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Theorem mule1 20923
Description: The Möbius function takes on values in magnitude at most 
1. (Together with mucl 20916, this implies that it takes a value in  { -u 1 ,  0 ,  1 } for every natural number.) (Contributed by Mario Carneiro, 22-Sep-2014.)
Assertion
Ref Expression
mule1  |-  ( A  e.  NN  ->  ( abs `  ( mmu `  A ) )  <_ 
1 )

Proof of Theorem mule1
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 muval 20907 . . . . 5  |-  ( A  e.  NN  ->  (
mmu `  A )  =  if ( E. p  e.  Prime  ( p ^
2 )  ||  A ,  0 ,  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  A } ) ) ) )
2 iftrue 3737 . . . . 5  |-  ( E. p  e.  Prime  (
p ^ 2 ) 
||  A  ->  if ( E. p  e.  Prime  ( p ^ 2 ) 
||  A ,  0 ,  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) ) )  =  0 )
31, 2sylan9eq 2487 . . . 4  |-  ( ( A  e.  NN  /\  E. p  e.  Prime  (
p ^ 2 ) 
||  A )  -> 
( mmu `  A
)  =  0 )
43fveq2d 5724 . . 3  |-  ( ( A  e.  NN  /\  E. p  e.  Prime  (
p ^ 2 ) 
||  A )  -> 
( abs `  (
mmu `  A )
)  =  ( abs `  0 ) )
5 abs0 12082 . . . 4  |-  ( abs `  0 )  =  0
6 0le1 9543 . . . 4  |-  0  <_  1
75, 6eqbrtri 4223 . . 3  |-  ( abs `  0 )  <_ 
1
84, 7syl6eqbr 4241 . 2  |-  ( ( A  e.  NN  /\  E. p  e.  Prime  (
p ^ 2 ) 
||  A )  -> 
( abs `  (
mmu `  A )
)  <_  1 )
9 iffalse 3738 . . . . . 6  |-  ( -. 
E. p  e.  Prime  ( p ^ 2 ) 
||  A  ->  if ( E. p  e.  Prime  ( p ^ 2 ) 
||  A ,  0 ,  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) ) )  =  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  A } ) ) )
101, 9sylan9eq 2487 . . . . 5  |-  ( ( A  e.  NN  /\  -.  E. p  e.  Prime  ( p ^ 2 ) 
||  A )  -> 
( mmu `  A
)  =  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) ) )
1110fveq2d 5724 . . . 4  |-  ( ( A  e.  NN  /\  -.  E. p  e.  Prime  ( p ^ 2 ) 
||  A )  -> 
( abs `  (
mmu `  A )
)  =  ( abs `  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) ) ) )
12 neg1cn 10059 . . . . . . 7  |-  -u 1  e.  CC
13 prmdvdsfi 20882 . . . . . . . 8  |-  ( A  e.  NN  ->  { p  e.  Prime  |  p  ||  A }  e.  Fin )
14 hashcl 11631 . . . . . . . 8  |-  ( { p  e.  Prime  |  p 
||  A }  e.  Fin  ->  ( # `  {
p  e.  Prime  |  p 
||  A } )  e.  NN0 )
1513, 14syl 16 . . . . . . 7  |-  ( A  e.  NN  ->  ( # `
 { p  e. 
Prime  |  p  ||  A } )  e.  NN0 )
16 absexp 12101 . . . . . . 7  |-  ( (
-u 1  e.  CC  /\  ( # `  {
p  e.  Prime  |  p 
||  A } )  e.  NN0 )  -> 
( abs `  ( -u 1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) ) )  =  ( ( abs `  -u 1
) ^ ( # `  { p  e.  Prime  |  p  ||  A }
) ) )
1712, 15, 16sylancr 645 . . . . . 6  |-  ( A  e.  NN  ->  ( abs `  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) ) )  =  ( ( abs `  -u 1
) ^ ( # `  { p  e.  Prime  |  p  ||  A }
) ) )
18 ax-1cn 9040 . . . . . . . . . 10  |-  1  e.  CC
1918absnegi 12195 . . . . . . . . 9  |-  ( abs `  -u 1 )  =  ( abs `  1
)
20 abs1 12094 . . . . . . . . 9  |-  ( abs `  1 )  =  1
2119, 20eqtri 2455 . . . . . . . 8  |-  ( abs `  -u 1 )  =  1
2221oveq1i 6083 . . . . . . 7  |-  ( ( abs `  -u 1
) ^ ( # `  { p  e.  Prime  |  p  ||  A }
) )  =  ( 1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) )
2315nn0zd 10365 . . . . . . . 8  |-  ( A  e.  NN  ->  ( # `
 { p  e. 
Prime  |  p  ||  A } )  e.  ZZ )
24 1exp 11401 . . . . . . . 8  |-  ( (
# `  { p  e.  Prime  |  p  ||  A } )  e.  ZZ  ->  ( 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  A } ) )  =  1 )
2523, 24syl 16 . . . . . . 7  |-  ( A  e.  NN  ->  (
1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) )  =  1 )
2622, 25syl5eq 2479 . . . . . 6  |-  ( A  e.  NN  ->  (
( abs `  -u 1
) ^ ( # `  { p  e.  Prime  |  p  ||  A }
) )  =  1 )
2717, 26eqtrd 2467 . . . . 5  |-  ( A  e.  NN  ->  ( abs `  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) ) )  =  1 )
2827adantr 452 . . . 4  |-  ( ( A  e.  NN  /\  -.  E. p  e.  Prime  ( p ^ 2 ) 
||  A )  -> 
( abs `  ( -u 1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) ) )  =  1 )
2911, 28eqtrd 2467 . . 3  |-  ( ( A  e.  NN  /\  -.  E. p  e.  Prime  ( p ^ 2 ) 
||  A )  -> 
( abs `  (
mmu `  A )
)  =  1 )
30 1le1 9642 . . 3  |-  1  <_  1
3129, 30syl6eqbr 4241 . 2  |-  ( ( A  e.  NN  /\  -.  E. p  e.  Prime  ( p ^ 2 ) 
||  A )  -> 
( abs `  (
mmu `  A )
)  <_  1 )
328, 31pm2.61dan 767 1  |-  ( A  e.  NN  ->  ( abs `  ( mmu `  A ) )  <_ 
1 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698   {crab 2701   ifcif 3731   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Fincfn 7101   CCcc 8980   0cc0 8982   1c1 8983    <_ cle 9113   -ucneg 9284   NNcn 9992   2c2 10041   NN0cn0 10213   ZZcz 10274   ^cexp 11374   #chash 11610   abscabs 12031    || cdivides 12844   Primecprime 13071   mmucmu 20869
This theorem is referenced by:  dchrmusum2  21180  dchrvmasumlem3  21185  mudivsum  21216  mulogsumlem  21217  mulog2sumlem2  21221  selberglem2  21232
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845  df-prm 13072  df-mu 20875
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