Theorem pcdvdsb 12911

Description: P ^ A divides N if and only if A is at most the count of P. (Contributed by Mario Carneiro, 3-Oct-2014.)

Assertion

Ref	Expression
pcdvdsb	|- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( A <_ ( P pCnt N ) <-> ( P ^ A ) || N ) )

Proof of Theorem pcdvdsb

Step	Hyp	Ref	Expression
1		nn0re 9411	. . . . . . . . 9 |- ( A e. NN0 -> A e. RR )
2	1	3ad2ant3 1046	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> A e. RR )
3	2	rexrd 8229	. . . . . . 7 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> A e. RR* )
4		pnfge 10024	. . . . . . 7 |- ( A e. RR* -> A <_ +oo )
5	3, 4	syl 14	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> A <_ +oo )
6		pc0 12895	. . . . . . 7 |- ( P e. Prime -> ( P pCnt 0 ) = +oo )
7	6	3ad2ant1 1044	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( P pCnt 0 ) = +oo )
8	5, 7	breqtrrd 4116	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> A <_ ( P pCnt 0 ) )
9		prmnn 12700	. . . . . . . . 9 |- ( P e. Prime -> P e. NN )
10		nnexpcl 10815	. . . . . . . . 9 |- ( ( P e. NN /\ A e. NN0 ) -> ( P ^ A ) e. NN )
11	9, 10	sylan 283	. . . . . . . 8 |- ( ( P e. Prime /\ A e. NN0 ) -> ( P ^ A ) e. NN )
12	11	3adant2 1042	. . . . . . 7 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( P ^ A ) e. NN )
13	12	nnzd 9601	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( P ^ A ) e. ZZ )
14		dvds0 12385	. . . . . 6 |- ( ( P ^ A ) e. ZZ -> ( P ^ A ) || 0 )
15	13, 14	syl 14	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( P ^ A ) || 0 )
16	8, 15	2thd 175	. . . 4 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( A <_ ( P pCnt 0 ) <-> ( P ^ A ) || 0 ) )
17	16	adantr 276	. . 3 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N = 0 ) -> ( A <_ ( P pCnt 0 ) <-> ( P ^ A ) || 0 ) )
18		oveq2 6026	. . . . . 6 |- ( N = 0 -> ( P pCnt N ) = ( P pCnt 0 ) )
19	18	breq2d 4100	. . . . 5 |- ( N = 0 -> ( A <_ ( P pCnt N ) <-> A <_ ( P pCnt 0 ) ) )
20		breq2 4092	. . . . 5 |- ( N = 0 -> ( ( P ^ A ) || N <-> ( P ^ A ) || 0 ) )
21	19, 20	bibi12d 235	. . . 4 |- ( N = 0 -> ( ( A <_ ( P pCnt N ) <-> ( P ^ A ) || N ) <-> ( A <_ ( P pCnt 0 ) <-> ( P ^ A ) || 0 ) ) )
22	21	adantl 277	. . 3 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N = 0 ) -> ( ( A <_ ( P pCnt N ) <-> ( P ^ A ) || N ) <-> ( A <_ ( P pCnt 0 ) <-> ( P ^ A ) || 0 ) ) )
23	17, 22	mpbird 167	. 2 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N = 0 ) -> ( A <_ ( P pCnt N ) <-> ( P ^ A ) || N ) )
24		simpl3 1028	. . . . . . 7 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> A e. NN0 )
25	24	nn0zd 9600	. . . . . 6 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> A e. ZZ )
26		simpl1 1026	. . . . . . . 8 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> P e. Prime )
27		simpl2 1027	. . . . . . . 8 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> N e. ZZ )
