Theorem subrngringnsg 13837

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 13836	. 2 |- ( A e. (SubRng ` R ) -> A e. (SubGrp ` R ) )
2		subrngrcl 13835	. . . . . . . . 9 |- ( A e. (SubRng ` R ) -> R e. Rng )
3		rngabl 13567	. . . . . . . . 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 2196	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
8		eqid 2196	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
9	7, 8	ablcom 13509	. . . . . 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 2265	. . . 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 2579	. 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 13409	. 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 2167   A.wral 2475   ` cfv 5259  (class class class)co 5925   Basecbs 12703   +g cplusg 12780  SubGrpcsubg 13373  NrmSGrpcnsg 13374   Abelcabl 13491  Rngcrng 13564  SubRngcsubrng 13829

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-ov 5928  df-inn 9008  df-2 9066  df-3 9067  df-ndx 12706  df-slot 12707  df-base 12709  df-plusg 12793  df-mulr 12794  df-subg 13376  df-nsg 13377  df-cmn 13492  df-abl 13493  df-rng 13565  df-subrng 13830

This theorem is referenced by:  rng2idlnsg  14150  rng2idlsubgnsg  14153