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Theorem pcdvdstr 12273
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 9216 . . . . . . 7  |-  0  e.  ZZ
2 zq 9578 . . . . . . 7  |-  ( 0  e.  ZZ  ->  0  e.  QQ )
31, 2ax-mp 5 . . . . . 6  |-  0  e.  QQ
4 pcxcl 12258 . . . . . 6  |-  ( ( P  e.  Prime  /\  0  e.  QQ )  ->  ( P  pCnt  0 )  e. 
RR* )
53, 4mpan2 423 . . . . 5  |-  ( P  e.  Prime  ->  ( P 
pCnt  0 )  e. 
RR* )
65xrleidd 9751 . . . 4  |-  ( P  e.  Prime  ->  ( P 
pCnt  0 )  <_ 
( P  pCnt  0
) )
76ad2antrr 485 . . 3  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  ( P  pCnt  0 )  <_  ( P  pCnt  0 ) )
8 simpr 109 . . . 4  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  A  = 
0 )
98oveq2d 5867 . . 3  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  ( P  pCnt  A )  =  ( P  pCnt  0 ) )
10 simplr3 1036 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  A  ||  B
)
118, 10eqbrtrrd 4011 . . . . 5  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  0  ||  B )
12 simplr2 1035 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  B  e.  ZZ )
13 0dvds 11766 . . . . . 6  |-  ( B  e.  ZZ  ->  (
0  ||  B  <->  B  = 
0 ) )
1412, 13syl 14 . . . . 5  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  ( 0 
||  B  <->  B  = 
0 ) )
1511, 14mpbid 146 . . . 4  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  B  = 
0 )
1615oveq2d 5867 . . 3  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  ( P  pCnt  B )  =  ( P  pCnt  0 ) )
177, 9, 163brtr4d 4019 . 2  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =  0 )  ->  ( P  pCnt  A )  <_  ( P  pCnt  B ) )
18 prmnn 12057 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  NN )
1918ad2antrr 485 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  P  e.  NN )
20 simpll 524 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  P  e.  Prime )
21 simplr1 1034 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  A  e.  ZZ )
22 simpr 109 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  A  =/=  0 )
23 pczcl 12245 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 ) )  -> 
( P  pCnt  A
)  e.  NN0 )
2420, 21, 22, 23syl12anc 1231 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  ( P  pCnt  A )  e.  NN0 )
2519, 24nnexpcld 10624 . . . . 5  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  ( P ^ ( P  pCnt  A ) )  e.  NN )
2625nnzd 9326 . . . 4  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  ( P ^ ( P  pCnt  A ) )  e.  ZZ )
27 simplr2 1035 . . . 4  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  B  e.  ZZ )
28 pczdvds 12260 . . . . 5  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 ) )  -> 
( P ^ ( P  pCnt  A ) ) 
||  A )
2920, 21, 22, 28syl12anc 1231 . . . 4  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  ( P ^ ( P  pCnt  A ) )  ||  A
)
30 simplr3 1036 . . . 4  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  A  ||  B
)
3126, 21, 27, 29, 30dvdstrd 11785 . . 3  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  ( P ^ ( P  pCnt  A ) )  ||  B
)
32 pcdvdsb 12266 . . . 4  |-  ( ( P  e.  Prime  /\  B  e.  ZZ  /\  ( P 
pCnt  A )  e.  NN0 )  ->  ( ( P 
pCnt  A )  <_  ( P  pCnt  B )  <->  ( P ^ ( P  pCnt  A ) )  ||  B
) )
3320, 27, 24, 32syl3anc 1233 . . 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
) )
3431, 33mpbird 166 . 2  |-  ( ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  /\  A  =/=  0
)  ->  ( P  pCnt  A )  <_  ( P  pCnt  B ) )
35 simpr1 998 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  ->  A  e.  ZZ )
36 zdceq 9280 . . . 4  |-  ( ( A  e.  ZZ  /\  0  e.  ZZ )  -> DECID  A  =  0 )
3735, 1, 36sylancl 411 . . 3  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  -> DECID  A  =  0 )
38 dcne 2351 . . 3  |-  (DECID  A  =  0  <->  ( A  =  0  \/  A  =/=  0 ) )
3937, 38sylib 121 . 2  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  A  ||  B ) )  -> 
( A  =  0  \/  A  =/=  0
) )
4017, 34, 39mpjaodan 793 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    /\ wa 103    <-> wb 104    \/ wo 703  DECID wdc 829    /\ w3a 973    = wceq 1348    e. wcel 2141    =/= wne 2340   class class class wbr 3987  (class class class)co 5851   0cc0 7767   RR*cxr 7946    <_ cle 7948   NNcn 8871   NN0cn0 9128   ZZcz 9205   QQcq 9571   ^cexp 10468    || cdvds 11742   Primecprime 12054    pCnt cpc 12231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7858  ax-resscn 7859  ax-1cn 7860  ax-1re 7861  ax-icn 7862  ax-addcl 7863  ax-addrcl 7864  ax-mulcl 7865  ax-mulrcl 7866  ax-addcom 7867  ax-mulcom 7868  ax-addass 7869  ax-mulass 7870  ax-distr 7871  ax-i2m1 7872  ax-0lt1 7873  ax-1rid 7874  ax-0id 7875  ax-rnegex 7876  ax-precex 7877  ax-cnre 7878  ax-pre-ltirr 7879  ax-pre-ltwlin 7880  ax-pre-lttrn 7881  ax-pre-apti 7882  ax-pre-ltadd 7883  ax-pre-mulgt0 7884  ax-pre-mulext 7885  ax-arch 7886  ax-caucvg 7887
This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-isom 5205  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-recs 6282  df-frec 6368  df-1o 6393  df-2o 6394  df-er 6511  df-en 6717  df-sup 6959  df-inf 6960  df-pnf 7949  df-mnf 7950  df-xr 7951  df-ltxr 7952  df-le 7953  df-sub 8085  df-neg 8086  df-reap 8487  df-ap 8494  df-div 8583  df-inn 8872  df-2 8930  df-3 8931  df-4 8932  df-n0 9129  df-z 9206  df-uz 9481  df-q 9572  df-rp 9604  df-fz 9959  df-fzo 10092  df-fl 10219  df-mod 10272  df-seqfrec 10395  df-exp 10469  df-cj 10799  df-re 10800  df-im 10801  df-rsqrt 10955  df-abs 10956  df-dvds 11743  df-gcd 11891  df-prm 12055  df-pc 12232
This theorem is referenced by:  pcgcd1  12274  pc2dvds  12276
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