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Theorem sqf11 20790
Description: A squarefree number is completely determined by the set of its prime divisors. (Contributed by Mario Carneiro, 1-Jul-2015.)
Assertion
Ref Expression
sqf11  |-  ( ( ( A  e.  NN  /\  ( mmu `  A
)  =/=  0 )  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  ->  ( A  =  B  <->  A. p  e.  Prime  ( p  ||  A 
<->  p  ||  B ) ) )
Distinct variable groups:    A, p    B, p

Proof of Theorem sqf11
StepHypRef Expression
1 nnnn0 10161 . . . 4  |-  ( A  e.  NN  ->  A  e.  NN0 )
2 nnnn0 10161 . . . 4  |-  ( B  e.  NN  ->  B  e.  NN0 )
3 pc11 13181 . . . 4  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( A  =  B  <->  A. p  e.  Prime  ( p  pCnt  A )  =  ( p  pCnt  B ) ) )
41, 2, 3syl2an 464 . . 3  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  =  B  <->  A. p  e.  Prime  ( p  pCnt  A )  =  ( p  pCnt  B ) ) )
54ad2ant2r 728 . 2  |-  ( ( ( A  e.  NN  /\  ( mmu `  A
)  =/=  0 )  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  ->  ( A  =  B  <->  A. p  e.  Prime  ( p  pCnt  A )  =  ( p 
pCnt  B ) ) )
6 eleq1 2448 . . . . 5  |-  ( ( p  pCnt  A )  =  ( p  pCnt  B )  ->  ( (
p  pCnt  A )  e.  NN  <->  ( p  pCnt  B )  e.  NN ) )
7 dfbi3 864 . . . . . 6  |-  ( ( ( p  pCnt  A
)  e.  NN  <->  ( p  pCnt  B )  e.  NN ) 
<->  ( ( ( p 
pCnt  A )  e.  NN  /\  ( p  pCnt  B
)  e.  NN )  \/  ( -.  (
p  pCnt  A )  e.  NN  /\  -.  (
p  pCnt  B )  e.  NN ) ) )
8 simpll 731 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  ( mmu `  A
)  =/=  0 )  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  ->  A  e.  NN )
98adantr 452 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  A  e.  NN )
10 simpllr 736 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
mmu `  A )  =/=  0 )
11 simpr 448 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  p  e.  Prime )
12 sqfpc 20788 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0  /\  p  e.  Prime )  ->  (
p  pCnt  A )  <_  1 )
139, 10, 11, 12syl3anc 1184 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
p  pCnt  A )  <_  1 )
14 nnle1eq1 9961 . . . . . . . . . 10  |-  ( ( p  pCnt  A )  e.  NN  ->  ( (
p  pCnt  A )  <_  1  <->  ( p  pCnt  A )  =  1 ) )
1513, 14syl5ibcom 212 . . . . . . . . 9  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
( p  pCnt  A
)  e.  NN  ->  ( p  pCnt  A )  =  1 ) )
16 simprl 733 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  ( mmu `  A
)  =/=  0 )  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  ->  B  e.  NN )
1716adantr 452 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  B  e.  NN )
18 simplrr 738 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
mmu `  B )  =/=  0 )
19 sqfpc 20788 . . . . . . . . . . 11  |-  ( ( B  e.  NN  /\  ( mmu `  B )  =/=  0  /\  p  e.  Prime )  ->  (
p  pCnt  B )  <_  1 )
2017, 18, 11, 19syl3anc 1184 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
p  pCnt  B )  <_  1 )
21 nnle1eq1 9961 . . . . . . . . . 10  |-  ( ( p  pCnt  B )  e.  NN  ->  ( (
p  pCnt  B )  <_  1  <->  ( p  pCnt  B )  =  1 ) )
2220, 21syl5ibcom 212 . . . . . . . . 9  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
( p  pCnt  B
)  e.  NN  ->  ( p  pCnt  B )  =  1 ) )
2315, 22anim12d 547 . . . . . . . 8  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
( ( p  pCnt  A )  e.  NN  /\  ( p  pCnt  B )  e.  NN )  -> 
( ( p  pCnt  A )  =  1  /\  ( p  pCnt  B
)  =  1 ) ) )
24 eqtr3 2407 . . . . . . . 8  |-  ( ( ( p  pCnt  A
)  =  1  /\  ( p  pCnt  B
)  =  1 )  ->  ( p  pCnt  A )  =  ( p 
pCnt  B ) )
2523, 24syl6 31 . . . . . . 7  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
( ( p  pCnt  A )  e.  NN  /\  ( p  pCnt  B )  e.  NN )  -> 
( p  pCnt  A
)  =  ( p 
pCnt  B ) ) )
26 id 20 . . . . . . . . . . . 12  |-  ( p  e.  Prime  ->  p  e. 
