Theorem subrngsubg 14162

Description: A subring is a subgroup. (Contributed by AV, 14-Feb-2025.)

Assertion

Ref	Expression
subrngsubg	|- ( A e. (SubRng ` R ) -> A e. (SubGrp ` R ) )

Proof of Theorem subrngsubg

Step	Hyp	Ref	Expression
1		subrngrcl 14161	. . 3 |- ( A e. (SubRng ` R ) -> R e. Rng )
2		rnggrp 13896	. . 3 |- ( R e. Rng -> R e. Grp )
3	1, 2	syl 14	. 2 |- ( A e. (SubRng ` R ) -> R e. Grp )
4		eqid 2229	. . 3 |- ( Base ` R ) = ( Base ` R )
5	4	subrngss 14158	. 2 |- ( A e. (SubRng ` R ) -> A C_ ( Base ` R ) )
6		eqid 2229	. . . 4 |- ( R ↾s A ) = ( R ↾s A )
7	6	subrngrng 14160	. . 3 |- ( A e. (SubRng ` R ) -> ( R ↾s A ) e. Rng )
8		rnggrp 13896	. . 3 |- ( ( R ↾s A ) e. Rng -> ( R ↾s A ) e. Grp )
9	7, 8	syl 14	. 2 |- ( A e. (SubRng ` R ) -> ( R ↾s A ) e. Grp )
10	4	issubg 13705	. 2 |- ( A e. (SubGrp ` R ) <-> ( R e. Grp /\ A C_ ( Base ` R ) /\ ( R ↾s A ) e. Grp ) )
11	3, 5, 9, 10	syl3anbrc 1205	1 |- ( A e. (SubRng ` R ) -> A e. (SubGrp ` R ) )

Syntax hints:   -> wi 4   e. wcel 2200   C_ wss 3197   ` cfv 5317  (class class class)co 6000   Basecbs 13027   ↾_s cress 13028   Grpcgrp 13528  SubGrpcsubg 13699  Rngcrng 13890  SubRngcsubrng 14155

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-cnex 8086  ax-resscn 8087  ax-1re 8089  ax-addrcl 8092

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325  df-ov 6003  df-inn 9107  df-2 9165  df-3 9166  df-ndx 13030  df-slot 13031  df-base 13033  df-plusg 13118  df-mulr 13119  df-subg 13702  df-abl 13819  df-rng 13891  df-subrng 14156

This theorem is referenced by:  subrngringnsg  14163  subrngbas  14164  subrng0  14165  subrngacl  14166  issubrng2  14168  subrngintm  14170  rng2idl0  14477  rng2idlsubg0  14480