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Theorem muval1 20777
Description: The value of the Möbius function at a non-squarefree number. (Contributed by Mario Carneiro, 21-Sep-2014.)
Assertion
Ref Expression
muval1  |-  ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  ->  (
mmu `  A )  =  0 )

Proof of Theorem muval1
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 muval 20776 . . 3  |-  ( A  e.  NN  ->  (
mmu `  A )  =  if ( E. p  e.  Prime  ( p ^
2 )  ||  A ,  0 ,  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  A } ) ) ) )
213ad2ant1 978 . 2  |-  ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  ->  (
mmu `  A )  =  if ( E. p  e.  Prime  ( p ^
2 )  ||  A ,  0 ,  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  A } ) ) ) )
3 exprmfct 13031 . . . . 5  |-  ( P  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  P
)
433ad2ant2 979 . . . 4  |-  ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  ->  E. p  e.  Prime  p  ||  P
)
5 prmnn 13003 . . . . . . . 8  |-  ( p  e.  Prime  ->  p  e.  NN )
65adantl 453 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  p  e.  NN )
7 simpl2 961 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  P  e.  ( ZZ>= `  2 )
)
8 eluz2b2 10474 . . . . . . . . 9  |-  ( P  e.  ( ZZ>= `  2
)  <->  ( P  e.  NN  /\  1  < 
P ) )
97, 8sylib 189 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  ( P  e.  NN  /\  1  <  P ) )
109simpld 446 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  P  e.  NN )
11 dvdssqlem 12980 . . . . . . 7  |-  ( ( p  e.  NN  /\  P  e.  NN )  ->  ( p  ||  P  <->  ( p ^ 2 ) 
||  ( P ^
2 ) ) )
126, 10, 11syl2anc 643 . . . . . 6  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  (
p  ||  P  <->  ( p ^ 2 )  ||  ( P ^ 2 ) ) )
13 simpl3 962 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  ( P ^ 2 )  ||  A )
14 prmz 13004 . . . . . . . . . 10  |-  ( p  e.  Prime  ->  p  e.  ZZ )
1514adantl 453 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  p  e.  ZZ )
16 zsqcl 11373 . . . . . . . . 9  |-  ( p  e.  ZZ  ->  (
p ^ 2 )  e.  ZZ )
1715, 16syl 16 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  (
p ^ 2 )  e.  ZZ )
18 eluzelz 10422 . . . . . . . . 9  |-  ( P  e.  ( ZZ>= `  2
)  ->  P  e.  ZZ )
19 zsqcl 11373 . . . . . . . . 9  |-  ( P  e.  ZZ  ->  ( P ^ 2 )  e.  ZZ )
207, 18, 193syl 19 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  ( P ^ 2 )  e.  ZZ )
21 simpl1 960 . . . . . . . . 9  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  A  e.  NN )
2221nnzd 10300 . . . . . . . 8  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  A  e.  ZZ )
23 dvdstr 12804 . . . . . . . 8  |-  ( ( ( p ^ 2 )  e.  ZZ  /\  ( P ^ 2 )  e.  ZZ  /\  A  e.  ZZ )  ->  (
( ( p ^
2 )  ||  ( P ^ 2 )  /\  ( P ^ 2 ) 
||  A )  -> 
( p ^ 2 )  ||  A ) )
2417, 20, 22, 23syl3anc 1184 . . . . . . 7  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  (
( ( p ^
2 )  ||  ( P ^ 2 )  /\  ( P ^ 2 ) 
||  A )  -> 
( p ^ 2 )  ||  A ) )
2513, 24mpan2d 656 . . . . . 6  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  (
( p ^ 2 )  ||  ( P ^ 2 )  -> 
( p ^ 2 )  ||  A ) )
2612, 25sylbid 207 . . . . 5  |-  ( ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  /\  p  e.  Prime )  ->  (
p  ||  P  ->  ( p ^ 2 ) 
||  A ) )
2726reximdva 2755 . . . 4  |-  ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  ->  ( E. p  e.  Prime  p 
||  P  ->  E. p  e.  Prime  ( p ^
2 )  ||  A
) )
284, 27mpd 15 . . 3  |-  ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  ->  E. p  e.  Prime  ( p ^
2 )  ||  A
)
29 iftrue 3682 . . 3  |-  ( E. p  e.  Prime  (
p ^ 2 ) 
||  A  ->  if ( E. p  e.  Prime  ( p ^ 2 ) 
||  A ,  0 ,  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) ) )  =  0 )
3028, 29syl 16 . 2  |-  ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  ->  if ( E. p  e.  Prime  ( p ^ 2 ) 
||  A ,  0 ,  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) ) )  =  0 )
312, 30eqtrd 2413 1  |-  ( ( A  e.  NN  /\  P  e.  ( ZZ>= ` 
2 )  /\  ( P ^ 2 )  ||  A )  ->  (
mmu `  A )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   E.wrex 2644   {crab 2647   ifcif 3676   class class class wbr 4147   ` cfv 5388  (class class class)co 6014   0cc0 8917   1c1 8918    < clt 9047   -ucneg 9218   NNcn 9926   2c2 9975   ZZcz 10208   ZZ>=cuz 10414   ^cexp 11303   #chash 11539    || cdivides 12773   Primecprime 13000   mmucmu 20738
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338  ax-un 4635  ax-cnex 8973  ax-resscn 8974  ax-1cn 8975  ax-icn 8976  ax-addcl 8977  ax-addrcl 8978  ax-mulcl 8979  ax-mulrcl 8980  ax-mulcom 8981  ax-addass 8982  ax-mulass 8983  ax-distr 8984  ax-i2m1 8985  ax-1ne0 8986  ax-1rid 8987  ax-rnegex 8988  ax-rrecex 8989  ax-cnre 8990  ax-pre-lttri 8991  ax-pre-lttrn 8992  ax-pre-ltadd 8993  ax-pre-mulgt0 8994  ax-pre-sup 8995
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 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-nel 2547  df-ral 2648  df-rex 2649  df-reu 2650  df-rmo 2651  df-rab 2652  df-v 2895  df-sbc 3099  df-csb 3189  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-pss 3273  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-tp 3759  df-op 3760  df-uni 3952  df-int 3987  df-iun 4031  df-br 4148  df-opab 4202  df-mpt 4203  df-tr 4238  df-eprel 4429  df-id 4433  df-po 4438  df-so 4439  df-fr 4476  df-we 4478  df-ord 4519  df-on 4520  df-lim 4521  df-suc 4522  df-om 4780  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5352  df-fun 5390  df-fn 5391  df-f 5392  df-f1 5393  df-fo 5394  df-f1o 5395  df-fv 5396  df-ov 6017  df-oprab 6018  df-mpt2 6019  df-1st 6282  df-2nd 6283  df-riota 6479  df-recs 6563  df-rdg 6598  df-1o 6654  df-2o 6655  df-oadd 6658  df-er 6835  df-en 7040  df-dom 7041  df-sdom 7042  df-fin 7043  df-sup 7375  df-pnf 9049  df-mnf 9050  df-xr 9051  df-ltxr 9052  df-le 9053  df-sub 9219  df-neg 9220  df-div 9604  df-nn 9927  df-2 9984  df-3 9985  df-n0 10148  df-z 10209  df-uz 10415  df-rp 10539  df-fz 10970  df-fl 11123  df-mod 11172  df-seq 11245  df-exp 11304  df-cj 11825  df-re 11826  df-im 11827  df-sqr 11961  df-abs 11962  df-dvds 12774  df-gcd 12928  df-prm 13001  df-mu 20744
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