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Theorem subrg1cl 13288
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 2177 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
2 subrg1cl.a . . . 4  |-  .1.  =  ( 1r `  R )
31, 2issubrg 13280 . . 3  |-  ( A  e.  (SubRing `  R
)  <->  ( ( R  e.  Ring  /\  ( Rs  A )  e.  Ring )  /\  ( A  C_  ( Base `  R )  /\  .1.  e.  A ) ) )
43simprbi 275 . 2  |-  ( A  e.  (SubRing `  R
)  ->  ( A  C_  ( Base `  R
)  /\  .1.  e.  A ) )
54simprd 114 1  |-  ( A  e.  (SubRing `  R
)  ->  .1.  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148    C_ wss 3129   ` cfv 5215  (class class class)co 5872   Basecbs 12454   ↾s cress 12455   1rcur 13073   Ringcrg 13110  SubRingcsubrg 13276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-cnex 7899  ax-resscn 7900  ax-1re 7902  ax-addrcl 7905
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-fv 5223  df-ov 5875  df-inn 8916  df-ndx 12457  df-slot 12458  df-base 12460  df-subrg 13278
This theorem is referenced by:  subrg1  13290  subrgsubm  13293  issubrg2  13300  subrgintm  13302  subsubrg  13304  zsssubrg  13348
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