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

Theorem pcdvdstr 14247
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 10864 . . . . . . 7  |-  0  e.  ZZ
2 zq 11177 . . . . . . 7  |-  ( 0  e.  ZZ  ->  0  e.  QQ )
31, 2ax-mp 5 . . . . . 6  |-  0  e.  QQ
4 pcxcl 14232 . . . . . 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 11345 . . . . 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 6291 . . 3  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  ( P  pCnt  A )  =  ( P  pCnt  0 ) )
11 simplr3 1035 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  A  ||  B
)
129, 11eqbrtrrd 4462 . . . . 5  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  0  ||  B )
13 simplr2 1034 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  B  e.  ZZ )
14 0dvds 13854 . . . . . 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 6291 . . 3  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  ( P  pCnt  B )  =  ( P  pCnt  0 ) )
188, 10, 173brtr4d 4470 . 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 1033 . . . . 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 14234 . . . . 5  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 ) )  -> 
( P ^ ( P  pCnt  A ) ) 
||  A )
2319, 20, 21, 22syl12anc 1221 . . . 4  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  ( P ^ ( P  pCnt  A ) )  ||  A
)
24 simplr3 1035 . . . 4  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  A  ||  B
)
25 prmnn 14068 . . . . . . . 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 14220 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 ) )  -> 
( P  pCnt  A
)  e.  NN0 )
2819, 20, 21, 27syl12anc 1221 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  ( P  pCnt  A )  e.  NN0 )
2926, 28nnexpcld 12286 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  ( P ^ ( P  pCnt  A ) )  e.  NN )
3029nnzd 10954 . . . . 5  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  ( P ^ ( P  pCnt  A ) )  e.  ZZ )
31 simplr2 1034 . . . . 5  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  B  e.  ZZ )
32 dvdstr 13867 . . . . 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 1223 . . . 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 14240 . . . 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 1223 . . 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 2778 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 968    = wceq 1374    e. wcel 1762    =/= wne 2655   class class class wbr 4440  (class class class)co 6275   0cc0 9481   RR*cxr 9616    <_ cle 9618   NNcn 10525   NN0cn0 10784   ZZcz 10853   QQcq 11171   ^cexp 12122    || cdivides 13836   Primecprime 14065    pCnt cpc 14208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  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 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-q 11172  df-rp 11210  df-fl 11886  df-mod 11953  df-seq 12064  df-exp 12123  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-dvds 13837  df-gcd 13993  df-prm 14066  df-pc 14209
This theorem is referenced by:  pcgcd1  14248  pc2dvds  14250  dvdsppwf1o  23183
  Copyright terms: Public domain W3C validator