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Theorem pcmul 12244
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 2170 . . 3  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  A } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  A } ,  RR ,  <  )
2 eqid 2170 . . 3  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  B } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  B } ,  RR ,  <  )
3 eqid 2170 . . 3  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  ( A  x.  B ) } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  ( A  x.  B
) } ,  RR ,  <  )
41, 2, 3pcpremul 12236 . 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 12240 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 ) )  -> 
( P  pCnt  A
)  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  A } ,  RR ,  <  )
)
653adant3 1012 . . 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 12240 . . . 4  |-  ( ( P  e.  Prime  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( P  pCnt  B
)  =  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  B } ,  RR ,  <  )
)
873adant2 1011 . . 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 5869 . 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 9254 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  x.  B
)  e.  ZZ )
1110ad2ant2r 506 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( A  x.  B
)  e.  ZZ )
12 zcn 9206 . . . . . . . 8  |-  ( A  e.  ZZ  ->  A  e.  CC )
1312ad2antrr 485 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  ->  A  e.  CC )
14 zcn 9206 . . . . . . . 8  |-  ( B  e.  ZZ  ->  B  e.  CC )
1514ad2antrl 487 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  ->  B  e.  CC )
16 simplr 525 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  ->  A  =/=  0 )
17 simpll 524 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  ->  A  e.  ZZ )
18 0zd 9213 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
0  e.  ZZ )
19 zapne 9275 . . . . . . . . 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 527 . . . . . . . 8  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  ->  B  =/=  0 )
23 simprl 526 . . . . . . . . 9  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  ->  B  e.  ZZ )
24 zapne 9275 . . . . . . . . 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 8565 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  A  =/=  0 )  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( A  x.  B
) #  0 )
28 zapne 9275 . . . . . . 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 12240 . . . 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 1194 . 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 2214 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 973    = wceq 1348    e. wcel 2141    =/= wne 2340   {crab 2452   class class class wbr 3987  (class class class)co 5851   supcsup 6956   CCcc 7761   RRcr 7762   0cc0 7763    + caddc 7766    x. cmul 7768    < clt 7943   # cap 8489   NN0cn0 9124   ZZcz 9201   ^cexp 10464    || cdvds 11738   Primecprime 12050    pCnt cpc 12227
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7854  ax-resscn 7855  ax-1cn 7856  ax-1re 7857  ax-icn 7858  ax-addcl 7859  ax-addrcl 7860  ax-mulcl 7861  ax-mulrcl 7862  ax-addcom 7863  ax-mulcom 7864  ax-addass 7865  ax-mulass 7866  ax-distr 7867  ax-i2m1 7868  ax-0lt1 7869  ax-1rid 7870  ax-0id 7871  ax-rnegex 7872  ax-precex 7873  ax-cnre 7874  ax-pre-ltirr 7875  ax-pre-ltwlin 7876  ax-pre-lttrn 7877  ax-pre-apti 7878  ax-pre-ltadd 7879  ax-pre-mulgt0 7880  ax-pre-mulext 7881  ax-arch 7882  ax-caucvg 7883
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-isom 5205  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-recs 6282  df-frec 6368  df-1o 6393  df-2o 6394  df-er 6510  df-en 6716  df-sup 6958  df-inf 6959  df-pnf 7945  df-mnf 7946  df-xr 7947  df-ltxr 7948  df-le 7949  df-sub 8081  df-neg 8082  df-reap 8483  df-ap 8490  df-div 8579  df-inn 8868  df-2 8926  df-3 8927  df-4 8928  df-n0 9125  df-z 9202  df-uz 9477  df-q 9568  df-rp 9600  df-fz 9955  df-fzo 10088  df-fl 10215  df-mod 10268  df-seqfrec 10391  df-exp 10465  df-cj 10795  df-re 10796  df-im 10797  df-rsqrt 10951  df-abs 10952  df-dvds 11739  df-gcd 11887  df-prm 12051  df-pc 12228
This theorem is referenced by:  pcqmul  12246  pcaddlem  12281  pcmpt  12284  pcfac  12291  pcbc  12292  lgsdi  13693
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