Theorem rhmopp 14155

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 2229	. 2 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
2		eqid 2229	. 2 |- ( 1r ` (oppr ` R ) ) = ( 1r ` (oppr ` R ) )
3		eqid 2229	. 2 |- ( 1r ` (oppr ` S ) ) = ( 1r ` (oppr ` S ) )
4		eqid 2229	. 2 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
5		eqid 2229	. 2 |- ( .r ` (oppr ` S ) ) = ( .r ` (oppr ` S ) )
6		rhmrcl1 14134	. . 3 |- ( F e. ( R RingHom S ) -> R e. Ring )
7		eqid 2229	. . . . 5 |- (oppr ` R ) = (oppr ` R )
8	7	opprringbg 14058	. . . 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 14135	. . 3 |- ( F e. ( R RingHom S ) -> S e. Ring )
12		eqid 2229	. . . . 5 |- (oppr ` S ) = (oppr ` S )
13	12	opprringbg 14058	. . . 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 2229	. . . 4 |- ( 1r ` R ) = ( 1r ` R )
17		eqid 2229	. . . 4 |- ( 1r ` S ) = ( 1r ` S )
18	16, 17	rhm1 14146	. . 3 |- ( F e. ( R RingHom S ) -> ( F ` ( 1r ` R ) ) = ( 1r ` S ) )
19	7, 16	oppr1g 14060	. . . . . 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 2235	. . . 4 |- ( F e. ( R RingHom S ) -> ( 1r ` (oppr ` R ) ) = ( 1r ` R ) )
22	21	fveq2d 5633	. . 3 |- ( F e. ( R RingHom S ) -> ( F ` ( 1r ` (oppr ` R ) ) ) = ( F ` ( 1r ` R ) ) )
23	12, 17	oppr1g 14060	. . . . 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 2235	. . 3 |- ( F e. ( R RingHom S ) -> ( 1r ` (oppr ` S ) ) = ( 1r ` S ) )
26	18, 22, 25	3eqtr4d 2272	. 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 2229	. . . . . . . 8 |- ( Base ` R ) = ( Base ` R )
30	7, 29	opprbasg 14053	. . . . . . 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 2309	. . . 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 2309	. . . 4 |- ( ( F e. ( R RingHom S ) /\ ( x e. ( Base ` (oppr ` R ) ) /\ y e. ( Base ` (oppr ` R ) ) ) ) -> x e. ( Base ` R ) )
36		eqid 2229	. . . . 5 |- ( .r ` R ) = ( .r ` R )
37		eqid 2229	. . . . 5 |- ( .r ` S ) = ( .r ` S )
38	29, 36, 37	rhmmul 14143	. . . 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 1271	. . 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 14049	. . . . 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 1271	. . . 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 5633	. . 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 2229	. . . . . . 7 |- ( Base ` S ) = ( Base ` S )
46	29, 45	rhmf 14142	. . . . . 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 5773	. . . 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 5773	. . . 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 14049	. . . 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 1271	. . 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 2272	. 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 13983	. . . . 5 |- ( F e. ( R RingHom S ) -> (oppr ` R ) e. Grp )
54	15	ringgrpd 13983	. . . . 5 |- ( F e. ( R RingHom S ) -> (oppr ` S ) e. Grp )
55		rhmghm 14141	. . . . . . . . . 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 2229	. . . . . . . . . 10 |- ( +g ` R ) = ( +g ` R )
60		eqid 2229	. . . . . . . . . 10 |- ( +g ` S ) = ( +g ` S )
61	29, 59, 60	ghmlin 13800	. . . . . . . . 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 1271	. . . . . . . 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 2603	. . . . . . 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 2603	. . . . . 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 14053	. . . . . . . 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 5469	. . . . . 6 |- ( F e. ( R RingHom S ) -> ( F : ( Base ` R ) --> ( Base ` S ) <-> F : ( Base ` (oppr ` R ) ) --> ( Base ` (oppr ` S ) ) ) )
70	7, 59	oppraddg 14054	. . . . . . . . . . . 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 6024	. . . . . . . . . 10 |- ( F e. ( R RingHom S ) -> ( x ( +g ` R ) y ) = ( x ( +g ` (oppr ` R ) ) y ) )
73	72	fveq2d 5633	. . . . . . . . 9 |- ( F e. ( R RingHom S ) -> ( F ` ( x ( +g ` R ) y ) ) = ( F ` ( x ( +g ` (oppr ` R ) ) y ) ) )
74	12, 60	oppraddg 14054	. . . . . . . . . . 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 6024	. . . . . . . . 9 |- ( F e. ( R RingHom S ) -> ( ( F ` x ) ( +g ` S ) ( F ` y ) ) = ( ( F ` x ) ( +g ` (oppr ` S ) ) ( F ` y ) ) )
77	73, 76	eqeq12d 2244	. . . . . . . 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 2744	. . . . . . 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 2744	. . . . . 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 2229	. . . 4 |- ( Base ` (oppr ` S ) ) = ( Base ` (oppr ` S ) )
84		eqid 2229	. . . 4 |- ( +g ` (oppr ` R ) ) = ( +g ` (oppr ` R ) )
85		eqid 2229	. . . 4 |- ( +g ` (oppr ` S ) ) = ( +g ` (oppr ` S ) )
86	1, 83, 84, 85	isghm 13795	. . 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 14144	1 |- ( F e. ( R RingHom S ) -> F e. ( (oppr ` R ) RingHom (oppr ` S ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   A.wral 2508   -->wf 5314   ` cfv 5318  (class class class)co 6007   Basecbs 13047   +g cplusg 13125   .rcmulr 13126   Grpcgrp 13548   GrpHom cghm 13792   1rcur 13937   Ringcrg 13974  opp_rcoppr 14045   RingHom crh 14129

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 617  ax-in2 618  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-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-pre-ltirr 8122  ax-pre-lttrn 8124  ax-pre-ltadd 8126

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  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-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  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-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-tpos 6397  df-map 6805  df-pnf 8194  df-mnf 8195  df-ltxr 8197  df-inn 9122  df-2 9180  df-3 9181  df-ndx 13050  df-slot 13051  df-base 13053  df-sets 13054  df-plusg 13138  df-mulr 13139  df-0g 13306  df-mgm 13404  df-sgrp 13450  df-mnd 13465  df-mhm 13507  df-grp 13551  df-ghm 13793  df-mgp 13899  df-ur 13938  df-ring 13976  df-oppr 14046  df-rhm 14131

This theorem is referenced by:  elrhmunit  14156