Theorem pcprmpw2 14791

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 760	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> A e. NN )
2	1	nnnn0d 10914	. . . 4 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> A e. NN0 )
3		prmnn 14585	. . . . . . 7 |- ( P e. Prime -> P e. NN )
4	3	ad2antrr 730	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> P e. NN )
5		pccl 14759	. . . . . . 7 |- ( ( P e. Prime /\ A e. NN ) -> ( P pCnt A ) e. NN0 )
6	5	adantr 466	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P pCnt A ) e. NN0 )
7	4, 6	nnexpcld 12423	. . . . 5 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P ^ ( P pCnt A ) ) e. NN )
8	7	nnnn0d 10914	. . . 4 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P ^ ( P pCnt A ) ) e. NN0 )
9	6	nn0red 10915	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P pCnt A ) e. RR )
10	9	leidd 10169	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P pCnt A ) <_ ( P pCnt A ) )
11		simpll 758	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> P e. Prime )
12	6	nn0zd 11027	. . . . . . . . . . 11 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P pCnt A ) e. ZZ )
13		pcid 14782	. . . . . . . . . . 11 |- ( ( P e. Prime /\ ( P pCnt A ) e. ZZ ) -> ( P pCnt ( P ^ ( P pCnt A ) ) ) = ( P pCnt A ) )
14	11, 12, 13	syl2anc 665	. . . . . . . . . 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 4443	. . . . . . . . 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 730	. . . . . . . 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 462	. . . . . . . . 9 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p = P ) -> p = P )
18	17	oveq1d 6311	. . . . . . . 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 6311	. . . . . . . 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 4447	. . . . . . 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 769	. . . . . . . . . . . . 13 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> A || ( P ^ n ) )
22		prmz 14586	. . . . . . . . . . . . . . 15 |- ( p e. Prime -> p e. ZZ )
23	22	adantl 467	. . . . . . . . . . . . . 14 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> p e. ZZ )
24	1	adantr 466	. . . . . . . . . . . . . . 15 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> A e. NN )
25	24	nnzd 11028	. . . . . . . . . . . . . 14 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> A e. ZZ )
26		simprl 762	. . . . . . . . . . . . . . . . 17 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> n e. NN0 )
27	4, 26	nnexpcld 12423	. . . . . . . . . . . . . . . 16 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P ^ n ) e. NN )
28	27	adantr 466	. . . . . . . . . . . . . . 15 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( P ^ n ) e. NN )
29	28	nnzd 11028	. . . . . . . . . . . . . 14 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( P ^ n ) e. ZZ )
30		dvdstr 14304	. . . . . . . . . . . . . 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 1264	. . . . . . . . . . . . 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 678	. . . . . . . . . . . 12 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( p || A -> p || ( P ^ n ) ) )
33		simpr 462	. . . . . . . . . . . . 13 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> p e. Prime )
34	11	adantr 466	. . . . . . . . . . . . 13 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> P e. Prime )
35		simplrl 768	. . . . . . . . . . . . 13 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> n e. NN0 )
36		prmdvdsexpr 14629	. . . . . . . . . . . . 13 |- ( ( p e. Prime /\ P e. Prime /\ n e. NN0 ) -> ( p || ( P ^ n ) -> p = P ) )
37	33, 34, 35, 36	syl3anc 1264	. . . . . . . . . . . 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 2632	. . . . . . . . . 10 |- ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) -> ( p =/= P -> -. p || A ) )
40	39	imp 430	. . . . . . . . 9 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p =/= P ) -> -. p || A )
