Theorem subrngsubg 14051

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 14050	. . 3 |- ( A e. (SubRng ` R ) -> R e. Rng )
2		rnggrp 13785	. . 3 |- ( R e. Rng -> R e. Grp )
3	1, 2	syl 14	. 2 |- ( A e. (SubRng ` R ) -> R e. Grp )
4		eqid 2206	. . 3 |- ( Base ` R ) = ( Base ` R )
5	4	subrngss 14047	. 2 |- ( A e. (SubRng ` R ) -> A C_ ( Base ` R ) )
6		eqid 2206	. . . 4 |- ( R ↾s A ) = ( R ↾s A )
7	6	subrngrng 14049	. . 3 |- ( A e. (SubRng ` R ) -> ( R ↾s A ) e. Rng )
8		rnggrp 13785	. . 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 13594	. 2 |- ( A e. (SubGrp ` R ) <-> ( R e. Grp /\ A C_ ( Base ` R ) /\ ( R ↾s A ) e. Grp ) )
11	3, 5, 9, 10	syl3anbrc 1184	1 |- ( A e. (SubRng ` R ) -> A e. (SubGrp ` R ) )

Syntax hints:   -> wi 4   e. wcel 2177   C_ wss 3170   ` cfv 5285  (class class class)co 5962   Basecbs 12917   ↾_s cress 12918   Grpcgrp 13417  SubGrpcsubg 13588  Rngcrng 13779  SubRngcsubrng 14044

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-cnex 8046  ax-resscn 8047  ax-1re 8049  ax-addrcl 8052

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-br 4055  df-opab 4117  df-mpt 4118  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-fv 5293  df-ov 5965  df-inn 9067  df-2 9125  df-3 9126  df-ndx 12920  df-slot 12921  df-base 12923  df-plusg 13007  df-mulr 13008  df-subg 13591  df-abl 13708  df-rng 13780  df-subrng 14045

This theorem is referenced by:  subrngringnsg  14052  subrngbas  14053  subrng0  14054  subrngacl  14055  issubrng2  14057  subrngintm  14059  rng2idl0  14366  rng2idlsubg0  14369