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Theorem abvres 20447
Description: The restriction of an absolute value to a subring is an absolute value. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
abvres.a 𝐴 = (AbsValβ€˜π‘…)
abvres.s 𝑆 = (𝑅 β†Ύs 𝐢)
abvres.b 𝐡 = (AbsValβ€˜π‘†)
Assertion
Ref Expression
abvres ((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) β†’ (𝐹 β†Ύ 𝐢) ∈ 𝐡)

Proof of Theorem abvres
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 abvres.b . . 3 𝐡 = (AbsValβ€˜π‘†)
21a1i 11 . 2 ((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) β†’ 𝐡 = (AbsValβ€˜π‘†))
3 abvres.s . . . 4 𝑆 = (𝑅 β†Ύs 𝐢)
43subrgbas 20328 . . 3 (𝐢 ∈ (SubRingβ€˜π‘…) β†’ 𝐢 = (Baseβ€˜π‘†))
54adantl 483 . 2 ((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) β†’ 𝐢 = (Baseβ€˜π‘†))
6 eqid 2733 . . . 4 (+gβ€˜π‘…) = (+gβ€˜π‘…)
73, 6ressplusg 17235 . . 3 (𝐢 ∈ (SubRingβ€˜π‘…) β†’ (+gβ€˜π‘…) = (+gβ€˜π‘†))
87adantl 483 . 2 ((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) β†’ (+gβ€˜π‘…) = (+gβ€˜π‘†))
9 eqid 2733 . . . 4 (.rβ€˜π‘…) = (.rβ€˜π‘…)
103, 9ressmulr 17252 . . 3 (𝐢 ∈ (SubRingβ€˜π‘…) β†’ (.rβ€˜π‘…) = (.rβ€˜π‘†))
1110adantl 483 . 2 ((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) β†’ (.rβ€˜π‘…) = (.rβ€˜π‘†))
12 subrgsubg 20325 . . . 4 (𝐢 ∈ (SubRingβ€˜π‘…) β†’ 𝐢 ∈ (SubGrpβ€˜π‘…))
1312adantl 483 . . 3 ((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) β†’ 𝐢 ∈ (SubGrpβ€˜π‘…))
14 eqid 2733 . . . 4 (0gβ€˜π‘…) = (0gβ€˜π‘…)
153, 14subg0 19012 . . 3 (𝐢 ∈ (SubGrpβ€˜π‘…) β†’ (0gβ€˜π‘…) = (0gβ€˜π‘†))
1613, 15syl 17 . 2 ((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) β†’ (0gβ€˜π‘…) = (0gβ€˜π‘†))
173subrgring 20322 . . 3 (𝐢 ∈ (SubRingβ€˜π‘…) β†’ 𝑆 ∈ Ring)
1817adantl 483 . 2 ((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) β†’ 𝑆 ∈ Ring)
19 abvres.a . . . 4 𝐴 = (AbsValβ€˜π‘…)
20 eqid 2733 . . . 4 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
2119, 20abvf 20431 . . 3 (𝐹 ∈ 𝐴 β†’ 𝐹:(Baseβ€˜π‘…)βŸΆβ„)
2220subrgss 20320 . . 3 (𝐢 ∈ (SubRingβ€˜π‘…) β†’ 𝐢 βŠ† (Baseβ€˜π‘…))
23 fssres 6758 . . 3 ((𝐹:(Baseβ€˜π‘…)βŸΆβ„ ∧ 𝐢 βŠ† (Baseβ€˜π‘…)) β†’ (𝐹 β†Ύ 𝐢):πΆβŸΆβ„)
2421, 22, 23syl2an 597 . 2 ((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) β†’ (𝐹 β†Ύ 𝐢):πΆβŸΆβ„)
