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Theorem evlslem4 19710
 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 2902 . . . . . 6 𝑖(((𝑥𝐼𝑋)‘𝑥) · ((𝑦𝐽𝑌)‘𝑦))
2 nfcv 2902 . . . . . 6 𝑗(((𝑥𝐼𝑋)‘𝑥) · ((𝑦𝐽𝑌)‘𝑦))
3 nffvmpt1 6360 . . . . . . 7 𝑥((𝑥𝐼𝑋)‘𝑖)
4 nfcv 2902 . . . . . . 7 𝑥 ·
5 nfcv 2902 . . . . . . 7 𝑥((𝑦𝐽𝑌)‘𝑗)
63, 4, 5nfov 6839 . . . . . 6 𝑥(((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗))
7 nfcv 2902 . . . . . . 7 𝑦((𝑥𝐼𝑋)‘𝑖)
8 nfcv 2902 . . . . . . 7 𝑦 ·
9 nffvmpt1 6360 . . . . . . 7 𝑦((𝑦𝐽𝑌)‘𝑗)
107, 8, 9nfov 6839 . . . . . 6 𝑦(((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗))
11 fveq2 6352 . . . . . . 7 (𝑥 = 𝑖 → ((𝑥𝐼𝑋)‘𝑥) = ((𝑥𝐼𝑋)‘𝑖))
12 fveq2 6352 . . . . . . 7 (𝑦 = 𝑗 → ((𝑦𝐽𝑌)‘𝑦) = ((𝑦𝐽𝑌)‘𝑗))
1311, 12oveqan12d 6832 . . . . . 6 ((𝑥 = 𝑖𝑦 = 𝑗) → (((𝑥𝐼𝑋)‘𝑥) · ((𝑦𝐽𝑌)‘𝑦)) = (((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗)))
141, 2, 6, 10, 13cbvmpt2 6899 . . . . 5 (𝑥𝐼, 𝑦𝐽 ↦ (((𝑥𝐼𝑋)‘𝑥) · ((𝑦𝐽𝑌)‘𝑦))) = (𝑖𝐼, 𝑗𝐽 ↦ (((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗)))
15 vex 3343 . . . . . . . 8 𝑖 ∈ V
16 vex 3343 . . . . . . . 8 𝑗 ∈ V
1715, 16eqop2 7376 . . . . . . 7 (𝑧 = ⟨𝑖, 𝑗⟩ ↔ (𝑧 ∈ (V × V) ∧ ((1st𝑧) = 𝑖 ∧ (2nd𝑧) = 𝑗)))
18 fveq2 6352 . . . . . . . . 9 ((1st𝑧) = 𝑖 → ((𝑥𝐼𝑋)‘(1st𝑧)) = ((𝑥𝐼𝑋)‘𝑖))
19 fveq2 6352 . . . . . . . . 9 ((2nd𝑧) = 𝑗 → ((𝑦𝐽𝑌)‘(2nd𝑧)) = ((𝑦𝐽𝑌)‘𝑗))
2018, 19oveqan12d 6832 . . . . . . . 8 (((1st𝑧) = 𝑖 ∧ (2nd𝑧) = 𝑗) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = (((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗)))
2120adantl 473 . . . . . . 7 ((𝑧 ∈ (V × V) ∧ ((1st𝑧) = 𝑖 ∧ (2nd𝑧) = 𝑗)) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = (((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗)))
2217, 21sylbi 207 . . . . . 6 (𝑧 = ⟨𝑖, 𝑗⟩ → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = (((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗)))
2322mpt2mpt 6917 . . . . 5 (𝑧 ∈ (𝐼 × 𝐽) ↦ (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧)))) = (𝑖𝐼, 𝑗𝐽 ↦ (((𝑥𝐼𝑋)‘𝑖) · ((𝑦𝐽𝑌)‘𝑗)))
2414, 23eqtr4i 2785 . . . 4 (𝑥𝐼, 𝑦𝐽 ↦ (((𝑥𝐼𝑋)‘𝑥) · ((𝑦𝐽𝑌)‘𝑦))) = (𝑧 ∈ (𝐼 × 𝐽) ↦ (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))))
