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Theorem muval1 22474
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 22473 . . 3  |-  ( A  e.  NN  ->  (
mmu `  A )  =  if ( E. p  e.  Prime  ( p ^
2 )  ||  A ,  0 ,  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  A } ) ) ) )
213ad2ant1 1009 . 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 13799 . . . . 5  |-  ( P  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  P
)
433ad2ant2 1010 . . . 4  |-  ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  ->  E. p  e.  Prime  p  ||  P
)
5 prmnn 13769 . . . . . . . 8  |-  ( p  e.  Prime  ->  p  e.  NN )
65adantl 466 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  p  e.  NN )
7 simpl2 992 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  P  e.  ( ZZ>= `  2 )
)
8 eluz2b2 10930 . . . . . . . . 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 459 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  P  e.  NN )
11 dvdssqlem 13746 . . . . . . 7  |-  ( ( p  e.  NN  /\  P  e.  NN )  ->  ( p  ||  P  <->  ( p ^ 2 ) 
||  ( P ^
2 ) ) )
126, 10, 11syl2anc 661 . . . . . 6  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  (
p  ||  P  <->  ( p ^ 2 )  ||  ( P ^ 2 ) ) )
13 simpl3 993 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  ( P ^ 2 )  ||  A )
14 prmz 13770 . . . . . . . . . 10  |-  ( p  e.  Prime  ->  p  e.  ZZ )
1514adantl 466 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  p  e.  ZZ )
16 zsqcl 11939 . . . . . . . . 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 10873 . . . . . . . . 9  |-  ( P  e.  ( ZZ>= `  2
)  ->  P  e.  ZZ )
19 zsqcl 11939 . . . . . . . . 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 991 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  A  e.  NN )
2221nnzd 10749 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  A  e.  ZZ )
23 dvdstr 13569 . . . . . . . 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 1218 . . . . . . 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 674 . . . . . 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 2831 . . . 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 3800 . . 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 2475 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 965    = wceq 1369    e. wcel 1756   E.wrex 2719   {crab 2722   ifcif 3794   class class class wbr 4295   ` cfv 5421  (class class class)co 6094   0cc0 9285   1c1 9286    < clt 9421   -ucneg 9599   NNcn 10325   2c2 10374   ZZcz 10649   ZZ>=cuz 10864   ^cexp 11868   #chash 12106    || cdivides 13538   Primecprime 13766   mmucmu 22435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-1o 6923  df-2o 6924  df-oadd 6927  df-er 7104  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-sup 7694  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-nn 10326  df-2 10383  df-3 10384  df-n0 10583  df-z 10650  df-uz 10865  df-rp 10995  df-fz 11441  df-fl 11645  df-mod 11712  df-seq 11810  df-exp 11869  df-cj 12591  df-re 12592  df-im 12593  df-sqr 12727  df-abs 12728  df-dvds 13539  df-gcd 13694  df-prm 13767  df-mu 22441
This theorem is referenced by: (None)
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