41		simplr 760	. . . . . . . . . 10 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p =/= P ) -> p e. Prime )
42	1	ad2antrr 730	. . . . . . . . . 10 |- ( ( ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) /\ p e. Prime ) /\ p =/= P ) -> A e. NN )
43		pceq0 14780	. . . . . . . . . 10 |- ( ( p e. Prime /\ A e. NN ) -> ( ( p pCnt A ) = 0 <-> -. p || A ) )
44	41, 42, 43	syl2anc 665	. . . . . . . . 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 235	. . . . . . . 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 730	. . . . . . . . . 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 14760	. . . . . . . . 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 10917	. . . . . . . 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 4437	. . . . . . 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 2740	. . . . . 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 2837	. . . . 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 11028	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> A e. ZZ )
53	7	nnzd 11028	. . . . . 6 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P ^ ( P pCnt A ) ) e. ZZ )
54		pc2dvds 14788	. . . . . 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 665	. . . . 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 235	. . . 4 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> A || ( P ^ ( P pCnt A ) ) )
57		pcdvds 14773	. . . . 5 |- ( ( P e. Prime /\ A e. NN ) -> ( P ^ ( P pCnt A ) ) || A )
58	57	adantr 466	. . . 4 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> ( P ^ ( P pCnt A ) ) || A )
59		dvdseq 14319	. . . 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 1265	. . 3 |- ( ( ( P e. Prime /\ A e. NN ) /\ ( n e. NN0 /\ A || ( P ^ n ) ) ) -> A = ( P ^ ( P pCnt A ) ) )
61	60	rexlimdvaa 2916	. 2 |- ( ( P e. Prime /\ A e. NN ) -> ( E. n e. NN0 A || ( P ^ n ) -> A = ( P ^ ( P pCnt A ) ) ) )
62	3	adantr 466	. . . . . . 7 |- ( ( P e. Prime /\ A e. NN ) -> P e. NN )
63	62, 5	nnexpcld 12423	. . . . . 6 |- ( ( P e. Prime /\ A e. NN ) -> ( P ^ ( P pCnt A ) ) e. NN )
64	63	nnzd 11028	. . . . 5 |- ( ( P e. Prime /\ A e. NN ) -> ( P ^ ( P pCnt A ) ) e. ZZ )
65		iddvds 14283	. . . . 5 |- ( ( P ^ ( P pCnt A ) ) e. ZZ -> ( P ^ ( P pCnt A ) ) || ( P ^ ( P pCnt A ) ) )
66	64, 65	syl 17	. . . 4 |- ( ( P e. Prime /\ A e. NN ) -> ( P ^ ( P pCnt A ) ) || ( P ^ ( P pCnt A ) ) )
67		oveq2 6304	. . . . . 6 |- ( n = ( P pCnt A ) -> ( P ^ n ) = ( P ^ ( P pCnt A ) ) )
68	67	breq2d 4429	. . . . 5 |- ( n = ( P pCnt A ) -> ( ( P ^ ( P pCnt A ) ) || ( P ^ n ) <-> ( P ^ ( P pCnt A ) ) || ( P ^ ( P pCnt A ) ) ) )
69	68	rspcev 3179	. . . 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 665	. . 3 |- ( ( P e. Prime /\ A e. NN ) -> E. n e. NN0 ( P ^ ( P pCnt A ) ) || ( P ^ n ) )
71		breq1 4420	. . . 4 |- ( A = ( P ^ ( P pCnt A ) ) -> ( A || ( P ^ n ) <-> ( P ^ ( P pCnt A ) ) || ( P ^ n ) ) )
72	71	rexbidv 2937	. . 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 225	. 2 |- ( ( P e. Prime /\ A e. NN ) -> ( A = ( P ^ ( P pCnt A ) ) -> E. n e. NN0 A || ( P ^ n ) ) )
74	61, 73	impbid 193	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 187   /\ wa 370   = wceq 1437   e. wcel 1867   =/= wne 2616   A.wral 2773   E.wrex 2774   class class class wbr 4417  (class class class)co 6296   0cc0 9528   <_ cle 9665   NNcn 10598   NN0cn0 10858   ZZcz 10926   ^cexp 12258   || cdvds 14272   Primecprime 14582   pCnt cpc 14746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-2o 7182  df-oadd 7185  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-sup 7953  df-inf 7954  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-n0 10859  df-z 10927  df-uz 11149  df-q 11254  df-rp 11292  df-fz 11772  df-fl 12014  df-mod 12083  df-seq 12200  df-exp 12259  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-dvds 14273  df-gcd 14432  df-prm 14583  df-pc 14747

This theorem is referenced by:  pcprmpw  14792  pgpfi1  17188  pgpfi  17198  sylow2alem2  17211  lt6abl  17470  pgpfac1lem3a  17650  dvdsppwf1o  24017