Theorem rhmopp 13675

Description: A ring homomorphism is also a ring homomorphism for the opposite rings. (Contributed by Thierry Arnoux, 27-Oct-2017.)

Assertion

Ref	Expression
rhmopp	|- ( F e. ( R RingHom S ) -> F e. ( (oppr ` R ) RingHom (oppr ` S ) ) )

Proof of Theorem rhmopp

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

Step	Hyp	Ref	Expression
1		eqid 2193	. 2 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
2		eqid 2193	. 2 |- ( 1r ` (oppr ` R ) ) = ( 1r ` (oppr ` R ) )
3		eqid 2193	. 2 |- ( 1r ` (oppr ` S ) ) = ( 1r ` (oppr ` S ) )
4		eqid 2193	. 2 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
5		eqid 2193	. 2 |- ( .r ` (oppr ` S ) ) = ( .r ` (oppr ` S ) )
6		rhmrcl1 13654	. . 3 |- ( F e. ( R RingHom S ) -> R e. Ring )
7		eqid 2193	. . . . 5 |- (oppr ` R ) = (oppr ` R )
8	7	opprringbg 13579	. . . 4 |- ( R e. Ring -> ( R e. Ring <-> (oppr ` R ) e. Ring ) )
9	6, 8	syl 14	. . 3 |- ( F e. ( R RingHom S ) -> ( R e. Ring <-> (oppr ` R ) e. Ring ) )
10	6, 9	mpbid 147	. 2 |- ( F e. ( R RingHom S ) -> (oppr ` R ) e. Ring )
11		rhmrcl2 13655	. . 3 |- ( F e. ( R RingHom S ) -> S e. Ring )
12		eqid 2193	. . . . 5 |- (oppr ` S ) = (oppr ` S )
13	12	opprringbg 13579	. . . 4 |- ( S e. Ring -> ( S e. Ring <-> (oppr ` S ) e. Ring ) )
14	11, 13	syl 14	. . 3 |- ( F e. ( R RingHom S ) -> ( S e. Ring <-> (oppr ` S ) e. Ring ) )
15	11, 14	mpbid 147	. 2 |- ( F e. ( R RingHom S ) -> (oppr ` S ) e. Ring )
16		eqid 2193	. . . 4 |- ( 1r ` R ) = ( 1r ` R )
17		eqid 2193	. . . 4 |- ( 1r ` S ) = ( 1r ` S )
18	16, 17	rhm1 13666	. . 3 |- ( F e. ( R RingHom S ) -> ( F ` ( 1r ` R ) ) = ( 1r ` S ) )
19	7, 16	oppr1g 13581	. . . . . 6 |- ( R e. Ring -> ( 1r ` R ) = ( 1r ` (oppr ` R ) ) )
20	6, 19	syl 14	. . . . 5 |- ( F e. ( R RingHom S ) -> ( 1r ` R ) = ( 1r ` (oppr ` R ) ) )
21	20	eqcomd 2199	. . . 4 |- ( F e. ( R RingHom S ) -> ( 1r ` (oppr ` R ) ) = ( 1r ` R ) )
22	21	fveq2d 5559	. . 3 |- ( F e. ( R RingHom S ) -> ( F ` ( 1r ` (oppr ` R ) ) ) = ( F ` ( 1r ` R ) ) )
23	12, 17	oppr1g 13581	. . . . 5 |- ( S e. Ring -> ( 1r ` S ) = ( 1r ` (oppr ` S ) ) )
24	11, 23	syl 14	. . . 4 |- ( F e. ( R RingHom S ) -> ( 1r ` S ) = ( 1r ` (oppr ` S ) ) )
25	24	eqcomd 2199	. . 3 |- ( F e. ( R RingHom S ) -> ( 1r ` (oppr ` S ) ) = ( 1r ` S ) )
26	18, 22, 25	3eqtr4d 2236	. 2 |- ( F e. ( R RingHom S ) -> ( F ` ( 1r ` (oppr ` R ) ) ) = ( 1r ` (oppr ` S ) ) )
27		simpl 109	. . . 4 |- ( ( F e. ( R RingHom S ) /\ ( x e. ( Base ` (oppr ` R ) ) /\ y e. ( Base ` (oppr ` R ) ) ) ) -> F e. ( R RingHom S ) )
28		simprr 531	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ ( x e. ( Base ` (oppr ` R ) ) /\ y e. ( Base ` (oppr ` R ) ) ) ) -> y e. ( Base ` (oppr ` R ) ) )
