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Theorem evlslem4 19904
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 2934 . . . . . 6 𝑖(((𝑥𝐼𝑋)‘𝑥) · ((𝑦𝐽𝑌)‘𝑦))
2 nfcv 2934 . . . . . 6 𝑗(((𝑥𝐼𝑋)‘𝑥) · ((𝑦𝐽𝑌)‘𝑦))
3 nffvmpt1 6457 . . . . . . 7 𝑥((𝑥𝐼𝑋)‘𝑖)
4 nfcv 2934 . . . . . . 7 𝑥 ·
5 nfcv 2934 . . . . . . 7 𝑥((𝑦𝐽𝑌)‘𝑗)
63, 4, 5nfov 6952 . . . . . 6 𝑥(((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗))
7 nfcv 2934 . . . . . . 7 𝑦((𝑥𝐼𝑋)‘𝑖)
8 nfcv 2934 . . . . . . 7 𝑦 ·
9 nffvmpt1 6457 . . . . . . 7 𝑦((𝑦𝐽𝑌)‘𝑗)
107, 8, 9nfov 6952 . . . . . 6 𝑦(((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗))
11 fveq2 6446 . . . . . . 7 (𝑥 = 𝑖 → ((𝑥𝐼𝑋)‘𝑥) = ((𝑥𝐼𝑋)‘𝑖))
12 fveq2 6446 . . . . . . 7 (𝑦 = 𝑗 → ((𝑦𝐽𝑌)‘𝑦) = ((𝑦𝐽𝑌)‘𝑗))
1311, 12oveqan12d 6941 . . . . . 6 ((𝑥 = 𝑖𝑦 = 𝑗) → (((𝑥𝐼𝑋)‘𝑥) · ((𝑦𝐽𝑌)‘𝑦)) = (((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗)))
141, 2, 6, 10, 13cbvmpt2 7011 . . . . 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 7488 . . . . . . 7 (𝑧 = ⟨𝑖, 𝑗⟩ ↔ (𝑧 ∈ (V × V) ∧ ((1st𝑧) = 𝑖 ∧ (2nd𝑧) = 𝑗)))
18 fveq2 6446 . . . . . . . . 9 ((1st𝑧) = 𝑖 → ((𝑥𝐼𝑋)‘(1st𝑧)) = ((𝑥𝐼𝑋)‘𝑖))
19 fveq2 6446 . . . . . . . . 9 ((2nd𝑧) = 𝑗 → ((𝑦𝐽𝑌)‘(2nd𝑧)) = ((𝑦𝐽𝑌)‘𝑗))
2018, 19oveqan12d 6941 . . . . . . . 8 (((1st𝑧) = 𝑖 ∧ (2nd𝑧) = 𝑗) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = (((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗)))
2120adantl 475 . . . . . . 7 ((𝑧 ∈ (V × V) ∧ ((1st𝑧) = 𝑖 ∧ (2nd𝑧) = 𝑗)) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = (((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗)))
2217, 21sylbi 209 . . . . . 6 (𝑧 = ⟨𝑖, 𝑗⟩ → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = (((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗)))
2322mpt2mpt 7029 . . . . 5 (𝑧 ∈ (𝐼 × 𝐽) ↦ (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧)))) = (𝑖𝐼, 𝑗𝐽 ↦ (((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗)))
2414, 23eqtr4i 2805 . . . 4 (𝑥𝐼, 𝑦𝐽 ↦ (((𝑥𝐼𝑋)‘𝑥) · ((𝑦𝐽𝑌)‘𝑦))) = (𝑧 ∈ (𝐼 × 𝐽) ↦ (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))))
25 simp2 1128 . . . . . . 7 ((𝜑𝑥𝐼𝑦𝐽) → 𝑥𝐼)
26 evlslem4.x . . . . . . . 8 ((𝜑𝑥𝐼) → 𝑋𝐵)
27263adant3 1123 . . . . . . 7 ((𝜑𝑥𝐼𝑦𝐽) → 𝑋𝐵)
28 eqid 2778 . . . . . . . 8 (𝑥𝐼𝑋) = (𝑥𝐼𝑋)
2928fvmpt2 6552 . . . . . . 7 ((𝑥𝐼𝑋𝐵) → ((𝑥𝐼𝑋)‘𝑥) = 𝑋)
3025, 27, 29syl2anc 579 . . . . . 6 ((𝜑𝑥𝐼𝑦𝐽) → ((𝑥𝐼𝑋)‘𝑥) = 𝑋)
31 simp3 1129 . . . . . . 7 ((𝜑𝑥𝐼𝑦𝐽) → 𝑦𝐽)
32 evlslem4.y . . . . . . . 8 ((𝜑𝑦𝐽) → 𝑌𝐵)
33323adant2 1122 . . . . . . 7 ((𝜑𝑥𝐼𝑦𝐽) → 𝑌𝐵)
34 eqid 2778 . . . . . . . 8 (𝑦𝐽𝑌) = (𝑦𝐽𝑌)
3534fvmpt2 6552 . . . . . . 7 ((𝑦𝐽𝑌𝐵) → ((𝑦𝐽𝑌)‘𝑦) = 𝑌)
3631, 33, 35syl2anc 579 . . . . . 6 ((𝜑𝑥𝐼𝑦𝐽) → ((𝑦𝐽𝑌)‘𝑦) = 𝑌)
3730, 36oveq12d 6940 . . . . 5 ((𝜑𝑥𝐼𝑦𝐽) → (((𝑥𝐼𝑋)‘𝑥) · ((𝑦𝐽𝑌)‘𝑦)) = (𝑋 · 𝑌))
3837mpt2eq3dva 6996 . . . 4 (𝜑 → (𝑥𝐼, 𝑦𝐽 ↦ (((𝑥𝐼𝑋)‘𝑥) · ((𝑦𝐽𝑌)‘𝑦))) = (𝑥𝐼, 𝑦𝐽 ↦ (𝑋 · 𝑌)))
