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Theorem evlslem4 21948
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 simp2 1136 . . . . . . 7 ((𝜑𝑥𝐼𝑦𝐽) → 𝑥𝐼)
2 evlslem4.x . . . . . . . 8 ((𝜑𝑥𝐼) → 𝑋𝐵)
323adant3 1131 . . . . . . 7 ((𝜑𝑥𝐼𝑦𝐽) → 𝑋𝐵)
4 eqid 2731 . . . . . . . 8 (𝑥𝐼𝑋) = (𝑥𝐼𝑋)
54fvmpt2 7009 . . . . . . 7 ((𝑥𝐼𝑋𝐵) → ((𝑥𝐼𝑋)‘𝑥) = 𝑋)
61, 3, 5syl2anc 583 . . . . . 6 ((𝜑𝑥𝐼𝑦𝐽) → ((𝑥𝐼𝑋)‘𝑥) = 𝑋)
7 simp3 1137 . . . . . . 7 ((𝜑𝑥𝐼𝑦𝐽) → 𝑦𝐽)
8 evlslem4.y . . . . . . 7 ((𝜑𝑦𝐽) → 𝑌𝐵)
9 eqid 2731 . . . . . . . 8 (𝑦𝐽𝑌) = (𝑦𝐽𝑌)
109fvmpt2 7009 . . . . . . 7 ((𝑦𝐽𝑌𝐵) → ((𝑦𝐽𝑌)‘𝑦) = 𝑌)
117, 8, 103imp3i2an 1344 . . . . . 6 ((𝜑𝑥𝐼𝑦𝐽) → ((𝑦𝐽𝑌)‘𝑦) = 𝑌)
126, 11oveq12d 7430 . . . . 5 ((𝜑𝑥𝐼𝑦𝐽) → (((𝑥𝐼𝑋)‘𝑥) · ((𝑦𝐽𝑌)‘𝑦)) = (𝑋 · 𝑌))
1312mpoeq3dva 7489 . . . 4 (𝜑 → (𝑥𝐼, 𝑦𝐽 ↦ (((𝑥𝐼𝑋)‘𝑥) · ((𝑦𝐽𝑌)‘𝑦))) = (𝑥𝐼, 𝑦𝐽 ↦ (𝑋 · 𝑌)))
14 nfcv 2902 . . . . . 6 𝑖(((𝑥𝐼𝑋)‘𝑥) · ((𝑦𝐽𝑌)‘𝑦))
15 nfcv 2902 . . . . . 6 𝑗(((𝑥𝐼𝑋)‘𝑥) · ((𝑦𝐽𝑌)‘𝑦))
16 nffvmpt1 6902 . . . . . . 7 𝑥((𝑥𝐼𝑋)‘𝑖)
17 nfcv 2902 . . . . . . 7 𝑥 ·
18 nfcv 2902 . . . . . . 7 𝑥((𝑦𝐽𝑌)‘𝑗)
1916, 17, 18nfov 7442 . . . . . 6 𝑥(((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗))
20 nfcv 2902 . . . . . . 7 𝑦((𝑥𝐼𝑋)‘𝑖)
21 nfcv 2902 . . . . . . 7 𝑦 ·
22 nffvmpt1 6902 . . . . . . 7 𝑦((𝑦𝐽𝑌)‘𝑗)
2320, 21, 22nfov 7442 . . . . . 6 𝑦(((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗))
24 fveq2 6891 . . . . . . 7 (𝑥 = 𝑖 → ((𝑥𝐼𝑋)‘𝑥) = ((𝑥𝐼𝑋)‘𝑖))
25 fveq2 6891 . . . . . . 7 (𝑦 = 𝑗 → ((𝑦𝐽𝑌)‘𝑦) = ((𝑦𝐽𝑌)‘𝑗))
2624, 25oveqan12d 7431 . . . . . 6 ((𝑥 = 𝑖𝑦 = 𝑗) → (((𝑥𝐼𝑋)‘𝑥) · ((𝑦𝐽𝑌)‘𝑦)) = (((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗)))
2714, 15, 19, 23, 26cbvmpo 7506 . . . . 5 (𝑥𝐼, 𝑦𝐽 ↦ (((𝑥𝐼𝑋)‘𝑥) · ((𝑦𝐽𝑌)‘𝑦))) = (𝑖𝐼, 𝑗𝐽 ↦ (((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗)))
28 vex 3477 . . . . . . . 8 𝑖 ∈ V
29 vex 3477 . . . . . . . 8 𝑗 ∈ V
3028, 29eqop2 8022 . . . . . . 7 (𝑧 = ⟨𝑖, 𝑗⟩ ↔ (𝑧 ∈ (V × V) ∧ ((1st𝑧) = 𝑖 ∧ (2nd𝑧) = 𝑗)))
31 fveq2 6891 . . . . . . . 8 ((1st𝑧) = 𝑖 → ((𝑥𝐼𝑋)‘(1st𝑧)) = ((𝑥𝐼𝑋)‘𝑖))
32 fveq2 6891 . . . . . . . 8 ((2nd𝑧) = 𝑗 → ((𝑦𝐽𝑌)‘(2nd𝑧)) = ((𝑦𝐽𝑌)‘𝑗))
3331, 32oveqan12d 7431 . . . . . . 7 (((1st𝑧) = 𝑖 ∧ (2nd𝑧) = 𝑗) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = (((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗)))
3430, 33simplbiim 504 . . . . . 6 (𝑧 = ⟨𝑖, 𝑗⟩ → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = (((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗)))
3534mpompt 7525 . . . . 5 (𝑧 ∈ (𝐼 × 𝐽) ↦ (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧)))) = (𝑖𝐼, 𝑗𝐽 ↦ (((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗)))
