Theorem pcprmpw2 13944

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 749	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> A e. NN )
2	1	nnnn0d 10632	. . . 4 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> A e. NN0 )
3		prmnn 13762	. . . . . . 7 |- ( P e. Prime -> P e. NN )
4	3	ad2antrr 720	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> P e. NN )
5		pccl 13912	. . . . . . 7 |- ( ( P e. Prime /\ A e. NN ) -> ( P pCnt A ) e. NN0 )
6	5	adantr 462	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P pCnt A ) e. NN0 )
7	4, 6	nnexpcld 12025	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P ^ ( P pCnt A ) ) e. NN )
8	7	nnnn0d 10632	. . . 4 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P ^ ( P pCnt A ) ) e. NN0 )
9	6	nn0red 10633	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P pCnt A ) e. RR )
10	9	leidd 9902	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P pCnt A ) <_ ( P pCnt A ) )
11		simpll 748	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> P e. Prime )
12	6	nn0zd 10741	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P pCnt A ) e. ZZ )
13		pcid 13935	. . . . . . . . . . 11 |- ( ( P e. Prime /\ ( P pCnt A ) e. ZZ ) -> ( P pCnt ( P ^ ( P pCnt A ) ) ) = ( P pCnt A ) )
14	11, 12, 13	syl2anc 656	. . . . . . . . . 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 4315	. . . . . . . . 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 720	. . . . . . . 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 458	. . . . . . . . 9 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p = P ) -> p = P )
18	17	oveq1d 6105	. . . . . . . 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 6105	. . . . . . . 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 4319	. . . . . . 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 755	. . . . . . . . . . . . 13 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> A || ( P ^ n ) )
22		prmz 13763	. . . . . . . . . . . . . . 15 |- ( p e. Prime -> p e. ZZ )
23	22	adantl 463	. . . . . . . . . . . . . 14 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> p e. ZZ )
24	1	adantr 462	. . . . . . . . . . . . . . 15 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> A e. NN )
25	24	nnzd 10742	. . . . . . . . . . . . . 14 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> A e. ZZ )
26		simprl 750	. . . . . . . . . . . . . . . . 17 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> n e. NN0 )
27	4, 26	nnexpcld 12025	. . . . . . . . . . . . . . . 16 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P ^ n ) e. NN )
28	27	adantr 462	. . . . . . . . . . . . . . 15 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( P ^ n ) e. NN )
29	28	nnzd 10742	. . . . . . . . . . . . . 14 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( P ^ n ) e. ZZ )
30		dvdstr 13562	. . . . . . . . . . . . . 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 1213	. . . . . . . . . . . . 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 669	. . . . . . . . . . . 12 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( p || A -> p || ( P ^ n ) ) )
33		simpr 458	. . . . . . . . . . . . 13 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> p e. Prime )
34	11	adantr 462	. . . . . . . . . . . . 13 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> P e. Prime )
35		simplrl 754	. . . . . . . . . . . . 13 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> n e. NN0 )
36		prmdvdsexpr 13798	. . . . . . . . . . . . 13 |- ( ( p e. Prime /\ P e. Prime /\ n e. NN0 ) -> ( p || ( P ^ n ) -> p = P ) )
37	33, 34, 35, 36	syl3anc 1213	. . . . . . . . . . . 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 44	. . . . . . . . . . 11 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( p || A -> p = P ) )
39	38	necon3ad 2642	. . . . . . . . . 10 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( p =/= P -> -. p || A ) )
40	39	imp 429	. . . . . . . . 9 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p =/= P ) -> -. p || A )
