MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pcmul Unicode version

Theorem pcmul 13180
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 2404 . . 3  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  A } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  A } ,  RR ,  <  )
2 eqid 2404 . . 3  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  B } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  B } ,  RR ,  <  )
3 eqid 2404 . . 3  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  ( A  x.  B ) } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  ( A  x.  B
) } ,  RR ,  <  )
41, 2, 3pcpremul 13172 . 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 13176 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 ) )  -> 
( P  pCnt  A
)  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  A } ,  RR ,  <  )
)
653adant3 977 . . 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 13176 . . . 4  |-  ( ( P  e.  Prime  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( P  pCnt  B
)  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  B } ,  RR ,  <  )
)
873adant2 976 . . 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 6058 . 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 10280 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  x.  B
)  e.  ZZ )
1110ad2ant2r 728 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( A  x.  B
)  e.  ZZ )
12 zcn 10243 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  CC )
1312anim1i 552 . . . . . 6  |-  ( ( A  e.  ZZ  /\  A  =/=  0 )  -> 
( A  e.  CC  /\  A  =/=  0 ) )
14 zcn 10243 . . . . . . 7  |-  ( B  e.  ZZ  ->  B  e.  CC )
1514anim1i 552 . . . . . 6  |-  ( ( B  e.  ZZ  /\  B  =/=  0 )  -> 
( B  e.  CC  /\  B  =/=  0 ) )
16 mulne0 9620 . . . . . 6  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( A  x.  B
)  =/=  0 )
1713, 15, 16syl2an 464 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( A  x.  B
)  =/=  0 )
1811, 17jca 519 . . . 4  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( ( A  x.  B )  e.  ZZ  /\  ( A  x.  B
)  =/=  0 ) )
193pczpre 13176 . . . 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 461 . . 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 1149 . 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 2447 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   {crab 2670   class class class wbr 4172  (class class class)co 6040   supcsup 7403   CCcc 8944   RRcr 8945   0cc0 8946    + caddc 8949    x. cmul 8951    < clt 9076   NN0cn0 10177   ZZcz 10238   ^cexp 11337    || cdivides 12807   Primecprime 13034    pCnt cpc 13165
This theorem is referenced by:  pcqmul  13182  pcaddlem  13212  pcmpt  13216  pcfac  13223  pcbc  13224  sylow1lem1  15187  sylow1lem5  15191  mumullem2  20916  chtublem  20948  lgsdi  21069
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962  df-prm 13035  df-pc 13166
  Copyright terms: Public domain W3C validator