28		simpr 110	. . . . . . . 8 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> N =/= 0 )
29		pczcl 12889	. . . . . . . 8 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P pCnt N ) e. NN0 )
30	26, 27, 28, 29	syl12anc 1271	. . . . . . 7 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P pCnt N ) e. NN0 )
31	30	nn0zd 9600	. . . . . 6 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P pCnt N ) e. ZZ )
32		eluz 9769	. . . . . 6 |- ( ( A e. ZZ /\ ( P pCnt N ) e. ZZ ) -> ( ( P pCnt N ) e. ( ZZ>= ` A ) <-> A <_ ( P pCnt N ) ) )
33	25, 31, 32	syl2anc 411	. . . . 5 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( ( P pCnt N ) e. ( ZZ>= ` A ) <-> A <_ ( P pCnt N ) ) )
34	26, 9	syl 14	. . . . . . 7 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> P e. NN )
35	34	nnzd 9601	. . . . . 6 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> P e. ZZ )
36		dvdsexp 12440	. . . . . . 7 |- ( ( P e. ZZ /\ A e. NN0 /\ ( P pCnt N ) e. ( ZZ>= ` A ) ) -> ( P ^ A ) || ( P ^ ( P pCnt N ) ) )
37	36	3expia 1231	. . . . . 6 |- ( ( P e. ZZ /\ A e. NN0 ) -> ( ( P pCnt N ) e. ( ZZ>= ` A ) -> ( P ^ A ) || ( P ^ ( P pCnt N ) ) ) )
38	35, 24, 37	syl2anc 411	. . . . 5 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( ( P pCnt N ) e. ( ZZ>= ` A ) -> ( P ^ A ) || ( P ^ ( P pCnt N ) ) ) )
39	33, 38	sylbird 170	. . . 4 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( A <_ ( P pCnt N ) -> ( P ^ A ) || ( P ^ ( P pCnt N ) ) ) )
40		pczdvds 12905	. . . . . 6 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( P pCnt N ) ) || N )
41	26, 27, 28, 40	syl12anc 1271	. . . . 5 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P ^ ( P pCnt N ) ) || N )
42	13	adantr 276	. . . . . 6 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P ^ A ) e. ZZ )
43	34, 30	nnexpcld 10958	. . . . . . 7 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P ^ ( P pCnt N ) ) e. NN )
44	43	nnzd 9601	. . . . . 6 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P ^ ( P pCnt N ) ) e. ZZ )
45		dvdstr 12407	. . . . . 6 |- ( ( ( P ^ A ) e. ZZ /\ ( P ^ ( P pCnt N ) ) e. ZZ /\ N e. ZZ ) -> ( ( ( P ^ A ) || ( P ^ ( P pCnt N ) ) /\ ( P ^ ( P pCnt N ) ) || N ) -> ( P ^ A ) || N ) )
46	42, 44, 27, 45	syl3anc 1273	. . . . 5 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( ( ( P ^ A ) || ( P ^ ( P pCnt N ) ) /\ ( P ^ ( P pCnt N ) ) || N ) -> ( P ^ A ) || N ) )
47	41, 46	mpan2d 428	. . . 4 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( ( P ^ A ) || ( P ^ ( P pCnt N ) ) -> ( P ^ A ) || N ) )
48	39, 47	syld 45	. . 3 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( A <_ ( P pCnt N ) -> ( P ^ A ) || N ) )
49		zdcle 9556	. . . . 5 |- ( ( A e. ZZ /\ ( P pCnt N ) e. ZZ ) -> DECID A <_ ( P pCnt N ) )
50	25, 31, 49	syl2anc 411	. . . 4 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> DECID A <_ ( P pCnt N ) )
51		nn0z 9499	. . . . . . . 8 |- ( ( P pCnt N ) e. NN0 -> ( P pCnt N ) e. ZZ )
