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Theorem sqf11 20377
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 9972 . . . 4  |-  ( A  e.  NN  ->  A  e.  NN0 )
2 nnnn0 9972 . . . 4  |-  ( B  e.  NN  ->  B  e.  NN0 )
3 pc11 12932 . . . 4  |-  ( ( A  e.  NN0  /\  B  e.  NN0 )  -> 
( A  =  B  <->  A. p  e.  Prime  ( p  pCnt  A )  =  ( p  pCnt  B ) ) )
41, 2, 3syl2an 463 . . 3  |-  ( ( A  e.  NN  /\  B  e.  NN )  ->  ( A  =  B  <->  A. p  e.  Prime  ( p  pCnt  A )  =  ( p  pCnt  B ) ) )
54ad2ant2r 727 . 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 2343 . . . . 5  |-  ( ( p  pCnt  A )  =  ( p  pCnt  B )  ->  ( (
p  pCnt  A )  e.  NN  <->  ( p  pCnt  B )  e.  NN ) )
7 dfbi3 863 . . . . . 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 730 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  ( mmu `  A
)  =/=  0 )  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  ->  A  e.  NN )
98adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  A  e.  NN )
10 simpllr 735 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
mmu `  A )  =/=  0 )
11 simpr 447 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  p  e.  Prime )
12 sqfpc 20375 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0  /\  p  e.  Prime )  ->  (
p  pCnt  A )  <_  1 )
139, 10, 11, 12syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
p  pCnt  A )  <_  1 )
14 nnle1eq1 9774 . . . . . . . . . 10  |-  ( ( p  pCnt  A )  e.  NN  ->  ( (
p  pCnt  A )  <_  1  <->  ( p  pCnt  A )  =  1 ) )
1513, 14syl5ibcom 211 . . . . . . . . 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 732 . . . . . . . . . . . 12  |-  ( ( ( A  e.  NN  /\  ( mmu `  A
)  =/=  0 )  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  ->  B  e.  NN )
1716adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  B  e.  NN )
18 simplrr 737 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
mmu `  B )  =/=  0 )
19 sqfpc 20375 . . . . . . . . . . 11  |-  ( ( B  e.  NN  /\  ( mmu `  B )  =/=  0  /\  p  e.  Prime )  ->  (
p  pCnt  B )  <_  1 )
2017, 18, 11, 19syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
p  pCnt  B )  <_  1 )
21 nnle1eq1 9774 . . . . . . . . . 10  |-  ( ( p  pCnt  B )  e.  NN  ->  ( (
p  pCnt  B )  <_  1  <->  ( p  pCnt  B )  =  1 ) )
2220, 21syl5ibcom 211 . . . . . . . . 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 546 . . . . . . . 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 2302 . . . . . . . 8  |-  ( ( ( p  pCnt  A
)  =  1  /\  ( p  pCnt  B
)  =  1 )  ->  ( p  pCnt  A )  =  ( p 
pCnt  B ) )
2523, 24syl6 29 . . . . . . 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 19 . . . . . . . . . . . 12  |-  ( p  e.  Prime  ->  p  e. 
Prime )
27 pccl 12902 . . . . . . . . . . . 12  |-  ( ( p  e.  Prime  /\  A  e.  NN )  ->  (
p  pCnt  A )  e.  NN0 )
2826, 8, 27syl2anr 464 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
p  pCnt  A )  e.  NN0 )
29 elnn0 9967 . . . . . . . . . . 11  |-  ( ( p  pCnt  A )  e.  NN0  <->  ( ( p 
pCnt  A )  e.  NN  \/  ( p  pCnt  A
)  =  0 ) )
3028, 29sylib 188 . . . . . . . . . 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 366 . . . . . . . . 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 12902 . . . . . . . . . . . 12  |-  ( ( p  e.  Prime  /\  B  e.  NN )  ->  (
p  pCnt  B )  e.  NN0 )
3326, 16, 32syl2anr 464 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  NN  /\  ( mmu `  A )  =/=  0
)  /\  ( B  e.  NN  /\  ( mmu `  B )  =/=  0
) )  /\  p  e.  Prime )  ->  (
p  pCnt  B )  e.  NN0 )
34 elnn0 9967 . . . . . . . . . . 11  |-  ( ( p  pCnt  B )  e.  NN0  <->  ( ( p 
pCnt  B )  e.  NN  \/  ( p  pCnt  B
)  =  0 ) )
3533, 34sylib 188 . . . . . . . . . 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 366 . . . . . . . . 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 546 . . . . . . . 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 2302 . . . . . . . 8  |-  ( ( ( p  pCnt  A
)  =  0  /\  ( p  pCnt  B
)  =  0 )  ->  ( p  pCnt  A )  =  ( p 
pCnt  B ) )
3937, 38syl6 29 . . . . . . 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 369 . . . . . 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 208 . . . . 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 195 . . . 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 12922 . . . . . 6  |-  ( ( p  e.  Prime  /\  A  e.  NN )  ->  (
( p  pCnt  A
)  e.  NN  <->  p  ||  A
) )
4426, 8, 43syl2anr 464 . . . . 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 12922 . . . . . 6  |-  ( ( p  e.  Prime  /\  B  e.  NN )  ->  (
( p  pCnt  B
)  e.  NN  <->  p  ||  B
) )
4626, 16, 45syl2anr 464 . . . . 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 312 . . . 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 244 . . 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 2559 . 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 244 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 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   0cc0 8737   1c1 8738    <_ cle 8868   NNcn 9746   NN0cn0 9965    || cdivides 12531   Primecprime 12758    pCnt cpc 12889   mmucmu 20332
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-fz 10783  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890  df-mu 20338
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