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Theorem subrg1cl 20359
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 2733 . . . 4 (Base‘𝑅) = (Base‘𝑅)
2 subrg1cl.a . . . 4 1 = (1r𝑅)
31, 2issubrg 20351 . . 3 (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ (Base‘𝑅) ∧ 1𝐴)))
43simprbi 498 . 2 (𝐴 ∈ (SubRing‘𝑅) → (𝐴 ⊆ (Base‘𝑅) ∧ 1𝐴))
54simprd 497 1 (𝐴 ∈ (SubRing‘𝑅) → 1𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wss 3947  cfv 6540  (class class class)co 7404  Basecbs 17140  s cress 17169  1rcur 19996  Ringcrg 20047  SubRingcsubrg 20347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7407  df-subrg 20349
This theorem is referenced by:  subrg1  20361  subrgsubm  20364  issubrg2  20371  subrgint  20374  subsubrg  20378  primefld1cl  20411  zsssubrg  20988  issubassa2  21428  subrgpsr  21521  mplassa  21563  mplbas2  21579  ply1assa  21705  isclmp  24595  taylply2  25862  0ringsubrg  32353  subrgchr  32361  fldgensdrg  32373  primefldgen1  32380  asclply1subcl  32607  drgextlsp  32627  evls1maprhm  32703  evlsmaprhm  41092  cnsrexpcl  41840  rngunsnply  41848
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