2514subg0cl 19014 . . . 4 (𝐢 ∈ (SubGrpβ€˜π‘…) β†’ (0gβ€˜π‘…) ∈ 𝐢)
26 fvres 6911 . . . 4 ((0gβ€˜π‘…) ∈ 𝐢 β†’ ((𝐹 β†Ύ 𝐢)β€˜(0gβ€˜π‘…)) = (πΉβ€˜(0gβ€˜π‘…)))
2713, 25, 263syl 18 . . 3 ((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) β†’ ((𝐹 β†Ύ 𝐢)β€˜(0gβ€˜π‘…)) = (πΉβ€˜(0gβ€˜π‘…)))
2819, 14abv0 20439 . . . 4 (𝐹 ∈ 𝐴 β†’ (πΉβ€˜(0gβ€˜π‘…)) = 0)
2928adantr 482 . . 3 ((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) β†’ (πΉβ€˜(0gβ€˜π‘…)) = 0)
3027, 29eqtrd 2773 . 2 ((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) β†’ ((𝐹 β†Ύ 𝐢)β€˜(0gβ€˜π‘…)) = 0)
31 simp1l 1198 . . . 4 (((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) ∧ π‘₯ ∈ 𝐢 ∧ π‘₯ β‰  (0gβ€˜π‘…)) β†’ 𝐹 ∈ 𝐴)
3222adantl 483 . . . . . 6 ((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) β†’ 𝐢 βŠ† (Baseβ€˜π‘…))
3332sselda 3983 . . . . 5 (((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) ∧ π‘₯ ∈ 𝐢) β†’ π‘₯ ∈ (Baseβ€˜π‘…))
34333adant3 1133 . . . 4 (((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) ∧ π‘₯ ∈ 𝐢 ∧ π‘₯ β‰  (0gβ€˜π‘…)) β†’ π‘₯ ∈ (Baseβ€˜π‘…))
35 simp3 1139 . . . 4 (((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) ∧ π‘₯ ∈ 𝐢 ∧ π‘₯ β‰  (0gβ€˜π‘…)) β†’ π‘₯ β‰  (0gβ€˜π‘…))
3619, 20, 14abvgt0 20436 . . . 4 ((𝐹 ∈ 𝐴 ∧ π‘₯ ∈ (Baseβ€˜π‘…) ∧ π‘₯ β‰  (0gβ€˜π‘…)) β†’ 0 < (πΉβ€˜π‘₯))
3731, 34, 35, 36syl3anc 1372 . . 3 (((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) ∧ π‘₯ ∈ 𝐢 ∧ π‘₯ β‰  (0gβ€˜π‘…)) β†’ 0 < (πΉβ€˜π‘₯))
38 fvres 6911 . . . 4 (π‘₯ ∈ 𝐢 β†’ ((𝐹 β†Ύ 𝐢)β€˜π‘₯) = (πΉβ€˜π‘₯))
39383ad2ant2 1135 . . 3 (((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) ∧ π‘₯ ∈ 𝐢 ∧ π‘₯ β‰  (0gβ€˜π‘…)) β†’ ((𝐹 β†Ύ 𝐢)β€˜π‘₯) = (πΉβ€˜π‘₯))
4037, 39breqtrrd 5177 . 2 (((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) ∧ π‘₯ ∈ 𝐢 ∧ π‘₯ β‰  (0gβ€˜π‘…)) β†’ 0 < ((𝐹 β†Ύ 𝐢)β€˜π‘₯))
41 simp1l 1198 . . . 4 (((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ 𝐢 ∧ π‘₯ β‰  (0gβ€˜π‘…)) ∧ (𝑦 ∈ 𝐢 ∧ 𝑦 β‰  (0gβ€˜π‘…))) β†’ 𝐹 ∈ 𝐴)
42 simp1r 1199 . . . . . 6 (((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ 𝐢 ∧ π‘₯ β‰  (0gβ€˜π‘…)) ∧ (𝑦 ∈ 𝐢 ∧ 𝑦 β‰  (0gβ€˜π‘…))) β†’ 𝐢 ∈ (SubRingβ€˜π‘…))
4342, 22syl 17 . . . . 5 (((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ 𝐢 ∧ π‘₯ β‰  (0gβ€˜π‘…)) ∧ (𝑦 ∈ 𝐢 ∧ 𝑦 β‰  (0gβ€˜π‘…))) β†’ 𝐢 βŠ† (Baseβ€˜π‘…))
