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Theorem mule1 23247
Description: The Möbius function takes on values in magnitude at most 
1. (Together with mucl 23240, 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 23231 . . . . 5  |-  ( A  e.  NN  ->  (
mmu `  A )  =  if ( E. p  e.  Prime  ( p ^
2 )  ||  A ,  0 ,  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  A } ) ) ) )
2 iftrue 3945 . . . . 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 2528 . . . 4  |-  ( ( A  e.  NN  /\  E. p  e.  Prime  (
p ^ 2 ) 
||  A )  -> 
( mmu `  A
)  =  0 )
43fveq2d 5870 . . 3  |-  ( ( A  e.  NN  /\  E. p  e.  Prime  (
p ^ 2 ) 
||  A )  -> 
( abs `  (
mmu `  A )
)  =  ( abs `  0 ) )
5 abs0 13084 . . . 4  |-  ( abs `  0 )  =  0
6 0le1 10077 . . . 4  |-  0  <_  1
75, 6eqbrtri 4466 . . 3  |-  ( abs `  0 )  <_ 
1
84, 7syl6eqbr 4484 . 2  |-  ( ( A  e.  NN  /\  E. p  e.  Prime  (
p ^ 2 ) 
||  A )  -> 
( abs `  (
mmu `  A )
)  <_  1 )
9 iffalse 3948 . . . . . 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 2528 . . . . 5  |-  ( ( A  e.  NN  /\  -.  E. p  e.  Prime  ( p ^ 2 ) 
||  A )  -> 
( mmu `  A
)  =  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) ) )
1110fveq2d 5870 . . . 4  |-  ( ( A  e.  NN  /\  -.  E. p  e.  Prime  ( p ^ 2 ) 
||  A )  -> 
( abs `  (
mmu `  A )
)  =  ( abs `  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) ) ) )
12 neg1cn 10640 . . . . . . 7  |-  -u 1  e.  CC
13 prmdvdsfi 23206 . . . . . . . 8  |-  ( A  e.  NN  ->  { p  e.  Prime  |  p  ||  A }  e.  Fin )
14 hashcl 12397 . . . . . . . 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 13103 . . . . . . 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 9551 . . . . . . . . . 10  |-  1  e.  CC
1918absnegi 13198 . . . . . . . . 9  |-  ( abs `  -u 1 )  =  ( abs `  1
)
20 abs1 13096 . . . . . . . . 9  |-  ( abs `  1 )  =  1
2119, 20eqtri 2496 . . . . . . . 8  |-  ( abs `  -u 1 )  =  1
2221oveq1i 6295 . . . . . . 7  |-  ( ( abs `  -u 1
) ^ ( # `  { p  e.  Prime  |  p  ||  A }
) )  =  ( 1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) )
2315nn0zd 10965 . . . . . . . 8  |-  ( A  e.  NN  ->  ( # `
 { p  e. 
Prime  |  p  ||  A } )  e.  ZZ )
24 1exp 12164 . . . . . . . 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 2520 . . . . . 6  |-  ( A  e.  NN  ->  (
( abs `  -u 1
) ^ ( # `  { p  e.  Prime  |  p  ||  A }
) )  =  1 )
2717, 26eqtrd 2508 . . . . 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 2508 . . 3  |-  ( ( A  e.  NN  /\  -.  E. p  e.  Prime  ( p ^ 2 ) 
||  A )  -> 
( abs `  (
mmu `  A )
)  =  1 )
30 1le1 10178 . . 3  |-  1  <_  1
3129, 30syl6eqbr 4484 . 2  |-  ( ( A  e.  NN  /\  -.  E. p  e.  Prime  ( p ^ 2 ) 
||  A )  -> 
( abs `  (
mmu `  A )
)  <_  1 )
328, 31pm2.61dan 789 1  |-  ( A  e.  NN  ->  ( abs `  ( mmu `  A ) )  <_ 
1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815   {crab 2818   ifcif 3939   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   Fincfn 7517   CCcc 9491   0cc0 9493   1c1 9494    <_ cle 9630   -ucneg 9807   NNcn 10537   2c2 10586   NN0cn0 10796   ZZcz 10865   ^cexp 12135   #chash 12374   abscabs 13033    || cdivides 13850   Primecprime 14079   mmucmu 23193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-sup 7902  df-card 8321  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11084  df-rp 11222  df-fz 11674  df-seq 12077  df-exp 12136  df-hash 12375  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-dvds 13851  df-prm 14080  df-mu 23199
This theorem is referenced by:  dchrmusum2  23504  dchrvmasumlem3  23509  mudivsum  23540  mulogsumlem  23541  mulog2sumlem2  23545  selberglem2  23556
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