Theorem subrgring 20483

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 2729	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		eqid 2729	. . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅)
4	2, 3	issubrg 20480	. . . 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 2832	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ⊆ wss 3914  ‘cfv 6511  (class class class)co 7387  Basecbs 17179   ↾_s cress 17200  1_rcur 20090  Ringcrg 20142  SubRingcsubrg 20478

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-subrg 20479

This theorem is referenced by:  subrgcrng  20484  subrgsubg  20486  subrg1  20491  subrgsubm  20494  subrguss  20496  subrginv  20497  subrgunit  20499  subrgugrp  20500  subrgnzr  20503  subsubrg  20507  resrhm  20510  resrhm2b  20511  issubdrg  20689  imadrhmcl  20706  subdrgint  20712  abvres  20740  sralmod  21094  ring2idlqus  21219  gzrngunitlem  21349  gzrngunit  21350  issubassa3  21775  subrgpsr  21887  mplring  21928  subrgmvrf  21941  subrgascl  21973  subrgasclcl  21974  evlssca  21996  evlsvar  21997  evlsgsumadd  21998  evlsvarpw  22001  mpfconst  22008  mpfproj  22009  mpfsubrg  22010  gsumply1subr  22118  ply1ring  22132  evls1sca  22210  evls1gsumadd  22211  evls1varpw  22214  evls1varpwval  22255  evls1fpws  22256  evls1addd  22258  evls1muld  22259  asclply1subcl  22261  evls1maplmhm  22264  dmatcrng  22389  scmatcrng  22408  scmatsgrp1  22409  scmatsrng1  22410  scmatmhm  22421  scmatrhm  22422  m2cpmrhm  22633  isclmp  24997  reefgim  26360  amgmlem  26900  cntrcrng  33010  ressply1evls1  33534  ressply10g  33536  evls1subd  33541  vr1nz  33559  evls1fldgencl  33665  0ringirng  33684  ply1annnr  33693  irngnminplynz  33702  minplyelirng  33705  algextdeglem6  33712  imacrhmcl  42502  evlsscaval  42552  evlsvarval  42553  evlsbagval  42554  evlsmaprhm  42558  amgmwlem  49791