Theorem subrngsubg 14621

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 14620	. . 3 |- ( A e. (SubRng ` R ) -> R e. Rng )
2		rnggrp 14344	. . 3 |- ( R e. Rng -> R e. Grp )
3	1, 2	syl 14	. 2 |- ( A e. (SubRng ` R ) -> R e. Grp )
4		eqid 2238	. . 3 |- ( Base ` R ) = ( Base ` R )
5	4	subrngss 14617	. 2 |- ( A e. (SubRng ` R ) -> A C_ ( Base ` R ) )
6		eqid 2238	. . . 4 |- ( R ↾s A ) = ( R ↾s A )
7	6	subrngrng 14619	. . 3 |- ( A e. (SubRng ` R ) -> ( R ↾s A ) e. Rng )
8		rnggrp 14344	. . 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 14035	. 2 |- ( A e. (SubGrp ` R ) <-> ( R e. Grp /\ A C_ ( Base ` R ) /\ ( R ↾s A ) e. Grp ) )
11	3, 5, 9, 10	syl3anbrc 1212	1 |- ( A e. (SubRng ` R ) -> A e. (SubGrp ` R ) )

This proof depends on syntax axioms:   -> wi 4   e. wcel 2209   C_ wss 3220   ` cfv 5380  (class class class)co 6088   Basecbs 13410   ↾_s cress 13411   Grpcgrp 13864  SubGrpcsubg 14029  Rngcrng 14338  SubRngcsubrng 14614

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-cnex 8276  ax-resscn 8277  ax-1re 8279  ax-addrcl 8282

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-fv 5388  df-ov 6091  df-inn 9314  df-2 9372  df-3 9373  df-ndx 13413  df-slot 13414  df-base 13416  df-plusg 13503  df-mulr 13504  df-subg 14032  df-abl 14180  df-rng 14339  df-subrng 14615

This theorem is used by:  subrngringnsg  14622  subrngbas  14623  subrng0  14624  subrngacl  14625  issubrng2  14627  subrngintm  14629  rng2idl0  14965  rng2idlsubg0  14968