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Theorem pcfaclem 14272
Description: Lemma for pcfac 14273. (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 10817 . . . 4  |-  ( N  e.  NN0  ->  0  <_  N )
213ad2ant1 1017 . . 3  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  0  <_  N )
3 nn0re 10800 . . . . 5  |-  ( N  e.  NN0  ->  N  e.  RR )
433ad2ant1 1017 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  N  e.  RR )
5 prmnn 14075 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  NN )
653ad2ant3 1019 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  P  e.  NN )
7 eluznn0 11147 . . . . . . 7  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N ) )  ->  M  e.  NN0 )
873adant3 1016 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  M  e.  NN0 )
96, 8nnexpcld 12295 . . . . 5  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( P ^ M )  e.  NN )
109nnred 10547 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( P ^ M )  e.  RR )
119nngt0d 10575 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  0  <  ( P ^ M
) )
12 ge0div 10405 . . . 4  |-  ( ( N  e.  RR  /\  ( P ^ M )  e.  RR  /\  0  <  ( P ^ M
) )  ->  (
0  <_  N  <->  0  <_  ( N  /  ( P ^ M ) ) ) )
134, 10, 11, 12syl3anc 1228 . . 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 10849 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  M  e.  RR )
16 eluzle 11090 . . . . . . 7  |-  ( M  e.  ( ZZ>= `  N
)  ->  N  <_  M )
17163ad2ant2 1018 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  N  <_  M )
18 prmuz2 14090 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
19183ad2ant3 1019 . . . . . . 7  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  P  e.  ( ZZ>= `  2 )
)
20 bernneq3 12258 . . . . . . 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 9735 . . . . 5  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  N  <  ( P ^ M
) )
239nncnd 10548 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( P ^ M )  e.  CC )
2423mulid1d 9609 . . . . 5  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  (
( P ^ M
)  x.  1 )  =  ( P ^ M ) )
2522, 24breqtrrd 4473 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  N  <  ( ( P ^ M )  x.  1 ) )
26 1red 9607 . . . . 5  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  1  e.  RR )
27 ltdivmul 10413 . . . . 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 1232 . . . 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 10643 . . 3  |-  ( 0  +  1 )  =  1
3129, 30syl6breqr 4487 . 2  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( N  /  ( P ^ M ) )  < 
( 0  +  1 ) )
324, 9nndivred 10580 . . 3  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( N  /  ( P ^ M ) )  e.  RR )
33 0z 10871 . . 3  |-  0  e.  ZZ
34 flbi 11916 . . 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 920 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 973    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   RRcr 9487   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493    < clt 9624    <_ cle 9625    / cdiv 10202   NNcn 10532   2c2 10581   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11078   |_cfl 11891   ^cexp 12130   Primecprime 14072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-fl 11893  df-seq 12072  df-exp 12131  df-dvds 13844  df-prm 14073
This theorem is referenced by:  pcfac  14273
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