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Theorem subrgring 19537
Description: A subring is a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
Hypothesis
Ref Expression
subrgring.1 𝑆 = (𝑅s 𝐴)
Assertion
Ref Expression
subrgring (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)

Proof of Theorem subrgring
StepHypRef Expression
1 subrgring.1 . 2 𝑆 = (𝑅s 𝐴)
2 eqid 2821 . . . . 5 (Base‘𝑅) = (Base‘𝑅)
3 eqid 2821 . . . . 5 (1r𝑅) = (1r𝑅)
42, 3issubrg 19534 . . . 4 (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ (Base‘𝑅) ∧ (1r𝑅) ∈ 𝐴)))
54simplbi 500 . . 3 (𝐴 ∈ (SubRing‘𝑅) → (𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring))
65simprd 498 . 2 (𝐴 ∈ (SubRing‘𝑅) → (𝑅s 𝐴) ∈ Ring)
71, 6eqeltrid 2917 1 (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1533  wcel 2110  wss 3935  cfv 6354  (class class class)co 7155  Basecbs 16482  s cress 16483  1rcur 19250  Ringcrg 19296  SubRingcsubrg 19530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fv 6362  df-ov 7158  df-subrg 19532
This theorem is referenced by:  subrgcrng  19538  subrgsubg  19540  subrg1  19544  subrgmcl  19546  subrgsubm  19547  subrguss  19549  subrginv  19550  subrgunit  19552  subrgugrp  19553  issubdrg  19559  subsubrg  19560  resrhm  19563  subdrgint  19581  abvres  19609  sralmod  19958  subrgnzr  20040  issubassa3  20096  subrgpsr  20198  mplring  20231  subrgmvrf  20242  subrgascl  20277  subrgasclcl  20278  evlssca  20301  evlsvar  20302  evlsgsumadd  20303  evlsvarpw  20306  mpfconst  20313  mpfproj  20314  mpfsubrg  20315  gsumply1subr  20401  ply1ring  20415  evls1sca  20485  evls1gsumadd  20486  evls1varpw  20489  gzrngunitlem  20609  gzrngunit  20610  dmatcrng  21110  scmatcrng  21129  scmatsgrp1  21130  scmatsrng1  21131  scmatmhm  21142  scmatrhm  21143  scmatrngiso  21144  m2cpmrhm  21353  isclmp  23700  reefgim  25037  amgmlem  25566  cntrcrng  30697  amgmwlem  44904
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