Theorem vmappw 24842

Description: Value of the von Mangoldt function at a prime power. (Contributed by Mario Carneiro, 7-Apr-2016.)

Assertion

Ref	Expression
vmappw	|- ( ( P e. Prime /\ K e. NN ) -> (Λ ` ( P ^ K ) ) = ( log ` P ) )

Proof of Theorem vmappw

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prmnn 15388	. . . 4 |- ( P e. Prime -> P e. NN )
2		nnnn0 11299	. . . 4 |- ( K e. NN -> K e. NN0 )
3		nnexpcl 12873	. . . 4 |- ( ( P e. NN /\ K e. NN0 ) -> ( P ^ K ) e. NN )
4	1, 2, 3	syl2an 494	. . 3 |- ( ( P e. Prime /\ K e. NN ) -> ( P ^ K ) e. NN )
5		eqid 2622	. . . 4 |- { p e. Prime | p || ( P ^ K ) } = { p e. Prime | p || ( P ^ K ) }
6	5	vmaval 24839	. . 3 |- ( ( P ^ K ) e. NN -> (Λ ` ( P ^ K ) ) = if ( ( # ` { p e. Prime | p || ( P ^ K ) } ) = 1 , ( log ` U. { p e. Prime | p || ( P ^ K ) } ) , 0 ) )
7	4, 6	syl 17	. 2 |- ( ( P e. Prime /\ K e. NN ) -> (Λ ` ( P ^ K ) ) = if ( ( # ` { p e. Prime | p || ( P ^ K ) } ) = 1 , ( log ` U. { p e. Prime | p || ( P ^ K ) } ) , 0 ) )
8		df-rab 2921	. . . . . 6 |- { p e. Prime | p || ( P ^ K ) } = { p | ( p e. Prime /\ p || ( P ^ K ) ) }
9		prmdvdsexpb 15428	. . . . . . . . . . . . 13 |- ( ( p e. Prime /\ P e. Prime /\ K e. NN ) -> ( p || ( P ^ K ) <-> p = P ) )
10	9	biimpd 219	. . . . . . . . . . . 12 |- ( ( p e. Prime /\ P e. Prime /\ K e. NN ) -> ( p || ( P ^ K ) -> p = P ) )
11	10	3coml 1272	. . . . . . . . . . 11 |- ( ( P e. Prime /\ K e. NN /\ p e. Prime ) -> ( p || ( P ^ K ) -> p = P ) )
12	11	3expa 1265	. . . . . . . . . 10 |- ( ( ( P e. Prime /\ K e. NN ) /\ p e. Prime ) -> ( p || ( P ^ K ) -> p = P ) )
13	12	expimpd 629	. . . . . . . . 9 |- ( ( P e. Prime /\ K e. NN ) -> ( ( p e. Prime /\ p || ( P ^ K ) ) -> p = P ) )
14		simpl 473	. . . . . . . . . . 11 |- ( ( P e. Prime /\ K e. NN ) -> P e. Prime )
15		prmz 15389	. . . . . . . . . . . 12 |- ( P e. Prime -> P e. ZZ )
16		iddvdsexp 15005	. . . . . . . . . . . 12 |- ( ( P e. ZZ /\ K e. NN ) -> P || ( P ^ K ) )
17	15, 16	sylan 488	. . . . . . . . . . 11 |- ( ( P e. Prime /\ K e. NN ) -> P || ( P ^ K ) )
18	14, 17	jca 554	. . . . . . . . . 10 |- ( ( P e. Prime /\ K e. NN ) -> ( P e. Prime /\ P || ( P ^ K ) ) )
19		eleq1 2689	. . . . . . . . . . 11 |- ( p = P -> ( p e. Prime <-> P e. Prime ) )
20		breq1 4656	. . . . . . . . . . 11 |- ( p = P -> ( p || ( P ^ K ) <-> P || ( P ^ K ) ) )
21	19, 20	anbi12d 747	. . . . . . . . . 10 |- ( p = P -> ( ( p e. Prime /\ p || ( P ^ K ) ) <-> ( P e. Prime /\ P || ( P ^ K ) ) ) )
22	18, 21	syl5ibrcom 237	. . . . . . . . 9 |- ( ( P e. Prime /\ K e. NN ) -> ( p = P -> ( p e. Prime /\ p || ( P ^ K ) ) ) )
23	13, 22	impbid 202	. . . . . . . 8 |- ( ( P e. Prime /\ K e. NN ) -> ( ( p e. Prime /\ p || ( P ^ K ) ) <-> p = P ) )
24		velsn 4193	. . . . . . . 8 |- ( p e. { P } <-> p = P )
25	23, 24	syl6bbr 278	. . . . . . 7 |- ( ( P e. Prime /\ K e. NN ) -> ( ( p e. Prime /\ p || ( P ^ K ) ) <-> p e. { P } ) )
26	25	abbi1dv 2743	. . . . . 6 |- ( ( P e. Prime /\ K e. NN ) -> { p | ( p e. Prime /\ p || ( P ^ K ) ) } = { P } )
27	8, 26	syl5eq 2668	. . . . 5 |- ( ( P e. Prime /\ K e. NN ) -> { p e. Prime | p || ( P ^ K ) } = { P } )
28	27	fveq2d 6195	. . . 4 |- ( ( P e. Prime /\ K e. NN ) -> ( # ` { p e. Prime | p || ( P ^ K ) } ) = ( # ` { P } ) )
29		hashsng 13159	. . . . 5 |- ( P e. Prime -> ( # ` { P } ) = 1 )
30	29	adantr 481	. . . 4 |- ( ( P e. Prime /\ K e. NN ) -> ( # ` { P } ) = 1 )
31	28, 30	eqtrd 2656	. . 3 |- ( ( P e. Prime /\ K e. NN ) -> ( # ` { p e. Prime | p || ( P ^ K ) } ) = 1 )
32	31	iftrued 4094	. 2 |- ( ( P e. Prime /\ K e. NN ) -> if ( ( # ` { p e. Prime | p || ( P ^ K ) } ) = 1 , ( log ` U. { p e. Prime | p || ( P ^ K ) } ) , 0 ) = ( log ` U. { p e. Prime | p || ( P ^ K ) } ) )
33	27	unieqd 4446	. . . 4 |- ( ( P e. Prime /\ K e. NN ) -> U. { p e. Prime | p || ( P ^ K ) } = U. { P } )
34		unisng 4452	. . . . 5 |- ( P e. Prime -> U. { P } = P )
35	34	adantr 481	. . . 4 |- ( ( P e. Prime /\ K e. NN ) -> U. { P } = P )
36	33, 35	eqtrd 2656	. . 3 |- ( ( P e. Prime /\ K e. NN ) -> U. { p e. Prime | p || ( P ^ K ) } = P )
37	36	fveq2d 6195	. 2 |- ( ( P e. Prime /\ K e. NN ) -> ( log ` U. { p e. Prime | p || ( P ^ K ) } ) = ( log ` P ) )
38	7, 32, 37	3eqtrd 2660	1 |- ( ( P e. Prime /\ K e. NN ) -> (Λ ` ( P ^ K ) ) = ( log ` P ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   {crab 2916   ifcif 4086   {csn 4177   U.cuni 4436   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   NNcn 11020   NN0cn0 11292   ZZcz 11377   ^cexp 12860   #chash 13117   || cdvds 14983   Primecprime 15385   logclog 24301  Λcvma 24818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-vma 24824

This theorem is referenced by:  vmaprm  24843  vmacl  24844  efvmacl  24846  vmalelog  24930  vmasum  24941  chpval2  24943  rplogsumlem2  25174  rpvmasumlem  25176