Theorem subrngringnsg 14184

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 14183	. 2 |- ( A e. (SubRng ` R ) -> A e. (SubGrp ` R ) )
2		subrngrcl 14182	. . . . . . . . 9 |- ( A e. (SubRng ` R ) -> R e. Rng )
3		rngabl 13913	. . . . . . . . 9 |- ( R e. Rng -> R e. Abel )
4	2, 3	syl 14	. . . . . . . 8 |- ( A e. (SubRng ` R ) -> R e. Abel )
5	4	3anim1i 1209	. . . . . . 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 1228	. . . . . 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 2229	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
8		eqid 2229	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
9	7, 8	ablcom 13855	. . . . . 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 2298	. . . 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 2612	. 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 13755	. 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 1002   = wceq 1395   e. wcel 2200   A.wral 2508   ` cfv 5318  (class class class)co 6007   Basecbs 13047   +g cplusg 13125  SubGrpcsubg 13719  NrmSGrpcnsg 13720   Abelcabl 13837  Rngcrng 13910  SubRngcsubrng 14176

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 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-cnex 8101  ax-resscn 8102  ax-1re 8104  ax-addrcl 8107

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 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-ov 6010  df-inn 9122  df-2 9180  df-3 9181  df-ndx 13050  df-slot 13051  df-base 13053  df-plusg 13138  df-mulr 13139  df-subg 13722  df-nsg 13723  df-cmn 13838  df-abl 13839  df-rng 13911  df-subrng 14177

This theorem is referenced by:  rng2idlnsg  14497  rng2idlsubgnsg  14500