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Theorem subrgrcl 17644
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 2400 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2400 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  R
)
31, 2issubrg 17639 . . 3  |-  ( A  e.  (SubRing `  R
)  <->  ( ( R  e.  Ring  /\  ( Rs  A )  e.  Ring )  /\  ( A  C_  ( Base `  R )  /\  ( 1r `  R
)  e.  A ) ) )
43simplbi 458 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( R  e.  Ring  /\  ( Rs  A
)  e.  Ring )
)
54simpld 457 1  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1840    C_ wss 3411   ` cfv 5523  (class class class)co 6232   Basecbs 14731   ↾s cress 14732   1rcur 17363   Ringcrg 17408  SubRingcsubrg 17635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fv 5531  df-ov 6235  df-subrg 17637
This theorem is referenced by:  subrgsubg  17645  subrg1  17649  subrgsubm  17652  subrginv  17655  subrgunit  17657  subrgugrp  17658  opprsubrg  17660  subrgint  17661  subsubrg  17665  sralmod  18043  subrgpsr  18284  subrgmpl  18332  subrgmvr  18333  subrgmvrf  18334  subrgascl  18373  subrgasclcl  18374
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