Theorem subrgring 20519

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 2737	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		eqid 2737	. . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅)
4	2, 3	issubrg 20516	. . . 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 2841	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ⊆ wss 3903  ‘cfv 6500  (class class class)co 7368  Basecbs 17148   ↾_s cress 17169  1_rcur 20128  Ringcrg 20180  SubRingcsubrg 20514

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 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-subrg 20515

This theorem is referenced by:  subrgcrng  20520  subrgsubg  20522  subrg1  20527  subrgsubm  20530  subrguss  20532  subrginv  20533  subrgunit  20535  subrgugrp  20536  subrgnzr  20539  subsubrg  20543  resrhm  20546  resrhm2b  20547  issubdrg  20725  imadrhmcl  20742  subdrgint  20748  abvres  20776  sralmod  21151  ring2idlqus  21276  gzrngunitlem  21399  gzrngunit  21400  issubassa3  21833  subrgpsr  21945  mplring  21986  subrgmvrf  22001  subrgascl  22033  subrgasclcl  22034  evlssca  22061  evlsvar  22062  evlsgsumadd  22063  evlsvarpw  22066  mpfconst  22076  mpfproj  22077  mpfsubrg  22078  gsumply1subr  22186  ply1ring  22200  evls1sca  22279  evls1gsumadd  22280  evls1varpw  22283  evls1varpwval  22324  evls1fpws  22325  evls1addd  22327  evls1muld  22328  asclply1subcl  22330  evls1maplmhm  22333  dmatcrng  22458  scmatcrng  22477  scmatsgrp1  22478  scmatsrng1  22479  scmatmhm  22490  scmatrhm  22491  m2cpmrhm  22702  isclmp  25065  reefgim  26428  amgmlem  26968  cntrcrng  33174  ressply1evls1  33657  ressply10g  33659  evls1subd  33664  evls1monply1  33671  vr1nz  33685  evls1fldgencl  33847  0ringirng  33866  extdgfialglem2  33870  ply1annnr  33880  irngnminplynz  33889  minplyelirng  33892  algextdeglem6  33899  imacrhmcl  42881  evlsscaval  42922  evlsvarval  42923  evlsbagval  42924  evlsmaprhm  42928  amgmwlem  50158