25 simp2 1132 . . . . . . 7 ((𝜑𝑥𝐼𝑦𝐽) → 𝑥𝐼)
26 evlslem4.x . . . . . . . 8 ((𝜑𝑥𝐼) → 𝑋𝐵)
27263adant3 1127 . . . . . . 7 ((𝜑𝑥𝐼𝑦𝐽) → 𝑋𝐵)
28 eqid 2760 . . . . . . . 8 (𝑥𝐼𝑋) = (𝑥𝐼𝑋)
2928fvmpt2 6453 . . . . . . 7 ((𝑥𝐼𝑋𝐵) → ((𝑥𝐼𝑋)‘𝑥) = 𝑋)
3025, 27, 29syl2anc 696 . . . . . 6 ((𝜑𝑥𝐼𝑦𝐽) → ((𝑥𝐼𝑋)‘𝑥) = 𝑋)
31 simp3 1133 . . . . . . 7 ((𝜑𝑥𝐼𝑦𝐽) → 𝑦𝐽)
32 evlslem4.y . . . . . . . 8 ((𝜑𝑦𝐽) → 𝑌𝐵)
33323adant2 1126 . . . . . . 7 ((𝜑𝑥𝐼𝑦𝐽) → 𝑌𝐵)
34 eqid 2760 . . . . . . . 8 (𝑦𝐽𝑌) = (𝑦𝐽𝑌)
3534fvmpt2 6453 . . . . . . 7 ((𝑦𝐽𝑌𝐵) → ((𝑦𝐽𝑌)‘𝑦) = 𝑌)
3631, 33, 35syl2anc 696 . . . . . 6 ((𝜑𝑥𝐼𝑦𝐽) → ((𝑦𝐽𝑌)‘𝑦) = 𝑌)
3730, 36oveq12d 6831 . . . . 5 ((𝜑𝑥𝐼𝑦𝐽) → (((𝑥𝐼𝑋)‘𝑥) · ((𝑦𝐽𝑌)‘𝑦)) = (𝑋 · 𝑌))
3837mpt2eq3dva 6884 . . . 4 (𝜑 → (𝑥𝐼, 𝑦𝐽 ↦ (((𝑥𝐼𝑋)‘𝑥) · ((𝑦𝐽𝑌)‘𝑦))) = (𝑥𝐼, 𝑦𝐽 ↦ (𝑋 · 𝑌)))
3924, 38syl5reqr 2809 . . 3 (𝜑 → (𝑥𝐼, 𝑦𝐽 ↦ (𝑋 · 𝑌)) = (𝑧 ∈ (𝐼 × 𝐽) ↦ (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧)))))
4039oveq1d 6828 . 2 (𝜑 → ((𝑥𝐼, 𝑦𝐽 ↦ (𝑋 · 𝑌)) supp 0 ) = ((𝑧 ∈ (𝐼 × 𝐽) ↦ (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧)))) supp 0 ))
41 difxp 5716 . . . . . 6 ((𝐼 × 𝐽) ∖ (((𝑥𝐼𝑋) supp 0 ) × ((𝑦𝐽𝑌) supp 0 ))) = (((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽) ∪ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 ))))
4241eleq2i 2831 . . . . 5 (𝑧 ∈ ((𝐼 × 𝐽) ∖ (((𝑥𝐼𝑋) supp 0 ) × ((𝑦𝐽𝑌) supp 0 ))) ↔ 𝑧 ∈ (((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽) ∪ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))))
43 elun 3896 . . . . 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 264 . . . 4 (𝑧 ∈ ((𝐼 × 𝐽) ∖ (((𝑥𝐼𝑋) supp 0 ) × ((𝑦𝐽𝑌) supp 0 ))) ↔ (𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽) ∨ 𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))))
45 xp1st 7365 . . . . . . . 8 (𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽) → (1st𝑧) ∈ (𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )))
4626, 28fmptd 6548 . . . . . . . . 9 (𝜑 → (𝑥𝐼𝑋):𝐼𝐵)
47 ssid 3765 . . . . . . . . . 10 ((𝑥𝐼𝑋) supp 0 ) ⊆ ((𝑥𝐼𝑋) supp 0 )
4847a1i 11 . . . . . . . . 9 (𝜑 → ((𝑥𝐼𝑋) supp 0 ) ⊆ ((𝑥𝐼𝑋) supp 0 ))
49 evlslem4.i . . . . . . . . 9 (𝜑𝐼𝑉)
50 evlslem4.z . . . . . . . . . . 11 0 = (0g𝑅)
