Theorem subrgring 20592

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 2752	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		eqid 2752	. . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅)
4	2, 3	issubrg 20589	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ (Base‘𝑅) ∧ (1r‘𝑅) ∈ 𝐴)))
5	4	simplbi 499	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring))
6	5	simprd 498	. 2 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑅 ↾s 𝐴) ∈ Ring)
7	1, 6	eqeltrid 2856	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1550   ∈ wcel 2132   ⊆ wss 3895  ‘cfv 6506  (class class class)co 7381  Basecbs 17217   ↾_s cress 17238  1_rcur 20199  Ringcrg 20251  SubRingcsubrg 20587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-rab 3405  df-v 3446  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-br 5091  df-opab 5153  df-mpt 5172  df-id 5531  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-iota 6462  df-fun 6508  df-fv 6514  df-ov 7384  df-subrg 20588

This theorem is referenced by:  subrgcrng  20593  subrgsubg  20595  subrg1  20600  subrgsubm  20603  subrguss  20605  subrginv  20606  subrgunit  20608  subrgugrp  20609  subrgnzr  20612  subsubrg  20616  resrhm  20619  resrhm2b  20620  issubdrg  20798  imadrhmcl  20815  subdrgint  20821  abvres  20849  sralmod  21223  ring2idlqus  21348  gzrngunitlem  21453  gzrngunit  21454  issubassa3  21887  subrgpsr  21998  mplring  22039  subrgmvrf  22056  subrgascl  22088  subrgasclcl  22089  evlssca  22116  evlsvar  22117  evlsgsumadd  22118  evlsvarpw  22121  mpfconst  22131  mpfproj  22132  mpfsubrg  22133  evlsscaval  22148  evlsvarval  22149  evlsmaprhm  22153  gsumply1subr  22264  ply1ring  22278  evls1sca  22355  evls1gsumadd  22356  evls1varpw  22359  evls1varpwval  22400  evls1fpws  22401  evls1addd  22403  evls1muld  22404  asclply1subcl  22406  evls1maplmhm  22409  dmatcrng  22531  scmatcrng  22550  scmatsgrp1  22551  scmatsrng1  22552  scmatmhm  22563  scmatrhm  22564  m2cpmrhm  22775  isclmp  25128  reefgim  26479  amgmlem  27020  cntrcrng  33211  ressply1evls1  33705  ressply10g  33707  evls1subd  33712  evls1monply1  33719  vr1nz  33733  evls1fldgencl  33911  0ringirng  33930  extdgfialglem2  33934  ply1annnr  33944  irngnminplynz  33953  minplyelirng  33956  algextdeglem6  33963  imacrhmcl  43074  evlsbagval  43106  amgmwlem  50361