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Theorem subrgring 19101
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 2799 . . . . 5 (Base‘𝑅) = (Base‘𝑅)
3 eqid 2799 . . . . 5 (1r𝑅) = (1r𝑅)
42, 3issubrg 19098 . . . 4 (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ (Base‘𝑅) ∧ (1r𝑅) ∈ 𝐴)))
54simplbi 492 . . 3 (𝐴 ∈ (SubRing‘𝑅) → (𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring))
65simprd 490 . 2 (𝐴 ∈ (SubRing‘𝑅) → (𝑅s 𝐴) ∈ Ring)
71, 6syl5eqel 2882 1 (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wss 3769  cfv 6101  (class class class)co 6878  Basecbs 16184  s cress 16185  1rcur 18817  Ringcrg 18863  SubRingcsubrg 19094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6881  df-subrg 19096
This theorem is referenced by:  subrgcrng  19102  subrgsubg  19104  subrg1  19108  subrgmcl  19110  subrgsubm  19111  subrguss  19113  subrginv  19114  subrgunit  19116  subrgugrp  19117  issubdrg  19123  subsubrg  19124  resrhm  19127  abvres  19157  sralmod  19510  subrgnzr  19591  issubassa  19647  subrgpsr  19742  mplring  19775  subrgmvrf  19785  subrgascl  19820  subrgasclcl  19821  evlssca  19844  evlsvar  19845  mpfconst  19852  mpfproj  19853  mpfsubrg  19854  gsumply1subr  19926  ply1ring  19940  evls1sca  20010  evls1gsumadd  20011  evls1varpw  20013  gzrngunitlem  20133  gzrngunit  20134  dmatcrng  20634  scmatcrng  20653  scmatsgrp1  20654  scmatsrng1  20655  scmatmhm  20666  scmatrhm  20667  scmatrngiso  20668  m2cpmrhm  20879  isclmp  23224  reefgim  24545  amgmlem  25068  amgmwlem  43350
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