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Theorem subrg1cl 13310
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 13302 . . 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 5216  (class class class)co 5874   Basecbs 12456   ↾s cress 12457   1rcur 13095   Ringcrg 13132  SubRingcsubrg 13298
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 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-cnex 7901  ax-resscn 7902  ax-1re 7904  ax-addrcl 7907
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 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224  df-ov 5877  df-inn 8918  df-ndx 12459  df-slot 12460  df-base 12462  df-subrg 13300
This theorem is referenced by:  subrg1  13312  subrgsubm  13315  issubrg2  13322  subrgintm  13324  subsubrg  13326  zsssubrg  13370
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