29		eqid 2193	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
30	7, 29	opprbasg 13574	. . . . . . 7 |- ( R e. Ring -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
31	6, 30	syl 14	. . . . . 6 |- ( F e. ( R RingHom S ) -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
32	27, 31	syl 14	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ ( x e. ( Base ` (oppr ` R ) ) /\ y e. ( Base ` (oppr ` R ) ) ) ) -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
33	28, 32	eleqtrrd 2273	. . . 4 |- ( ( F e. ( R RingHom S ) /\ ( x e. ( Base ` (oppr ` R ) ) /\ y e. ( Base ` (oppr ` R ) ) ) ) -> y e. ( Base ` R ) )
34		simprl 529	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ ( x e. ( Base ` (oppr ` R ) ) /\ y e. ( Base ` (oppr ` R ) ) ) ) -> x e. ( Base ` (oppr ` R ) ) )
35	34, 32	eleqtrrd 2273	. . . 4 |- ( ( F e. ( R RingHom S ) /\ ( x e. ( Base ` (oppr ` R ) ) /\ y e. ( Base ` (oppr ` R ) ) ) ) -> x e. ( Base ` R ) )
36		eqid 2193	. . . . 5 |- ( .r ` R ) = ( .r ` R )
37		eqid 2193	. . . . 5 |- ( .r ` S ) = ( .r ` S )
38	29, 36, 37	rhmmul 13663	. . . 4 |- ( ( F e. ( R RingHom S ) /\ y e. ( Base ` R ) /\ x e. ( Base ` R ) ) -> ( F ` ( y ( .r ` R ) x ) ) = ( ( F ` y ) ( .r ` S ) ( F ` x ) ) )
39	27, 33, 35, 38	syl3anc 1249	. . 3 |- ( ( F e. ( R RingHom S ) /\ ( x e. ( Base ` (oppr ` R ) ) /\ y e. ( Base ` (oppr ` R ) ) ) ) -> ( F ` ( y ( .r ` R ) x ) ) = ( ( F ` y ) ( .r ` S ) ( F ` x ) ) )
40	27, 6	syl 14	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ ( x e. ( Base ` (oppr ` R ) ) /\ y e. ( Base ` (oppr ` R ) ) ) ) -> R e. Ring )
41	29, 36, 7, 4	opprmulg 13570	. . . . 5 |- ( ( R e. Ring /\ x e. ( Base ` (oppr ` R ) ) /\ y e. ( Base ` (oppr ` R ) ) ) -> ( x ( .r ` (oppr ` R ) ) y ) = ( y ( .r ` R ) x ) )
42	40, 34, 28, 41	syl3anc 1249	. . . 4 |- ( ( F e. ( R RingHom S ) /\ ( x e. ( Base ` (oppr ` R ) ) /\ y e. ( Base ` (oppr ` R ) ) ) ) -> ( x ( .r ` (oppr ` R ) ) y ) = ( y ( .r ` R ) x ) )
43	42	fveq2d 5559	. . 3 |- ( ( F e. ( R RingHom S ) /\ ( x e. ( Base ` (oppr ` R ) ) /\ y e. ( Base ` (oppr ` R ) ) ) ) -> ( F ` ( x ( .r ` (oppr ` R ) ) y ) ) = ( F ` ( y ( .r ` R ) x ) ) )
44	27, 11	syl 14	. . . 4 |- ( ( F e. ( R RingHom S ) /\ ( x e. ( Base ` (oppr ` R ) ) /\ y e. ( Base ` (oppr ` R ) ) ) ) -> S e. Ring )
45		eqid 2193	. . . . . . 7 |- ( Base ` S ) = ( Base ` S )
46	29, 45	rhmf 13662	. . . . . 6 |- ( F e. ( R RingHom S ) -> F : ( Base ` R ) --> ( Base ` S ) )
47	27, 46	syl 14	. . . . 5 |- ( ( F e. ( R RingHom S ) /\ ( x e. ( Base ` (oppr ` R ) ) /\ y e. ( Base ` (oppr ` R ) ) ) ) -> F : ( Base ` R ) --> ( Base ` S ) )
