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Theorem pcmul 13910
Description: Multiplication property of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)
Assertion
Ref Expression
pcmul  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( P  pCnt  ( A  x.  B )
)  =  ( ( P  pCnt  A )  +  ( P  pCnt  B ) ) )

Proof of Theorem pcmul
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 eqid 2438 . . 3  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  A } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  A } ,  RR ,  <  )
2 eqid 2438 . . 3  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  B } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  B } ,  RR ,  <  )
3 eqid 2438 . . 3  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  ( A  x.  B ) } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  ( A  x.  B
) } ,  RR ,  <  )
41, 2, 3pcpremul 13902 . 2  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( sup ( { n  e.  NN0  | 
( P ^ n
)  ||  A } ,  RR ,  <  )  +  sup ( { n  e.  NN0  |  ( P ^ n )  ||  B } ,  RR ,  <  ) )  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  ( A  x.  B ) } ,  RR ,  <  ) )
51pczpre 13906 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 ) )  -> 
( P  pCnt  A
)  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  A } ,  RR ,  <  )
)
653adant3 1008 . . 3  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( P  pCnt  A
)  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  A } ,  RR ,  <  )
)
72pczpre 13906 . . . 4  |-  ( ( P  e.  Prime  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( P  pCnt  B
)  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  B } ,  RR ,  <  )
)
873adant2 1007 . . 3  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( P  pCnt  B
)  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  B } ,  RR ,  <  )
)
96, 8oveq12d 6104 . 2  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( ( P  pCnt  A )  +  ( P 
pCnt  B ) )  =  ( sup ( { n  e.  NN0  | 
( P ^ n
)  ||  A } ,  RR ,  <  )  +  sup ( { n  e.  NN0  |  ( P ^ n )  ||  B } ,  RR ,  <  ) ) )
10 zmulcl 10685 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  x.  B
)  e.  ZZ )
1110ad2ant2r 746 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( A  x.  B
)  e.  ZZ )
12 zcn 10643 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  CC )
1312anim1i 568 . . . . . 6  |-  ( ( A  e.  ZZ  /\  A  =/=  0 )  -> 
( A  e.  CC  /\  A  =/=  0 ) )
14 zcn 10643 . . . . . . 7  |-  ( B  e.  ZZ  ->  B  e.  CC )
1514anim1i 568 . . . . . 6  |-  ( ( B  e.  ZZ  /\  B  =/=  0 )  -> 
( B  e.  CC  /\  B  =/=  0 ) )
16 mulne0 9970 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( A  x.  B
)  =/=  0 )
1713, 15, 16syl2an 477 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( A  x.  B
)  =/=  0 )
1811, 17jca 532 . . . 4  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( ( A  x.  B )  e.  ZZ  /\  ( A  x.  B
)  =/=  0 ) )
193pczpre 13906 . . . 4  |-  ( ( P  e.  Prime  /\  (
( A  x.  B
)  e.  ZZ  /\  ( A  x.  B
)  =/=  0 ) )  ->  ( P  pCnt  ( A  x.  B
) )  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  ( A  x.  B ) } ,  RR ,  <  ) )
2018, 19sylan2 474 . . 3  |-  ( ( P  e.  Prime  /\  (
( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) ) )  ->  ( P  pCnt  ( A  x.  B ) )  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  ( A  x.  B ) } ,  RR ,  <  ) )
21203impb 1183 . 2  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( P  pCnt  ( A  x.  B )
)  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  ( A  x.  B ) } ,  RR ,  <  ) )
224, 9, 213eqtr4rd 2481 1  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( P  pCnt  ( A  x.  B )
)  =  ( ( P  pCnt  A )  +  ( P  pCnt  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   {crab 2714   class class class wbr 4287  (class class class)co 6086   supcsup 7682   CCcc 9272   RRcr 9273   0cc0 9274    + caddc 9277    x. cmul 9279    < clt 9410   NN0cn0 10571   ZZcz 10638   ^cexp 11857    || cdivides 13527   Primecprime 13755    pCnt cpc 13895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-q 10946  df-rp 10984  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-dvds 13528  df-gcd 13683  df-prm 13756  df-pc 13896
This theorem is referenced by:  pcqmul  13912  pcaddlem  13942  pcmpt  13946  pcfac  13953  pcbc  13954  sylow1lem1  16088  sylow1lem5  16092  mumullem2  22493  chtublem  22525  lgsdi  22646
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