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Theorem pcgcd 14063
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 14062 . . 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 3906 . . . 4  |-  ( ( P  pCnt  A )  <_  ( P  pCnt  B
)  ->  if (
( P  pCnt  A
)  <_  ( P  pCnt  B ) ,  ( P  pCnt  A ) ,  ( P  pCnt  B ) )  =  ( P  pCnt  A )
)
32adantl 466 . . 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 2498 . 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 13823 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  =  ( B  gcd  A ) )
653adant1 1006 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B )  =  ( B  gcd  A
) )
76adantr 465 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( P  pCnt  A )  <_  ( P  pCnt  B ) )  -> 
( A  gcd  B
)  =  ( B  gcd  A ) )
87oveq2d 6217 . . 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 3908 . . . . 5  |-  ( -.  ( P  pCnt  A
)  <_  ( P  pCnt  B )  ->  if ( ( P  pCnt  A )  <_  ( P  pCnt  B ) ,  ( P  pCnt  A ) ,  ( P  pCnt  B ) )  =  ( P  pCnt  B )
)
109adantl 466 . . . 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 11071 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  A  e.  QQ )
12 pcxcl 14046 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  QQ )  ->  ( P  pCnt  A )  e. 
RR* )
1311, 12sylan2 474 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  pCnt  A )  e. 
RR* )
14133adant3 1008 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( P  pCnt  A )  e. 
RR* )
15 zq 11071 . . . . . . . . 9  |-  ( B  e.  ZZ  ->  B  e.  QQ )
16 pcxcl 14046 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  B  e.  QQ )  ->  ( P  pCnt  B )  e. 
RR* )
1715, 16sylan2 474 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  B  e.  ZZ )  ->  ( P  pCnt  B )  e. 
RR* )
18173adant2 1007 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( P  pCnt  B )  e. 
RR* )
19 xrletri 11240 . . . . . . 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 661 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( P  pCnt  A
)  <_  ( P  pCnt  B )  \/  ( P  pCnt  B )  <_ 
( P  pCnt  A
) ) )
2120orcanai 904 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( P  pCnt  A )  <_  ( P  pCnt  B ) )  -> 
( P  pCnt  B
)  <_  ( P  pCnt  A ) )
22 3ancomb 974 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  <->  ( P  e.  Prime  /\  B  e.  ZZ  /\  A  e.  ZZ ) )
23 pcgcd1 14062 . . . . . 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 472 . . . . 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 470 . . . 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 2498 . . 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 2498 . 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 789 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   ifcif 3900   class class class wbr 4401  (class class class)co 6201   RR*cxr 9529    <_ cle 9531   ZZcz 10758   QQcq 11065    gcd cgcd 13809   Primecprime 13882    pCnt cpc 14022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-pre-sup 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-2o 7032  df-oadd 7035  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-sup 7803  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-div 10106  df-nn 10435  df-2 10492  df-3 10493  df-n0 10692  df-z 10759  df-uz 10974  df-q 11066  df-rp 11104  df-fl 11760  df-mod 11827  df-seq 11925  df-exp 11984  df-cj 12707  df-re 12708  df-im 12709  df-sqr 12843  df-abs 12844  df-dvds 13655  df-gcd 13810  df-prm 13883  df-pc 14023
This theorem is referenced by:  pc2dvds  14064  mumullem2  22652
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