48	47, 35	ffvelcdmd 5695	. . . 4 |- ( ( F e. ( R RingHom S ) /\ ( x e. ( Base ` (oppr ` R ) ) /\ y e. ( Base ` (oppr ` R ) ) ) ) -> ( F ` x ) e. ( Base ` S ) )
49	47, 33	ffvelcdmd 5695	. . . 4 |- ( ( F e. ( R RingHom S ) /\ ( x e. ( Base ` (oppr ` R ) ) /\ y e. ( Base ` (oppr ` R ) ) ) ) -> ( F ` y ) e. ( Base ` S ) )
50	45, 37, 12, 5	opprmulg 13570	. . . 4 |- ( ( S e. Ring /\ ( F ` x ) e. ( Base ` S ) /\ ( F ` y ) e. ( Base ` S ) ) -> ( ( F ` x ) ( .r ` (oppr ` S ) ) ( F ` y ) ) = ( ( F ` y ) ( .r ` S ) ( F ` x ) ) )
51	44, 48, 49, 50	syl3anc 1249	. . 3 |- ( ( F e. ( R RingHom S ) /\ ( x e. ( Base ` (oppr ` R ) ) /\ y e. ( Base ` (oppr ` R ) ) ) ) -> ( ( F ` x ) ( .r ` (oppr ` S ) ) ( F ` y ) ) = ( ( F ` y ) ( .r ` S ) ( F ` x ) ) )
52	39, 43, 51	3eqtr4d 2236	. 2 |- ( ( F e. ( R RingHom S ) /\ ( x e. ( Base ` (oppr ` R ) ) /\ y e. ( Base ` (oppr ` R ) ) ) ) -> ( F ` ( x ( .r ` (oppr ` R ) ) y ) ) = ( ( F ` x ) ( .r ` (oppr ` S ) ) ( F ` y ) ) )
53	10	ringgrpd 13504	. . . . 5 |- ( F e. ( R RingHom S ) -> (oppr ` R ) e. Grp )
54	15	ringgrpd 13504	. . . . 5 |- ( F e. ( R RingHom S ) -> (oppr ` S ) e. Grp )
55		rhmghm 13661	. . . . . . . . . 10 |- ( F e. ( R RingHom S ) -> F e. ( R GrpHom S ) )
56	55	ad2antrr 488	. . . . . . . . 9 |- ( ( ( F e. ( R RingHom S ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> F e. ( R GrpHom S ) )
57		simplr 528	. . . . . . . . 9 |- ( ( ( F e. ( R RingHom S ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> x e. ( Base ` R ) )
58		simpr 110	. . . . . . . . 9 |- ( ( ( F e. ( R RingHom S ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> y e. ( Base ` R ) )
59		eqid 2193	. . . . . . . . . 10 |- ( +g ` R ) = ( +g ` R )
60		eqid 2193	. . . . . . . . . 10 |- ( +g ` S ) = ( +g ` S )
61	29, 59, 60	ghmlin 13321	. . . . . . . . 9 |- ( ( F e. ( R GrpHom S ) /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( F ` ( x ( +g ` R ) y ) ) = ( ( F ` x ) ( +g ` S ) ( F ` y ) ) )
62	56, 57, 58, 61	syl3anc 1249	. . . . . . . 8 |- ( ( ( F e. ( R RingHom S ) /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> ( F ` ( x ( +g ` R ) y ) ) = ( ( F ` x ) ( +g ` S ) ( F ` y ) ) )
63	62	ralrimiva 2567	. . . . . . 7 |- ( ( F e. ( R RingHom S ) /\ x e. ( Base ` R ) ) -> A. y e. ( Base ` R ) ( F ` ( x ( +g ` R ) y ) ) = ( ( F ` x ) ( +g ` S ) ( F ` y ) ) )
64	63	ralrimiva 2567	. . . . . 6 |- ( F e. ( R RingHom S ) -> A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( +g ` R ) y ) ) = ( ( F ` x ) ( +g ` S ) ( F ` y ) ) )
65	46, 64	jca 306	. . . . 5 |- ( F e. ( R RingHom S ) -> ( F : ( Base ` R ) --> ( Base ` S ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( +g ` R ) y ) ) = ( ( F ` x ) ( +g ` S ) ( F ` y ) ) ) )
