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Theorem pcmul 12259
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 2171 . . 3  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  A } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  A } ,  RR ,  <  )
2 eqid 2171 . . 3  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  B } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  B } ,  RR ,  <  )
3 eqid 2171 . . 3  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  ( A  x.  B ) } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  ( A  x.  B
) } ,  RR ,  <  )
41, 2, 3pcpremul 12251 . 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 12255 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 ) )  -> 
( P  pCnt  A
)  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  A } ,  RR ,  <  )
)
653adant3 1013 . . 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 12255 . . . 4  |-  ( ( P  e.  Prime  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( P  pCnt  B
)  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  B } ,  RR ,  <  )
)
873adant2 1012 . . 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 5875 . 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 9269 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  x.  B
)  e.  ZZ )
1110ad2ant2r 507 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( A  x.  B
)  e.  ZZ )
12 zcn 9221 . . . . . . . 8  |-  ( A  e.  ZZ  ->  A  e.  CC )
1312ad2antrr 486 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  ->  A  e.  CC )
14 zcn 9221 . . . . . . . 8  |-  ( B  e.  ZZ  ->  B  e.  CC )
1514ad2antrl 488 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  ->  B  e.  CC )
16 simplr 526 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  ->  A  =/=  0 )
17 simpll 525 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  ->  A  e.  ZZ )
18 0zd 9228 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
0  e.  ZZ )
19 zapne 9290 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  0  e.  ZZ )  ->  ( A #  0  <->  A  =/=  0 ) )
2017, 18, 19syl2anc 409 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( A #  0  <->  A  =/=  0 ) )
2116, 20mpbird 166 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  ->  A #  0 )
22 simprr 528 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  ->  B  =/=  0 )
23 simprl 527 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  ->  B  e.  ZZ )
24 zapne 9290 . . . . . . . . 9  |-  ( ( B  e.  ZZ  /\  0  e.  ZZ )  ->  ( B #  0  <->  B  =/=  0 ) )
2523, 18, 24syl2anc 409 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( B #  0  <->  B  =/=  0 ) )
2622, 25mpbird 166 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  ->  B #  0 )
2713, 15, 21, 26mulap0d 8580 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( A  x.  B
) #  0 )
28 zapne 9290 . . . . . . 7  |-  ( ( ( A  x.  B
)  e.  ZZ  /\  0  e.  ZZ )  ->  ( ( A  x.  B ) #  0  <->  ( A  x.  B )  =/=  0
) )
2911, 18, 28syl2anc 409 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( ( A  x.  B ) #  0  <->  ( A  x.  B )  =/=  0
) )
3027, 29mpbid 146 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( A  x.  B
)  =/=  0 )
3111, 30jca 304 . . . 4  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( ( A  x.  B )  e.  ZZ  /\  ( A  x.  B
)  =/=  0 ) )
323pczpre 12255 . . . 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 ,  <  ) )
3331, 32sylan2 284 . . 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 ,  <  ) )
34333impb 1195 . 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 ,  <  ) )
354, 9, 343eqtr4rd 2215 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 103    <-> wb 104    /\ w3a 974    = wceq 1349    e. wcel 2142    =/= wne 2341   {crab 2453   class class class wbr 3990  (class class class)co 5857   supcsup 6963   CCcc 7776   RRcr 7777   0cc0 7778    + caddc 7781    x. cmul 7783    < clt 7958   # cap 8504   NN0cn0 9139   ZZcz 9216   ^cexp 10479    || cdvds 11753   Primecprime 12065    pCnt cpc 12242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 610  ax-in2 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-13 2144  ax-14 2145  ax-ext 2153  ax-coll 4105  ax-sep 4108  ax-nul 4116  ax-pow 4161  ax-pr 4195  ax-un 4419  ax-setind 4522  ax-iinf 4573  ax-cnex 7869  ax-resscn 7870  ax-1cn 7871  ax-1re 7872  ax-icn 7873  ax-addcl 7874  ax-addrcl 7875  ax-mulcl 7876  ax-mulrcl 7877  ax-addcom 7878  ax-mulcom 7879  ax-addass 7880  ax-mulass 7881  ax-distr 7882  ax-i2m1 7883  ax-0lt1 7884  ax-1rid 7885  ax-0id 7886  ax-rnegex 7887  ax-precex 7888  ax-cnre 7889  ax-pre-ltirr 7890  ax-pre-ltwlin 7891  ax-pre-lttrn 7892  ax-pre-apti 7893  ax-pre-ltadd 7894  ax-pre-mulgt0 7895  ax-pre-mulext 7896  ax-arch 7897  ax-caucvg 7898
This theorem depends on definitions:  df-bi 116  df-dc 831  df-3or 975  df-3an 976  df-tru 1352  df-fal 1355  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ne 2342  df-nel 2437  df-ral 2454  df-rex 2455  df-reu 2456  df-rmo 2457  df-rab 2458  df-v 2733  df-sbc 2957  df-csb 3051  df-dif 3124  df-un 3126  df-in 3128  df-ss 3135  df-nul 3416  df-if 3528  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-uni 3798  df-int 3833  df-iun 3876  df-br 3991  df-opab 4052  df-mpt 4053  df-tr 4089  df-id 4279  df-po 4282  df-iso 4283  df-iord 4352  df-on 4354  df-ilim 4355  df-suc 4357  df-iom 4576  df-xp 4618  df-rel 4619  df-cnv 4620  df-co 4621  df-dm 4622  df-rn 4623  df-res 4624  df-ima 4625  df-iota 5162  df-fun 5202  df-fn 5203  df-f 5204  df-f1 5205  df-fo 5206  df-f1o 5207  df-fv 5208  df-isom 5209  df-riota 5813  df-ov 5860  df-oprab 5861  df-mpo 5862  df-1st 6123  df-2nd 6124  df-recs 6288  df-frec 6374  df-1o 6399  df-2o 6400  df-er 6517  df-en 6723  df-sup 6965  df-inf 6966  df-pnf 7960  df-mnf 7961  df-xr 7962  df-ltxr 7963  df-le 7964  df-sub 8096  df-neg 8097  df-reap 8498  df-ap 8505  df-div 8594  df-inn 8883  df-2 8941  df-3 8942  df-4 8943  df-n0 9140  df-z 9217  df-uz 9492  df-q 9583  df-rp 9615  df-fz 9970  df-fzo 10103  df-fl 10230  df-mod 10283  df-seqfrec 10406  df-exp 10480  df-cj 10810  df-re 10811  df-im 10812  df-rsqrt 10966  df-abs 10967  df-dvds 11754  df-gcd 11902  df-prm 12066  df-pc 12243
This theorem is referenced by:  pcqmul  12261  pcaddlem  12296  pcmpt  12299  pcfac  12306  pcbc  12307  lgsdi  13817
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