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Theorem subrgrcl 19532
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 2819 . . . 4 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2819 . . . 4 (1r𝑅) = (1r𝑅)
31, 2issubrg 19527 . . 3 (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ (Base‘𝑅) ∧ (1r𝑅) ∈ 𝐴)))
43simplbi 500 . 2 (𝐴 ∈ (SubRing‘𝑅) → (𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring))
54simpld 497 1 (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wcel 2107  wss 3934  cfv 6348  (class class class)co 7148  Basecbs 16475  s cress 16476  1rcur 19243  Ringcrg 19289  SubRingcsubrg 19523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7151  df-subrg 19525
This theorem is referenced by:  subrgsubg  19533  subrg1  19537  subrgsubm  19540  subrginv  19543  subrgunit  19545  subrgugrp  19546  opprsubrg  19548  subrgint  19549  subsubrg  19553  sralmod  19951  subrgpsr  20191  subrgmpl  20233  subrgmvr  20234  subrgmvrf  20235  subrgascl  20270  subrgasclcl  20271
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