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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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