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

Theorem pcfaclem 12962
Description: Lemma for pcfac 12963. (Contributed by Mario Carneiro, 20-May-2014.)
Assertion
Ref Expression
pcfaclem  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( |_ `  ( N  / 
( P ^ M
) ) )  =  0 )

Proof of Theorem pcfaclem
StepHypRef Expression
1 nn0ge0 10007 . . . 4  |-  ( N  e.  NN0  ->  0  <_  N )
213ad2ant1 976 . . 3  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  0  <_  N )
3 nn0re 9990 . . . . 5  |-  ( N  e.  NN0  ->  N  e.  RR )
433ad2ant1 976 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  N  e.  RR )
5 prmnn 12777 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  NN )
653ad2ant3 978 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  P  e.  NN )
7 eluznn0 10304 . . . . . . 7  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N ) )  ->  M  e.  NN0 )
873adant3 975 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  M  e.  NN0 )
96, 8nnexpcld 11282 . . . . 5  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( P ^ M )  e.  NN )
109nnred 9777 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( P ^ M )  e.  RR )
119nngt0d 9805 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  0  <  ( P ^ M
) )
12 ge0div 9639 . . . 4  |-  ( ( N  e.  RR  /\  ( P ^ M )  e.  RR  /\  0  <  ( P ^ M
) )  ->  (
0  <_  N  <->  0  <_  ( N  /  ( P ^ M ) ) ) )
134, 10, 11, 12syl3anc 1182 . . 3  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  (
0  <_  N  <->  0  <_  ( N  /  ( P ^ M ) ) ) )
142, 13mpbid 201 . 2  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  0  <_  ( N  /  ( P ^ M ) ) )
158nn0red 10035 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  M  e.  RR )
16 eluzle 10256 . . . . . . 7  |-  ( M  e.  ( ZZ>= `  N
)  ->  N  <_  M )
17163ad2ant2 977 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  N  <_  M )
18 prmuz2 12792 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
19183ad2ant3 978 . . . . . . 7  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  P  e.  ( ZZ>= `  2 )
)
20 bernneq3 11245 . . . . . . 7  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  M  e.  NN0 )  ->  M  <  ( P ^ M
) )
2119, 8, 20syl2anc 642 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  M  <  ( P ^ M
) )
224, 15, 10, 17, 21lelttrd 8990 . . . . 5  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  N  <  ( P ^ M
) )
239nncnd 9778 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( P ^ M )  e.  CC )
2423mulid1d 8868 . . . . 5  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  (
( P ^ M
)  x.  1 )  =  ( P ^ M ) )
2522, 24breqtrrd 4065 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  N  <  ( ( P ^ M )  x.  1 ) )
26 1re 8853 . . . . . 6  |-  1  e.  RR
2726a1i 10 . . . . 5  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  1  e.  RR )
28 ltdivmul 9644 . . . . 5  |-  ( ( N  e.  RR  /\  1  e.  RR  /\  (
( P ^ M
)  e.  RR  /\  0  <  ( P ^ M ) ) )  ->  ( ( N  /  ( P ^ M ) )  <  1  <->  N  <  ( ( P ^ M )  x.  1 ) ) )
294, 27, 10, 11, 28syl112anc 1186 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  (
( N  /  ( P ^ M ) )  <  1  <->  N  <  ( ( P ^ M
)  x.  1 ) ) )
3025, 29mpbird 223 . . 3  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( N  /  ( P ^ M ) )  <  1 )
31 0p1e1 9855 . . 3  |-  ( 0  +  1 )  =  1
3230, 31syl6breqr 4079 . 2  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( N  /  ( P ^ M ) )  < 
( 0  +  1 ) )
334, 9nndivred 9810 . . 3  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( N  /  ( P ^ M ) )  e.  RR )
34 0z 10051 . . 3  |-  0  e.  ZZ
35 flbi 10962 . . 3  |-  ( ( ( N  /  ( P ^ M ) )  e.  RR  /\  0  e.  ZZ )  ->  (
( |_ `  ( N  /  ( P ^ M ) ) )  =  0  <->  ( 0  <_  ( N  / 
( P ^ M
) )  /\  ( N  /  ( P ^ M ) )  < 
( 0  +  1 ) ) ) )
3633, 34, 35sylancl 643 . 2  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  (
( |_ `  ( N  /  ( P ^ M ) ) )  =  0  <->  ( 0  <_  ( N  / 
( P ^ M
) )  /\  ( N  /  ( P ^ M ) )  < 
( 0  +  1 ) ) ) )
3714, 32, 36mpbir2and 888 1  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( |_ `  ( N  / 
( P ^ M
) ) )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884    / cdiv 9439   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   |_cfl 10940   ^cexp 11120   Primecprime 12774
This theorem is referenced by:  pcfac  12963
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fl 10941  df-seq 11063  df-exp 11121  df-dvds 12548  df-prm 12775
  Copyright terms: Public domain W3C validator