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Theorem pcdvdstr 14271
Description: The prime count increases under the divisibility relation. (Contributed by Mario Carneiro, 13-Mar-2014.)
Assertion
Ref Expression
pcdvdstr  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  -> 
( P  pCnt  A
)  <_  ( P  pCnt  B ) )

Proof of Theorem pcdvdstr
StepHypRef Expression
1 0z 10876 . . . . . . 7  |-  0  e.  ZZ
2 zq 11192 . . . . . . 7  |-  ( 0  e.  ZZ  ->  0  e.  QQ )
31, 2ax-mp 5 . . . . . 6  |-  0  e.  QQ
4 pcxcl 14256 . . . . . 6  |-  ( ( P  e.  Prime  /\  0  e.  QQ )  ->  ( P  pCnt  0 )  e. 
RR* )
53, 4mpan2 671 . . . . 5  |-  ( P  e.  Prime  ->  ( P 
pCnt  0 )  e. 
RR* )
6 xrleid 11360 . . . . 5  |-  ( ( P  pCnt  0 )  e.  RR*  ->  ( P 
pCnt  0 )  <_ 
( P  pCnt  0
) )
75, 6syl 16 . . . 4  |-  ( P  e.  Prime  ->  ( P 
pCnt  0 )  <_ 
( P  pCnt  0
) )
87ad2antrr 725 . . 3  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  ( P  pCnt  0 )  <_  ( P  pCnt  0 ) )
9 simpr 461 . . . 4  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  A  = 
0 )
109oveq2d 6293 . . 3  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  ( P  pCnt  A )  =  ( P  pCnt  0 ) )
11 simplr3 1039 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  A  ||  B
)
129, 11eqbrtrrd 4455 . . . . 5  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  0  ||  B )
13 simplr2 1038 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  B  e.  ZZ )
14 0dvds 13876 . . . . . 6  |-  ( B  e.  ZZ  ->  (
0  ||  B  <->  B  = 
0 ) )
1513, 14syl 16 . . . . 5  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  ( 0 
||  B  <->  B  = 
0 ) )
1612, 15mpbid 210 . . . 4  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  B  = 
0 )
1716oveq2d 6293 . . 3  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  ( P  pCnt  B )  =  ( P  pCnt  0 ) )
188, 10, 173brtr4d 4463 . 2  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  ( P  pCnt  A )  <_  ( P  pCnt  B ) )
19 simpll 753 . . . . 5  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  P  e.  Prime )
20 simplr1 1037 . . . . 5  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  A  e.  ZZ )
21 simpr 461 . . . . 5  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  A  =/=  0 )
22 pczdvds 14258 . . . . 5  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 ) )  -> 
( P ^ ( P  pCnt  A ) ) 
||  A )
2319, 20, 21, 22syl12anc 1225 . . . 4  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  ( P ^ ( P  pCnt  A ) )  ||  A
)
24 simplr3 1039 . . . 4  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  A  ||  B
)
25 prmnn 14092 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  NN )
2619, 25syl 16 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  P  e.  NN )
27 pczcl 14244 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 ) )  -> 
( P  pCnt  A
)  e.  NN0 )
2819, 20, 21, 27syl12anc 1225 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  ( P  pCnt  A )  e.  NN0 )
2926, 28nnexpcld 12305 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  ( P ^ ( P  pCnt  A ) )  e.  NN )
3029nnzd 10968 . . . . 5  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  ( P ^ ( P  pCnt  A ) )  e.  ZZ )
31 simplr2 1038 . . . . 5  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  B  e.  ZZ )
32 dvdstr 13890 . . . . 5  |-  ( ( ( P ^ ( P  pCnt  A ) )  e.  ZZ  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( ( P ^
( P  pCnt  A
) )  ||  A  /\  A  ||  B )  ->  ( P ^
( P  pCnt  A
) )  ||  B
) )
3330, 20, 31, 32syl3anc 1227 . . . 4  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  ( (
( P ^ ( P  pCnt  A ) ) 
||  A  /\  A  ||  B )  ->  ( P ^ ( P  pCnt  A ) )  ||  B
) )
3423, 24, 33mp2and 679 . . 3  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  ( P ^ ( P  pCnt  A ) )  ||  B
)
35 pcdvdsb 14264 . . . 4  |-  ( ( P  e.  Prime  /\  B  e.  ZZ  /\  ( P 
pCnt  A )  e.  NN0 )  ->  ( ( P 
pCnt  A )  <_  ( P  pCnt  B )  <->  ( P ^ ( P  pCnt  A ) )  ||  B
) )
3619, 31, 28, 35syl3anc 1227 . . 3  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  ( ( P  pCnt  A )  <_ 
( P  pCnt  B
)  <->  ( P ^
( P  pCnt  A
) )  ||  B
) )
3734, 36mpbird 232 . 2  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  ( P  pCnt  A )  <_  ( P  pCnt  B ) )
3818, 37pm2.61dane 2759 1  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  -> 
( P  pCnt  A
)  <_  ( P  pCnt  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802    =/= wne 2636   class class class wbr 4433  (class class class)co 6277   0cc0 9490   RR*cxr 9625    <_ cle 9627   NNcn 10537   NN0cn0 10796   ZZcz 10865   QQcq 11186   ^cexp 12140    || cdvds 13858   Primecprime 14089    pCnt cpc 14232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-2o 7129  df-oadd 7132  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-sup 7899  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11086  df-q 11187  df-rp 11225  df-fl 11903  df-mod 11971  df-seq 12082  df-exp 12141  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-dvds 13859  df-gcd 14017  df-prm 14090  df-pc 14233
This theorem is referenced by:  pcgcd1  14272  pc2dvds  14274  dvdsppwf1o  23327
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