Theorem subrgring 20575

Description: A subring is a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)

Hypothesis

Ref	Expression
subrgring.1	⊢ 𝑆 = (𝑅 ↾s 𝐴)

Assertion

Ref	Expression
subrgring	⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)

Proof of Theorem subrgring

Step	Hyp	Ref	Expression
1		subrgring.1	. 2 ⊢ 𝑆 = (𝑅 ↾s 𝐴)
2		eqid 2736	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		eqid 2736	. . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅)
4	2, 3	issubrg 20572	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ (Base‘𝑅) ∧ (1r‘𝑅) ∈ 𝐴)))
5	4	simplbi 497	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring))
6	5	simprd 495	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑅 ↾s 𝐴) ∈ Ring)
7	1, 6	eqeltrid 2844	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ⊆ wss 3950  ‘cfv 6560  (class class class)co 7432  Basecbs 17248   ↾_s cress 17275  1_rcur 20179  Ringcrg 20231  SubRingcsubrg 20570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-subrg 20571

This theorem is referenced by:  subrgcrng  20576  subrgsubg  20578  subrg1  20583  subrgsubm  20586  subrguss  20588  subrginv  20589  subrgunit  20591  subrgugrp  20592  subrgnzr  20595  subsubrg  20599  resrhm  20602  resrhm2b  20603  issubdrg  20782  imadrhmcl  20799  subdrgint  20805  abvres  20833  sralmod  21195  ring2idlqus  21320  gzrngunitlem  21451  gzrngunit  21452  issubassa3  21887  subrgpsr  21999  mplring  22040  subrgmvrf  22053  subrgascl  22091  subrgasclcl  22092  evlssca  22114  evlsvar  22115  evlsgsumadd  22116  evlsvarpw  22119  mpfconst  22126  mpfproj  22127  mpfsubrg  22128  gsumply1subr  22236  ply1ring  22250  evls1sca  22328  evls1gsumadd  22329  evls1varpw  22332  evls1varpwval  22373  evls1fpws  22374  evls1addd  22376  evls1muld  22377  asclply1subcl  22379  evls1maplmhm  22382  dmatcrng  22509  scmatcrng  22528  scmatsgrp1  22529  scmatsrng1  22530  scmatmhm  22541  scmatrhm  22542  m2cpmrhm  22753  isclmp  25131  reefgim  26495  amgmlem  27034  cntrcrng  33074  ressply10g  33593  evls1subd  33598  evls1fldgencl  33721  0ringirng  33740  ply1annnr  33747  irngnminplynz  33756  algextdeglem6  33764  imacrhmcl  42529  evlsscaval  42579  evlsvarval  42580  evlsbagval  42581  evlsmaprhm  42585  amgmwlem  49376