Theorem subrgring 20477

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 20474	. . . 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 3905  ‘cfv 6486  (class class class)co 7353  Basecbs 17138   ↾_s cress 17159  1_rcur 20084  Ringcrg 20136  SubRingcsubrg 20472

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 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374

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 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356  df-subrg 20473

This theorem is referenced by:  subrgcrng  20478  subrgsubg  20480  subrg1  20485  subrgsubm  20488  subrguss  20490  subrginv  20491  subrgunit  20493  subrgugrp  20494  subrgnzr  20497  subsubrg  20501  resrhm  20504  resrhm2b  20505  issubdrg  20683  imadrhmcl  20700  subdrgint  20706  abvres  20734  sralmod  21109  ring2idlqus  21234  gzrngunitlem  21357  gzrngunit  21358  issubassa3  21791  subrgpsr  21903  mplring  21944  subrgmvrf  21957  subrgascl  21989  subrgasclcl  21990  evlssca  22012  evlsvar  22013  evlsgsumadd  22014  evlsvarpw  22017  mpfconst  22024  mpfproj  22025  mpfsubrg  22026  gsumply1subr  22134  ply1ring  22148  evls1sca  22226  evls1gsumadd  22227  evls1varpw  22230  evls1varpwval  22271  evls1fpws  22272  evls1addd  22274  evls1muld  22275  asclply1subcl  22277  evls1maplmhm  22280  dmatcrng  22405  scmatcrng  22424  scmatsgrp1  22425  scmatsrng1  22426  scmatmhm  22437  scmatrhm  22438  m2cpmrhm  22649  isclmp  25013  reefgim  26376  amgmlem  26916  cntrcrng  33036  ressply1evls1  33513  ressply10g  33515  evls1subd  33520  vr1nz  33538  evls1fldgencl  33644  0ringirng  33663  ply1annnr  33672  irngnminplynz  33681  minplyelirng  33684  algextdeglem6  33691  imacrhmcl  42490  evlsscaval  42540  evlsvarval  42541  evlsbagval  42542  evlsmaprhm  42546  amgmwlem  49791