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Theorem subrg1cl 20471
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 2731 . . . 4 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
2 subrg1cl.a . . . 4 1 = (1rβ€˜π‘…)
31, 2issubrg 20462 . . 3 (𝐴 ∈ (SubRingβ€˜π‘…) ↔ ((𝑅 ∈ Ring ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ (𝐴 βŠ† (Baseβ€˜π‘…) ∧ 1 ∈ 𝐴)))
43simprbi 496 . 2 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (𝐴 βŠ† (Baseβ€˜π‘…) ∧ 1 ∈ 𝐴))
54simprd 495 1 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 1 ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105   βŠ† wss 3948  β€˜cfv 6543  (class class class)co 7412  Basecbs 17149   β†Ύs cress 17178  1rcur 20076  Ringcrg 20128  SubRingcsubrg 20458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  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 7415  df-subrg 20460
This theorem is referenced by:  subrg1  20473  subrgsubm  20476  issubrg2  20483  subrgint  20486  subsubrg  20489  primefld1cl  20567  zsssubrg  21204  issubassa2  21666  subrgpsr  21759  mplassa  21801  mplbas2  21817  ply1assa  21943  isclmp  24845  taylply2  26117  0ringsubrg  32650  subrgchr  32657  fldgensdrg  32675  primefldgen1  32682  asclply1subcl  32935  drgextlsp  32969  evls1fldgencl  33034  evls1maprhm  33049  ply1annnr  33054  algextdeglem4  33066  evlsmaprhm  41445  cnsrexpcl  42210  rngunsnply  42218
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