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Theorem subrgrcl 18994
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 2770 . . . 4 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2770 . . . 4 (1r𝑅) = (1r𝑅)
31, 2issubrg 18989 . . 3 (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ (Base‘𝑅) ∧ (1r𝑅) ∈ 𝐴)))
43simplbi 479 . 2 (𝐴 ∈ (SubRing‘𝑅) → (𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring))
54simpld 476 1 (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 2144  wss 3721  cfv 6031  (class class class)co 6792  Basecbs 16063  s cress 16064  1rcur 18708  Ringcrg 18754  SubRingcsubrg 18985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-subrg 18987
This theorem is referenced by:  subrgsubg  18995  subrg1  18999  subrgsubm  19002  subrginv  19005  subrgunit  19007  subrgugrp  19008  opprsubrg  19010  subrgint  19011  subsubrg  19015  sralmod  19401  subrgpsr  19633  subrgmpl  19674  subrgmvr  19675  subrgmvrf  19676  subrgascl  19712  subrgasclcl  19713
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