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Theorem mule1 20892
Description: The Möbius function takes on values in magnitude at most 
1. (Together with mucl 20885, 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 20876 . . . . 5  |-  ( A  e.  NN  ->  (
mmu `  A )  =  if ( E. p  e.  Prime  ( p ^
2 )  ||  A ,  0 ,  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  A } ) ) ) )
2 iftrue 3713 . . . . 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 2464 . . . 4  |-  ( ( A  e.  NN  /\  E. p  e.  Prime  (
p ^ 2 ) 
||  A )  -> 
( mmu `  A
)  =  0 )
43fveq2d 5699 . . 3  |-  ( ( A  e.  NN  /\  E. p  e.  Prime  (
p ^ 2 ) 
||  A )  -> 
( abs `  (
mmu `  A )
)  =  ( abs `  0 ) )
5 abs0 12053 . . . 4  |-  ( abs `  0 )  =  0
6 0le1 9515 . . . 4  |-  0  <_  1
75, 6eqbrtri 4199 . . 3  |-  ( abs `  0 )  <_ 
1
84, 7syl6eqbr 4217 . 2  |-  ( ( A  e.  NN  /\  E. p  e.  Prime  (
p ^ 2 ) 
||  A )  -> 
( abs `  (
mmu `  A )
)  <_  1 )
9 iffalse 3714 . . . . . 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 2464 . . . . 5  |-  ( ( A  e.  NN  /\  -.  E. p  e.  Prime  ( p ^ 2 ) 
||  A )  -> 
( mmu `  A
)  =  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) ) )
1110fveq2d 5699 . . . 4  |-  ( ( A  e.  NN  /\  -.  E. p  e.  Prime  ( p ^ 2 ) 
||  A )  -> 
( abs `  (
mmu `  A )
)  =  ( abs `  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) ) ) )
12 neg1cn 10031 . . . . . . 7  |-  -u 1  e.  CC
13 prmdvdsfi 20851 . . . . . . . 8  |-  ( A  e.  NN  ->  { p  e.  Prime  |  p  ||  A }  e.  Fin )
14 hashcl 11602 . . . . . . . 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 12072 . . . . . . 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 9012 . . . . . . . . . 10  |-  1  e.  CC
1918absnegi 12166 . . . . . . . . 9  |-  ( abs `  -u 1 )  =  ( abs `  1
)
20 abs1 12065 . . . . . . . . 9  |-  ( abs `  1 )  =  1
2119, 20eqtri 2432 . . . . . . . 8  |-  ( abs `  -u 1 )  =  1
2221oveq1i 6058 . . . . . . 7  |-  ( ( abs `  -u 1
) ^ ( # `  { p  e.  Prime  |  p  ||  A }
) )  =  ( 1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) )
2315nn0zd 10337 . . . . . . . 8  |-  ( A  e.  NN  ->  ( # `
 { p  e. 
Prime  |  p  ||  A } )  e.  ZZ )
24 1exp 11372 . . . . . . . 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 2456 . . . . . 6  |-  ( A  e.  NN  ->  (
( abs `  -u 1
) ^ ( # `  { p  e.  Prime  |  p  ||  A }
) )  =  1 )
2717, 26eqtrd 2444 . . . . 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 2444 . . 3  |-  ( ( A  e.  NN  /\  -.  E. p  e.  Prime  ( p ^ 2 ) 
||  A )  -> 
( abs `  (
mmu `  A )
)  =  1 )
30 1le1 9614 . . 3  |-  1  <_  1
3129, 30syl6eqbr 4217 . 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 1649    e. wcel 1721   E.wrex 2675   {crab 2678   ifcif 3707   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   Fincfn 7076   CCcc 8952   0cc0 8954   1c1 8955    <_ cle 9085   -ucneg 9256   NNcn 9964   2c2 10013   NN0cn0 10185   ZZcz 10246   ^cexp 11345   #chash 11581   abscabs 12002    || cdivides 12815   Primecprime 13042   mmucmu 20838
This theorem is referenced by:  dchrmusum2  21149  dchrvmasumlem3  21154  mudivsum  21185  mulogsumlem  21186  mulog2sumlem2  21190  selberglem2  21201
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-card 7790  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-n0 10186  df-z 10247  df-uz 10453  df-rp 10577  df-fz 11008  df-seq 11287  df-exp 11346  df-hash 11582  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-dvds 12816  df-prm 13043  df-mu 20844
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