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Theorem pcfaclem 14797
Description: Lemma for pcfac 14798. (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 10895 . . . 4  |-  ( N  e.  NN0  ->  0  <_  N )
213ad2ant1 1026 . . 3  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  0  <_  N )
3 nn0re 10878 . . . . 5  |-  ( N  e.  NN0  ->  N  e.  RR )
433ad2ant1 1026 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  N  e.  RR )
5 prmnn 14587 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  NN )
653ad2ant3 1028 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  P  e.  NN )
7 eluznn0 11228 . . . . . . 7  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N ) )  ->  M  e.  NN0 )
873adant3 1025 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  M  e.  NN0 )
96, 8nnexpcld 12434 . . . . 5  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( P ^ M )  e.  NN )
109nnred 10624 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( P ^ M )  e.  RR )
119nngt0d 10653 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  0  <  ( P ^ M
) )
12 ge0div 10471 . . . 4  |-  ( ( N  e.  RR  /\  ( P ^ M )  e.  RR  /\  0  <  ( P ^ M
) )  ->  (
0  <_  N  <->  0  <_  ( N  /  ( P ^ M ) ) ) )
134, 10, 11, 12syl3anc 1264 . . 3  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  (
0  <_  N  <->  0  <_  ( N  /  ( P ^ M ) ) ) )
142, 13mpbid 213 . 2  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  0  <_  ( N  /  ( P ^ M ) ) )
158nn0red 10926 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  M  e.  RR )
16 eluzle 11171 . . . . . . 7  |-  ( M  e.  ( ZZ>= `  N
)  ->  N  <_  M )
17163ad2ant2 1027 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  N  <_  M )
18 prmuz2 14604 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
19183ad2ant3 1028 . . . . . . 7  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  P  e.  ( ZZ>= `  2 )
)
20 bernneq3 12397 . . . . . . 7  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  M  e.  NN0 )  ->  M  <  ( P ^ M
) )
2119, 8, 20syl2anc 665 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  M  <  ( P ^ M
) )
224, 15, 10, 17, 21lelttrd 9792 . . . . 5  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  N  <  ( P ^ M
) )
239nncnd 10625 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( P ^ M )  e.  CC )
2423mulid1d 9659 . . . . 5  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  (
( P ^ M
)  x.  1 )  =  ( P ^ M ) )
2522, 24breqtrrd 4452 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  N  <  ( ( P ^ M )  x.  1 ) )
26 1red 9657 . . . . 5  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  1  e.  RR )
27 ltdivmul 10479 . . . . 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 1268 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  (
( N  /  ( P ^ M ) )  <  1  <->  N  <  ( ( P ^ M
)  x.  1 ) ) )
2925, 28mpbird 235 . . 3  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( N  /  ( P ^ M ) )  <  1 )
30 0p1e1 10721 . . 3  |-  ( 0  +  1 )  =  1
3129, 30syl6breqr 4466 . 2  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( N  /  ( P ^ M ) )  < 
( 0  +  1 ) )
324, 9nndivred 10658 . . 3  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( N  /  ( P ^ M ) )  e.  RR )
33 0z 10948 . . 3  |-  0  e.  ZZ
34 flbi 12048 . . 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 666 . 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 930 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   RRcr 9537   0cc0 9538   1c1 9539    + caddc 9541    x. cmul 9543    < clt 9674    <_ cle 9675    / cdiv 10268   NNcn 10609   2c2 10659   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   |_cfl 12023   ^cexp 12269   Primecprime 14584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fl 12025  df-seq 12211  df-exp 12270  df-dvds 14284  df-prm 14585
This theorem is referenced by:  pcfac  14798
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