Theorem subrngringnsg 14454

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 14453	. 2 |- ( A e. (SubRng ` R ) -> A e. (SubGrp ` R ) )
2		subrngrcl 14452	. . . . . . . . 9 |- ( A e. (SubRng ` R ) -> R e. Rng )
3		rngabl 14177	. . . . . . . . 9 |- ( R e. Rng -> R e. Abel )
4	2, 3	syl 14	. . . . . . . 8 |- ( A e. (SubRng ` R ) -> R e. Abel )
5	4	3anim1i 1212	. . . . . . 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 1231	. . . . . 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 2234	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
8		eqid 2234	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
9	7, 8	ablcom 14059	. . . . . 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 2303	. . . 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 2626	. 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 13959	. 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 1005   = wceq 1398   e. wcel 2205   A.wral 2522   ` cfv 5357  (class class class)co 6058   Basecbs 13299   +g cplusg 13377  SubGrpcsubg 13923  NrmSGrpcnsg 13924   Abelcabl 14041  Rngcrng 14174  SubRngcsubrng 14446

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-ov 6061  df-inn 9258  df-2 9316  df-3 9317  df-ndx 13302  df-slot 13303  df-base 13305  df-plusg 13390  df-mulr 13391  df-subg 13926  df-nsg 13927  df-cmn 14042  df-abl 14043  df-rng 14175  df-subrng 14447

This theorem is referenced by:  rng2idlnsg  14795  rng2idlsubgnsg  14798