Theorem pcprmpw2 12905

Description: Self-referential expression for a prime power. (Contributed by Mario Carneiro, 16-Jan-2015.)

Assertion

Ref	Expression
pcprmpw2	|- ( ( P e. Prime /\ A e. NN ) -> ( E. n e. NN0 A || ( P ^ n ) <-> A = ( P ^ ( P pCnt A ) ) ) )

Distinct variable groups:   A, n   P, n

Proof of Theorem pcprmpw2

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 529	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> A e. NN )
2	1	nnnn0d 9454	. . . 4 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> A e. NN0 )
3		prmnn 12681	. . . . . . 7 |- ( P e. Prime -> P e. NN )
4	3	ad2antrr 488	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> P e. NN )
5		pccl 12871	. . . . . . 7 |- ( ( P e. Prime /\ A e. NN ) -> ( P pCnt A ) e. NN0 )
6	5	adantr 276	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P pCnt A ) e. NN0 )
7	4, 6	nnexpcld 10956	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P ^ ( P pCnt A ) ) e. NN )
8	7	nnnn0d 9454	. . . 4 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P ^ ( P pCnt A ) ) e. NN0 )
9	6	nn0red 9455	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P pCnt A ) e. RR )
10	9	leidd 8693	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P pCnt A ) <_ ( P pCnt A ) )
11		simpll 527	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> P e. Prime )
12	6	nn0zd 9599	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P pCnt A ) e. ZZ )
13		pcid 12896	. . . . . . . . . . 11 |- ( ( P e. Prime /\ ( P pCnt A ) e. ZZ ) -> ( P pCnt ( P ^ ( P pCnt A ) ) ) = ( P pCnt A ) )
14	11, 12, 13	syl2anc 411	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P pCnt ( P ^ ( P pCnt A ) ) ) = ( P pCnt A ) )
15	10, 14	breqtrrd 4116	. . . . . . . . 9 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P pCnt A ) <_ ( P pCnt ( P ^ ( P pCnt A ) ) ) )
16	15	ad2antrr 488	. . . . . . . 8 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p = P ) -> ( P pCnt A ) <_ ( P pCnt ( P ^ ( P pCnt A ) ) ) )
17		simpr 110	. . . . . . . . 9 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p = P ) -> p = P )
18	17	oveq1d 6032	. . . . . . . 8 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p = P ) -> ( p pCnt A ) = ( P pCnt A ) )
19	17	oveq1d 6032	. . . . . . . 8 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p = P ) -> ( p pCnt ( P ^ ( P pCnt A ) ) ) = ( P pCnt ( P ^ ( P pCnt A ) ) ) )
20	16, 18, 19	3brtr4d 4120	. . . . . . 7 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p = P ) -> ( p pCnt A ) <_ ( p pCnt ( P ^ ( P pCnt A ) ) ) )
21		simplrr 538	. . . . . . . . . . . . 13 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> A || ( P ^ n ) )
22		prmz 12682	. . . . . . . . . . . . . . 15 |- ( p e. Prime -> p e. ZZ )
23	22	adantl 277	. . . . . . . . . . . . . 14 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> p e. ZZ )
24	1	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> A e. NN )
25	24	nnzd 9600	. . . . . . . . . . . . . 14 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> A e. ZZ )
26		simprl 531	. . . . . . . . . . . . . . . . 17 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> n e. NN0 )
27	4, 26	nnexpcld 10956	. . . . . . . . . . . . . . . 16 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P ^ n ) e. NN )
28	27	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( P ^ n ) e. NN )
29	28	nnzd 9600	. . . . . . . . . . . . . 14 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( P ^ n ) e. ZZ )
30		dvdstr 12388	. . . . . . . . . . . . . 14 |- ( ( p e. ZZ /\ A e. ZZ /\ ( P ^ n ) e. ZZ ) -> ( ( p || A /\ A || ( P ^ n ) ) -> p || ( P ^ n ) ) )
