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Theorem sqf11 20927
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 10233 . . . 4  |-  ( A  e.  NN  ->  A  e.  NN0 )
2 nnnn0 10233 . . . 4  |-  ( B  e.  NN  ->  B  e.  NN0 )
3 pc11 13258 . . . 4  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( A  =  B  <->  A. p  e.  Prime  ( p  pCnt  A )  =  ( p  pCnt  B ) ) )
41, 2, 3syl2an 465 . . 3  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  =  B  <->  A. p  e.  Prime  ( p  pCnt  A )  =  ( p  pCnt  B ) ) )
54ad2ant2r 729 . 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 2498 . . . . 5  |-  ( ( p  pCnt  A )  =  ( p  pCnt  B )  ->  ( (
p  pCnt  A )  e.  NN  <->  ( p  pCnt  B )  e.  NN ) )
7 dfbi3 865 . . . . . 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 732 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  ( mmu `  A
)  =/=  0 )  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  ->  A  e.  NN )
98adantr 453 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  A  e.  NN )
10 simpllr 737 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
mmu `  A )  =/=  0 )
11 simpr 449 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  p  e.  Prime )
12 sqfpc 20925 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0  /\  p  e.  Prime )  ->  (
p  pCnt  A )  <_  1 )
139, 10, 11, 12syl3anc 1185 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
p  pCnt  A )  <_  1 )
14 nnle1eq1 10033 . . . . . . . . . 10  |-  ( ( p  pCnt  A )  e.  NN  ->  ( (
p  pCnt  A )  <_  1  <->  ( p  pCnt  A )  =  1 ) )
1513, 14syl5ibcom 213 . . . . . . . . 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 734 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  ( mmu `  A
)  =/=  0 )  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  ->  B  e.  NN )
1716adantr 453 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  B  e.  NN )
18 simplrr 739 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
mmu `  B )  =/=  0 )
19 sqfpc 20925 . . . . . . . . . . 11  |-  ( ( B  e.  NN  /\  ( mmu `  B )  =/=  0  /\  p  e.  Prime )  ->  (
p  pCnt  B )  <_  1 )
2017, 18, 11, 19syl3anc 1185 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
p  pCnt  B )  <_  1 )
21 nnle1eq1 10033 . . . . . . . . . 10  |-  ( ( p  pCnt  B )  e.  NN  ->  ( (
p  pCnt  B )  <_  1  <->  ( p  pCnt  B )  =  1 ) )
2220, 21syl5ibcom 213 . . . . . . . . 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 548 . . . . . . . 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 2457 . . . . . . . 8  |-  ( ( ( p  pCnt  A
)  =  1  /\  ( p  pCnt  B
)  =  1 )  ->  ( p  pCnt  A )  =  ( p 
pCnt  B ) )
2523, 24syl6 32 . . . . . . 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 21 . . . . . . . . . . . 12  |-  ( p  e.  Prime  ->  p  e. 
Prime )
27 pccl 13228 . . . . . . . . . . . 12  |-  ( ( p  e.  Prime  /\  A  e.  NN )  ->  (
p  pCnt  A )  e.  NN0 )
2826, 8, 27syl2anr 466 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
p  pCnt  A )  e.  NN0 )
29 elnn0 10228 . . . . . . . . . . 11  |-  ( ( p  pCnt  A )  e.  NN0  <->  ( ( p 
pCnt  A )  e.  NN  \/  ( p  pCnt  A
)  =  0 ) )
3028, 29sylib 190 . . . . . . . . . 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 368 . . . . . . . . 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 13228 . . . . . . . . . . . 12  |-  ( ( p  e.  Prime  /\  B  e.  NN )  ->  (
p  pCnt  B )  e.  NN0 )
3326, 16, 32syl2anr 466 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
p  pCnt  B )  e.  NN0 )
34 elnn0 10228 . . . . . . . . . . 11  |-  ( ( p  pCnt  B )  e.  NN0  <->  ( ( p 
pCnt  B )  e.  NN  \/  ( p  pCnt  B
)  =  0 ) )
3533, 34sylib 190 . . . . . . . . . 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 368 . . . . . . . . 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 548 . . . . . . . 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 2457 . . . . . . . 8  |-  ( ( ( p  pCnt  A
)  =  0  /\  ( p  pCnt  B
)  =  0 )  ->  ( p  pCnt  A )  =  ( p 
pCnt  B ) )
3937, 38syl6 32 . . . . . . 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 371 . . . . . 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 210 . . . . 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 197 . . . 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 13248 . . . . . 6  |-  ( ( p  e.  Prime  /\  A  e.  NN )  ->  (
( p  pCnt  A
)  e.  NN  <->  p  ||  A
) )
4426, 8, 43syl2anr 466 . . . . 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 13248 . . . . . 6  |-  ( ( p  e.  Prime  /\  B  e.  NN )  ->  (
( p  pCnt  B
)  e.  NN  <->  p  ||  B
) )
4626, 16, 45syl2anr 466 . . . . 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 314 . . . 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 246 . . 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 2723 . 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 246 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 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   0cc0 8995   1c1 8996    <_ cle 9126   NNcn 10005   NN0cn0 10226    || cdivides 12857   Primecprime 13084    pCnt cpc 13215   mmucmu 20882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-q 10580  df-rp 10618  df-fz 11049  df-fl 11207  df-mod 11256  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-dvds 12858  df-gcd 13012  df-prm 13085  df-pc 13216  df-mu 20888
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