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Theorem subrg1cl 15868
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 2435 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
2 subrg1cl.a . . . 4  |-  .1.  =  ( 1r `  R )
31, 2issubrg 15860 . . 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 1652    e. wcel 1725    C_ wss 3312   ` cfv 5446  (class class class)co 6073   Basecbs 13461   ↾s cress 13462   Ringcrg 15652   1rcur 15654  SubRingcsubrg 15856
This theorem is referenced by:  subrg1  15870  subrgsubm  15873  issubrg2  15880  subrgint  15882  subsubrg  15886  issubassa2  16395  subrgpsr  16474  mplassa  16509  mplbas2  16523  ply1assa  16589  zsssubrg  16749  taylply2  20276  subrgchr  24222  cnsrexpcl  27328  rngunsnply  27336
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-subrg 15858
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