31	23, 25, 29, 30	syl3anc 1273	. . . . . . . . . . . . 13 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( ( p || A /\ A || ( P ^ n ) ) -> p || ( P ^ n ) ) )
32	21, 31	mpan2d 428	. . . . . . . . . . . 12 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( p || A -> p || ( P ^ n ) ) )
33		simpr 110	. . . . . . . . . . . . 13 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> p e. Prime )
34	11	adantr 276	. . . . . . . . . . . . 13 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> P e. Prime )
35		simplrl 537	. . . . . . . . . . . . 13 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> n e. NN0 )
36		prmdvdsexpr 12721	. . . . . . . . . . . . 13 |- ( ( p e. Prime /\ P e. Prime /\ n e. NN0 ) -> ( p || ( P ^ n ) -> p = P ) )
37	33, 34, 35, 36	syl3anc 1273	. . . . . . . . . . . 12 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( p || ( P ^ n ) -> p = P ) )
38	32, 37	syld 45	. . . . . . . . . . 11 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( p || A -> p = P ) )
39	38	necon3ad 2444	. . . . . . . . . 10 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( p =/= P -> -. p || A ) )
40	39	imp 124	. . . . . . . . 9 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p =/= P ) -> -. p || A )
41		simplr 529	. . . . . . . . . 10 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p =/= P ) -> p e. Prime )
42	1	ad2antrr 488	. . . . . . . . . 10 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p =/= P ) -> A e. NN )
43		pceq0 12894	. . . . . . . . . 10 |- ( ( p e. Prime /\ A e. NN ) -> ( ( p pCnt A ) = 0 <-> -. p || A ) )
44	41, 42, 43	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p =/= P ) -> ( ( p pCnt A ) = 0 <-> -. p || A ) )
45	40, 44	mpbird 167	. . . . . . . 8 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p =/= P ) -> ( p pCnt A ) = 0 )
46	7	ad2antrr 488	. . . . . . . . . 10 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p =/= P ) -> ( P ^ ( P pCnt A ) ) e. NN )
47	41, 46	pccld 12872	. . . . . . . . 9 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p =/= P ) -> ( p pCnt ( P ^ ( P pCnt A ) ) ) e. NN0 )
48	47	nn0ge0d 9457	. . . . . . . 8 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p =/= P ) -> 0 <_ ( p pCnt ( P ^ ( P pCnt A ) ) ) )
49	45, 48	eqbrtrd 4110	. . . . . . 7 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p =/= P ) -> ( p pCnt A ) <_ ( p pCnt ( P ^ ( P pCnt A ) ) ) )
50		prmz 12682	. . . . . . . . . . 11 |- ( P e. Prime -> P e. ZZ )
51	50	adantr 276	. . . . . . . . . 10 |- ( ( P e. Prime /\ A e. NN ) -> P e. ZZ )
52	51	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> P e. ZZ )
53		zdceq 9554	. . . . . . . . 9 |- ( ( p e. ZZ /\ P e. ZZ ) -> DECID p = P )
54	23, 52, 53	syl2anc 411	. . . . . . . 8 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> DECID p = P )
55		dcne 2413	. . . . . . . 8 |- (DECID p = P <-> ( p = P \/ p =/= P ) )
56	54, 55	sylib 122	. . . . . . 7 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( p = P \/ p =/= P ) )
57	20, 49, 56	mpjaodan 805	. . . . . 6 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( p pCnt A ) <_ ( p pCnt ( P ^ ( P pCnt A ) ) ) )
58	57	ralrimiva 2605	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> A. p e. Prime ( p pCnt A ) <_ ( p pCnt ( P ^ ( P pCnt A ) ) ) )
59	1	nnzd 9600	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> A e. ZZ )
60	7	nnzd 9600	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P ^ ( P pCnt A ) ) e. ZZ )
