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Theorem pcdvdstr 13176
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 10225 . . . . . . 7  |-  0  e.  ZZ
2 zq 10512 . . . . . . 7  |-  ( 0  e.  ZZ  ->  0  e.  QQ )
31, 2ax-mp 8 . . . . . 6  |-  0  e.  QQ
4 pcxcl 13161 . . . . . 6  |-  ( ( P  e.  Prime  /\  0  e.  QQ )  ->  ( P  pCnt  0 )  e. 
RR* )
53, 4mpan2 653 . . . . 5  |-  ( P  e.  Prime  ->  ( P 
pCnt  0 )  e. 
RR* )
6 xrleid 10675 . . . . 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 707 . . 3  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  ( P  pCnt  0 )  <_  ( P  pCnt  0 ) )
9 simpr 448 . . . 4  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  A  = 
0 )
109oveq2d 6036 . . 3  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  ( P  pCnt  A )  =  ( P  pCnt  0 ) )
11 simplr3 1001 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  A  ||  B
)
129, 11eqbrtrrd 4175 . . . . 5  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  0  ||  B )
13 simplr2 1000 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  B  e.  ZZ )
14 0dvds 12797 . . . . . 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 202 . . . 4  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  B  = 
0 )
1716oveq2d 6036 . . 3  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  ( P  pCnt  B )  =  ( P  pCnt  0 ) )
188, 10, 173brtr4d 4183 . 2  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  ( P  pCnt  A )  <_  ( P  pCnt  B ) )
19 simpll 731 . . . . 5  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  P  e.  Prime )
20 simplr1 999 . . . . 5  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  A  e.  ZZ )
21 simpr 448 . . . . 5  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  A  =/=  0 )
22 pczdvds 13163 . . . . 5  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 ) )  -> 
( P ^ ( P  pCnt  A ) ) 
||  A )
2319, 20, 21, 22syl12anc 1182 . . . 4  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  ( P ^ ( P  pCnt  A ) )  ||  A
)
24 simplr3 1001 . . . 4  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  A  ||  B
)
25 prmnn 13009 . . . . . . . 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 13149 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 ) )  -> 
( P  pCnt  A
)  e.  NN0 )
2819, 20, 21, 27syl12anc 1182 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  ( P  pCnt  A )  e.  NN0 )
2926, 28nnexpcld 11471 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  ( P ^ ( P  pCnt  A ) )  e.  NN )
3029nnzd 10306 . . . . 5  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  ( P ^ ( P  pCnt  A ) )  e.  ZZ )
31 simplr2 1000 . . . . 5  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  B  e.  ZZ )
32 dvdstr 12810 . . . . 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 1184 . . . 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 661 . . 3  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  ( P ^ ( P  pCnt  A ) )  ||  B
)
35 pcdvdsb 13169 . . . 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 1184 . . 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 224 . 2  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  ( P  pCnt  A )  <_  ( P  pCnt  B ) )
3818, 37pm2.61dane 2628 1  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  -> 
( P  pCnt  A
)  <_  ( P  pCnt  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   class class class wbr 4153  (class class class)co 6020   0cc0 8923   RR*cxr 9052    <_ cle 9054   NNcn 9932   NN0cn0 10153   ZZcz 10214   QQcq 10506   ^cexp 11309    || cdivides 12779   Primecprime 13006    pCnt cpc 13137
This theorem is referenced by:  pcgcd1  13177  pc2dvds  13179  dvdsppwf1o  20838
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-q 10507  df-rp 10545  df-fl 11129  df-mod 11178  df-seq 11251  df-exp 11310  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-dvds 12780  df-gcd 12934  df-prm 13007  df-pc 13138
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