Theorem subrgring 20551

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 20548	. . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ (Base‘𝑅) ∧ (1r‘𝑅) ∈ 𝐴)))
5	4	simplbi 496	. . 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 2840	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ⊆ wss 3889  ‘cfv 6498  (class class class)co 7367  Basecbs 17179   ↾_s cress 17200  1_rcur 20162  Ringcrg 20214  SubRingcsubrg 20546

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fv 6506  df-ov 7370  df-subrg 20547

This theorem is referenced by:  subrgcrng  20552  subrgsubg  20554  subrg1  20559  subrgsubm  20562  subrguss  20564  subrginv  20565  subrgunit  20567  subrgugrp  20568  subrgnzr  20571  subsubrg  20575  resrhm  20578  resrhm2b  20579  issubdrg  20757  imadrhmcl  20774  subdrgint  20780  abvres  20808  sralmod  21182  ring2idlqus  21307  gzrngunitlem  21412  gzrngunit  21413  issubassa3  21846  subrgpsr  21956  mplring  21997  subrgmvrf  22012  subrgascl  22044  subrgasclcl  22045  evlssca  22072  evlsvar  22073  evlsgsumadd  22074  evlsvarpw  22077  mpfconst  22087  mpfproj  22088  mpfsubrg  22089  gsumply1subr  22197  ply1ring  22211  evls1sca  22288  evls1gsumadd  22289  evls1varpw  22292  evls1varpwval  22333  evls1fpws  22334  evls1addd  22336  evls1muld  22337  asclply1subcl  22339  evls1maplmhm  22342  dmatcrng  22467  scmatcrng  22486  scmatsgrp1  22487  scmatsrng1  22488  scmatmhm  22499  scmatrhm  22500  m2cpmrhm  22711  isclmp  25064  reefgim  26415  amgmlem  26953  cntrcrng  33142  ressply1evls1  33625  ressply10g  33627  evls1subd  33632  evls1monply1  33639  vr1nz  33653  evls1fldgencl  33814  0ringirng  33833  extdgfialglem2  33837  ply1annnr  33847  irngnminplynz  33856  minplyelirng  33859  algextdeglem6  33866  imacrhmcl  42959  evlsscaval  43000  evlsvarval  43001  evlsbagval  43002  evlsmaprhm  43006  amgmwlem  50277