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Theorem isnsqf 20786
Description: Two ways to say that a number is not squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)
Assertion
Ref Expression
isnsqf  |-  ( A  e.  NN  ->  (
( mmu `  A
)  =  0  <->  E. p  e.  Prime  ( p ^ 2 )  ||  A ) )
Distinct variable group:    A, p

Proof of Theorem isnsqf
StepHypRef Expression
1 prmdvdsfi 20758 . . . . . . . 8  |-  ( A  e.  NN  ->  { p  e.  Prime  |  p  ||  A }  e.  Fin )
2 hashcl 11567 . . . . . . . 8  |-  ( { p  e.  Prime  |  p 
||  A }  e.  Fin  ->  ( # `  {
p  e.  Prime  |  p 
||  A } )  e.  NN0 )
31, 2syl 16 . . . . . . 7  |-  ( A  e.  NN  ->  ( # `
 { p  e. 
Prime  |  p  ||  A } )  e.  NN0 )
43nn0zd 10306 . . . . . 6  |-  ( A  e.  NN  ->  ( # `
 { p  e. 
Prime  |  p  ||  A } )  e.  ZZ )
5 neg1cn 10000 . . . . . . 7  |-  -u 1  e.  CC
6 ax-1cn 8982 . . . . . . . 8  |-  1  e.  CC
7 ax-1ne0 8993 . . . . . . . 8  |-  1  =/=  0
86, 7negne0i 9308 . . . . . . 7  |-  -u 1  =/=  0
9 expne0i 11340 . . . . . . 7  |-  ( (
-u 1  e.  CC  /\  -u 1  =/=  0  /\  ( # `  {
p  e.  Prime  |  p 
||  A } )  e.  ZZ )  -> 
( -u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  A } ) )  =/=  0 )
105, 8, 9mp3an12 1269 . . . . . 6  |-  ( (
# `  { p  e.  Prime  |  p  ||  A } )  e.  ZZ  ->  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) )  =/=  0 )
114, 10syl 16 . . . . 5  |-  ( A  e.  NN  ->  ( -u 1 ^ ( # `  { p  e.  Prime  |  p  ||  A }
) )  =/=  0
)
12 iffalse 3690 . . . . . 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 } ) ) )
1312neeq1d 2564 . . . . 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  <->  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) )  =/=  0 ) )
1411, 13syl5ibrcom 214 . . . 4  |-  ( A  e.  NN  ->  ( -.  E. p  e.  Prime  ( p ^ 2 ) 
||  A  ->  if ( E. p  e.  Prime  ( p ^ 2 ) 
||  A ,  0 ,  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) ) )  =/=  0
) )
15 muval 20783 . . . . 5  |-  ( A  e.  NN  ->  (
mmu `  A )  =  if ( E. p  e.  Prime  ( p ^
2 )  ||  A ,  0 ,  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  A } ) ) ) )
1615neeq1d 2564 . . . 4  |-  ( A  e.  NN  ->  (
( mmu `  A
)  =/=  0  <->  if ( E. p  e.  Prime  ( p ^ 2 ) 
||  A ,  0 ,  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) ) )  =/=  0
) )
1714, 16sylibrd 226 . . 3  |-  ( A  e.  NN  ->  ( -.  E. p  e.  Prime  ( p ^ 2 ) 
||  A  ->  (
mmu `  A )  =/=  0 ) )
1817necon4bd 2613 . 2  |-  ( A  e.  NN  ->  (
( mmu `  A
)  =  0  ->  E. p  e.  Prime  ( p ^ 2 ) 
||  A ) )
19 iftrue 3689 . . 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 )
2015eqeq1d 2396 . . 3  |-  ( A  e.  NN  ->  (
( mmu `  A
)  =  0  <->  if ( E. p  e.  Prime  ( p ^ 2 ) 
||  A ,  0 ,  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  A } ) ) )  =  0 ) )
2119, 20syl5ibr 213 . 2  |-  ( A  e.  NN  ->  ( E. p  e.  Prime  ( p ^ 2 ) 
||  A  ->  (
mmu `  A )  =  0 ) )
2218, 21impbid 184 1  |-  ( A  e.  NN  ->  (
( mmu `  A
)  =  0  <->  E. p  e.  Prime  ( p ^ 2 )  ||  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717    =/= wne 2551   E.wrex 2651   {crab 2654   ifcif 3683   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   Fincfn 7046   CCcc 8922   0cc0 8924   1c1 8925   -ucneg 9225   NNcn 9933   2c2 9982   NN0cn0 10154   ZZcz 10215   ^cexp 11310   #chash 11546    || cdivides 12780   Primecprime 13007   mmucmu 20745
This theorem is referenced by:  issqf  20787  dvdssqf  20789  mumullem1  20830
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 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-card 7760  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-n0 10155  df-z 10216  df-uz 10422  df-fz 10977  df-seq 11252  df-exp 11311  df-hash 11547  df-dvds 12781  df-prm 13008  df-mu 20751
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