66	53, 54, 65	jca31 309	. . . 4 |- ( F e. ( R RingHom S ) -> ( ( (oppr ` R ) e. Grp /\ (oppr ` S ) e. Grp ) /\ ( F : ( Base ` R ) --> ( Base ` S ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( +g ` R ) y ) ) = ( ( F ` x ) ( +g ` S ) ( F ` y ) ) ) ) )
67	12, 45	opprbasg 13574	. . . . . . . 8 |- ( S e. Ring -> ( Base ` S ) = ( Base ` (oppr ` S ) ) )
68	11, 67	syl 14	. . . . . . 7 |- ( F e. ( R RingHom S ) -> ( Base ` S ) = ( Base ` (oppr ` S ) ) )
69	31, 68	feq23d 5400	. . . . . 6 |- ( F e. ( R RingHom S ) -> ( F : ( Base ` R ) --> ( Base ` S ) <-> F : ( Base ` (oppr ` R ) ) --> ( Base ` (oppr ` S ) ) ) )
70	7, 59	oppraddg 13575	. . . . . . . . . . . 12 |- ( R e. Ring -> ( +g ` R ) = ( +g ` (oppr ` R ) ) )
71	6, 70	syl 14	. . . . . . . . . . 11 |- ( F e. ( R RingHom S ) -> ( +g ` R ) = ( +g ` (oppr ` R ) ) )
72	71	oveqd 5936	. . . . . . . . . 10 |- ( F e. ( R RingHom S ) -> ( x ( +g ` R ) y ) = ( x ( +g ` (oppr ` R ) ) y ) )
73	72	fveq2d 5559	. . . . . . . . 9 |- ( F e. ( R RingHom S ) -> ( F ` ( x ( +g ` R ) y ) ) = ( F ` ( x ( +g ` (oppr ` R ) ) y ) ) )
74	12, 60	oppraddg 13575	. . . . . . . . . . 11 |- ( S e. Ring -> ( +g ` S ) = ( +g ` (oppr ` S ) ) )
75	11, 74	syl 14	. . . . . . . . . 10 |- ( F e. ( R RingHom S ) -> ( +g ` S ) = ( +g ` (oppr ` S ) ) )
76	75	oveqd 5936	. . . . . . . . 9 |- ( F e. ( R RingHom S ) -> ( ( F ` x ) ( +g ` S ) ( F ` y ) ) = ( ( F ` x ) ( +g ` (oppr ` S ) ) ( F ` y ) ) )
77	73, 76	eqeq12d 2208	. . . . . . . 8 |- ( F e. ( R RingHom S ) -> ( ( F ` ( x ( +g ` R ) y ) ) = ( ( F ` x ) ( +g ` S ) ( F ` y ) ) <-> ( F ` ( x ( +g ` (oppr ` R ) ) y ) ) = ( ( F ` x ) ( +g ` (oppr ` S ) ) ( F ` y ) ) ) )
78	31, 77	raleqbidv 2706	. . . . . . 7 |- ( F e. ( R RingHom S ) -> ( A. y e. ( Base ` R ) ( F ` ( x ( +g ` R ) y ) ) = ( ( F ` x ) ( +g ` S ) ( F ` y ) ) <-> A. y e. ( Base ` (oppr ` R ) ) ( F ` ( x ( +g ` (oppr ` R ) ) y ) ) = ( ( F ` x ) ( +g ` (oppr ` S ) ) ( F ` y ) ) ) )
79	31, 78	raleqbidv 2706	. . . . . 6 |- ( F e. ( R RingHom S ) -> ( A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( +g ` R ) y ) ) = ( ( F ` x ) ( +g ` S ) ( F ` y ) ) <-> A. x e. ( Base ` (oppr ` R ) ) A. y e. ( Base ` (oppr ` R ) ) ( F ` ( x ( +g ` (oppr ` R ) ) y ) ) = ( ( F ` x ) ( +g ` (oppr ` S ) ) ( F ` y ) ) ) )
80	69, 79	anbi12d 473	. . . . 5 |- ( F e. ( R RingHom S ) -> ( ( F : ( Base ` R ) --> ( Base ` S ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( +g ` R ) y ) ) = ( ( F ` x ) ( +g ` S ) ( F ` y ) ) ) <-> ( F : ( Base ` (oppr ` R ) ) --> ( Base ` (oppr ` S ) ) /\ A. x e. ( Base ` (oppr ` R ) ) A. y e. ( Base ` (oppr ` R ) ) ( F ` ( x ( +g ` (oppr ` R ) ) y ) ) = ( ( F ` x ) ( +g ` (oppr ` S ) ) ( F ` y ) ) ) ) )