51 fvex 6362 . . . . . . . . . . 11 (0g𝑅) ∈ V
5250, 51eqeltri 2835 . . . . . . . . . 10 0 ∈ V
5352a1i 11 . . . . . . . . 9 (𝜑0 ∈ V)
5446, 48, 49, 53suppssr 7495 . . . . . . . 8 ((𝜑 ∧ (1st𝑧) ∈ (𝐼 ∖ ((𝑥𝐼𝑋) supp 0 ))) → ((𝑥𝐼𝑋)‘(1st𝑧)) = 0 )
5545, 54sylan2 492 . . . . . . 7 ((𝜑𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽)) → ((𝑥𝐼𝑋)‘(1st𝑧)) = 0 )
5655oveq1d 6828 . . . . . 6 ((𝜑𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽)) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = ( 0 · ((𝑦𝐽𝑌)‘(2nd𝑧))))
57 evlslem4.r . . . . . . . 8 (𝜑𝑅 ∈ Ring)
5857adantr 472 . . . . . . 7 ((𝜑𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽)) → 𝑅 ∈ Ring)
5932, 34fmptd 6548 . . . . . . . 8 (𝜑 → (𝑦𝐽𝑌):𝐽𝐵)
60 xp2nd 7366 . . . . . . . 8 (𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽) → (2nd𝑧) ∈ 𝐽)
61 ffvelrn 6520 . . . . . . . 8 (((𝑦𝐽𝑌):𝐽𝐵 ∧ (2nd𝑧) ∈ 𝐽) → ((𝑦𝐽𝑌)‘(2nd𝑧)) ∈ 𝐵)
6259, 60, 61syl2an 495 . . . . . . 7 ((𝜑𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽)) → ((𝑦𝐽𝑌)‘(2nd𝑧)) ∈ 𝐵)
63 evlslem4.b . . . . . . . 8 𝐵 = (Base‘𝑅)
64 evlslem4.t . . . . . . . 8 · = (.r𝑅)
6563, 64, 50ringlz 18787 . . . . . . 7 ((𝑅 ∈ Ring ∧ ((𝑦𝐽𝑌)‘(2nd𝑧)) ∈ 𝐵) → ( 0 · ((𝑦𝐽𝑌)‘(2nd𝑧))) = 0 )
6658, 62, 65syl2anc 696 . . . . . 6 ((𝜑𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽)) → ( 0 · ((𝑦𝐽𝑌)‘(2nd𝑧))) = 0 )
6756, 66eqtrd 2794 . . . . 5 ((𝜑𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽)) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = 0 )
68 xp2nd 7366 . . . . . . . 8 (𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 ))) → (2nd𝑧) ∈ (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))
69 ssid 3765 . . . . . . . . . 10 ((𝑦𝐽𝑌) supp 0 ) ⊆ ((𝑦𝐽𝑌) supp 0 )
7069a1i 11 . . . . . . . . 9 (𝜑 → ((𝑦𝐽𝑌) supp 0 ) ⊆ ((𝑦𝐽𝑌) supp 0 ))
71 evlslem4.j . . . . . . . . 9 (𝜑𝐽𝑊)
7259, 70, 71, 53suppssr 7495 . . . . . . . 8 ((𝜑 ∧ (2nd𝑧) ∈ (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 ))) → ((𝑦𝐽𝑌)‘(2nd𝑧)) = 0 )
7368, 72sylan2 492 . . . . . . 7 ((𝜑𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))) → ((𝑦𝐽𝑌)‘(2nd𝑧)) = 0 )
7473oveq2d 6829 . . . . . 6 ((𝜑𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = (((𝑥𝐼𝑋)‘(1st𝑧)) · 0 ))