3627, 35eqtr4i 2762 . . . 4 (𝑥𝐼, 𝑦𝐽 ↦ (((𝑥𝐼𝑋)‘𝑥) · ((𝑦𝐽𝑌)‘𝑦))) = (𝑧 ∈ (𝐼 × 𝐽) ↦ (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))))
3713, 36eqtr3di 2786 . . 3 (𝜑 → (𝑥𝐼, 𝑦𝐽 ↦ (𝑋 · 𝑌)) = (𝑧 ∈ (𝐼 × 𝐽) ↦ (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧)))))
3837oveq1d 7427 . 2 (𝜑 → ((𝑥𝐼, 𝑦𝐽 ↦ (𝑋 · 𝑌)) supp 0 ) = ((𝑧 ∈ (𝐼 × 𝐽) ↦ (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧)))) supp 0 ))
39 difxp 6163 . . . . . 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 4148 . . . . 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 275 . . . 4 (𝑧 ∈ ((𝐼 × 𝐽) ∖ (((𝑥𝐼𝑋) supp 0 ) × ((𝑦𝐽𝑌) supp 0 ))) ↔ (𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽) ∨ 𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))))
43 xp1st 8011 . . . . . . . 8 (𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽) → (1st𝑧) ∈ (𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )))
442fmpttd 7116 . . . . . . . . 9 (𝜑 → (𝑥𝐼𝑋):𝐼𝐵)
45 ssidd 4005 . . . . . . . . 9 (𝜑 → ((𝑥𝐼𝑋) supp 0 ) ⊆ ((𝑥𝐼𝑋) supp 0 ))
46 evlslem4.i . . . . . . . . 9 (𝜑𝐼𝑉)
47 evlslem4.z . . . . . . . . . . 11 0 = (0g𝑅)
4847fvexi 6905 . . . . . . . . . 10 0 ∈ V
4948a1i 11 . . . . . . . . 9 (𝜑0 ∈ V)
5044, 45, 46, 49suppssr 8186 . . . . . . . 8 ((𝜑 ∧ (1st𝑧) ∈ (𝐼 ∖ ((𝑥𝐼𝑋) supp 0 ))) → ((𝑥𝐼𝑋)‘(1st𝑧)) = 0 )
5143, 50sylan2 592 . . . . . . 7 ((𝜑𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽)) → ((𝑥𝐼𝑋)‘(1st𝑧)) = 0 )
5251oveq1d 7427 . . . . . 6 ((𝜑𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽)) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = ( 0 · ((𝑦𝐽𝑌)‘(2nd𝑧))))
53 evlslem4.r . . . . . . 7 (𝜑𝑅 ∈ Ring)
548fmpttd 7116 . . . . . . . 8 (𝜑 → (𝑦𝐽𝑌):𝐽𝐵)
55 xp2nd 8012 . . . . . . . 8 (𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽) → (2nd𝑧) ∈ 𝐽)
56 ffvelcdm 7083 . . . . . . . 8 (((𝑦𝐽𝑌):𝐽𝐵 ∧ (2nd𝑧) ∈ 𝐽) → ((𝑦𝐽𝑌)‘(2nd𝑧)) ∈ 𝐵)
5754, 55, 56syl2an 595 . . . . . . 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 20188 . . . . . . 7 ((𝑅 ∈ Ring ∧ ((𝑦𝐽𝑌)‘(2nd𝑧)) ∈ 𝐵) → ( 0 · ((𝑦𝐽𝑌)‘(2nd𝑧))) = 0 )
6153, 57, 60syl2an2r 682 . . . . . 6 ((𝜑𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽)) → ( 0 · ((𝑦𝐽𝑌)‘(2nd𝑧))) = 0 )
6252, 61eqtrd 2771 . . . . 5 ((𝜑𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽)) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = 0 )
63 xp2nd 8012 . . . . . . . 8 (𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 ))) → (2nd𝑧) ∈ (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))