3924, 38syl5reqr 2829 . . 3 (𝜑 → (𝑥𝐼, 𝑦𝐽 ↦ (𝑋 · 𝑌)) = (𝑧 ∈ (𝐼 × 𝐽) ↦ (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧)))))
4039oveq1d 6937 . 2 (𝜑 → ((𝑥𝐼, 𝑦𝐽 ↦ (𝑋 · 𝑌)) supp 0 ) = ((𝑧 ∈ (𝐼 × 𝐽) ↦ (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧)))) supp 0 ))
41 difxp 5812 . . . . . 6 ((𝐼 × 𝐽) ∖ (((𝑥𝐼𝑋) supp 0 ) × ((𝑦𝐽𝑌) supp 0 ))) = (((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽) ∪ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 ))))
4241eleq2i 2851 . . . . 5 (𝑧 ∈ ((𝐼 × 𝐽) ∖ (((𝑥𝐼𝑋) supp 0 ) × ((𝑦𝐽𝑌) supp 0 ))) ↔ 𝑧 ∈ (((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽) ∪ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))))
43 elun 3976 . . . . 5 (𝑧 ∈ (((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽) ∪ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))) ↔ (𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽) ∨ 𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))))
4442, 43bitri 267 . . . 4 (𝑧 ∈ ((𝐼 × 𝐽) ∖ (((𝑥𝐼𝑋) supp 0 ) × ((𝑦𝐽𝑌) supp 0 ))) ↔ (𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽) ∨ 𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))))
45 xp1st 7477 . . . . . . . 8 (𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽) → (1st𝑧) ∈ (𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )))
4626fmpttd 6649 . . . . . . . . 9 (𝜑 → (𝑥𝐼𝑋):𝐼𝐵)
47 ssidd 3843 . . . . . . . . 9 (𝜑 → ((𝑥𝐼𝑋) supp 0 ) ⊆ ((𝑥𝐼𝑋) supp 0 ))
48 evlslem4.i . . . . . . . . 9 (𝜑𝐼𝑉)
49 evlslem4.z . . . . . . . . . . 11 0 = (0g𝑅)
5049fvexi 6460 . . . . . . . . . 10 0 ∈ V
5150a1i 11 . . . . . . . . 9 (𝜑0 ∈ V)
5246, 47, 48, 51suppssr 7608 . . . . . . . 8 ((𝜑 ∧ (1st𝑧) ∈ (𝐼 ∖ ((𝑥𝐼𝑋) supp 0 ))) → ((𝑥𝐼𝑋)‘(1st𝑧)) = 0 )
5345, 52sylan2 586 . . . . . . 7 ((𝜑𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽)) → ((𝑥𝐼𝑋)‘(1st𝑧)) = 0 )
5453oveq1d 6937 . . . . . 6 ((𝜑𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽)) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = ( 0 · ((𝑦𝐽𝑌)‘(2nd𝑧))))
55 evlslem4.r . . . . . . . 8 (𝜑𝑅 ∈ Ring)
5655adantr 474 . . . . . . 7 ((𝜑𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽)) → 𝑅 ∈ Ring)
5732fmpttd 6649 . . . . . . . 8 (𝜑 → (𝑦𝐽𝑌):𝐽𝐵)
58 xp2nd 7478 . . . . . . . 8 (𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽) → (2nd𝑧) ∈ 𝐽)
59 ffvelrn 6621 . . . . . . . 8 (((𝑦𝐽𝑌):𝐽𝐵 ∧ (2nd𝑧) ∈ 𝐽) → ((𝑦𝐽𝑌)‘(2nd𝑧)) ∈ 𝐵)
6057, 58, 59syl2an 589 . . . . . . 7 ((𝜑𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽)) → ((𝑦𝐽𝑌)‘(2nd𝑧)) ∈ 𝐵)
61 evlslem4.b . . . . . . . 8 𝐵 = (Base‘𝑅)
62 evlslem4.t . . . . . . . 8 · = (.r𝑅)
6361, 62, 49ringlz 18974 . . . . . . 7 ((𝑅 ∈ Ring ∧ ((𝑦𝐽𝑌)‘(2nd𝑧)) ∈ 𝐵) → ( 0 · ((𝑦𝐽𝑌)‘(2nd𝑧))) = 0 )
6456, 60, 63syl2anc 579 . . . . . 6 ((𝜑𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽)) → ( 0 · ((𝑦𝐽𝑌)‘(2nd𝑧))) = 0 )
