Theorem subrgsubg 18786

Description: A subring is a subgroup. (Contributed by Mario Carneiro, 3-Dec-2014.)

Assertion

Ref	Expression
subrgsubg	|- ( A e. (SubRing ` R ) -> A e. (SubGrp ` R ) )

Proof of Theorem subrgsubg

Step	Hyp	Ref	Expression
1		subrgrcl 18785	. . 3 |- ( A e. (SubRing ` R ) -> R e. Ring )
2		ringgrp 18552	. . 3 |- ( R e. Ring -> R e. Grp )
3	1, 2	syl 17	. 2 |- ( A e. (SubRing ` R ) -> R e. Grp )
4		eqid 2622	. . 3 |- ( Base ` R ) = ( Base ` R )
5	4	subrgss 18781	. 2 |- ( A e. (SubRing ` R ) -> A C_ ( Base ` R ) )
6		eqid 2622	. . . 4 |- ( R ↾s A ) = ( R ↾s A )
7	6	subrgring 18783	. . 3 |- ( A e. (SubRing ` R ) -> ( R ↾s A ) e. Ring )
8		ringgrp 18552	. . 3 |- ( ( R ↾s A ) e. Ring -> ( R ↾s A ) e. Grp )
9	7, 8	syl 17	. 2 |- ( A e. (SubRing ` R ) -> ( R ↾s A ) e. Grp )
10	4	issubg 17594	. 2 |- ( A e. (SubGrp ` R ) <-> ( R e. Grp /\ A C_ ( Base ` R ) /\ ( R ↾s A ) e. Grp ) )
11	3, 5, 9, 10	syl3anbrc 1246	1 |- ( A e. (SubRing ` R ) -> A e. (SubGrp ` R ) )

Syntax hints:   -> wi 4   e. wcel 1990   C_ wss 3574   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858   Grpcgrp 17422  SubGrpcsubg 17588   Ringcrg 18547  SubRingcsubrg 18776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-subg 17591  df-ring 18549  df-subrg 18778

This theorem is referenced by:  subrg0  18787  subrgbas  18789  subrgacl  18791  issubrg2  18800  subrgint  18802  resrhm  18809  rhmima  18811  abvres  18839  issubassa2  19345  resspsrmul  19417  subrgpsr  19419  mplbas2  19470  gsumply1subr  19604  zsssubrg  19804  gzrngunitlem  19811  zringlpirlem1  19832  zringcyg  19839  prmirred  19843  zndvds  19898  resubgval  19955  subrgnrg  22477  sranlm  22488  clmsub  22880  clmneg  22881  clmabs  22883  clmsubcl  22886  isncvsngp  22949  cphsqrtcl3  22987  tchcph  23036  plypf1  23968  dvply2g  24040  taylply2  24122  circgrp  24298  circsubm  24299  rzgrp  24300  jensenlem2  24714  amgmlem  24716  lgseisenlem4  25103  qrng0  25310  qrngneg  25312  subrgchr  29794  nn0archi  29843  rezh  30015  qqhcn  30035  qqhucn  30036  fsumcnsrcl  37736  cnsrplycl  37737  rngunsnply  37743  zringsubgval  42183  amgmwlem  42548