7557adantr 472 . . . . . . 7 ((𝜑𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))) → 𝑅 ∈ Ring)
76 xp1st 7365 . . . . . . . 8 (𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 ))) → (1st𝑧) ∈ 𝐼)
77 ffvelrn 6520 . . . . . . . 8 (((𝑥𝐼𝑋):𝐼𝐵 ∧ (1st𝑧) ∈ 𝐼) → ((𝑥𝐼𝑋)‘(1st𝑧)) ∈ 𝐵)
7846, 76, 77syl2an 495 . . . . . . 7 ((𝜑𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))) → ((𝑥𝐼𝑋)‘(1st𝑧)) ∈ 𝐵)
7963, 64, 50ringrz 18788 . . . . . . 7 ((𝑅 ∈ Ring ∧ ((𝑥𝐼𝑋)‘(1st𝑧)) ∈ 𝐵) → (((𝑥𝐼𝑋)‘(1st𝑧)) · 0 ) = 0 )
8075, 78, 79syl2anc 696 . . . . . 6 ((𝜑𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))) → (((𝑥𝐼𝑋)‘(1st𝑧)) · 0 ) = 0 )
8174, 80eqtrd 2794 . . . . 5 ((𝜑𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 )))) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = 0 )
8267, 81jaodan 861 . . . 4 ((𝜑 ∧ (𝑧 ∈ ((𝐼 ∖ ((𝑥𝐼𝑋) supp 0 )) × 𝐽) ∨ 𝑧 ∈ (𝐼 × (𝐽 ∖ ((𝑦𝐽𝑌) supp 0 ))))) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = 0 )
8344, 82sylan2b 493 . . 3 ((𝜑𝑧 ∈ ((𝐼 × 𝐽) ∖ (((𝑥𝐼𝑋) supp 0 ) × ((𝑦𝐽𝑌) supp 0 )))) → (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧))) = 0 )
84 xpexg 7125 . . . 4 ((𝐼𝑉𝐽𝑊) → (𝐼 × 𝐽) ∈ V)
8549, 71, 84syl2anc 696 . . 3 (𝜑 → (𝐼 × 𝐽) ∈ V)
8683, 85suppss2 7498 . 2 (𝜑 → ((𝑧 ∈ (𝐼 × 𝐽) ↦ (((𝑥𝐼𝑋)‘(1st𝑧)) · ((𝑦𝐽𝑌)‘(2nd𝑧)))) supp 0 ) ⊆ (((𝑥𝐼𝑋) supp 0 ) × ((𝑦𝐽𝑌) supp 0 )))
8740, 86eqsstrd 3780 1 (𝜑 → ((𝑥𝐼, 𝑦𝐽 ↦ (𝑋 · 𝑌)) supp 0 ) ⊆ (((𝑥𝐼𝑋) supp 0 ) × ((𝑦𝐽𝑌) supp 0 )))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  Vcvv 3340   ∖ cdif 3712   ∪ cun 3713   ⊆ wss 3715  ⟨cop 4327   ↦ cmpt 4881   × cxp 5264  ⟶wf 6045  ‘cfv 6049  (class class class)co 6813   ↦ cmpt2 6815  1st c1st 7331  2nd c2nd 7332   supp csupp 7463  Basecbs 16059  .rcmulr 16144  0gc0g 16302  Ringcrg 18747 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-supp 7464  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-plusg 16156  df-0g 16304  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-grp 17626  df-minusg 17627  df-mgp 18690  df-ring 18749 This theorem is referenced by:  evlslem2  19714
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