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Theorem sqf11 20914
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 10220 . . . 4  |-  ( A  e.  NN  ->  A  e.  NN0 )
2 nnnn0 10220 . . . 4  |-  ( B  e.  NN  ->  B  e.  NN0 )
3 pc11 13245 . . . 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 2495 . . . . 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 20912 . . . . . . . . . . 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 10020 . . . . . . . . . 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 20912 . . . . . . . . . . 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 10020 . . . . . . . . . 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 2454 . . . . . . . 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 13215 . . . . . . . . . . . 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 10215 . . . . . . . . . . 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 13215 . . . . . . . . . . . 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 10215 . . . . . . . . . . 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 2454 . . . . . . . 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 13235 . . . . . 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 13235 . . . . . 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 2713 . 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 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   0cc0 8982   1c1 8983    <_ cle 9113   NNcn 9992   NN0cn0 10213    || cdivides 12844   Primecprime 13071    pCnt cpc 13202   mmucmu 20869
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-fz 11036  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845  df-gcd 12999  df-prm 13072  df-pc 13203  df-mu 20875
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