Theorem subrgring 20490

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 2731	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		eqid 2731	. . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅)
4	2, 3	issubrg 20487	. . . 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 2835	1 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ⊆ wss 3902  ‘cfv 6481  (class class class)co 7346  Basecbs 17120   ↾_s cress 17141  1_rcur 20100  Ringcrg 20152  SubRingcsubrg 20485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-subrg 20486

This theorem is referenced by:  subrgcrng  20491  subrgsubg  20493  subrg1  20498  subrgsubm  20501  subrguss  20503  subrginv  20504  subrgunit  20506  subrgugrp  20507  subrgnzr  20510  subsubrg  20514  resrhm  20517  resrhm2b  20518  issubdrg  20696  imadrhmcl  20713  subdrgint  20719  abvres  20747  sralmod  21122  ring2idlqus  21247  gzrngunitlem  21370  gzrngunit  21371  issubassa3  21804  subrgpsr  21916  mplring  21957  subrgmvrf  21970  subrgascl  22002  subrgasclcl  22003  evlssca  22025  evlsvar  22026  evlsgsumadd  22027  evlsvarpw  22030  mpfconst  22037  mpfproj  22038  mpfsubrg  22039  gsumply1subr  22147  ply1ring  22161  evls1sca  22239  evls1gsumadd  22240  evls1varpw  22243  evls1varpwval  22284  evls1fpws  22285  evls1addd  22287  evls1muld  22288  asclply1subcl  22290  evls1maplmhm  22293  dmatcrng  22418  scmatcrng  22437  scmatsgrp1  22438  scmatsrng1  22439  scmatmhm  22450  scmatrhm  22451  m2cpmrhm  22662  isclmp  25025  reefgim  26388  amgmlem  26928  cntrcrng  33048  ressply1evls1  33526  ressply10g  33528  evls1subd  33533  evls1monply1  33540  vr1nz  33552  evls1fldgencl  33681  0ringirng  33700  extdgfialglem2  33704  ply1annnr  33714  irngnminplynz  33723  minplyelirng  33726  algextdeglem6  33733  imacrhmcl  42553  evlsscaval  42603  evlsvarval  42604  evlsbagval  42605  evlsmaprhm  42609  amgmwlem  49840