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Theorem subrg1cl 15803
Description: A subring contains the multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
Hypothesis
Ref Expression
subrg1cl.a  |-  .1.  =  ( 1r `  R )
Assertion
Ref Expression
subrg1cl  |-  ( A  e.  (SubRing `  R
)  ->  .1.  e.  A )

Proof of Theorem subrg1cl
StepHypRef Expression
1 eqid 2387 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
2 subrg1cl.a . . . 4  |-  .1.  =  ( 1r `  R )
31, 2issubrg 15795 . . 3  |-  ( A  e.  (SubRing `  R
)  <->  ( ( R  e.  Ring  /\  ( Rs  A )  e.  Ring )  /\  ( A  C_  ( Base `  R )  /\  .1.  e.  A ) ) )
43simprbi 451 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( A  C_  ( Base `  R
)  /\  .1.  e.  A ) )
54simprd 450 1  |-  ( A  e.  (SubRing `  R
)  ->  .1.  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3263   ` cfv 5394  (class class class)co 6020   Basecbs 13396   ↾s cress 13397   Ringcrg 15587   1rcur 15589  SubRingcsubrg 15791
This theorem is referenced by:  subrg1  15805  subrgsubm  15808  issubrg2  15815  subrgint  15817  subsubrg  15821  issubassa2  16330  subrgpsr  16409  mplassa  16444  mplbas2  16458  ply1assa  16524  zsssubrg  16680  taylply2  20151  cnsrexpcl  27039  rngunsnply  27047
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fv 5402  df-ov 6023  df-subrg 15793
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