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Theorem muval1 22430
Description: The value of the Möbius function at a non-squarefree number. (Contributed by Mario Carneiro, 21-Sep-2014.)
Assertion
Ref Expression
muval1  |-  ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  ->  (
mmu `  A )  =  0 )

Proof of Theorem muval1
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 muval 22429 . . 3  |-  ( A  e.  NN  ->  (
mmu `  A )  =  if ( E. p  e.  Prime  ( p ^
2 )  ||  A ,  0 ,  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  A } ) ) ) )
213ad2ant1 1004 . 2  |-  ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  ->  (
mmu `  A )  =  if ( E. p  e.  Prime  ( p ^
2 )  ||  A ,  0 ,  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  A } ) ) ) )
3 exprmfct 13792 . . . . 5  |-  ( P  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  P
)
433ad2ant2 1005 . . . 4  |-  ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  ->  E. p  e.  Prime  p  ||  P
)
5 prmnn 13762 . . . . . . . 8  |-  ( p  e.  Prime  ->  p  e.  NN )
65adantl 463 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  p  e.  NN )
7 simpl2 987 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  P  e.  ( ZZ>= `  2 )
)
8 eluz2b2 10923 . . . . . . . . 9  |-  ( P  e.  ( ZZ>= `  2
)  <->  ( P  e.  NN  /\  1  < 
P ) )
97, 8sylib 196 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  ( P  e.  NN  /\  1  <  P ) )
109simpld 456 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  P  e.  NN )
11 dvdssqlem 13739 . . . . . . 7  |-  ( ( p  e.  NN  /\  P  e.  NN )  ->  ( p  ||  P  <->  ( p ^ 2 ) 
||  ( P ^
2 ) ) )
126, 10, 11syl2anc 656 . . . . . 6  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  (
p  ||  P  <->  ( p ^ 2 )  ||  ( P ^ 2 ) ) )
13 simpl3 988 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  ( P ^ 2 )  ||  A )
14 prmz 13763 . . . . . . . . . 10  |-  ( p  e.  Prime  ->  p  e.  ZZ )
1514adantl 463 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  p  e.  ZZ )
16 zsqcl 11932 . . . . . . . . 9  |-  ( p  e.  ZZ  ->  (
p ^ 2 )  e.  ZZ )
1715, 16syl 16 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  (
p ^ 2 )  e.  ZZ )
18 eluzelz 10866 . . . . . . . . 9  |-  ( P  e.  ( ZZ>= `  2
)  ->  P  e.  ZZ )
19 zsqcl 11932 . . . . . . . . 9  |-  ( P  e.  ZZ  ->  ( P ^ 2 )  e.  ZZ )
207, 18, 193syl 20 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  ( P ^ 2 )  e.  ZZ )
21 simpl1 986 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  A  e.  NN )
2221nnzd 10742 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  A  e.  ZZ )
23 dvdstr 13562 . . . . . . . 8  |-  ( ( ( p ^ 2 )  e.  ZZ  /\  ( P ^ 2 )  e.  ZZ  /\  A  e.  ZZ )  ->  (
( ( p ^
2 )  ||  ( P ^ 2 )  /\  ( P ^ 2 ) 
||  A )  -> 
( p ^ 2 )  ||  A ) )
2417, 20, 22, 23syl3anc 1213 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  (
( ( p ^
2 )  ||  ( P ^ 2 )  /\  ( P ^ 2 ) 
||  A )  -> 
( p ^ 2 )  ||  A ) )
2513, 24mpan2d 669 . . . . . 6  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  (
( p ^ 2 )  ||  ( P ^ 2 )  -> 
( p ^ 2 )  ||  A ) )
2612, 25sylbid 215 . . . . 5  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  (
p  ||  P  ->  ( p ^ 2 ) 
||  A ) )
2726reximdva 2826 . . . 4  |-  ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  ->  ( E. p  e.  Prime  p 
||  P  ->  E. p  e.  Prime  ( p ^
2 )  ||  A
) )
284, 27mpd 15 . . 3  |-  ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  ->  E. p  e.  Prime  ( p ^
2 )  ||  A
)
29 iftrue 3794 . . 3  |-  ( E. p  e.  Prime  (
p ^ 2 ) 
||  A  ->  if ( E. p  e.  Prime  ( p ^ 2 ) 
||  A ,  0 ,  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) ) )  =  0 )
3028, 29syl 16 . 2  |-  ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  ->  if ( E. p  e.  Prime  ( p ^ 2 ) 
||  A ,  0 ,  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) ) )  =  0 )
312, 30eqtrd 2473 1  |-  ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  ->  (
mmu `  A )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   E.wrex 2714   {crab 2717   ifcif 3788   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   0cc0 9278   1c1 9279    < clt 9414   -ucneg 9592   NNcn 10318   2c2 10367   ZZcz 10642   ZZ>=cuz 10857   ^cexp 11861   #chash 12099    || cdivides 13531   Primecprime 13759   mmucmu 22391
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-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-mu 22397
This theorem is referenced by: (None)
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