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Theorem subrgrcl 20593
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 2735 . . . 4 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2735 . . . 4 (1r𝑅) = (1r𝑅)
31, 2issubrg 20588 . . 3 (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ (Base‘𝑅) ∧ (1r𝑅) ∈ 𝐴)))
43simplbi 497 . 2 (𝐴 ∈ (SubRing‘𝑅) → (𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring))
54simpld 494 1 (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  wss 3963  cfv 6563  (class class class)co 7431  Basecbs 17245  s cress 17274  1rcur 20199  Ringcrg 20251  SubRingcsubrg 20586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-subrg 20587
This theorem is referenced by:  subrgsubg  20594  subrg1  20599  subrgsubm  20602  subrginv  20605  subrgunit  20607  subrgugrp  20608  opprsubrg  20610  subrgint  20612  subsubrg  20615  resrhm2b  20619  sralmod  21212  subrgpsr  22016  subrgmpl  22068  subrgmvr  22069  subrgmvrf  22070  subrgascl  22108  subrgasclcl  22109  asclply1subcl  22394  subrdom  33269  idlinsubrg  33439  ressply10g  33572  ressply1invg  33574
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