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Theorem mule1 20492
Description: The Möbius function takes on values in magnitude at most 
1. (Together with mucl 20485, 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 20476 . . . . 5  |-  ( A  e.  NN  ->  (
mmu `  A )  =  if ( E. p  e.  Prime  ( p ^
2 )  ||  A ,  0 ,  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  A } ) ) ) )
2 iftrue 3647 . . . . 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 2410 . . . 4  |-  ( ( A  e.  NN  /\  E. p  e.  Prime  (
p ^ 2 ) 
||  A )  -> 
( mmu `  A
)  =  0 )
43fveq2d 5609 . . 3  |-  ( ( A  e.  NN  /\  E. p  e.  Prime  (
p ^ 2 ) 
||  A )  -> 
( abs `  (
mmu `  A )
)  =  ( abs `  0 ) )
5 abs0 11860 . . . 4  |-  ( abs `  0 )  =  0
6 0le1 9384 . . . 4  |-  0  <_  1
75, 6eqbrtri 4121 . . 3  |-  ( abs `  0 )  <_ 
1
84, 7syl6eqbr 4139 . 2  |-  ( ( A  e.  NN  /\  E. p  e.  Prime  (
p ^ 2 ) 
||  A )  -> 
( abs `  (
mmu `  A )
)  <_  1 )
9 iffalse 3648 . . . . . 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 2410 . . . . 5  |-  ( ( A  e.  NN  /\  -.  E. p  e.  Prime  ( p ^ 2 ) 
||  A )  -> 
( mmu `  A
)  =  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) ) )
1110fveq2d 5609 . . . 4  |-  ( ( A  e.  NN  /\  -.  E. p  e.  Prime  ( p ^ 2 ) 
||  A )  -> 
( abs `  (
mmu `  A )
)  =  ( abs `  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) ) ) )
12 neg1cn 9900 . . . . . . 7  |-  -u 1  e.  CC
13 prmdvdsfi 20451 . . . . . . . 8  |-  ( A  e.  NN  ->  { p  e.  Prime  |  p  ||  A }  e.  Fin )
14 hashcl 11440 . . . . . . . 8  |-  ( { p  e.  Prime  |  p 
||  A }  e.  Fin  ->  ( # `  {
p  e.  Prime  |  p 
||  A } )  e.  NN0 )
1513, 14syl 15 . . . . . . 7  |-  ( A  e.  NN  ->  ( # `
 { p  e. 
Prime  |  p  ||  A } )  e.  NN0 )
16 absexp 11879 . . . . . . 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 644 . . . . . 6  |-  ( A  e.  NN  ->  ( abs `  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) ) )  =  ( ( abs `  -u 1
) ^ ( # `  { p  e.  Prime  |  p  ||  A }
) ) )
18 ax-1cn 8882 . . . . . . . . . 10  |-  1  e.  CC
1918absnegi 11973 . . . . . . . . 9  |-  ( abs `  -u 1 )  =  ( abs `  1
)
20 abs1 11872 . . . . . . . . 9  |-  ( abs `  1 )  =  1
2119, 20eqtri 2378 . . . . . . . 8  |-  ( abs `  -u 1 )  =  1
2221oveq1i 5952 . . . . . . 7  |-  ( ( abs `  -u 1
) ^ ( # `  { p  e.  Prime  |  p  ||  A }
) )  =  ( 1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) )
2315nn0zd 10204 . . . . . . . 8  |-  ( A  e.  NN  ->  ( # `
 { p  e. 
Prime  |  p  ||  A } )  e.  ZZ )
24 1exp 11221 . . . . . . . 8  |-  ( (
# `  { p  e.  Prime  |  p  ||  A } )  e.  ZZ  ->  ( 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  A } ) )  =  1 )
2523, 24syl 15 . . . . . . 7  |-  ( A  e.  NN  ->  (
1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) )  =  1 )
2622, 25syl5eq 2402 . . . . . 6  |-  ( A  e.  NN  ->  (
( abs `  -u 1
) ^ ( # `  { p  e.  Prime  |  p  ||  A }
) )  =  1 )
2717, 26eqtrd 2390 . . . . 5  |-  ( A  e.  NN  ->  ( abs `  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) ) )  =  1 )
2827adantr 451 . . . 4  |-  ( ( A  e.  NN  /\  -.  E. p  e.  Prime  ( p ^ 2 ) 
||  A )  -> 
( abs `  ( -u 1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) ) )  =  1 )
2911, 28eqtrd 2390 . . 3  |-  ( ( A  e.  NN  /\  -.  E. p  e.  Prime  ( p ^ 2 ) 
||  A )  -> 
( abs `  (
mmu `  A )
)  =  1 )
30 1le1 9483 . . 3  |-  1  <_  1
3129, 30syl6eqbr 4139 . 2  |-  ( ( A  e.  NN  /\  -.  E. p  e.  Prime  ( p ^ 2 ) 
||  A )  -> 
( abs `  (
mmu `  A )
)  <_  1 )
328, 31pm2.61dan 766 1  |-  ( A  e.  NN  ->  ( abs `  ( mmu `  A ) )  <_ 
1 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   E.wrex 2620   {crab 2623   ifcif 3641   class class class wbr 4102   ` cfv 5334  (class class class)co 5942   Fincfn 6948   CCcc 8822   0cc0 8824   1c1 8825    <_ cle 8955   -ucneg 9125   NNcn 9833   2c2 9882   NN0cn0 10054   ZZcz 10113   ^cexp 11194   #chash 11427   abscabs 11809    || cdivides 12622   Primecprime 12849   mmucmu 20438
This theorem is referenced by:  dchrmusum2  20749  dchrvmasumlem3  20754  mudivsum  20785  mulogsumlem  20786  mulog2sumlem2  20790  selberglem2  20801
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-sup 7281  df-card 7659  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-n0 10055  df-z 10114  df-uz 10320  df-rp 10444  df-fz 10872  df-seq 11136  df-exp 11195  df-hash 11428  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811  df-dvds 12623  df-prm 12850  df-mu 20444
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