41		simplr 749	. . . . . . . . . 10 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p =/= P ) -> p e. Prime )
42	1	ad2antrr 720	. . . . . . . . . 10 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p =/= P ) -> A e. NN )
43		pceq0 13933	. . . . . . . . . 10 |- ( ( p e. Prime /\ A e. NN ) -> ( ( p pCnt A ) = 0 <-> -. p || A ) )
44	41, 42, 43	syl2anc 656	. . . . . . . . 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 232	. . . . . . . 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 720	. . . . . . . . . 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 13913	. . . . . . . . 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 10635	. . . . . . . 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 4309	. . . . . . 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	20, 49	pm2.61dane 2687	. . . . . 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 ) ) ) )
51	50	ralrimiva 2797	. . . . 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 ) ) ) )
52	1	nnzd 10742	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> A e. ZZ )
53	7	nnzd 10742	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P ^ ( P pCnt A ) ) e. ZZ )
54		pc2dvds 13941	. . . . . 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 ) ) ) ) )
55	52, 53, 54	syl2anc 656	. . . . 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 ) ) ) ) )
56	51, 55	mpbird 232	. . . 4 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> A || ( P ^ ( P pCnt A ) ) )
57		pcdvds 13926	. . . . 5 |- ( ( P e. Prime /\ A e. NN ) -> ( P ^ ( P pCnt A ) ) || A )
58	57	adantr 462	. . . 4 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P ^ ( P pCnt A ) ) || A )
59		dvdseq 13576	. . . 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 ) ) )
60	2, 8, 56, 58, 59	syl22anc 1214	. . 3 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> A = ( P ^ ( P pCnt A ) ) )
61	60	rexlimdvaa 2840	. 2 |- ( ( P e. Prime /\ A e. NN ) -> ( E. n e. NN0 A || ( P ^ n ) -> A = ( P ^ ( P pCnt A ) ) ) )
62	3	adantr 462	. . . . . . 7 |- ( ( P e. Prime /\ A e. NN ) -> P e. NN )
63	62, 5	nnexpcld 12025	. . . . . 6 |- ( ( P e. Prime /\ A e. NN ) -> ( P ^ ( P pCnt A ) ) e. NN )
64	63	nnzd 10742	. . . . 5 |- ( ( P e. Prime /\ A e. NN ) -> ( P ^ ( P pCnt A ) ) e. ZZ )
65		iddvds 13542	. . . . 5 |- ( ( P ^ ( P pCnt A ) ) e. ZZ -> ( P ^ ( P pCnt A ) ) || ( P ^ ( P pCnt A ) ) )
66	64, 65	syl 16	. . . 4 |- ( ( P e. Prime /\ A e. NN ) -> ( P ^ ( P pCnt A ) ) || ( P ^ ( P pCnt A ) ) )
67		oveq2 6098	. . . . . 6 |- ( n = ( P pCnt A ) -> ( P ^ n ) = ( P ^ ( P pCnt A ) ) )
68	67	breq2d 4301	. . . . 5 |- ( n = ( P pCnt A ) -> ( ( P ^ ( P pCnt A ) ) || ( P ^ n ) <-> ( P ^ ( P pCnt A ) ) || ( P ^ ( P pCnt A ) ) ) )
69	68	rspcev 3070	. . . 4 |- ( ( ( P pCnt A ) e. NN0 /\ ( P ^ ( P pCnt A ) ) || ( P ^ ( P pCnt A ) ) ) -> E. n e. NN0 ( P ^ ( P pCnt A ) ) || ( P ^ n ) )
70	5, 66, 69	syl2anc 656	. . 3 |- ( ( P e. Prime /\ A e. NN ) -> E. n e. NN0 ( P ^ ( P pCnt A ) ) || ( P ^ n ) )
71		breq1 4292	. . . 4 |- ( A = ( P ^ ( P pCnt A ) ) -> ( A || ( P ^ n ) <-> ( P ^ ( P pCnt A ) ) || ( P ^ n ) ) )
72	71	rexbidv 2734	. . 3 |- ( A = ( P ^ ( P pCnt A ) ) -> ( E. n e. NN0 A || ( P ^ n ) <-> E. n e. NN0 ( P ^ ( P pCnt A ) ) || ( P ^ n ) ) )
73	70, 72	syl5ibrcom 222	. 2 |- ( ( P e. Prime /\ A e. NN ) -> ( A = ( P ^ ( P pCnt A ) ) -> E. n e. NN0 A || ( P ^ n ) ) )
74	61, 73	impbid 191	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   <-> wb 184   /\ wa 369   = wceq 1364   e. wcel 1761   =/= wne 2604   A.wral 2713   E.wrex 2714   class class class wbr 4289  (class class class)co 6090   0cc0 9278   <_ cle 9415   NNcn 10318   NN0cn0 10575   ZZcz 10642   ^cexp 11861   || cdivides 13531   Primecprime 13759   pCnt cpc 13899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-q 10950  df-rp 10988  df-fz 11434  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-dvds 13532  df-gcd 13687  df-prm 13760  df-pc 13900

This theorem is referenced by:  pcprmpw  13945  pgpfi1  16087  pgpfi  16097  sylow2alem2  16110  lt6abl  16364  pgpfac1lem3a  16567  dvdsppwf1o  22485