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Theorem subrg1cl 16873
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 2443 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
2 subrg1cl.a . . . 4  |-  .1.  =  ( 1r `  R )
31, 2issubrg 16865 . . 3  |-  ( A  e.  (SubRing `  R
)  <->  ( ( R  e.  Ring  /\  ( Rs  A )  e.  Ring )  /\  ( A  C_  ( Base `  R )  /\  .1.  e.  A ) ) )
43simprbi 464 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( A  C_  ( Base `  R
)  /\  .1.  e.  A ) )
54simprd 463 1  |-  ( A  e.  (SubRing `  R
)  ->  .1.  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3328   ` cfv 5418  (class class class)co 6091   Basecbs 14174   ↾s cress 14175   1rcur 16603   Ringcrg 16645  SubRingcsubrg 16861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-subrg 16863
This theorem is referenced by:  subrg1  16875  subrgsubm  16878  issubrg2  16885  subrgint  16887  subsubrg  16891  issubassa2  17415  subrgpsr  17491  mplassa  17533  mplbas2  17551  mplbas2OLD  17552  ply1assa  17655  zsssubrg  17871  taylply2  21833  subrgchr  26262  cnsrexpcl  29522  rngunsnply  29530
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