52		nn0z 9499	. . . . . . . 8 |- ( A e. NN0 -> A e. ZZ )
53		zltnle 9525	. . . . . . . 8 |- ( ( ( P pCnt N ) e. ZZ /\ A e. ZZ ) -> ( ( P pCnt N ) < A <-> -. A <_ ( P pCnt N ) ) )
54	51, 52, 53	syl2an 289	. . . . . . 7 |- ( ( ( P pCnt N ) e. NN0 /\ A e. NN0 ) -> ( ( P pCnt N ) < A <-> -. A <_ ( P pCnt N ) ) )
55		nn0ltp1le 9542	. . . . . . 7 |- ( ( ( P pCnt N ) e. NN0 /\ A e. NN0 ) -> ( ( P pCnt N ) < A <-> ( ( P pCnt N ) + 1 ) <_ A ) )
56	54, 55	bitr3d 190	. . . . . 6 |- ( ( ( P pCnt N ) e. NN0 /\ A e. NN0 ) -> ( -. A <_ ( P pCnt N ) <-> ( ( P pCnt N ) + 1 ) <_ A ) )
57	30, 24, 56	syl2anc 411	. . . . 5 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( -. A <_ ( P pCnt N ) <-> ( ( P pCnt N ) + 1 ) <_ A ) )
58		peano2nn0 9442	. . . . . . . . . 10 |- ( ( P pCnt N ) e. NN0 -> ( ( P pCnt N ) + 1 ) e. NN0 )
59	30, 58	syl 14	. . . . . . . . 9 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( ( P pCnt N ) + 1 ) e. NN0 )
60	59	nn0zd 9600	. . . . . . . 8 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( ( P pCnt N ) + 1 ) e. ZZ )
61		eluz 9769	. . . . . . . 8 |- ( ( ( ( P pCnt N ) + 1 ) e. ZZ /\ A e. ZZ ) -> ( A e. ( ZZ>= ` ( ( P pCnt N ) + 1 ) ) <-> ( ( P pCnt N ) + 1 ) <_ A ) )
62	60, 25, 61	syl2anc 411	. . . . . . 7 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( A e. ( ZZ>= ` ( ( P pCnt N ) + 1 ) ) <-> ( ( P pCnt N ) + 1 ) <_ A ) )
63		dvdsexp 12440	. . . . . . . . 9 |- ( ( P e. ZZ /\ ( ( P pCnt N ) + 1 ) e. NN0 /\ A e. ( ZZ>= ` ( ( P pCnt N ) + 1 ) ) ) -> ( P ^ ( ( P pCnt N ) + 1 ) ) || ( P ^ A ) )
64	63	3expia 1231	. . . . . . . 8 |- ( ( P e. ZZ /\ ( ( P pCnt N ) + 1 ) e. NN0 ) -> ( A e. ( ZZ>= ` ( ( P pCnt N ) + 1 ) ) -> ( P ^ ( ( P pCnt N ) + 1 ) ) || ( P ^ A ) ) )
65	35, 59, 64	syl2anc 411	. . . . . . 7 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( A e. ( ZZ>= ` ( ( P pCnt N ) + 1 ) ) -> ( P ^ ( ( P pCnt N ) + 1 ) ) || ( P ^ A ) ) )
66	62, 65	sylbird 170	. . . . . 6 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( ( ( P pCnt N ) + 1 ) <_ A -> ( P ^ ( ( P pCnt N ) + 1 ) ) || ( P ^ A ) ) )
67		pczndvds 12907	. . . . . . . . 9 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. ( P ^ ( ( P pCnt N ) + 1 ) ) || N )
68	26, 27, 28, 67	syl12anc 1271	. . . . . . . 8 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> -. ( P ^ ( ( P pCnt N ) + 1 ) ) || N )
69	34, 59	nnexpcld 10958	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P ^ ( ( P pCnt N ) + 1 ) ) e. NN )
70	69	nnzd 9601	. . . . . . . . 9 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P ^ ( ( P pCnt N ) + 1 ) ) e. ZZ )
71		dvdstr 12407	. . . . . . . . 9 |- ( ( ( P ^ ( ( P pCnt N ) + 1 ) ) e. ZZ /\ ( P ^ A ) e. ZZ /\ N e. ZZ ) -> ( ( ( P ^ ( ( P pCnt N ) + 1 ) ) || ( P ^ A ) /\ ( P ^ A ) || N ) -> ( P ^ ( ( P pCnt N ) + 1 ) ) || N ) )
