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Theorem sqf11 23929
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 10876 . . . 4  |-  ( A  e.  NN  ->  A  e.  NN0 )
2 nnnn0 10876 . . . 4  |-  ( B  e.  NN  ->  B  e.  NN0 )
3 pc11 14792 . . . 4  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( A  =  B  <->  A. p  e.  Prime  ( p  pCnt  A )  =  ( p  pCnt  B ) ) )
41, 2, 3syl2an 479 . . 3  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  =  B  <->  A. p  e.  Prime  ( p  pCnt  A )  =  ( p  pCnt  B ) ) )
54ad2ant2r 751 . 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 2501 . . . . 5  |-  ( ( p  pCnt  A )  =  ( p  pCnt  B )  ->  ( (
p  pCnt  A )  e.  NN  <->  ( p  pCnt  B )  e.  NN ) )
7 dfbi3 901 . . . . . 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 758 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  ( mmu `  A
)  =/=  0 )  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  ->  A  e.  NN )
98adantr 466 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  A  e.  NN )
10 simpllr 767 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
mmu `  A )  =/=  0 )
11 simpr 462 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  p  e.  Prime )
12 sqfpc 23927 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0  /\  p  e.  Prime )  ->  (
p  pCnt  A )  <_  1 )
139, 10, 11, 12syl3anc 1264 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
p  pCnt  A )  <_  1 )
14 nnle1eq1 10637 . . . . . . . . . 10  |-  ( ( p  pCnt  A )  e.  NN  ->  ( (
p  pCnt  A )  <_  1  <->  ( p  pCnt  A )  =  1 ) )
1513, 14syl5ibcom 223 . . . . . . . . 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 762 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  ( mmu `  A
)  =/=  0 )  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  ->  B  e.  NN )
1716adantr 466 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  B  e.  NN )
18 simplrr 769 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
mmu `  B )  =/=  0 )
19 sqfpc 23927 . . . . . . . . . . 11  |-  ( ( B  e.  NN  /\  ( mmu `  B )  =/=  0  /\  p  e.  Prime )  ->  (
p  pCnt  B )  <_  1 )
2017, 18, 11, 19syl3anc 1264 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
p  pCnt  B )  <_  1 )
21 nnle1eq1 10637 . . . . . . . . . 10  |-  ( ( p  pCnt  B )  e.  NN  ->  ( (
p  pCnt  B )  <_  1  <->  ( p  pCnt  B )  =  1 ) )
2220, 21syl5ibcom 223 . . . . . . . . 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 565 . . . . . . . 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 34 . . . . . . 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 23 . . . . . . . . . . . 12  |-  ( p  e.  Prime  ->  p  e. 
Prime )
27 pccl 14762 . . . . . . . . . . . 12  |-  ( ( p  e.  Prime  /\  A  e.  NN )  ->  (
p  pCnt  A )  e.  NN0 )
2826, 8, 27syl2anr 480 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
p  pCnt  A )  e.  NN0 )
29 elnn0 10871 . . . . . . . . . . 11  |-  ( ( p  pCnt  A )  e.  NN0  <->  ( ( p 
pCnt  A )  e.  NN  \/  ( p  pCnt  A
)  =  0 ) )
3028, 29sylib 199 . . . . . . . . . 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 378 . . . . . . . . 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 14762 . . . . . . . . . . . 12  |-  ( ( p  e.  Prime  /\  B  e.  NN )  ->  (
p  pCnt  B )  e.  NN0 )
3326, 16, 32syl2anr 480 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
p  pCnt  B )  e.  NN0 )
34 elnn0 10871 . . . . . . . . . . 11  |-  ( ( p  pCnt  B )  e.  NN0  <->  ( ( p 
pCnt  B )  e.  NN  \/  ( p  pCnt  B
)  =  0 ) )
3533, 34sylib 199 . . . . . . . . . 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 378 . . . . . . . . 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 565 . . . . . . . 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 34 . . . . . . 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 381 . . . . . 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 220 . . . . 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 207 . . . 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 14782 . . . . . 6  |-  ( ( p  e.  Prime  /\  A  e.  NN )  ->  (
( p  pCnt  A
)  e.  NN  <->  p  ||  A
) )
4426, 8, 43syl2anr 480 . . . . 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 14782 . . . . . 6  |-  ( ( p  e.  Prime  /\  B  e.  NN )  ->  (
( p  pCnt  B
)  e.  NN  <->  p  ||  B
) )
4626, 16, 45syl2anr 480 . . . . 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 322 . . . 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 256 . . 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 2868 . 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 256 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   0cc0 9538   1c1 9539    <_ cle 9675   NNcn 10609   NN0cn0 10869    || cdvds 14283   Primecprime 14593    pCnt cpc 14749   mmucmu 23884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-inf 7963  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-fz 11783  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-dvds 14284  df-gcd 14443  df-prm 14594  df-pc 14750  df-mu 23890
This theorem is referenced by: (None)
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