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Theorem subrg1cl 18911
Description: A subring contains the multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
Hypothesis
Ref Expression
subrg1cl.a 1 = (1r𝑅)
Assertion
Ref Expression
subrg1cl (𝐴 ∈ (SubRing‘𝑅) → 1𝐴)

Proof of Theorem subrg1cl
StepHypRef Expression
1 eqid 2724 . . . 4 (Base‘𝑅) = (Base‘𝑅)
2 subrg1cl.a . . . 4 1 = (1r𝑅)
31, 2issubrg 18903 . . 3 (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ (Base‘𝑅) ∧ 1𝐴)))
43simprbi 483 . 2 (𝐴 ∈ (SubRing‘𝑅) → (𝐴 ⊆ (Base‘𝑅) ∧ 1𝐴))
54simprd 482 1 (𝐴 ∈ (SubRing‘𝑅) → 1𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1596  wcel 2103  wss 3680  cfv 6001  (class class class)co 6765  Basecbs 15980  s cress 15981  1rcur 18622  Ringcrg 18668  SubRingcsubrg 18899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fv 6009  df-ov 6768  df-subrg 18901
This theorem is referenced by:  subrg1  18913  subrgsubm  18916  issubrg2  18923  subrgint  18925  subsubrg  18929  issubassa2  19468  subrgpsr  19542  mplassa  19577  mplbas2  19593  ply1assa  19692  zsssubrg  19927  isclmp  23018  taylply2  24242  subrgchr  30024  cnsrexpcl  38154  rngunsnply  38162
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