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Theorem subrgrcl 15836
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 2412 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2412 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  R
)
31, 2issubrg 15831 . . 3  |-  ( A  e.  (SubRing `  R
)  <->  ( ( R  e.  Ring  /\  ( Rs  A )  e.  Ring )  /\  ( A  C_  ( Base `  R )  /\  ( 1r `  R
)  e.  A ) ) )
43simplbi 447 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( R  e.  Ring  /\  ( Rs  A
)  e.  Ring )
)
54simpld 446 1  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1721    C_ wss 3288   ` cfv 5421  (class class class)co 6048   Basecbs 13432   ↾s cress 13433   Ringcrg 15623   1rcur 15625  SubRingcsubrg 15827
This theorem is referenced by:  subrgsubg  15837  subrg1  15841  subrgsubm  15844  subrginv  15847  subrgunit  15849  subrgugrp  15850  opprsubrg  15852  subrgint  15853  subsubrg  15857  sralmod  16221  subrgpsr  16445  subrgmpl  16486  subrgmvr  16487  subrgmvrf  16488  subrgascl  16521  subrgasclcl  16522
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fv 5429  df-ov 6051  df-subrg 15829
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