44 simp2l 1200 . . . . 5 (((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ 𝐢 ∧ π‘₯ β‰  (0gβ€˜π‘…)) ∧ (𝑦 ∈ 𝐢 ∧ 𝑦 β‰  (0gβ€˜π‘…))) β†’ π‘₯ ∈ 𝐢)
4543, 44sseldd 3984 . . . 4 (((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ 𝐢 ∧ π‘₯ β‰  (0gβ€˜π‘…)) ∧ (𝑦 ∈ 𝐢 ∧ 𝑦 β‰  (0gβ€˜π‘…))) β†’ π‘₯ ∈ (Baseβ€˜π‘…))
46 simp3l 1202 . . . . 5 (((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ 𝐢 ∧ π‘₯ β‰  (0gβ€˜π‘…)) ∧ (𝑦 ∈ 𝐢 ∧ 𝑦 β‰  (0gβ€˜π‘…))) β†’ 𝑦 ∈ 𝐢)
4743, 46sseldd 3984 . . . 4 (((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ 𝐢 ∧ π‘₯ β‰  (0gβ€˜π‘…)) ∧ (𝑦 ∈ 𝐢 ∧ 𝑦 β‰  (0gβ€˜π‘…))) β†’ 𝑦 ∈ (Baseβ€˜π‘…))
4819, 20, 9abvmul 20437 . . . 4 ((𝐹 ∈ 𝐴 ∧ π‘₯ ∈ (Baseβ€˜π‘…) ∧ 𝑦 ∈ (Baseβ€˜π‘…)) β†’ (πΉβ€˜(π‘₯(.rβ€˜π‘…)𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)))
4941, 45, 47, 48syl3anc 1372 . . 3 (((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ 𝐢 ∧ π‘₯ β‰  (0gβ€˜π‘…)) ∧ (𝑦 ∈ 𝐢 ∧ 𝑦 β‰  (0gβ€˜π‘…))) β†’ (πΉβ€˜(π‘₯(.rβ€˜π‘…)𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)))
509subrgmcl 20331 . . . . 5 ((𝐢 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐢) β†’ (π‘₯(.rβ€˜π‘…)𝑦) ∈ 𝐢)
5142, 44, 46, 50syl3anc 1372 . . . 4 (((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ 𝐢 ∧ π‘₯ β‰  (0gβ€˜π‘…)) ∧ (𝑦 ∈ 𝐢 ∧ 𝑦 β‰  (0gβ€˜π‘…))) β†’ (π‘₯(.rβ€˜π‘…)𝑦) ∈ 𝐢)
5251fvresd 6912 . . 3 (((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ 𝐢 ∧ π‘₯ β‰  (0gβ€˜π‘…)) ∧ (𝑦 ∈ 𝐢 ∧ 𝑦 β‰  (0gβ€˜π‘…))) β†’ ((𝐹 β†Ύ 𝐢)β€˜(π‘₯(.rβ€˜π‘…)𝑦)) = (πΉβ€˜(π‘₯(.rβ€˜π‘…)𝑦)))
5344fvresd 6912 . . . 4 (((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ 𝐢 ∧ π‘₯ β‰  (0gβ€˜π‘…)) ∧ (𝑦 ∈ 𝐢 ∧ 𝑦 β‰  (0gβ€˜π‘…))) β†’ ((𝐹 β†Ύ 𝐢)β€˜π‘₯) = (πΉβ€˜π‘₯))
5446fvresd 6912 . . . 4 (((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ 𝐢 ∧ π‘₯ β‰  (0gβ€˜π‘…)) ∧ (𝑦 ∈ 𝐢 ∧ 𝑦 β‰  (0gβ€˜π‘…))) β†’ ((𝐹 β†Ύ 𝐢)β€˜π‘¦) = (πΉβ€˜π‘¦))
5553, 54oveq12d 7427 . . 3 (((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ 𝐢 ∧ π‘₯ β‰  (0gβ€˜π‘…)) ∧ (𝑦 ∈ 𝐢 ∧ 𝑦 β‰  (0gβ€˜π‘…))) β†’ (((𝐹 β†Ύ 𝐢)β€˜π‘₯) Β· ((𝐹 β†Ύ 𝐢)β€˜π‘¦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)))
5649, 52, 553eqtr4d 2783 . 2 (((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ 𝐢 ∧ π‘₯ β‰  (0gβ€˜π‘…)) ∧ (𝑦 ∈ 𝐢 ∧ 𝑦 β‰  (0gβ€˜π‘…))) β†’ ((𝐹 β†Ύ 𝐢)β€˜(π‘₯(.rβ€˜π‘…)𝑦)) = (((𝐹 β†Ύ 𝐢)β€˜π‘₯) Β· ((𝐹 β†Ύ 𝐢)β€˜π‘¦)))
