Theorem subrgring 20553

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 2740	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		eqid 2740	. . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅)
4	2, 3	issubrg 20550	. . . 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 496	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑅 ↾s 𝐴) ∈ Ring)
7	1, 6	eqeltrid 2844	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ⊆ wss 3890  ‘cfv 6492  (class class class)co 7363  Basecbs 17177   ↾_s cress 17198  1_rcur 20160  Ringcrg 20212  SubRingcsubrg 20548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7366  df-subrg 20549

This theorem is referenced by:  subrgcrng  20554  subrgsubg  20556  subrg1  20561  subrgsubm  20564  subrguss  20566  subrginv  20567  subrgunit  20569  subrgugrp  20570  subrgnzr  20573  subsubrg  20577  resrhm  20580  resrhm2b  20581  issubdrg  20759  imadrhmcl  20776  subdrgint  20782  abvres  20810  sralmod  21184  ring2idlqus  21309  gzrngunitlem  21414  gzrngunit  21415  issubassa3  21848  subrgpsr  21959  mplring  22000  subrgmvrf  22017  subrgascl  22049  subrgasclcl  22050  evlssca  22077  evlsvar  22078  evlsgsumadd  22079  evlsvarpw  22082  mpfconst  22092  mpfproj  22093  mpfsubrg  22094  evlsscaval  22109  evlsvarval  22110  evlsmaprhm  22114  gsumply1subr  22225  ply1ring  22239  evls1sca  22316  evls1gsumadd  22317  evls1varpw  22320  evls1varpwval  22361  evls1fpws  22362  evls1addd  22364  evls1muld  22365  asclply1subcl  22367  evls1maplmhm  22370  dmatcrng  22492  scmatcrng  22511  scmatsgrp1  22512  scmatsrng1  22513  scmatmhm  22524  scmatrhm  22525  m2cpmrhm  22736  isclmp  25089  reefgim  26440  amgmlem  26978  cntrcrng  33169  ressply1evls1  33655  ressply10g  33657  evls1subd  33662  evls1monply1  33669  vr1nz  33683  evls1fldgencl  33861  0ringirng  33880  extdgfialglem2  33884  ply1annnr  33894  irngnminplynz  33903  minplyelirng  33906  algextdeglem6  33913  imacrhmcl  43011  evlsbagval  43043  amgmwlem  50299