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Theorem pcgcd 14485
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 14484 . . 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 3935 . . . 4  |-  ( ( P  pCnt  A )  <_  ( P  pCnt  B
)  ->  if (
( P  pCnt  A
)  <_  ( P  pCnt  B ) ,  ( P  pCnt  A ) ,  ( P  pCnt  B ) )  =  ( P  pCnt  A )
)
32adantl 464 . . 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 14242 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  =  ( B  gcd  A ) )
653adant1 1012 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B )  =  ( B  gcd  A
) )
76adantr 463 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( P  pCnt  A )  <_  ( P  pCnt  B ) )  -> 
( A  gcd  B
)  =  ( B  gcd  A ) )
87oveq2d 6286 . . 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 3938 . . . . 5  |-  ( -.  ( P  pCnt  A
)  <_  ( P  pCnt  B )  ->  if ( ( P  pCnt  A )  <_  ( P  pCnt  B ) ,  ( P  pCnt  A ) ,  ( P  pCnt  B ) )  =  ( P  pCnt  B )
)
109adantl 464 . . . 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 11189 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  A  e.  QQ )
12 pcxcl 14468 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  QQ )  ->  ( P  pCnt  A )  e. 
RR* )
1311, 12sylan2 472 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  pCnt  A )  e. 
RR* )
14133adant3 1014 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( P  pCnt  A )  e. 
RR* )
15 zq 11189 . . . . . . . . 9  |-  ( B  e.  ZZ  ->  B  e.  QQ )
16 pcxcl 14468 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  B  e.  QQ )  ->  ( P  pCnt  B )  e. 
RR* )
1715, 16sylan2 472 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  B  e.  ZZ )  ->  ( P  pCnt  B )  e. 
RR* )
18173adant2 1013 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( P  pCnt  B )  e. 
RR* )
19 xrletri 11360 . . . . . . 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 659 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( P  pCnt  A
)  <_  ( P  pCnt  B )  \/  ( P  pCnt  B )  <_ 
( P  pCnt  A
) ) )
2120orcanai 911 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( P  pCnt  A )  <_  ( P  pCnt  B ) )  -> 
( P  pCnt  B
)  <_  ( P  pCnt  A ) )
22 3ancomb 980 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  <->  ( P  e.  Prime  /\  B  e.  ZZ  /\  A  e.  ZZ ) )
23 pcgcd1 14484 . . . . . 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 470 . . . . 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 468 . . . 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 366    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   ifcif 3929   class class class wbr 4439  (class class class)co 6270   RR*cxr 9616    <_ cle 9618   ZZcz 10860   QQcq 11183    gcd cgcd 14228   Primecprime 14301    pCnt cpc 14444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-q 11184  df-rp 11222  df-fl 11910  df-mod 11979  df-seq 12090  df-exp 12149  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-dvds 14071  df-gcd 14229  df-prm 14302  df-pc 14445
This theorem is referenced by:  pc2dvds  14486  mumullem2  23652
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