5719, 20, 6abvtri 20438 . . . 4 ((𝐹 ∈ 𝐴 ∧ π‘₯ ∈ (Baseβ€˜π‘…) ∧ 𝑦 ∈ (Baseβ€˜π‘…)) β†’ (πΉβ€˜(π‘₯(+gβ€˜π‘…)𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))
5841, 45, 47, 57syl3anc 1372 . . 3 (((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ 𝐢 ∧ π‘₯ β‰  (0gβ€˜π‘…)) ∧ (𝑦 ∈ 𝐢 ∧ 𝑦 β‰  (0gβ€˜π‘…))) β†’ (πΉβ€˜(π‘₯(+gβ€˜π‘…)𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))
596subrgacl 20330 . . . . 5 ((𝐢 ∈ (SubRingβ€˜π‘…) ∧ π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐢) β†’ (π‘₯(+gβ€˜π‘…)𝑦) ∈ 𝐢)
6042, 44, 46, 59syl3anc 1372 . . . 4 (((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ 𝐢 ∧ π‘₯ β‰  (0gβ€˜π‘…)) ∧ (𝑦 ∈ 𝐢 ∧ 𝑦 β‰  (0gβ€˜π‘…))) β†’ (π‘₯(+gβ€˜π‘…)𝑦) ∈ 𝐢)
6160fvresd 6912 . . 3 (((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ 𝐢 ∧ π‘₯ β‰  (0gβ€˜π‘…)) ∧ (𝑦 ∈ 𝐢 ∧ 𝑦 β‰  (0gβ€˜π‘…))) β†’ ((𝐹 β†Ύ 𝐢)β€˜(π‘₯(+gβ€˜π‘…)𝑦)) = (πΉβ€˜(π‘₯(+gβ€˜π‘…)𝑦)))
6253, 54oveq12d 7427 . . 3 (((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ 𝐢 ∧ π‘₯ β‰  (0gβ€˜π‘…)) ∧ (𝑦 ∈ 𝐢 ∧ 𝑦 β‰  (0gβ€˜π‘…))) β†’ (((𝐹 β†Ύ 𝐢)β€˜π‘₯) + ((𝐹 β†Ύ 𝐢)β€˜π‘¦)) = ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))
6358, 61, 623brtr4d 5181 . 2 (((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) ∧ (π‘₯ ∈ 𝐢 ∧ π‘₯ β‰  (0gβ€˜π‘…)) ∧ (𝑦 ∈ 𝐢 ∧ 𝑦 β‰  (0gβ€˜π‘…))) β†’ ((𝐹 β†Ύ 𝐢)β€˜(π‘₯(+gβ€˜π‘…)𝑦)) ≀ (((𝐹 β†Ύ 𝐢)β€˜π‘₯) + ((𝐹 β†Ύ 𝐢)β€˜π‘¦)))
642, 5, 8, 11, 16, 18, 24, 30, 40, 56, 63isabvd 20428 1 ((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) β†’ (𝐹 β†Ύ 𝐢) ∈ 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941   βŠ† wss 3949   class class class wbr 5149   β†Ύ cres 5679  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  β„cr 11109  0cc0 11110   + caddc 11113   Β· cmul 11115   < clt 11248   ≀ cle 11249  Basecbs 17144   β†Ύs cress 17173  +gcplusg 17197  .rcmulr 17198  0gc0g 17385  SubGrpcsubg 19000  Ringcrg 20056  SubRingcsubrg 20315  AbsValcabv 20424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-ico 13330  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-subg 19003  df-mgp 19988  df-ring 20058  df-subrg 20317  df-abv 20425
This theorem is referenced by:  subrgnrg  24190  qabsabv  27132
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