61		pc2dvds 12902	. . . . . 6 |- ( ( A e. ZZ /\ ( P ^ ( P pCnt A ) ) e. ZZ ) -> ( A || ( P ^ ( P pCnt A ) ) <-> A. p e. Prime ( p pCnt A ) <_ ( p pCnt ( P ^ ( P pCnt A ) ) ) ) )
62	59, 60, 61	syl2anc 411	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( A || ( P ^ ( P pCnt A ) ) <-> A. p e. Prime ( p pCnt A ) <_ ( p pCnt ( P ^ ( P pCnt A ) ) ) ) )
63	58, 62	mpbird 167	. . . 4 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> A || ( P ^ ( P pCnt A ) ) )
64		pcdvds 12887	. . . . 5 |- ( ( P e. Prime /\ A e. NN ) -> ( P ^ ( P pCnt A ) ) || A )
65	64	adantr 276	. . . 4 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P ^ ( P pCnt A ) ) || A )
66		dvdseq 12408	. . . 4 |- ( ( ( A e. NN0 /\ ( P ^ ( P pCnt A ) ) e. NN0 ) /\ ( A || ( P ^ ( P pCnt A ) ) /\ ( P ^ ( P pCnt A ) ) || A ) ) -> A = ( P ^ ( P pCnt A ) ) )
67	2, 8, 63, 65, 66	syl22anc 1274	. . 3 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> A = ( P ^ ( P pCnt A ) ) )
68	67	rexlimdvaa 2651	. 2 |- ( ( P e. Prime /\ A e. NN ) -> ( E. n e. NN0 A || ( P ^ n ) -> A = ( P ^ ( P pCnt A ) ) ) )
69	3	adantr 276	. . . . . . 7 |- ( ( P e. Prime /\ A e. NN ) -> P e. NN )
70	69, 5	nnexpcld 10956	. . . . . 6 |- ( ( P e. Prime /\ A e. NN ) -> ( P ^ ( P pCnt A ) ) e. NN )
71	70	nnzd 9600	. . . . 5 |- ( ( P e. Prime /\ A e. NN ) -> ( P ^ ( P pCnt A ) ) e. ZZ )
72		iddvds 12364	. . . . 5 |- ( ( P ^ ( P pCnt A ) ) e. ZZ -> ( P ^ ( P pCnt A ) ) || ( P ^ ( P pCnt A ) ) )
73	71, 72	syl 14	. . . 4 |- ( ( P e. Prime /\ A e. NN ) -> ( P ^ ( P pCnt A ) ) || ( P ^ ( P pCnt A ) ) )
74		oveq2 6025	. . . . . 6 |- ( n = ( P pCnt A ) -> ( P ^ n ) = ( P ^ ( P pCnt A ) ) )
75	74	breq2d 4100	. . . . 5 |- ( n = ( P pCnt A ) -> ( ( P ^ ( P pCnt A ) ) || ( P ^ n ) <-> ( P ^ ( P pCnt A ) ) || ( P ^ ( P pCnt A ) ) ) )
76	75	rspcev 2910	. . . 4 |- ( ( ( P pCnt A ) e. NN0 /\ ( P ^ ( P pCnt A ) ) || ( P ^ ( P pCnt A ) ) ) -> E. n e. NN0 ( P ^ ( P pCnt A ) ) || ( P ^ n ) )
77	5, 73, 76	syl2anc 411	. . 3 |- ( ( P e. Prime /\ A e. NN ) -> E. n e. NN0 ( P ^ ( P pCnt A ) ) || ( P ^ n ) )
78		breq1 4091	. . . 4 |- ( A = ( P ^ ( P pCnt A ) ) -> ( A || ( P ^ n ) <-> ( P ^ ( P pCnt A ) ) || ( P ^ n ) ) )
79	78	rexbidv 2533	. . 3 |- ( A = ( P ^ ( P pCnt A ) ) -> ( E. n e. NN0 A || ( P ^ n ) <-> E. n e. NN0 ( P ^ ( P pCnt A ) ) || ( P ^ n ) ) )
80	77, 79	syl5ibrcom 157	. 2 |- ( ( P e. Prime /\ A e. NN ) -> ( A = ( P ^ ( P pCnt A ) ) -> E. n e. NN0 A || ( P ^ n ) ) )
81	68, 80	impbid 129	1 |- ( ( P e. Prime /\ A e. NN ) -> ( E. n e. NN0 A || ( P ^ n ) <-> A = ( P ^ ( P pCnt A ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715  DECID wdc 841   = wceq 1397   e. wcel 2202   =/= wne 2402   A.wral 2510   E.wrex 2511   class class class wbr 4088  (class class class)co 6017   0cc0 8031   <_ cle 8214   NNcn 9142   NN0cn0 9401   ZZcz 9478   ^cexp 10799   || cdvds 12347   Primecprime 12678   pCnt cpc 12856

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 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

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 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-1o 6581  df-2o 6582  df-er 6701  df-en 6909  df-sup 7182  df-inf 7183  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-xnn0 9465  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-fz 10243  df-fzo 10377  df-fl 10529  df-mod 10584  df-seqfrec 10709  df-exp 10800  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-dvds 12348  df-gcd 12524  df-prm 12679  df-pc 12857

This theorem is referenced by:  pcprmpw  12906  dvdsprmpweq  12907  dvdsppwf1o  15712