64 ssidd 4005 . . . . . . . . 9 (𝜑 → ((𝑦𝐽𝑌) supp 0 ) ⊆ ((𝑦𝐽𝑌) supp 0 ))
65 evlslem4.j . . . . . . . . 9 (𝜑𝐽𝑊)
6654, 64, 65, 49suppssr 8186 . . . . . . . 8 ((𝜑 ∧ (2nd𝑧) ∈ (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 ))) → ((𝑦𝐽𝑌)‘(2nd𝑧)) = 0 )
6763, 66sylan2 592 . . . . . . 7 ((𝜑𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))) → ((𝑦𝐽𝑌)‘(2nd𝑧)) = 0 )
6867oveq2d 7428 . . . . . 6 ((𝜑𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = (((𝑥𝐼𝑋)‘(1st𝑧)) · 0 ))
69 xp1st 8011 . . . . . . . 8 (𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 ))) → (1st𝑧) ∈ 𝐼)
70 ffvelcdm 7083 . . . . . . . 8 (((𝑥𝐼𝑋):𝐼𝐵 ∧ (1st𝑧) ∈ 𝐼) → ((𝑥𝐼𝑋)‘(1st𝑧)) ∈ 𝐵)
7144, 69, 70syl2an 595 . . . . . . 7 ((𝜑𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))) → ((𝑥𝐼𝑋)‘(1st𝑧)) ∈ 𝐵)
7258, 59, 47ringrz 20189 . . . . . . 7 ((𝑅 ∈ Ring ∧ ((𝑥𝐼𝑋)‘(1st𝑧)) ∈ 𝐵) → (((𝑥𝐼𝑋)‘(1st𝑧)) · 0 ) = 0 )
7353, 71, 72syl2an2r 682 . . . . . 6 ((𝜑𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))) → (((𝑥𝐼𝑋)‘(1st𝑧)) · 0 ) = 0 )
7468, 73eqtrd 2771 . . . . 5 ((𝜑𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = 0 )
7562, 74jaodan 955 . . . 4 ((𝜑 ∧ (𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽) ∨ 𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 ))))) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = 0 )
7642, 75sylan2b 593 . . 3 ((𝜑𝑧 ∈ ((𝐼 × 𝐽) ∖ (((𝑥𝐼𝑋) supp 0 ) × ((𝑦𝐽𝑌) supp 0 )))) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = 0 )
7746, 65xpexd 7742 . . 3 (𝜑 → (𝐼 × 𝐽) ∈ V)
7876, 77suppss2 8191 . 2 (𝜑 → ((𝑧 ∈ (𝐼 × 𝐽) ↦ (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧)))) supp 0 ) ⊆ (((𝑥𝐼𝑋) supp 0 ) × ((𝑦𝐽𝑌) supp 0 )))
7938, 78eqsstrd 4020 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 395  wo 844  w3a 1086   = wceq 1540  wcel 2105  Vcvv 3473  cdif 3945  cun 3946  wss 3948  cop 4634  cmpt 5231   × cxp 5674  wf 6539  cfv 6543  (class class class)co 7412  cmpo 7414  1st c1st 7977  2nd c2nd 7978   supp csupp 8151  Basecbs 17151  .rcmulr 17205  0gc0g 17392  Ringcrg 20134
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-supp 8152  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-plusg 17217  df-0g 17394  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-grp 18864  df-minusg 18865  df-cmn 19698  df-abl 19699  df-mgp 20036  df-rng 20054  df-ur 20083  df-ring 20136
This theorem is referenced by:  evlslem2  21953
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