Theorem subrngringnsg 14515

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 14514	. 2 |- ( A e. (SubRng ` R ) -> A e. (SubGrp ` R ) )
2		subrngrcl 14513	. . . . . . . . 9 |- ( A e. (SubRng ` R ) -> R e. Rng )
3		rngabl 14236	. . . . . . . . 9 |- ( R e. Rng -> R e. Abel )
4	2, 3	syl 14	. . . . . . . 8 |- ( A e. (SubRng ` R ) -> R e. Abel )
5	4	3anim1i 1216	. . . . . . 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 1235	. . . . . 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 2238	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
8		eqid 2238	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
9	7, 8	ablcom 14108	. . . . . 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 2307	. . . 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 2632	. 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 14008	. 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 421	1 |- ( A e. (SubRng ` R ) -> A e. (NrmSGrp ` R ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   ` cfv 5377  (class class class)co 6085   Basecbs 13354   +g cplusg 13433  SubGrpcsubg 13972  NrmSGrpcnsg 13973   Abelcabl 14090  Rngcrng 14233  SubRngcsubrng 14507

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 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

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 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-ov 6088  df-inn 9306  df-2 9364  df-3 9365  df-ndx 13357  df-slot 13358  df-base 13360  df-plusg 13446  df-mulr 13447  df-subg 13975  df-nsg 13976  df-cmn 14091  df-abl 14092  df-rng 14234  df-subrng 14508

This theorem is used by:  rng2idlnsg  14857  rng2idlsubgnsg  14860