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Theorem subrgrcl 18709
Description: Reverse closure for a subring predicate. (Contributed by Mario Carneiro, 3-Dec-2014.)
Assertion
Ref Expression
subrgrcl (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)

Proof of Theorem subrgrcl
StepHypRef Expression
1 eqid 2621 . . . 4 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2621 . . . 4 (1r𝑅) = (1r𝑅)
31, 2issubrg 18704 . . 3 (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ (Base‘𝑅) ∧ (1r𝑅) ∈ 𝐴)))
43simplbi 476 . 2 (𝐴 ∈ (SubRing‘𝑅) → (𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring))
54simpld 475 1 (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  wss 3556  cfv 5849  (class class class)co 6607  Basecbs 15784  s cress 15785  1rcur 18425  Ringcrg 18471  SubRingcsubrg 18700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fv 5857  df-ov 6610  df-subrg 18702
This theorem is referenced by:  subrgsubg  18710  subrg1  18714  subrgsubm  18717  subrginv  18720  subrgunit  18722  subrgugrp  18723  opprsubrg  18725  subrgint  18726  subsubrg  18730  sralmod  19109  subrgpsr  19341  subrgmpl  19382  subrgmvr  19383  subrgmvrf  19384  subrgascl  19420  subrgasclcl  19421
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