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Theorem mule1 23547
Description: The Möbius function takes on values in magnitude at most 
1. (Together with mucl 23540, this implies that it takes a value in  { -u 1 ,  0 ,  1 } for every positive integer.) (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 23531 . . . . 5  |-  ( A  e.  NN  ->  (
mmu `  A )  =  if ( E. p  e.  Prime  ( p ^
2 )  ||  A ,  0 ,  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  A } ) ) ) )
2 iftrue 3950 . . . . 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 2518 . . . 4  |-  ( ( A  e.  NN  /\  E. p  e.  Prime  (
p ^ 2 ) 
||  A )  -> 
( mmu `  A
)  =  0 )
43fveq2d 5876 . . 3  |-  ( ( A  e.  NN  /\  E. p  e.  Prime  (
p ^ 2 ) 
||  A )  -> 
( abs `  (
mmu `  A )
)  =  ( abs `  0 ) )
5 abs0 13129 . . . 4  |-  ( abs `  0 )  =  0
6 0le1 10097 . . . 4  |-  0  <_  1
75, 6eqbrtri 4475 . . 3  |-  ( abs `  0 )  <_ 
1
84, 7syl6eqbr 4493 . 2  |-  ( ( A  e.  NN  /\  E. p  e.  Prime  (
p ^ 2 ) 
||  A )  -> 
( abs `  (
mmu `  A )
)  <_  1 )
9 iffalse 3953 . . . . . 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 2518 . . . . 5  |-  ( ( A  e.  NN  /\  -.  E. p  e.  Prime  ( p ^ 2 ) 
||  A )  -> 
( mmu `  A
)  =  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) ) )
1110fveq2d 5876 . . . 4  |-  ( ( A  e.  NN  /\  -.  E. p  e.  Prime  ( p ^ 2 ) 
||  A )  -> 
( abs `  (
mmu `  A )
)  =  ( abs `  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) ) ) )
12 neg1cn 10660 . . . . . . 7  |-  -u 1  e.  CC
13 prmdvdsfi 23506 . . . . . . . 8  |-  ( A  e.  NN  ->  { p  e.  Prime  |  p  ||  A }  e.  Fin )
14 hashcl 12430 . . . . . . . 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 13148 . . . . . . 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 663 . . . . . 6  |-  ( A  e.  NN  ->  ( abs `  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) ) )  =  ( ( abs `  -u 1
) ^ ( # `  { p  e.  Prime  |  p  ||  A }
) ) )
18 ax-1cn 9567 . . . . . . . . . 10  |-  1  e.  CC
1918absnegi 13243 . . . . . . . . 9  |-  ( abs `  -u 1 )  =  ( abs `  1
)
20 abs1 13141 . . . . . . . . 9  |-  ( abs `  1 )  =  1
2119, 20eqtri 2486 . . . . . . . 8  |-  ( abs `  -u 1 )  =  1
2221oveq1i 6306 . . . . . . 7  |-  ( ( abs `  -u 1
) ^ ( # `  { p  e.  Prime  |  p  ||  A }
) )  =  ( 1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) )
2315nn0zd 10988 . . . . . . . 8  |-  ( A  e.  NN  ->  ( # `
 { p  e. 
Prime  |  p  ||  A } )  e.  ZZ )
24 1exp 12197 . . . . . . . 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 2510 . . . . . 6  |-  ( A  e.  NN  ->  (
( abs `  -u 1
) ^ ( # `  { p  e.  Prime  |  p  ||  A }
) )  =  1 )
2717, 26eqtrd 2498 . . . . 5  |-  ( A  e.  NN  ->  ( abs `  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) ) )  =  1 )
2827adantr 465 . . . 4  |-  ( ( A  e.  NN  /\  -.  E. p  e.  Prime  ( p ^ 2 ) 
||  A )  -> 
( abs `  ( -u 1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) ) )  =  1 )
2911, 28eqtrd 2498 . . 3  |-  ( ( A  e.  NN  /\  -.  E. p  e.  Prime  ( p ^ 2 ) 
||  A )  -> 
( abs `  (
mmu `  A )
)  =  1 )
30 1le1 10198 . . 3  |-  1  <_  1
3129, 30syl6eqbr 4493 . 2  |-  ( ( A  e.  NN  /\  -.  E. p  e.  Prime  ( p ^ 2 ) 
||  A )  -> 
( abs `  (
mmu `  A )
)  <_  1 )
328, 31pm2.61dan 791 1  |-  ( A  e.  NN  ->  ( abs `  ( mmu `  A ) )  <_ 
1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   E.wrex 2808   {crab 2811   ifcif 3944   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   Fincfn 7535   CCcc 9507   0cc0 9509   1c1 9510    <_ cle 9646   -ucneg 9825   NNcn 10556   2c2 10606   NN0cn0 10816   ZZcz 10885   ^cexp 12168   #chash 12407   abscabs 13078    || cdvds 13997   Primecprime 14228   mmucmu 23493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-seq 12110  df-exp 12169  df-hash 12408  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-dvds 13998  df-prm 14229  df-mu 23499
This theorem is referenced by:  dchrmusum2  23804  dchrvmasumlem3  23809  mudivsum  23840  mulogsumlem  23841  mulog2sumlem2  23845  selberglem2  23856
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