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Theorem pcmul 14252
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 2443 . . 3  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  A } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  A } ,  RR ,  <  )
2 eqid 2443 . . 3  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  B } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  B } ,  RR ,  <  )
3 eqid 2443 . . 3  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  ( A  x.  B ) } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  ( A  x.  B
) } ,  RR ,  <  )
41, 2, 3pcpremul 14244 . 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 14248 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 ) )  -> 
( P  pCnt  A
)  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  A } ,  RR ,  <  )
)
653adant3 1017 . . 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 14248 . . . 4  |-  ( ( P  e.  Prime  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( P  pCnt  B
)  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  B } ,  RR ,  <  )
)
873adant2 1016 . . 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 6299 . 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 10918 . . . . . 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 10875 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  CC )
1312anim1i 568 . . . . . 6  |-  ( ( A  e.  ZZ  /\  A  =/=  0 )  -> 
( A  e.  CC  /\  A  =/=  0 ) )
14 zcn 10875 . . . . . . 7  |-  ( B  e.  ZZ  ->  B  e.  CC )
1514anim1i 568 . . . . . 6  |-  ( ( B  e.  ZZ  /\  B  =/=  0 )  -> 
( B  e.  CC  /\  B  =/=  0 ) )
16 mulne0 10197 . . . . . 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 14248 . . . 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 1193 . 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 2495 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 974    = wceq 1383    e. wcel 1804    =/= wne 2638   {crab 2797   class class class wbr 4437  (class class class)co 6281   supcsup 7902   CCcc 9493   RRcr 9494   0cc0 9495    + caddc 9498    x. cmul 9500    < clt 9631   NN0cn0 10801   ZZcz 10870   ^cexp 12145    || cdvds 13863   Primecprime 14094    pCnt cpc 14237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-n0 10802  df-z 10871  df-uz 11091  df-q 11192  df-rp 11230  df-fl 11908  df-mod 11976  df-seq 12087  df-exp 12146  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-dvds 13864  df-gcd 14022  df-prm 14095  df-pc 14238
This theorem is referenced by:  pcqmul  14254  pcaddlem  14284  pcmpt  14288  pcfac  14295  pcbc  14296  sylow1lem1  16492  sylow1lem5  16496  mumullem2  23326  chtublem  23358  lgsdi  23479
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