81	80	anbi2d 464	. . . 4 |- ( F e. ( R RingHom S ) -> ( ( ( (oppr ` R ) e. Grp /\ (oppr ` S ) e. Grp ) /\ ( F : ( Base ` R ) --> ( Base ` S ) /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( F ` ( x ( +g ` R ) y ) ) = ( ( F ` x ) ( +g ` S ) ( F ` y ) ) ) ) <-> ( ( (oppr ` R ) e. Grp /\ (oppr ` S ) e. Grp ) /\ ( F : ( Base ` (oppr ` R ) ) --> ( Base ` (oppr ` S ) ) /\ A. x e. ( Base ` (oppr ` R ) ) A. y e. ( Base ` (oppr ` R ) ) ( F ` ( x ( +g ` (oppr ` R ) ) y ) ) = ( ( F ` x ) ( +g ` (oppr ` S ) ) ( F ` y ) ) ) ) ) )
82	66, 81	mpbid 147	. . 3 |- ( F e. ( R RingHom S ) -> ( ( (oppr ` R ) e. Grp /\ (oppr ` S ) e. Grp ) /\ ( F : ( Base ` (oppr ` R ) ) --> ( Base ` (oppr ` S ) ) /\ A. x e. ( Base ` (oppr ` R ) ) A. y e. ( Base ` (oppr ` R ) ) ( F ` ( x ( +g ` (oppr ` R ) ) y ) ) = ( ( F ` x ) ( +g ` (oppr ` S ) ) ( F ` y ) ) ) ) )
83		eqid 2193	. . . 4 |- ( Base ` (oppr ` S ) ) = ( Base ` (oppr ` S ) )
84		eqid 2193	. . . 4 |- ( +g ` (oppr ` R ) ) = ( +g ` (oppr ` R ) )
85		eqid 2193	. . . 4 |- ( +g ` (oppr ` S ) ) = ( +g ` (oppr ` S ) )
86	1, 83, 84, 85	isghm 13316	. . 3 |- ( F e. ( (oppr ` R ) GrpHom (oppr ` S ) ) <-> ( ( (oppr ` R ) e. Grp /\ (oppr ` S ) e. Grp ) /\ ( F : ( Base ` (oppr ` R ) ) --> ( Base ` (oppr ` S ) ) /\ A. x e. ( Base ` (oppr ` R ) ) A. y e. ( Base ` (oppr ` R ) ) ( F ` ( x ( +g ` (oppr ` R ) ) y ) ) = ( ( F ` x ) ( +g ` (oppr ` S ) ) ( F ` y ) ) ) ) )
87	82, 86	sylibr 134	. 2 |- ( F e. ( R RingHom S ) -> F e. ( (oppr ` R ) GrpHom (oppr ` S ) ) )
88	1, 2, 3, 4, 5, 10, 15, 26, 52, 87	isrhm2d 13664	1 |- ( F e. ( R RingHom S ) -> F e. ( (oppr ` R ) RingHom (oppr ` S ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   A.wral 2472   -->wf 5251   ` cfv 5255  (class class class)co 5919   Basecbs 12621   +g cplusg 12698   .rcmulr 12699   Grpcgrp 13075   GrpHom cghm 13313   1rcur 13458   Ringcrg 13495  opp_rcoppr 13566   RingHom crh 13649

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-in1 615  ax-in2 616  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 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-lttrn 7988  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-tpos 6300  df-map 6706  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-3 9044  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-plusg 12711  df-mulr 12712  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-mhm 13034  df-grp 13078  df-ghm 13314  df-mgp 13420  df-ur 13459  df-ring 13497  df-oppr 13567  df-rhm 13651

This theorem is referenced by:  elrhmunit  13676