Theorem quotcan 24064

Description: Exact division with a multiple. (Contributed by Mario Carneiro, 26-Jul-2014.)

Hypothesis

Ref	Expression
quotcan.1	|- H = ( F oF x. G )

Assertion

Ref	Expression
quotcan	|- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( H quot G ) = F )

Proof of Theorem quotcan

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

Step	Hyp	Ref	Expression
1		plyssc 23956	. . . . . . . . 9 |- (Poly ` S ) C_ (Poly ` CC )
2		simp2 1062	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> G e. (Poly ` S ) )
3	1, 2	sseldi 3601	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> G e. (Poly ` CC ) )
4		simp1 1061	. . . . . . . . . 10 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> F e. (Poly ` S ) )
5	1, 4	sseldi 3601	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> F e. (Poly ` CC ) )
6		quotcan.1	. . . . . . . . . . . 12 |- H = ( F oF x. G )
7		plymulcl 23977	. . . . . . . . . . . 12 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( F oF x. G ) e. (Poly ` CC ) )
8	6, 7	syl5eqel 2705	. . . . . . . . . . 11 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> H e. (Poly ` CC ) )
9	8	3adant3 1081	. . . . . . . . . 10 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> H e. (Poly ` CC ) )
10		simp3 1063	. . . . . . . . . 10 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> G =/= 0p )
11		quotcl2 24057	. . . . . . . . . 10 |- ( ( H e. (Poly ` CC ) /\ G e. (Poly ` CC ) /\ G =/= 0p ) -> ( H quot G ) e. (Poly ` CC ) )
12	9, 3, 10, 11	syl3anc 1326	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( H quot G ) e. (Poly ` CC ) )
13		plysubcl 23978	. . . . . . . . 9 |- ( ( F e. (Poly ` CC ) /\ ( H quot G ) e. (Poly ` CC ) ) -> ( F oF - ( H quot G ) ) e. (Poly ` CC ) )
14	5, 12, 13	syl2anc 693	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( F oF - ( H quot G ) ) e. (Poly ` CC ) )
15		plymul0or 24036	. . . . . . . 8 |- ( ( G e. (Poly ` CC ) /\ ( F oF - ( H quot G ) ) e. (Poly ` CC ) ) -> ( ( G oF x. ( F oF - ( H quot G ) ) ) = 0p <-> ( G = 0p \/ ( F oF - ( H quot G ) ) = 0p ) ) )
16	3, 14, 15	syl2anc 693	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( ( G oF x. ( F oF - ( H quot G ) ) ) = 0p <-> ( G = 0p \/ ( F oF - ( H quot G ) ) = 0p ) ) )
17		cnex 10017	. . . . . . . . . . . . 13 |- CC e. _V
18	17	a1i 11	. . . . . . . . . . . 12 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> CC e. _V )
19		plyf 23954	. . . . . . . . . . . . 13 |- ( F e. (Poly ` S ) -> F : CC --> CC )
20	4, 19	syl 17	. . . . . . . . . . . 12 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> F : CC --> CC )
21		plyf 23954	. . . . . . . . . . . . 13 |- ( G e. (Poly ` S ) -> G : CC --> CC )
22	2, 21	syl 17	. . . . . . . . . . . 12 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> G : CC --> CC )
23		mulcom 10022	. . . . . . . . . . . . 13 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) = ( y x. x ) )
24	23	adantl 482	. . . . . . . . . . . 12 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) = ( y x. x ) )
25	18, 20, 22, 24	caofcom 6929	. . . . . . . . . . 11 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( F oF x. G ) = ( G oF x. F ) )
26	6, 25	syl5eq 2668	. . . . . . . . . 10 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> H = ( G oF x. F ) )
27	26	oveq1d 6665	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( H oF - ( G oF x. ( H quot G ) ) ) = ( ( G oF x. F ) oF - ( G oF x. ( H quot G ) ) ) )
28		plyf 23954	. . . . . . . . . . 11 |- ( ( H quot G ) e. (Poly ` CC ) -> ( H quot G ) : CC --> CC )
29	12, 28	syl 17	. . . . . . . . . 10 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( H quot G ) : CC --> CC )
30		subdi 10463	. . . . . . . . . . 11 |- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( x x. ( y - z ) ) = ( ( x x. y ) - ( x x. z ) ) )