72	70, 42, 27, 71	syl3anc 1273	. . . . . . . 8 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( ( ( P ^ ( ( P pCnt N ) + 1 ) ) || ( P ^ A ) /\ ( P ^ A ) || N ) -> ( P ^ ( ( P pCnt N ) + 1 ) ) || N ) )
73	68, 72	mtod 669	. . . . . . 7 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> -. ( ( P ^ ( ( P pCnt N ) + 1 ) ) || ( P ^ A ) /\ ( P ^ A ) || N ) )
74		imnan 696	. . . . . . 7 |- ( ( ( P ^ ( ( P pCnt N ) + 1 ) ) || ( P ^ A ) -> -. ( P ^ A ) || N ) <-> -. ( ( P ^ ( ( P pCnt N ) + 1 ) ) || ( P ^ A ) /\ ( P ^ A ) || N ) )
75	73, 74	sylibr 134	. . . . . 6 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( ( P ^ ( ( P pCnt N ) + 1 ) ) || ( P ^ A ) -> -. ( P ^ A ) || N ) )
76	66, 75	syld 45	. . . . 5 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( ( ( P pCnt N ) + 1 ) <_ A -> -. ( P ^ A ) || N ) )
77	57, 76	sylbid 150	. . . 4 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( -. A <_ ( P pCnt N ) -> -. ( P ^ A ) || N ) )
78		condc 860	. . . 4 |- (DECID A <_ ( P pCnt N ) -> ( ( -. A <_ ( P pCnt N ) -> -. ( P ^ A ) || N ) -> ( ( P ^ A ) || N -> A <_ ( P pCnt N ) ) ) )
79	50, 77, 78	sylc 62	. . 3 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( ( P ^ A ) || N -> A <_ ( P pCnt N ) ) )
80	48, 79	impbid 129	. 2 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( A <_ ( P pCnt N ) <-> ( P ^ A ) || N ) )
81		simp2 1024	. . . 4 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> N e. ZZ )
82		0zd 9491	. . . 4 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> 0 e. ZZ )
83		zdceq 9555	. . . 4 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
84	81, 82, 83	syl2anc 411	. . 3 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> DECID N = 0 )
85		dcne 2413	. . 3 |- (DECID N = 0 <-> ( N = 0 \/ N =/= 0 ) )
86	84, 85	sylib 122	. 2 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( N = 0 \/ N =/= 0 ) )
87	23, 80, 86	mpjaodan 805	1 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( A <_ ( P pCnt N ) <-> ( P ^ A ) || N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715  DECID wdc 841   /\ w3a 1004   = wceq 1397   e. wcel 2202   =/= wne 2402   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   RRcr 8031   0cc0 8032   1c1 8033   + caddc 8035   +oocpnf 8211   RR*cxr 8213   < clt 8214   <_ cle 8215   NNcn 9143   NN0cn0 9402   ZZcz 9479   ZZ>=cuz 9755   ^cexp 10801   || cdvds 12366   Primecprime 12697   pCnt cpc 12875

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-1o 6582  df-2o 6583  df-er 6702  df-en 6910  df-sup 7183  df-inf 7184  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-fz 10244  df-fzo 10378  df-fl 10531  df-mod 10586  df-seqfrec 10711  df-exp 10802  df-cj 11420  df-re 11421  df-im 11422  df-rsqrt 11576  df-abs 11577  df-dvds 12367  df-gcd 12543  df-prm 12698  df-pc 12876

This theorem is referenced by:  pcelnn  12912  pcidlem  12914  pcdvdstr  12918  pcgcd1  12919  pcfac  12941  pockthlem  12947  pockthg  12948