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Theorem muval1 20909
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 20908 . . 3  |-  ( A  e.  NN  ->  (
mmu `  A )  =  if ( E. p  e.  Prime  ( p ^
2 )  ||  A ,  0 ,  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  A } ) ) ) )
213ad2ant1 978 . 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 13103 . . . . 5  |-  ( P  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  P
)
433ad2ant2 979 . . . 4  |-  ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  ->  E. p  e.  Prime  p  ||  P
)
5 prmnn 13075 . . . . . . . 8  |-  ( p  e.  Prime  ->  p  e.  NN )
65adantl 453 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  p  e.  NN )
7 simpl2 961 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  P  e.  ( ZZ>= `  2 )
)
8 eluz2b2 10541 . . . . . . . . 9  |-  ( P  e.  ( ZZ>= `  2
)  <->  ( P  e.  NN  /\  1  < 
P ) )
97, 8sylib 189 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  ( P  e.  NN  /\  1  <  P ) )
109simpld 446 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  P  e.  NN )
11 dvdssqlem 13052 . . . . . . 7  |-  ( ( p  e.  NN  /\  P  e.  NN )  ->  ( p  ||  P  <->  ( p ^ 2 ) 
||  ( P ^
2 ) ) )
126, 10, 11syl2anc 643 . . . . . 6  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  (
p  ||  P  <->  ( p ^ 2 )  ||  ( P ^ 2 ) ) )
13 simpl3 962 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  ( P ^ 2 )  ||  A )
14 prmz 13076 . . . . . . . . . 10  |-  ( p  e.  Prime  ->  p  e.  ZZ )
1514adantl 453 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  p  e.  ZZ )
16 zsqcl 11445 . . . . . . . . 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 10489 . . . . . . . . 9  |-  ( P  e.  ( ZZ>= `  2
)  ->  P  e.  ZZ )
19 zsqcl 11445 . . . . . . . . 9  |-  ( P  e.  ZZ  ->  ( P ^ 2 )  e.  ZZ )
207, 18, 193syl 19 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  ( P ^ 2 )  e.  ZZ )
21 simpl1 960 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  A  e.  NN )
2221nnzd 10367 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  A  e.  ZZ )
23 dvdstr 12876 . . . . . . . 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 1184 . . . . . . 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 656 . . . . . 6  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  (
( p ^ 2 )  ||  ( P ^ 2 )  -> 
( p ^ 2 )  ||  A ) )
2612, 25sylbid 207 . . . . 5  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  (
p  ||  P  ->  ( p ^ 2 ) 
||  A ) )
2726reximdva 2811 . . . 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 3738 . . 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 2468 1  |-  ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  ->  (
mmu `  A )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2699   {crab 2702   ifcif 3732   class class class wbr 4205   ` cfv 5447  (class class class)co 6074   0cc0 8983   1c1 8984    < clt 9113   -ucneg 9285   NNcn 9993   2c2 10042   ZZcz 10275   ZZ>=cuz 10481   ^cexp 11375   #chash 11611    || cdivides 12845   Primecprime 13072   mmucmu 20870
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060  ax-pre-sup 9061
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-int 4044  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-riota 6542  df-recs 6626  df-rdg 6661  df-1o 6717  df-2o 6718  df-oadd 6721  df-er 6898  df-en 7103  df-dom 7104  df-sdom 7105  df-fin 7106  df-sup 7439  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-div 9671  df-nn 9994  df-2 10051  df-3 10052  df-n0 10215  df-z 10276  df-uz 10482  df-rp 10606  df-fz 11037  df-fl 11195  df-mod 11244  df-seq 11317  df-exp 11376  df-cj 11897  df-re 11898  df-im 11899  df-sqr 12033  df-abs 12034  df-dvds 12846  df-gcd 13000  df-prm 13073  df-mu 20876
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