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Theorem pcfaclem 14071
Description: Lemma for pcfac 14072. (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 10709 . . . 4  |-  ( N  e.  NN0  ->  0  <_  N )
213ad2ant1 1009 . . 3  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  0  <_  N )
3 nn0re 10692 . . . . 5  |-  ( N  e.  NN0  ->  N  e.  RR )
433ad2ant1 1009 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  N  e.  RR )
5 prmnn 13877 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  NN )
653ad2ant3 1011 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  P  e.  NN )
7 eluznn0 11028 . . . . . . 7  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N ) )  ->  M  e.  NN0 )
873adant3 1008 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  M  e.  NN0 )
96, 8nnexpcld 12139 . . . . 5  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( P ^ M )  e.  NN )
109nnred 10441 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( P ^ M )  e.  RR )
119nngt0d 10469 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  0  <  ( P ^ M
) )
12 ge0div 10300 . . . 4  |-  ( ( N  e.  RR  /\  ( P ^ M )  e.  RR  /\  0  <  ( P ^ M
) )  ->  (
0  <_  N  <->  0  <_  ( N  /  ( P ^ M ) ) ) )
134, 10, 11, 12syl3anc 1219 . . 3  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  (
0  <_  N  <->  0  <_  ( N  /  ( P ^ M ) ) ) )
142, 13mpbid 210 . 2  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  0  <_  ( N  /  ( P ^ M ) ) )
158nn0red 10741 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  M  e.  RR )
16 eluzle 10977 . . . . . . 7  |-  ( M  e.  ( ZZ>= `  N
)  ->  N  <_  M )
17163ad2ant2 1010 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  N  <_  M )
18 prmuz2 13892 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
19183ad2ant3 1011 . . . . . . 7  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  P  e.  ( ZZ>= `  2 )
)
20 bernneq3 12102 . . . . . . 7  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  M  e.  NN0 )  ->  M  <  ( P ^ M
) )
2119, 8, 20syl2anc 661 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  M  <  ( P ^ M
) )
224, 15, 10, 17, 21lelttrd 9633 . . . . 5  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  N  <  ( P ^ M
) )
239nncnd 10442 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( P ^ M )  e.  CC )
2423mulid1d 9507 . . . . 5  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  (
( P ^ M
)  x.  1 )  =  ( P ^ M ) )
2522, 24breqtrrd 4419 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  N  <  ( ( P ^ M )  x.  1 ) )
26 1red 9505 . . . . 5  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  1  e.  RR )
27 ltdivmul 10308 . . . . 5  |-  ( ( N  e.  RR  /\  1  e.  RR  /\  (
( P ^ M
)  e.  RR  /\  0  <  ( P ^ M ) ) )  ->  ( ( N  /  ( P ^ M ) )  <  1  <->  N  <  ( ( P ^ M )  x.  1 ) ) )
284, 26, 10, 11, 27syl112anc 1223 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  (
( N  /  ( P ^ M ) )  <  1  <->  N  <  ( ( P ^ M
)  x.  1 ) ) )
2925, 28mpbird 232 . . 3  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( N  /  ( P ^ M ) )  <  1 )
30 0p1e1 10537 . . 3  |-  ( 0  +  1 )  =  1
3129, 30syl6breqr 4433 . 2  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( N  /  ( P ^ M ) )  < 
( 0  +  1 ) )
324, 9nndivred 10474 . . 3  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( N  /  ( P ^ M ) )  e.  RR )
33 0z 10761 . . 3  |-  0  e.  ZZ
34 flbi 11774 . . 3  |-  ( ( ( N  /  ( P ^ M ) )  e.  RR  /\  0  e.  ZZ )  ->  (
( |_ `  ( N  /  ( P ^ M ) ) )  =  0  <->  ( 0  <_  ( N  / 
( P ^ M
) )  /\  ( N  /  ( P ^ M ) )  < 
( 0  +  1 ) ) ) )
3532, 33, 34sylancl 662 . 2  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  (
( |_ `  ( N  /  ( P ^ M ) ) )  =  0  <->  ( 0  <_  ( N  / 
( P ^ M
) )  /\  ( N  /  ( P ^ M ) )  < 
( 0  +  1 ) ) ) )
3614, 31, 35mpbir2and 913 1  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( |_ `  ( N  / 
( P ^ M
) ) )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4393   ` cfv 5519  (class class class)co 6193   RRcr 9385   0cc0 9386   1c1 9387    + caddc 9389    x. cmul 9391    < clt 9522    <_ cle 9523    / cdiv 10097   NNcn 10426   2c2 10475   NN0cn0 10683   ZZcz 10750   ZZ>=cuz 10965   |_cfl 11750   ^cexp 11975   Primecprime 13874
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 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-sup 7795  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-n0 10684  df-z 10751  df-uz 10966  df-fl 11752  df-seq 11917  df-exp 11976  df-dvds 13647  df-prm 13875
This theorem is referenced by:  pcfac  14072
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