Prime )
27 pccl 13151 . . . . . . . . . . . 12  |-  ( ( p  e.  Prime  /\  A  e.  NN )  ->  (
p  pCnt  A )  e.  NN0 )
2826, 8, 27syl2anr 465 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
p  pCnt  A )  e.  NN0 )
29 elnn0 10156 . . . . . . . . . . 11  |-  ( ( p  pCnt  A )  e.  NN0  <->  ( ( p 
pCnt  A )  e.  NN  \/  ( p  pCnt  A
)  =  0 ) )
3028, 29sylib 189 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
( p  pCnt  A
)  e.  NN  \/  ( p  pCnt  A )  =  0 ) )
3130ord 367 . . . . . . . . 9  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  ( -.  ( p  pCnt  A
)  e.  NN  ->  ( p  pCnt  A )  =  0 ) )
32 pccl 13151 . . . . . . . . . . . 12  |-  ( ( p  e.  Prime  /\  B  e.  NN )  ->  (
p  pCnt  B )  e.  NN0 )
3326, 16, 32syl2anr 465 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
p  pCnt  B )  e.  NN0 )
34 elnn0 10156 . . . . . . . . . . 11  |-  ( ( p  pCnt  B )  e.  NN0  <->  ( ( p 
pCnt  B )  e.  NN  \/  ( p  pCnt  B
)  =  0 ) )
3533, 34sylib 189 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
( p  pCnt  B
)  e.  NN  \/  ( p  pCnt  B )  =  0 ) )
3635ord 367 . . . . . . . . 9  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  ( -.  ( p  pCnt  B
)  e.  NN  ->  ( p  pCnt  B )  =  0 ) )
3731, 36anim12d 547 . . . . . . . 8  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
( -.  ( p 
pCnt  A )  e.  NN  /\ 
-.  ( p  pCnt  B )  e.  NN )  ->  ( ( p 
pCnt  A )  =  0  /\  ( p  pCnt  B )  =  0 ) ) )
38 eqtr3 2407 . . . . . . . 8  |-  ( ( ( p  pCnt  A
)  =  0  /\  ( p  pCnt  B
)  =  0 )  ->  ( p  pCnt  A )  =  ( p 
pCnt  B ) )
3937, 38syl6 31 . . . . . . 7  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
( -.  ( p 
pCnt  A )  e.  NN  /\ 
-.  ( p  pCnt  B )  e.  NN )  ->  ( p  pCnt  A )  =  ( p 
pCnt  B ) ) )
4025, 39jaod 370 . . . . . 6  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
( ( ( p 
pCnt  A )  e.  NN  /\  ( p  pCnt  B
)  e.  NN )  \/  ( -.  (
p  pCnt  A )  e.  NN  /\  -.  (
p  pCnt  B )  e.  NN ) )  -> 
( p  pCnt  A
)  =  ( p 
pCnt  B ) ) )
417, 40syl5bi 209 . . . . 5  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
( ( p  pCnt  A )  e.  NN  <->  ( p  pCnt  B )  e.  NN )  ->  ( p  pCnt  A )  =  ( p 
pCnt  B ) ) )
426, 41impbid2 196 . . . 4  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
( p  pCnt  A
)  =  ( p 
pCnt  B )  <->  ( (
p  pCnt  A )  e.  NN  <->  ( p  pCnt  B )  e.  NN ) ) )
43 pcelnn 13171 . . . . . 6  |-  ( ( p  e.  Prime  /\  A  e.  NN )  ->  (
( p  pCnt  A
)  e.  NN  <->  p  ||  A
) )
4426, 8, 43syl2anr 465 . . . . 5  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
( p  pCnt  A
)  e.  NN  <->  p  ||  A
) )
45 pcelnn 13171 . . . . . 6  |-  ( ( p  e.  Prime  /\  B  e.  NN )  ->  (
( p  pCnt  B
)  e.  NN  <->  p  ||  B
) )
4626, 16, 45syl2anr 465 . . . . 5  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
( p  pCnt  B
)  e.  NN  <->  p  ||  B
) )
4744, 46bibi12d 313 . . . 4  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
( ( p  pCnt  A )  e.  NN  <->  ( p  pCnt  B )  e.  NN ) 
<->  ( p  ||  A  <->  p 
||  B ) ) )
4842, 47bitrd 245 . . 3  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
( p  pCnt  A
)  =  ( p 
pCnt  B )  <->  ( p  ||  A  <->  p  ||  B ) ) )
4948ralbidva 2666 . 2  |-  ( ( ( A  e.  NN  /\  ( mmu `  A
)  =/=  0 )  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  ->  ( A. p  e.  Prime  ( p  pCnt  A )  =  ( p  pCnt  B )  <->  A. p  e.  Prime  ( p  ||  A  <->  p  ||  B
) ) )
505, 49bitrd 245 1  |-  ( ( ( A  e.  NN  /\  ( mmu `  A
)  =/=  0 )  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  ->  ( A  =  B  <->  A. p  e.  Prime  ( p  ||  A 
<->  p  ||  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2551   A.wral 2650   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   0cc0 8924   1c1 8925    <_ cle 9055   NNcn 9933   NN0cn0 10154    || cdivides 12780   Primecprime 13007    pCnt cpc 13138   mmucmu 20745
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  ax-pre-sup 9002
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-2o 6662  df-oadd 6665  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-sup 7382  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-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-q 10508  df-rp 10546  df-fz 10977  df-fl 11130  df-mod 11179  df-seq 11252  df-exp 11311  df-hash 11547  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-dvds 12781  df-gcd 12935  df-prm 13008  df-pc 13139  df-mu 20751
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