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Theorem subrgrcl 16996
Description: Reverse closure for a subring predicate. (Contributed by Mario Carneiro, 3-Dec-2014.)
Assertion
Ref Expression
subrgrcl  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )

Proof of Theorem subrgrcl
StepHypRef Expression
1 eqid 2454 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2454 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  R
)
31, 2issubrg 16991 . . 3  |-  ( A  e.  (SubRing `  R
)  <->  ( ( R  e.  Ring  /\  ( Rs  A )  e.  Ring )  /\  ( A  C_  ( Base `  R )  /\  ( 1r `  R
)  e.  A ) ) )
43simplbi 460 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( R  e.  Ring  /\  ( Rs  A
)  e.  Ring )
)
54simpld 459 1  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1758    C_ wss 3439   ` cfv 5529  (class class class)co 6203   Basecbs 14295   ↾s cress 14296   1rcur 16728   Ringcrg 16771  SubRingcsubrg 16987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fv 5537  df-ov 6206  df-subrg 16989
This theorem is referenced by:  subrgsubg  16997  subrg1  17001  subrgsubm  17004  subrginv  17007  subrgunit  17009  subrgugrp  17010  opprsubrg  17012  subrgint  17013  subsubrg  17017  sralmod  17394  subrgpsr  17618  subrgmpl  17666  subrgmvr  17667  subrgmvrf  17668  subrgascl  17707  subrgasclcl  17708
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