6554, 64eqtrd 2814 . . . . 5 ((𝜑𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽)) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = 0 )
66 xp2nd 7478 . . . . . . . 8 (𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 ))) → (2nd𝑧) ∈ (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))
67 ssidd 3843 . . . . . . . . 9 (𝜑 → ((𝑦𝐽𝑌) supp 0 ) ⊆ ((𝑦𝐽𝑌) supp 0 ))
68 evlslem4.j . . . . . . . . 9 (𝜑𝐽𝑊)
6957, 67, 68, 51suppssr 7608 . . . . . . . 8 ((𝜑 ∧ (2nd𝑧) ∈ (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 ))) → ((𝑦𝐽𝑌)‘(2nd𝑧)) = 0 )
7066, 69sylan2 586 . . . . . . 7 ((𝜑𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))) → ((𝑦𝐽𝑌)‘(2nd𝑧)) = 0 )
7170oveq2d 6938 . . . . . 6 ((𝜑𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = (((𝑥𝐼𝑋)‘(1st𝑧)) · 0 ))
7255adantr 474 . . . . . . 7 ((𝜑𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))) → 𝑅 ∈ Ring)
73 xp1st 7477 . . . . . . . 8 (𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 ))) → (1st𝑧) ∈ 𝐼)
74 ffvelrn 6621 . . . . . . . 8 (((𝑥𝐼𝑋):𝐼𝐵 ∧ (1st𝑧) ∈ 𝐼) → ((𝑥𝐼𝑋)‘(1st𝑧)) ∈ 𝐵)
7546, 73, 74syl2an 589 . . . . . . 7 ((𝜑𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))) → ((𝑥𝐼𝑋)‘(1st𝑧)) ∈ 𝐵)
7661, 62, 49ringrz 18975 . . . . . . 7 ((𝑅 ∈ Ring ∧ ((𝑥𝐼𝑋)‘(1st𝑧)) ∈ 𝐵) → (((𝑥𝐼𝑋)‘(1st𝑧)) · 0 ) = 0 )
7772, 75, 76syl2anc 579 . . . . . 6 ((𝜑𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))) → (((𝑥𝐼𝑋)‘(1st𝑧)) · 0 ) = 0 )
7871, 77eqtrd 2814 . . . . 5 ((𝜑𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = 0 )
7965, 78jaodan 943 . . . 4 ((𝜑 ∧ (𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽) ∨ 𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 ))))) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = 0 )
8044, 79sylan2b 587 . . 3 ((𝜑𝑧 ∈ ((𝐼 × 𝐽) ∖ (((𝑥𝐼𝑋) supp 0 ) × ((𝑦𝐽𝑌) supp 0 )))) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = 0 )
8148, 68xpexd 7238 . . 3 (𝜑 → (𝐼 × 𝐽) ∈ V)
8280, 81suppss2 7611 . 2 (𝜑 → ((𝑧 ∈ (𝐼 × 𝐽) ↦ (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧)))) supp 0 ) ⊆ (((𝑥𝐼𝑋) supp 0 ) × ((𝑦𝐽𝑌) supp 0 )))
8340, 82eqsstrd 3858 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 386  wo 836  w3a 1071   = wceq 1601  wcel 2107  Vcvv 3398  cdif 3789  cun 3790  wss 3792  cop 4404  cmpt 4965   × cxp 5353  wf 6131  cfv 6135  (class class class)co 6922  cmpt2 6924  1st c1st 7443  2nd c2nd 7444   supp csupp 7576  Basecbs 16255  .rcmulr 16339  0gc0g 16486  Ringcrg 18934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-supp 7577  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-ndx 16258  df-slot 16259  df-base 16261  df-sets 16262  df-plusg 16351  df-0g 16488  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-grp 17812  df-minusg 17813  df-mgp 18877  df-ring 18936
This theorem is referenced by:  evlslem2  19908
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