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Theorem subrg1cl 20331
Description: A subring contains the multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
Hypothesis
Ref Expression
subrg1cl.a 1 = (1rβ€˜π‘…)
Assertion
Ref Expression
subrg1cl (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 1 ∈ 𝐴)

Proof of Theorem subrg1cl
StepHypRef Expression
1 eqid 2732 . . . 4 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
2 subrg1cl.a . . . 4 1 = (1rβ€˜π‘…)
31, 2issubrg 20323 . . 3 (𝐴 ∈ (SubRingβ€˜π‘…) ↔ ((𝑅 ∈ Ring ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ (𝐴 βŠ† (Baseβ€˜π‘…) ∧ 1 ∈ 𝐴)))
43simprbi 497 . 2 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (𝐴 βŠ† (Baseβ€˜π‘…) ∧ 1 ∈ 𝐴))
54simprd 496 1 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 1 ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   βŠ† wss 3948  β€˜cfv 6543  (class class class)co 7411  Basecbs 17146   β†Ύs cress 17175  1rcur 20006  Ringcrg 20058  SubRingcsubrg 20319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7414  df-subrg 20321
This theorem is referenced by:  subrg1  20333  subrgsubm  20336  issubrg2  20343  subrgint  20346  subsubrg  20349  primefld1cl  20427  zsssubrg  21009  issubassa2  21452  subrgpsr  21545  mplassa  21587  mplbas2  21603  ply1assa  21729  isclmp  24620  taylply2  25887  0ringsubrg  32420  subrgchr  32427  fldgensdrg  32445  primefldgen1  32452  asclply1subcl  32705  drgextlsp  32739  evls1fldgencl  32804  evls1maprhm  32819  ply1annnr  32824  algextdeglem4  32836  evlsmaprhm  41224  cnsrexpcl  41989  rngunsnply  41997
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