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Theorem pcgcd 12930
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 12929 . . 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 3571 . . . 4  |-  ( ( P  pCnt  A )  <_  ( P  pCnt  B
)  ->  if (
( P  pCnt  A
)  <_  ( P  pCnt  B ) ,  ( P  pCnt  A ) ,  ( P  pCnt  B ) )  =  ( P  pCnt  A )
)
32adantl 452 . . 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 2318 . 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 12699 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B
)  =  ( B  gcd  A ) )
653adant1 973 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  B )  =  ( B  gcd  A
) )
76adantr 451 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( P  pCnt  A )  <_  ( P  pCnt  B ) )  -> 
( A  gcd  B
)  =  ( B  gcd  A ) )
87oveq2d 5874 . . 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 3572 . . . . 5  |-  ( -.  ( P  pCnt  A
)  <_  ( P  pCnt  B )  ->  if ( ( P  pCnt  A )  <_  ( P  pCnt  B ) ,  ( P  pCnt  A ) ,  ( P  pCnt  B ) )  =  ( P  pCnt  B )
)
109adantl 452 . . . 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 10322 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  A  e.  QQ )
12 pcxcl 12913 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  QQ )  ->  ( P  pCnt  A )  e. 
RR* )
1311, 12sylan2 460 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  pCnt  A )  e. 
RR* )
14133adant3 975 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( P  pCnt  A )  e. 
RR* )
15 zq 10322 . . . . . . . . 9  |-  ( B  e.  ZZ  ->  B  e.  QQ )
16 pcxcl 12913 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  B  e.  QQ )  ->  ( P  pCnt  B )  e. 
RR* )
1715, 16sylan2 460 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  B  e.  ZZ )  ->  ( P  pCnt  B )  e. 
RR* )
18173adant2 974 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( P  pCnt  B )  e. 
RR* )
19 xrletri 10485 . . . . . . 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 642 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( P  pCnt  A
)  <_  ( P  pCnt  B )  \/  ( P  pCnt  B )  <_ 
( P  pCnt  A
) ) )
2120orcanai 879 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( P  pCnt  A )  <_  ( P  pCnt  B ) )  -> 
( P  pCnt  B
)  <_  ( P  pCnt  A ) )
22 3ancomb 943 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  <->  ( P  e.  Prime  /\  B  e.  ZZ  /\  A  e.  ZZ ) )
23 pcgcd1 12929 . . . . . 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 458 . . . . 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 456 . . . 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 2318 . . 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 2318 . 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 766 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 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ifcif 3565   class class class wbr 4023  (class class class)co 5858   RR*cxr 8866    <_ cle 8868   ZZcz 10024   QQcq 10316    gcd cgcd 12685   Primecprime 12758    pCnt cpc 12889
This theorem is referenced by:  pc2dvds  12931  mumullem2  20418
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890
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