Theorem subrgring 20602

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 20599	. . . 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 2848	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ⊆ wss 3976  ‘cfv 6573  (class class class)co 7448  Basecbs 17258   ↾_s cress 17287  1_rcur 20208  Ringcrg 20260  SubRingcsubrg 20595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-subrg 20597

This theorem is referenced by:  subrgcrng  20603  subrgsubg  20605  subrg1  20610  subrgsubm  20613  subrguss  20615  subrginv  20616  subrgunit  20618  subrgugrp  20619  subrgnzr  20622  subsubrg  20626  resrhm  20629  resrhm2b  20630  issubdrg  20803  imadrhmcl  20820  subdrgint  20826  abvres  20854  sralmod  21217  ring2idlqus  21342  gzrngunitlem  21473  gzrngunit  21474  issubassa3  21909  subrgpsr  22021  mplring  22062  subrgmvrf  22075  subrgascl  22113  subrgasclcl  22114  evlssca  22136  evlsvar  22137  evlsgsumadd  22138  evlsvarpw  22141  mpfconst  22148  mpfproj  22149  mpfsubrg  22150  gsumply1subr  22256  ply1ring  22270  evls1sca  22348  evls1gsumadd  22349  evls1varpw  22352  evls1varpwval  22393  evls1fpws  22394  evls1addd  22396  evls1muld  22397  asclply1subcl  22399  evls1maplmhm  22402  dmatcrng  22529  scmatcrng  22548  scmatsgrp1  22549  scmatsrng1  22550  scmatmhm  22561  scmatrhm  22562  m2cpmrhm  22773  isclmp  25149  reefgim  26512  amgmlem  27051  cntrcrng  33046  ressply10g  33557  evls1subd  33562  evls1fldgencl  33680  0ringirng  33689  ply1annnr  33696  irngnminplynz  33705  algextdeglem6  33713  imacrhmcl  42469  evlsscaval  42519  evlsvarval  42520  evlsbagval  42521  evlsmaprhm  42525  amgmwlem  48896