑧) = 𝑗) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = (((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗)))
2117, 20simplbiim 508 . . . . . 6 (𝑧 = ⟨𝑖, 𝑗⟩ → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = (((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗)))
2221mpompt 7274 . . . . 5 (𝑧 ∈ (𝐼 × 𝐽) ↦ (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧)))) = (𝑖𝐼, 𝑗𝐽 ↦ (((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗)))
2314, 22eqtr4i 2764 . . . 4 (𝑥𝐼, 𝑦𝐽 ↦ (((𝑥𝐼𝑋)‘𝑥) · ((𝑦𝐽𝑌)‘𝑦))) = (𝑧 ∈ (𝐼 × 𝐽) ↦ (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))))
24 simp2 1138 . . . . . . 7 ((𝜑𝑥𝐼𝑦𝐽) → 𝑥𝐼)
25 evlslem4.x . . . . . . . 8 ((𝜑𝑥𝐼) → 𝑋𝐵)
26253adant3 1133 . . . . . . 7 ((𝜑𝑥𝐼𝑦𝐽) → 𝑋𝐵)
27 eqid 2738 . . . . . . . 8 (𝑥𝐼𝑋) = (𝑥𝐼𝑋)
2827fvmpt2 6780 . . . . . . 7 ((𝑥𝐼𝑋𝐵) → ((𝑥𝐼𝑋)‘𝑥) = 𝑋)
2924, 26, 28syl2anc 587 . . . . . 6 ((𝜑𝑥𝐼𝑦𝐽) → ((𝑥𝐼𝑋)‘𝑥) = 𝑋)
30 simp3 1139 . . . . . . 7 ((𝜑𝑥𝐼𝑦𝐽) → 𝑦𝐽)
31 evlslem4.y . . . . . . 7 ((𝜑𝑦𝐽) → 𝑌𝐵)
32 eqid 2738 . . . . . . . 8 (𝑦𝐽𝑌) = (𝑦𝐽𝑌)
3332fvmpt2 6780 . . . . . . 7 ((𝑦𝐽𝑌𝐵) → ((𝑦𝐽𝑌)‘𝑦) = 𝑌)
3430, 31, 333imp3i2an 1346 . . . . . 6 ((𝜑𝑥𝐼𝑦𝐽) → ((𝑦𝐽𝑌)‘𝑦) = 𝑌)
3529, 34oveq12d 7182 . . . . 5 ((𝜑𝑥𝐼𝑦𝐽) → (((𝑥𝐼𝑋)‘𝑥) · ((𝑦𝐽𝑌)‘𝑦)) = (𝑋 · 𝑌))
3635mpoeq3dva 7239 . . . 4 (𝜑 → (𝑥𝐼, 𝑦𝐽 ↦ (((𝑥𝐼𝑋)‘𝑥) · ((𝑦𝐽𝑌)‘𝑦))) = (𝑥𝐼, 𝑦𝐽 ↦ (𝑋 · 𝑌)))
3723, 36syl5reqr 2788 . . 3 (𝜑 → (𝑥𝐼, 𝑦𝐽 ↦ (𝑋 · 𝑌)) = (𝑧 ∈ (𝐼 × 𝐽) ↦ (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧)))))
3837oveq1d 7179 . 2 (𝜑 → ((𝑥𝐼, 𝑦𝐽 ↦ (𝑋 · 𝑌)) supp 0 ) = ((𝑧 ∈ (𝐼 × 𝐽) ↦ (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧)))) supp 0 ))
39 difxp 5990 . . . . . 6 ((𝐼 × 𝐽) ∖ (((𝑥𝐼𝑋) supp 0 ) × ((𝑦𝐽𝑌) supp 0 ))) = (((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽) ∪ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 ))))
4039eleq2i 2824 . . . . 5 (𝑧 ∈ ((𝐼 × 𝐽) ∖ (((𝑥𝐼𝑋) supp 0 ) × ((𝑦𝐽𝑌) supp 0 ))) ↔ 𝑧 ∈ (((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽) ∪ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))))
41 elun 4037 . . . . 5 (𝑧 ∈ (((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽) ∪ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))) ↔ (𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽) ∨ 𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))))
4240, 41bitri 278 . . . 4 (𝑧 ∈ ((𝐼 × 𝐽) ∖ (((𝑥𝐼𝑋) supp 0 ) × ((𝑦𝐽𝑌) supp 0 ))) ↔ (𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽) ∨ 𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))))
43 xp1st 7739 . . . . . . . 8 (𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽) → (1st𝑧) ∈ (𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )))
4425fmpttd 6883 . . . . . . . . 9 (𝜑 → (𝑥𝐼𝑋):𝐼𝐵)
45 ssidd 3898 . . . . . . . . 9 (𝜑 → ((𝑥𝐼𝑋) supp 0 ) ⊆ ((𝑥𝐼𝑋) supp 0 ))
46 evlslem4.i . . . . . . . . 9 (𝜑𝐼𝑉)
47 evlslem4.z . . . . . . . . . . 11 0 = (0g𝑅)
4847fvexi 6682 . . . . . . . . . 10 0 ∈ V
4948a1i 11 . . . . . . . . 9 (𝜑0 ∈ V)
5044, 45, 46, 49suppssr 7884 . . . . . . . 8 ((𝜑 ∧ (1st𝑧) ∈ (𝐼 ∖ ((𝑥𝐼𝑋) supp 0 ))) → ((𝑥𝐼𝑋)‘(1st𝑧)) = 0 )
5143, 50sylan2 596 . . . . . . 7 ((𝜑𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽)) → ((𝑥𝐼𝑋)‘(1st𝑧)) = 0 )