31	30	adantl 482	. . . . . . . . . 10 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) /\ ( x e. CC /\ y e. CC /\ z e. CC ) ) -> ( x x. ( y - z ) ) = ( ( x x. y ) - ( x x. z ) ) )
32	18, 22, 20, 29, 31	caofdi 6933	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( G oF x. ( F oF - ( H quot G ) ) ) = ( ( G oF x. F ) oF - ( G oF x. ( H quot G ) ) ) )
33	27, 32	eqtr4d 2659	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( H oF - ( G oF x. ( H quot G ) ) ) = ( G oF x. ( F oF - ( H quot G ) ) ) )
34	33	eqeq1d 2624	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( ( H oF - ( G oF x. ( H quot G ) ) ) = 0p <-> ( G oF x. ( F oF - ( H quot G ) ) ) = 0p ) )
35	10	neneqd 2799	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> -. G = 0p )
36		biorf 420	. . . . . . . 8 |- ( -. G = 0p -> ( ( F oF - ( H quot G ) ) = 0p <-> ( G = 0p \/ ( F oF - ( H quot G ) ) = 0p ) ) )
37	35, 36	syl 17	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( ( F oF - ( H quot G ) ) = 0p <-> ( G = 0p \/ ( F oF - ( H quot G ) ) = 0p ) ) )
38	16, 34, 37	3bitr4d 300	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( ( H oF - ( G oF x. ( H quot G ) ) ) = 0p <-> ( F oF - ( H quot G ) ) = 0p ) )
39	38	biimpd 219	. . . . 5 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( ( H oF - ( G oF x. ( H quot G ) ) ) = 0p -> ( F oF - ( H quot G ) ) = 0p ) )
40		eqid 2622	. . . . . . . . . . 11 |- (deg ` G ) = (deg ` G )
41		eqid 2622	. . . . . . . . . . 11 |- (deg ` ( F oF - ( H quot G ) ) ) = (deg ` ( F oF - ( H quot G ) ) )
42	40, 41	dgrmul 24026	. . . . . . . . . 10 |- ( ( ( G e. (Poly ` CC ) /\ G =/= 0p ) /\ ( ( F oF - ( H quot G ) ) e. (Poly ` CC ) /\ ( F oF - ( H quot G ) ) =/= 0p ) ) -> (deg ` ( G oF x. ( F oF - ( H quot G ) ) ) ) = ( (deg ` G ) + (deg ` ( F oF - ( H quot G ) ) ) ) )
43	42	expr 643	. . . . . . . . 9 |- ( ( ( G e. (Poly ` CC ) /\ G =/= 0p ) /\ ( F oF - ( H quot G ) ) e. (Poly ` CC ) ) -> ( ( F oF - ( H quot G ) ) =/= 0p -> (deg ` ( G oF x. ( F oF - ( H quot G ) ) ) ) = ( (deg ` G ) + (deg ` ( F oF - ( H quot G ) ) ) ) ) )
44	3, 10, 14, 43	syl21anc 1325	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( ( F oF - ( H quot G ) ) =/= 0p -> (deg ` ( G oF x. ( F oF - ( H quot G ) ) ) ) = ( (deg ` G ) + (deg ` ( F oF - ( H quot G ) ) ) ) ) )
45		dgrcl 23989	. . . . . . . . . . . 12 |- ( G e. (Poly ` S ) -> (deg ` G ) e. NN0 )
46	2, 45	syl 17	. . . . . . . . . . 11 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> (deg ` G ) e. NN0 )
47	46	nn0red 11352	. . . . . . . . . 10 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> (deg ` G ) e. RR )
48		dgrcl 23989	. . . . . . . . . . 11 |- ( ( F oF - ( H quot G ) ) e. (Poly ` CC ) -> (deg ` ( F oF - ( H quot G ) ) ) e. NN0 )
49	14, 48	syl 17	. . . . . . . . . 10 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> (deg ` ( F oF - ( H quot G ) ) ) e. NN0 )
50		nn0addge1 11339	. . . . . . . . . 10 |- ( ( (deg ` G ) e. RR /\ (deg ` ( F oF - ( H quot G ) ) ) e. NN0 ) -> (deg ` G ) <_ ( (deg ` G ) + (deg ` ( F oF - ( H quot G ) ) ) ) )
51	47, 49, 50	syl2anc 693	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> (deg ` G ) <_ ( (deg ` G ) + (deg ` ( F oF - ( H quot G ) ) ) ) )
52		breq2 4657	. . . . . . . . 9 |- ( (deg ` ( G oF x. ( F oF - ( H quot G ) ) ) ) = ( (deg ` G ) + (deg ` ( F oF - ( H quot G ) ) ) ) -> ( (deg ` G ) <_ (deg ` ( G oF x. ( F oF - ( H quot G ) ) ) ) <-> (deg ` G ) <_ ( (deg ` G ) + (deg ` ( F oF - ( H quot G ) ) ) ) ) )
53	51, 52	syl5ibrcom 237	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( (deg ` ( G oF x. ( F oF - ( H quot G ) ) ) ) = ( (deg ` G ) + (deg ` ( F oF - ( H quot G ) ) ) ) -> (deg ` G ) <_ (deg ` ( G oF x. ( F oF - ( H quot G ) ) ) ) ) )
54	44, 53	syld 47	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( ( F oF - ( H quot G ) ) =/= 0p -> (deg ` G ) <_ (deg ` ( G oF x. ( F oF - ( H quot G ) ) ) ) ) )
