Theorem subrngringnsg 13549

Description: A subring is a normal subgroup. (Contributed by AV, 25-Feb-2025.)

Assertion

Ref	Expression
subrngringnsg	|- ( A e. (SubRng ` R ) -> A e. (NrmSGrp ` R ) )

Proof of Theorem subrngringnsg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		subrngsubg 13548	. 2 |- ( A e. (SubRng ` R ) -> A e. (SubGrp ` R ) )
2		subrngrcl 13547	. . . . . . . . 9 |- ( A e. (SubRng ` R ) -> R e. Rng )
3		rngabl 13286	. . . . . . . . 9 |- ( R e. Rng -> R e. Abel )
4	2, 3	syl 14	. . . . . . . 8 |- ( A e. (SubRng ` R ) -> R e. Abel )
5	4	3anim1i 1187	. . . . . . 7 |- ( ( A e. (SubRng ` R ) /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( R e. Abel /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) )
6	5	3expb 1206	. . . . . 6 |- ( ( A e. (SubRng ` R ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( R e. Abel /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) )
7		eqid 2189	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
8		eqid 2189	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
9	7, 8	ablcom 13239	. . . . . 6 |- ( ( R e. Abel /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x ( +g ` R ) y ) = ( y ( +g ` R ) x ) )
10	6, 9	syl 14	. . . . 5 |- ( ( A e. (SubRng ` R ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( x ( +g ` R ) y ) = ( y ( +g ` R ) x ) )
11	10	eleq1d 2258	. . . 4 |- ( ( A e. (SubRng ` R ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( ( x ( +g ` R ) y ) e. A <-> ( y ( +g ` R ) x ) e. A ) )
12	11	biimpd 144	. . 3 |- ( ( A e. (SubRng ` R ) /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( ( x ( +g ` R ) y ) e. A -> ( y ( +g ` R ) x ) e. A ) )
13	12	ralrimivva 2572	. 2 |- ( A e. (SubRng ` R ) -> A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( ( x ( +g ` R ) y ) e. A -> ( y ( +g ` R ) x ) e. A ) )
14	7, 8	isnsg2 13139	. 2 |- ( A e. (NrmSGrp ` R ) <-> ( A e. (SubGrp ` R ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( ( x ( +g ` R ) y ) e. A -> ( y ( +g ` R ) x ) e. A ) ) )
15	1, 13, 14	sylanbrc 417	1 |- ( A e. (SubRng ` R ) -> A e. (NrmSGrp ` R ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2160   A.wral 2468   ` cfv 5235  (class class class)co 5895   Basecbs 12511   +g cplusg 12586  SubGrpcsubg 13103  NrmSGrpcnsg 13104   Abelcabl 13221  Rngcrng 13283  SubRngcsubrng 13541

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-cnex 7931  ax-resscn 7932  ax-1re 7934  ax-addrcl 7937

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243  df-ov 5898  df-inn 8949  df-2 9007  df-3 9008  df-ndx 12514  df-slot 12515  df-base 12517  df-plusg 12599  df-mulr 12600  df-subg 13106  df-nsg 13107  df-cmn 13222  df-abl 13223  df-rng 13284  df-subrng 13542

This theorem is referenced by:  rng2idlnsg  13830  rng2idlsubgnsg  13833