MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subrgrcl Unicode version

Theorem subrgrcl 15649
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 2358 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2358 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  R
)
31, 2issubrg 15644 . . 3  |-  ( A  e.  (SubRing `  R
)  <->  ( ( R  e.  Ring  /\  ( Rs  A )  e.  Ring )  /\  ( A  C_  ( Base `  R )  /\  ( 1r `  R
)  e.  A ) ) )
43simplbi 446 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( R  e.  Ring  /\  ( Rs  A
)  e.  Ring )
)
54simpld 445 1  |-  ( A  e.  (SubRing `  R
)  ->  R  e.  Ring )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1710    C_ wss 3228   ` cfv 5337  (class class class)co 5945   Basecbs 13245   ↾s cress 13246   Ringcrg 15436   1rcur 15438  SubRingcsubrg 15640
This theorem is referenced by:  subrgsubg  15650  subrg1  15654  subrgsubm  15657  subrginv  15660  subrgunit  15662  subrgugrp  15663  opprsubrg  15665  subrgint  15666  subsubrg  15670  sralmod  16038  subrgpsr  16262  subrgmpl  16303  subrgmvr  16304  subrgmvrf  16305  subrgascl  16338  subrgasclcl  16339
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fv 5345  df-ov 5948  df-subrg 15642
  Copyright terms: Public domain W3C validator