Theorem pcdvdsb 12956

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 9453	. . . . . . . . 9 |- ( A e. NN0 -> A e. RR )
2	1	3ad2ant3 1047	. . . . . . . 8 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> A e. RR )
3	2	rexrd 8271	. . . . . . 7 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> A e. RR* )
4		pnfge 10068	. . . . . . 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 12940	. . . . . . 7 |- ( P e. Prime -> ( P pCnt 0 ) = +oo )
7	6	3ad2ant1 1045	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( P pCnt 0 ) = +oo )
8	5, 7	breqtrrd 4121	. . . . 5 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> A <_ ( P pCnt 0 ) )
9		prmnn 12745	. . . . . . . . 9 |- ( P e. Prime -> P e. NN )
10		nnexpcl 10860	. . . . . . . . 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 1043	. . . . . . 7 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( P ^ A ) e. NN )
13	12	nnzd 9645	. . . . . 6 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> ( P ^ A ) e. ZZ )
14		dvds0 12430	. . . . . 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 6036	. . . . . 6 |- ( N = 0 -> ( P pCnt N ) = ( P pCnt 0 ) )
19	18	breq2d 4105	. . . . 5 |- ( N = 0 -> ( A <_ ( P pCnt N ) <-> A <_ ( P pCnt 0 ) ) )
20		breq2 4097	. . . . 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 1029	. . . . . . 7 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> A e. NN0 )
25	24	nn0zd 9644	. . . . . 6 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> A e. ZZ )
26		simpl1 1027	. . . . . . . 8 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> P e. Prime )
27		simpl2 1028	. . . . . . . 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 12934	. . . . . . . 8 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P pCnt N ) e. NN0 )
30	26, 27, 28, 29	syl12anc 1272	. . . . . . 7 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P pCnt N ) e. NN0 )
31	30	nn0zd 9644	. . . . . 6 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P pCnt N ) e. ZZ )
32		eluz 9813	. . . . . 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 9645	. . . . . 6 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> P e. ZZ )
36		dvdsexp 12485	. . . . . . 7 |- ( ( P e. ZZ /\ A e. NN0 /\ ( P pCnt N ) e. ( ZZ>= ` A ) ) -> ( P ^ A ) || ( P ^ ( P pCnt N ) ) )
37	36	3expia 1232	. . . . . 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 12950	. . . . . 6 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( P pCnt N ) ) || N )
41	26, 27, 28, 40	syl12anc 1272	. . . . 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 11003	. . . . . . 7 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P ^ ( P pCnt N ) ) e. NN )
44	43	nnzd 9645	. . . . . 6 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P ^ ( P pCnt N ) ) e. ZZ )
45		dvdstr 12452	. . . . . 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 1274	. . . . 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 9600	. . . . 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 9543	. . . . . . . 8 |- ( ( P pCnt N ) e. NN0 -> ( P pCnt N ) e. ZZ )
52		nn0z 9543	. . . . . . . 8 |- ( A e. NN0 -> A e. ZZ )
53		zltnle 9569	. . . . . . . 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 9586	. . . . . . 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 9484	. . . . . . . . . 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 9644	. . . . . . . 8 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( ( P pCnt N ) + 1 ) e. ZZ )
61		eluz 9813	. . . . . . . 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 12485	. . . . . . . . 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 1232	. . . . . . . 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 12952	. . . . . . . . 9 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. ( P ^ ( ( P pCnt N ) + 1 ) ) || N )
68	26, 27, 28, 67	syl12anc 1272	. . . . . . . 8 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> -. ( P ^ ( ( P pCnt N ) + 1 ) ) || N )
69	34, 59	nnexpcld 11003	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P ^ ( ( P pCnt N ) + 1 ) ) e. NN )
70	69	nnzd 9645	. . . . . . . . 9 |- ( ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) /\ N =/= 0 ) -> ( P ^ ( ( P pCnt N ) + 1 ) ) e. ZZ )
71		dvdstr 12452	. . . . . . . . 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 1274	. . . . . . . 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 697	. . . . . . 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 861	. . . 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 1025	. . . 4 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> N e. ZZ )
82		0zd 9535	. . . 4 |- ( ( P e. Prime /\ N e. ZZ /\ A e. NN0 ) -> 0 e. ZZ )
83		zdceq 9599	. . . 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 2414	. . 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 806	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 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2202   =/= wne 2403   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   RRcr 8074   0cc0 8075   1c1 8076   + caddc 8078   +oocpnf 8253   RR*cxr 8255   < clt 8256   <_ cle 8257   NNcn 9185   NN0cn0 9444   ZZcz 9523   ZZ>=cuz 9799   ^cexp 10846   || cdvds 12411   Primecprime 12742   pCnt cpc 12920

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-2o 6626  df-er 6745  df-en 6953  df-sup 7226  df-inf 7227  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-n0 9445  df-z 9524  df-uz 9800  df-q 9898  df-rp 9933  df-fz 10289  df-fzo 10423  df-fl 10576  df-mod 10631  df-seqfrec 10756  df-exp 10847  df-cj 11465  df-re 11466  df-im 11467  df-rsqrt 11621  df-abs 11622  df-dvds 12412  df-gcd 12588  df-prm 12743  df-pc 12921

This theorem is referenced by:  pcelnn  12957  pcidlem  12959  pcdvdstr  12963  pcgcd1  12964  pcfac  12986  pockthlem  12992  pockthg  12993