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Theorem subrgring 20464
Description: A subring is a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
Hypothesis
Ref Expression
subrgring.1 𝑆 = (𝑅 β†Ύs 𝐴)
Assertion
Ref Expression
subrgring (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝑆 ∈ Ring)

Proof of Theorem subrgring
StepHypRef Expression
1 subrgring.1 . 2 𝑆 = (𝑅 β†Ύs 𝐴)
2 eqid 2730 . . . . 5 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
3 eqid 2730 . . . . 5 (1rβ€˜π‘…) = (1rβ€˜π‘…)
42, 3issubrg 20461 . . . 4 (𝐴 ∈ (SubRingβ€˜π‘…) ↔ ((𝑅 ∈ Ring ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) ∧ (𝐴 βŠ† (Baseβ€˜π‘…) ∧ (1rβ€˜π‘…) ∈ 𝐴)))
54simplbi 496 . . 3 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (𝑅 ∈ Ring ∧ (𝑅 β†Ύs 𝐴) ∈ Ring))
65simprd 494 . 2 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ (𝑅 β†Ύs 𝐴) ∈ Ring)
71, 6eqeltrid 2835 1 (𝐴 ∈ (SubRingβ€˜π‘…) β†’ 𝑆 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104   βŠ† wss 3947  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148   β†Ύs cress 17177  1rcur 20075  Ringcrg 20127  SubRingcsubrg 20457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  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 6494  df-fun 6544  df-fv 6550  df-ov 7414  df-subrg 20459
This theorem is referenced by:  subrgcrng  20465  subrgsubg  20467  subrg1  20472  subrgsubm  20475  subrguss  20477  subrginv  20478  subrgunit  20480  subrgugrp  20481  subrgnzr  20484  subsubrg  20488  resrhm  20491  resrhm2b  20492  issubdrg  20544  imadrhmcl  20556  subdrgint  20562  abvres  20590  sralmod  20954  ring2idlqus  21068  gzrngunitlem  21210  gzrngunit  21211  issubassa3  21639  subrgpsr  21758  mplring  21797  subrgmvrf  21808  subrgascl  21846  subrgasclcl  21847  evlssca  21871  evlsvar  21872  evlsgsumadd  21873  evlsvarpw  21876  mpfconst  21883  mpfproj  21884  mpfsubrg  21885  gsumply1subr  21976  ply1ring  21990  evls1sca  22062  evls1gsumadd  22063  evls1varpw  22066  dmatcrng  22224  scmatcrng  22243  scmatsgrp1  22244  scmatsrng1  22245  scmatmhm  22256  scmatrhm  22257  m2cpmrhm  22468  isclmp  24844  reefgim  26198  amgmlem  26730  cntrcrng  32484  evls1varpwval  32919  evls1fpws  32920  evls1addd  32922  evls1muld  32923  ressply10g  32930  asclply1subcl  32934  evls1fldgencl  33033  0ringirng  33042  evls1maplmhm  33049  ply1annnr  33053  irngnminplynz  33060  algextdeglem6  33067  imacrhmcl  41393  evlsscaval  41438  evlsvarval  41439  evlsbagval  41440  evlsmaprhm  41444  amgmwlem  47936
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