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Theorem pcdvdstr 14061
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 10769 . . . . . . 7  |-  0  e.  ZZ
2 zq 11071 . . . . . . 7  |-  ( 0  e.  ZZ  ->  0  e.  QQ )
31, 2ax-mp 5 . . . . . 6  |-  0  e.  QQ
4 pcxcl 14046 . . . . . 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 11239 . . . . 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 6217 . . 3  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  ( P  pCnt  A )  =  ( P  pCnt  0 ) )
11 simplr3 1032 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  A  ||  B
)
129, 11eqbrtrrd 4423 . . . . 5  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  0  ||  B )
13 simplr2 1031 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  B  e.  ZZ )
14 0dvds 13672 . . . . . 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 6217 . . 3  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  ( P  pCnt  B )  =  ( P  pCnt  0 ) )
188, 10, 173brtr4d 4431 . 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 1030 . . . . 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 14048 . . . . 5  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 ) )  -> 
( P ^ ( P  pCnt  A ) ) 
||  A )
2319, 20, 21, 22syl12anc 1217 . . . 4  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  ( P ^ ( P  pCnt  A ) )  ||  A
)
24 simplr3 1032 . . . 4  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  A  ||  B
)
25 prmnn 13885 . . . . . . . 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 14034 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 ) )  -> 
( P  pCnt  A
)  e.  NN0 )
2819, 20, 21, 27syl12anc 1217 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  ( P  pCnt  A )  e.  NN0 )
2926, 28nnexpcld 12147 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  ( P ^ ( P  pCnt  A ) )  e.  NN )
3029nnzd 10858 . . . . 5  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  ( P ^ ( P  pCnt  A ) )  e.  ZZ )
31 simplr2 1031 . . . . 5  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  B  e.  ZZ )
32 dvdstr 13685 . . . . 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 1219 . . . 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 14054 . . . 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 1219 . . 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 2770 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 965    = wceq 1370    e. wcel 1758    =/= wne 2648   class class class wbr 4401  (class class class)co 6201   0cc0 9394   RR*cxr 9529    <_ cle 9531   NNcn 10434   NN0cn0 10691   ZZcz 10758   QQcq 11065   ^cexp 11983    || cdivides 13654   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:  pcgcd1  14062  pc2dvds  14064  dvdsppwf1o  22660
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