55	33	fveq2d 6195	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> (deg ` ( H oF - ( G oF x. ( H quot G ) ) ) ) = (deg ` ( G oF x. ( F oF - ( H quot G ) ) ) ) )
56	55	breq2d 4665	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( (deg ` G ) <_ (deg ` ( H oF - ( G oF x. ( H quot G ) ) ) ) <-> (deg ` G ) <_ (deg ` ( G oF x. ( F oF - ( H quot G ) ) ) ) ) )
57		plymulcl 23977	. . . . . . . . . . . . 13 |- ( ( G e. (Poly ` CC ) /\ ( H quot G ) e. (Poly ` CC ) ) -> ( G oF x. ( H quot G ) ) e. (Poly ` CC ) )
58	3, 12, 57	syl2anc 693	. . . . . . . . . . . 12 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( G oF x. ( H quot G ) ) e. (Poly ` CC ) )
59		plysubcl 23978	. . . . . . . . . . . 12 |- ( ( H e. (Poly ` CC ) /\ ( G oF x. ( H quot G ) ) e. (Poly ` CC ) ) -> ( H oF - ( G oF x. ( H quot G ) ) ) e. (Poly ` CC ) )
60	9, 58, 59	syl2anc 693	. . . . . . . . . . 11 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( H oF - ( G oF x. ( H quot G ) ) ) e. (Poly ` CC ) )
61		dgrcl 23989	. . . . . . . . . . 11 |- ( ( H oF - ( G oF x. ( H quot G ) ) ) e. (Poly ` CC ) -> (deg ` ( H oF - ( G oF x. ( H quot G ) ) ) ) e. NN0 )
62	60, 61	syl 17	. . . . . . . . . 10 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> (deg ` ( H oF - ( G oF x. ( H quot G ) ) ) ) e. NN0 )
63	62	nn0red 11352	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> (deg ` ( H oF - ( G oF x. ( H quot G ) ) ) ) e. RR )
64	47, 63	lenltd 10183	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( (deg ` G ) <_ (deg ` ( H oF - ( G oF x. ( H quot G ) ) ) ) <-> -. (deg ` ( H oF - ( G oF x. ( H quot G ) ) ) ) < (deg ` G ) ) )
65	56, 64	bitr3d 270	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( (deg ` G ) <_ (deg ` ( G oF x. ( F oF - ( H quot G ) ) ) ) <-> -. (deg ` ( H oF - ( G oF x. ( H quot G ) ) ) ) < (deg ` G ) ) )
66	54, 65	sylibd 229	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( ( F oF - ( H quot G ) ) =/= 0p -> -. (deg ` ( H oF - ( G oF x. ( H quot G ) ) ) ) < (deg ` G ) ) )
67	66	necon4ad 2813	. . . . 5 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( (deg ` ( H oF - ( G oF x. ( H quot G ) ) ) ) < (deg ` G ) -> ( F oF - ( H quot G ) ) = 0p ) )
68		eqid 2622	. . . . . . 7 |- ( H oF - ( G oF x. ( H quot G ) ) ) = ( H oF - ( G oF x. ( H quot G ) ) )
69	68	quotdgr 24058	. . . . . 6 |- ( ( H e. (Poly ` CC ) /\ G e. (Poly ` CC ) /\ G =/= 0p ) -> ( ( H oF - ( G oF x. ( H quot G ) ) ) = 0p \/ (deg ` ( H oF - ( G oF x. ( H quot G ) ) ) ) < (deg ` G ) ) )
70	9, 3, 10, 69	syl3anc 1326	. . . . 5 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( ( H oF - ( G oF x. ( H quot G ) ) ) = 0p \/ (deg ` ( H oF - ( G oF x. ( H quot G ) ) ) ) < (deg ` G ) ) )
71	39, 67, 70	mpjaod 396	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( F oF - ( H quot G ) ) = 0p )
72		df-0p 23437	. . . 4 |- 0p = ( CC X. { 0 } )
73	71, 72	syl6eq 2672	. . 3 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( F oF - ( H quot G ) ) = ( CC X. { 0 } ) )
74		ofsubeq0 11017	. . . 4 |- ( ( CC e. _V /\ F : CC --> CC /\ ( H quot G ) : CC --> CC ) -> ( ( F oF - ( H quot G ) ) = ( CC X. { 0 } ) <-> F = ( H quot G ) ) )
75	18, 20, 29, 74	syl3anc 1326	. . 3 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( ( F oF - ( H quot G ) ) = ( CC X. { 0 } ) <-> F = ( H quot G ) ) )
76	73, 75	mpbid 222	. 2 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> F = ( H quot G ) )
77	76	eqcomd 2628	1 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ G =/= 0p ) -> ( H quot G ) = F )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   {csn 4177   class class class wbr 4653   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   NN0cn0 11292   0pc0p 23436  Polycply 23940  degcdgr 23943   quot cquot 24045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-0p 23437  df-ply 23944  df-coe 23946  df-dgr 23947  df-quot 24046

This theorem is referenced by: (None)