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Theorem subrgrcl 15904
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 2442 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2442 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  R
)
31, 2issubrg 15899 . . 3  |-  ( A  e.  (SubRing `  R
)  <->  ( ( R  e.  Ring  /\  ( Rs  A )  e.  Ring )  /\  ( A  C_  ( Base `  R )  /\  ( 1r `  R
)  e.  A ) ) )
43simplbi 448 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( R  e.  Ring  /\  ( Rs  A
)  e.  Ring )
)
54simpld 447 1  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1727    C_ wss 3306   ` cfv 5483  (class class class)co 6110   Basecbs 13500   ↾s cress 13501   Ringcrg 15691   1rcur 15693  SubRingcsubrg 15895
This theorem is referenced by:  subrgsubg  15905  subrg1  15909  subrgsubm  15912  subrginv  15915  subrgunit  15917  subrgugrp  15918  opprsubrg  15920  subrgint  15921  subsubrg  15925  sralmod  16289  subrgpsr  16513  subrgmpl  16554  subrgmvr  16555  subrgmvrf  16556  subrgascl  16589  subrgasclcl  16590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fv 5491  df-ov 6113  df-subrg 15897
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