5251oveq1d 7179 . . . . . 6 ((𝜑𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽)) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = ( 0 · ((𝑦𝐽𝑌)‘(2nd𝑧))))
53 evlslem4.r . . . . . . 7 (𝜑𝑅 ∈ Ring)
5431fmpttd 6883 . . . . . . . 8 (𝜑 → (𝑦𝐽𝑌):𝐽𝐵)
55 xp2nd 7740 . . . . . . . 8 (𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽) → (2nd𝑧) ∈ 𝐽)
56 ffvelrn 6853 . . . . . . . 8 (((𝑦𝐽𝑌):𝐽𝐵 ∧ (2nd𝑧) ∈ 𝐽) → ((𝑦𝐽𝑌)‘(2nd𝑧)) ∈ 𝐵)
5754, 55, 56syl2an 599 . . . . . . 7 ((𝜑𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽)) → ((𝑦𝐽𝑌)‘(2nd𝑧)) ∈ 𝐵)
58 evlslem4.b . . . . . . . 8 𝐵 = (Base‘𝑅)
59 evlslem4.t . . . . . . . 8 · = (.r𝑅)
6058, 59, 47ringlz 19452 . . . . . . 7 ((𝑅 ∈ Ring ∧ ((𝑦𝐽𝑌)‘(2nd𝑧)) ∈ 𝐵) → ( 0 · ((𝑦𝐽𝑌)‘(2nd𝑧))) = 0 )
6153, 57, 60syl2an2r 685 . . . . . 6 ((𝜑𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽)) → ( 0 · ((𝑦𝐽𝑌)‘(2nd𝑧))) = 0 )
6252, 61eqtrd 2773 . . . . 5 ((𝜑𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽)) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = 0 )
63 xp2nd 7740 . . . . . . . 8 (𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 ))) → (2nd𝑧) ∈ (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))
64 ssidd 3898 . . . . . . . . 9 (𝜑 → ((𝑦𝐽𝑌) supp 0 ) ⊆ ((𝑦𝐽𝑌) supp 0 ))
65 evlslem4.j . . . . . . . . 9 (𝜑𝐽𝑊)
6654, 64, 65, 49suppssr 7884 . . . . . . . 8 ((𝜑 ∧ (2nd𝑧) ∈ (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 ))) → ((𝑦𝐽𝑌)‘(2nd𝑧)) = 0 )
6763, 66sylan2 596 . . . . . . 7 ((𝜑𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))) → ((𝑦𝐽𝑌)‘(2nd𝑧)) = 0 )
6867oveq2d 7180 . . . . . 6 ((𝜑𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = (((𝑥𝐼𝑋)‘(1st𝑧)) · 0 ))
69 xp1st 7739 . . . . . . . 8 (𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 ))) → (1st𝑧) ∈ 𝐼)
70 ffvelrn 6853 . . . . . . . 8 (((𝑥𝐼𝑋):𝐼𝐵 ∧ (1st𝑧) ∈ 𝐼) → ((𝑥𝐼𝑋)‘(1st𝑧)) ∈ 𝐵)
7144, 69, 70syl2an 599 . . . . . . 7 ((𝜑𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))) → ((𝑥𝐼𝑋)‘(1st𝑧)) ∈ 𝐵)
7258, 59, 47ringrz 19453 . . . . . . 7 ((𝑅 ∈ Ring ∧ ((𝑥𝐼𝑋)‘(1st𝑧)) ∈ 𝐵) → (((𝑥𝐼𝑋)‘(1st𝑧)) · 0 ) = 0 )
7353, 71, 72syl2an2r 685 . . . . . 6 ((𝜑𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))) → (((𝑥𝐼𝑋)‘(1st𝑧)) · 0 ) = 0 )
7468, 73eqtrd 2773 . . . . 5 ((𝜑𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = 0 )
7562, 74jaodan 957 . . . 4 ((𝜑 ∧ (𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽) ∨ 𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 ))))) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = 0 )
7642, 75sylan2b 597 . . 3 ((𝜑𝑧 ∈ ((𝐼 × 𝐽) ∖ (((𝑥𝐼𝑋) supp 0 ) × ((𝑦𝐽𝑌) supp 0 )))) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = 0 )
7746, 65xpexd 7486 . . 3 (𝜑 → (𝐼 × 𝐽) ∈ V)
7876, 77suppss2 7888 . 2 (𝜑 → ((𝑧 ∈ (𝐼 × 𝐽) ↦ (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧)))) supp 0 ) ⊆ (((𝑥𝐼𝑋) supp 0 ) × ((𝑦𝐽𝑌) supp 0 )))
7938, 78eqsstrd 3913 1 (𝜑 → ((𝑥𝐼, 𝑦𝐽 ↦ (𝑋 · 𝑌)) supp 0 ) ⊆ (((𝑥𝐼𝑋) supp 0 ) × ((𝑦𝐽𝑌) supp 0 )))
Colors of variables: wff setvar class
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  1st c1st 7705  2nd c2nd 7706   supp csupp 7849  Basecbs 16579  .rcmulr 16662  0gc0g 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
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