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Theorem subrgrcl 20560
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 2726 . . . 4 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2726 . . . 4 (1r𝑅) = (1r𝑅)
31, 2issubrg 20555 . . 3 (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ (Base‘𝑅) ∧ (1r𝑅) ∈ 𝐴)))
43simplbi 496 . 2 (𝐴 ∈ (SubRing‘𝑅) → (𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring))
54simpld 493 1 (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2099  wss 3947  cfv 6554  (class class class)co 7424  Basecbs 17213  s cress 17242  1rcur 20164  Ringcrg 20216  SubRingcsubrg 20551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fv 6562  df-ov 7427  df-subrg 20553
This theorem is referenced by:  subrgsubg  20561  subrg1  20566  subrgsubm  20569  subrginv  20572  subrgunit  20574  subrgugrp  20575  opprsubrg  20577  subrgint  20579  subsubrg  20582  resrhm2b  20586  sralmod  21173  subrgpsr  21987  subrgmpl  22039  subrgmvr  22040  subrgmvrf  22041  subrgascl  22079  subrgasclcl  22080  asclply1subcl  22365  subrdom  33137  idlinsubrg  33306  ressply10g  33439  ressply1invg  33441
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