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

Theorem pcgcd 13243
Description: The prime count of a GCD is the minimum of the prime counts of the arguments. (Contributed by Mario Carneiro, 3-Oct-2014.)
Assertion
Ref Expression
pcgcd  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( P  pCnt  ( A  gcd  B ) )  =  if ( ( P  pCnt  A )  <_  ( P  pCnt  B ) ,  ( P  pCnt  A ) ,  ( P  pCnt  B ) ) )

Proof of Theorem pcgcd
StepHypRef Expression
1 pcgcd1 13242 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( P  pCnt  A
)  <_  ( P  pCnt  B ) )  -> 
( P  pCnt  ( A  gcd  B ) )  =  ( P  pCnt  A ) )
2 iftrue 3737 . . . 4  |-  ( ( P  pCnt  A )  <_  ( P  pCnt  B
)  ->  if (
( P  pCnt  A
)  <_  ( P  pCnt  B ) ,  ( P  pCnt  A ) ,  ( P  pCnt  B ) )  =  ( P  pCnt  A )
)
32adantl 453 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( P  pCnt  A
)  <_  ( P  pCnt  B ) )  ->  if ( ( P  pCnt  A )  <_  ( P  pCnt  B ) ,  ( P  pCnt  A ) ,  ( P  pCnt  B ) )  =  ( P  pCnt  A )
)
41, 3eqtr4d 2470 . 2  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( P  pCnt  A
)  <_  ( P  pCnt  B ) )  -> 
( P  pCnt  ( A  gcd  B ) )  =  if ( ( P  pCnt  A )  <_  ( P  pCnt  B
) ,  ( P 
pCnt  A ) ,  ( P  pCnt  B )
) )
5 gcdcom 13012 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  =  ( B  gcd  A ) )
653adant1 975 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B )  =  ( B  gcd  A
) )
76adantr 452 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( P  pCnt  A )  <_  ( P  pCnt  B ) )  -> 
( A  gcd  B
)  =  ( B  gcd  A ) )
87oveq2d 6089 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( P  pCnt  A )  <_  ( P  pCnt  B ) )  -> 
( P  pCnt  ( A  gcd  B ) )  =  ( P  pCnt  ( B  gcd  A ) ) )
9 iffalse 3738 . . . . 5  |-  ( -.  ( P  pCnt  A
)  <_  ( P  pCnt  B )  ->  if ( ( P  pCnt  A )  <_  ( P  pCnt  B ) ,  ( P  pCnt  A ) ,  ( P  pCnt  B ) )  =  ( P  pCnt  B )
)
109adantl 453 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( P  pCnt  A )  <_  ( P  pCnt  B ) )  ->  if ( ( P  pCnt  A )  <_  ( P  pCnt  B ) ,  ( P  pCnt  A ) ,  ( P  pCnt  B ) )  =  ( P  pCnt  B )
)
11 zq 10572 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  A  e.  QQ )
12 pcxcl 13226 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  QQ )  ->  ( P  pCnt  A )  e. 
RR* )
1311, 12sylan2 461 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  pCnt  A )  e. 
RR* )
14133adant3 977 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( P  pCnt  A )  e. 
RR* )
15 zq 10572 . . . . . . . . 9  |-  ( B  e.  ZZ  ->  B  e.  QQ )
16 pcxcl 13226 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  B  e.  QQ )  ->  ( P  pCnt  B )  e. 
RR* )
1715, 16sylan2 461 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  B  e.  ZZ )  ->  ( P  pCnt  B )  e. 
RR* )
18173adant2 976 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( P  pCnt  B )  e. 
RR* )
19 xrletri 10736 . . . . . . 7  |-  ( ( ( P  pCnt  A
)  e.  RR*  /\  ( P  pCnt  B )  e. 
RR* )  ->  (
( P  pCnt  A
)  <_  ( P  pCnt  B )  \/  ( P  pCnt  B )  <_ 
( P  pCnt  A
) ) )
2014, 18, 19syl2anc 643 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( P  pCnt  A
)  <_  ( P  pCnt  B )  \/  ( P  pCnt  B )  <_ 
( P  pCnt  A
) ) )
2120orcanai 880 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( P  pCnt  A )  <_  ( P  pCnt  B ) )  -> 
( P  pCnt  B
)  <_  ( P  pCnt  A ) )
22 3ancomb 945 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  <->  ( P  e.  Prime  /\  B  e.  ZZ  /\  A  e.  ZZ ) )
23 pcgcd1 13242 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  B  e.  ZZ  /\  A  e.  ZZ )  /\  ( P  pCnt  B
)  <_  ( P  pCnt  A ) )  -> 
( P  pCnt  ( B  gcd  A ) )  =  ( P  pCnt  B ) )
2422, 23sylanb 459 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( P  pCnt  B
)  <_  ( P  pCnt  A ) )  -> 
( P  pCnt  ( B  gcd  A ) )  =  ( P  pCnt  B ) )
2521, 24syldan 457 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( P  pCnt  A )  <_  ( P  pCnt  B ) )  -> 
( P  pCnt  ( B  gcd  A ) )  =  ( P  pCnt  B ) )
2610, 25eqtr4d 2470 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( P  pCnt  A )  <_  ( P  pCnt  B ) )  ->  if ( ( P  pCnt  A )  <_  ( P  pCnt  B ) ,  ( P  pCnt  A ) ,  ( P  pCnt  B ) )  =  ( P  pCnt  ( B  gcd  A ) ) )
278, 26eqtr4d 2470 . 2  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( P  pCnt  A )  <_  ( P  pCnt  B ) )  -> 
( P  pCnt  ( A  gcd  B ) )  =  if ( ( P  pCnt  A )  <_  ( P  pCnt  B
) ,  ( P 
pCnt  A ) ,  ( P  pCnt  B )
) )
284, 27pm2.61dan 767 1  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( P  pCnt  ( A  gcd  B ) )  =  if ( ( P  pCnt  A )  <_  ( P  pCnt  B ) ,  ( P  pCnt  A ) ,  ( P  pCnt  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ifcif 3731   class class class wbr 4204  (class class class)co 6073   RR*cxr 9111    <_ cle 9113   ZZcz 10274   QQcq 10566    gcd cgcd 12998   Primecprime 13071    pCnt cpc 13202
This theorem is referenced by:  pc2dvds  13244  mumullem2  20955
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845  df-gcd 12999  df-prm 13072  df-pc 13203
  Copyright terms: Public domain W3C validator