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Theorem sqf11 24865
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 11299 . . . 4 |- ( A e. NN -> A e. NN0 )
2 nnnn0 11299 . . . 4 |- ( B e. NN -> B e. NN0 )
3 pc11 15584 . . . 4 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( A = B <-> A. p e. Prime ( p pCnt A ) = ( p pCnt B ) ) )
41, 2, 3syl2an 494 . . 3 |- ( ( A e. NN /\ B e. NN ) -> ( A = B <-> A. p e. Prime ( p pCnt A ) = ( p pCnt B ) ) )
54ad2ant2r 783 . 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 2689 . . . . 5 |- ( ( p pCnt A ) = ( p pCnt B ) -> ( ( p pCnt A ) e. NN <-> ( p pCnt B ) e. NN ) )
7 dfbi3 994 . . . . . 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 790 . . . . . . . . . . . 12 |- ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) -> A e. NN )
98adantr 481 . . . . . . . . . . 11 |- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> A e. NN )
10 simpllr 799 . . . . . . . . . . 11 |- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( mmu ` A ) =/= 0 )
11 simpr 477 . . . . . . . . . . 11 |- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> p e. Prime )
12 sqfpc 24863 . . . . . . . . . . 11 |- ( ( A e. NN /\ ( mmu ` A ) =/= 0 /\ p e. Prime ) -> ( p pCnt A ) <_ 1 )
139, 10, 11, 12syl3anc 1326 . . . . . . . . . 10 |- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( p pCnt A ) <_ 1 )
14 nnle1eq1 11048 . . . . . . . . . 10 |- ( ( p pCnt A ) e. NN -> ( ( p pCnt A ) <_ 1 <-> ( p pCnt A ) = 1 ) )
1513, 14syl5ibcom 235 . . . . . . . . 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 794 . . . . . . . . . . . 12 |- ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) -> B e. NN )
1716adantr 481 . . . . . . . . . . 11 |- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> B e. NN )
18 simplrr 801 . . . . . . . . . . 11 |- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( mmu ` B ) =/= 0 )
19 sqfpc 24863 . . . . . . . . . . 11 |- ( ( B e. NN /\ ( mmu ` B ) =/= 0 /\ p e. Prime ) -> ( p pCnt B ) <_ 1 )
2017, 18, 11, 19syl3anc 1326 . . . . . . . . . 10 |- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( p pCnt B ) <_ 1 )
21 nnle1eq1 11048 . . . . . . . . . 10 |- ( ( p pCnt B ) e. NN -> ( ( p pCnt B ) <_ 1 <-> ( p pCnt B ) = 1 ) )
2220, 21syl5ibcom 235 . . . . . . . . 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 586 . . . . . . . 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 2643 . . . . . . . 8 |- ( ( ( p pCnt A ) = 1 /\ ( p pCnt B ) = 1 ) -> ( p pCnt A ) = ( p pCnt B ) )
2523, 24syl6 35 . . . . . . 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 22 . . . . . . . . . . . 12 |- ( p e. Prime -> p e. Prime )
27 pccl 15554 . . . . . . . . . . . 12 |- ( ( p e. Prime /\ A e. NN ) -> ( p pCnt A ) e. NN0 )
2826, 8, 27syl2anr 495 . . . . . . . . . . 11 |- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( p pCnt A ) e. NN0 )
29 elnn0 11294 . . . . . . . . . . 11 |- ( ( p pCnt A ) e. NN0 <-> ( ( p pCnt A ) e. NN \/ ( p pCnt A ) = 0 ) )
3028, 29sylib 208 . . . . . . . . . 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 392 . . . . . . . . 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 15554 . . . . . . . . . . . 12 |- ( ( p e. Prime /\ B e. NN ) -> ( p pCnt B ) e. NN0 )
3326, 16, 32syl2anr 495 . . . . . . . . . . 11 |- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( p pCnt B ) e. NN0 )
34 elnn0 11294 . . . . . . . . . . 11 |- ( ( p pCnt B ) e. NN0 <-> ( ( p pCnt B ) e. NN \/ ( p pCnt B ) = 0 ) )
3533, 34sylib 208 . . . . . . . . . 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 392 . . . . . . . . 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 586 . . . . . . . 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 2643 . . . . . . . 8 |- ( ( ( p pCnt A ) = 0 /\ ( p pCnt B ) = 0 ) -> ( p pCnt A ) = ( p pCnt B ) )
3937, 38syl6 35 . . . . . . 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 395 . . . . . 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 232 . . . . 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 216 . . . 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 15574 . . . . . 6 |- ( ( p e. Prime /\ A e. NN ) -> ( ( p pCnt A ) e. NN <-> p || A ) )
4426, 8, 43syl2anr 495 . . . . 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 15574 . . . . . 6 |- ( ( p e. Prime /\ B e. NN ) -> ( ( p pCnt B ) e. NN <-> p || B ) )
4626, 16, 45syl2anr 495 . . . . 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 335 . . . 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 268 . . 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 2985 . 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 268 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 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   <_ cle 10075   NNcn 11020   NN0cn0 11292   || cdvds 14983   Primecprime 15385   pCnt cpc 15541   mmucmu 24821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-pc 15542  df-mu 24827
This theorem is referenced by: (None)
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