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Theorem pcfaclem 13267
Description: Lemma for pcfac 13268. (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 10247 . . . 4  |-  ( N  e.  NN0  ->  0  <_  N )
213ad2ant1 978 . . 3  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  0  <_  N )
3 nn0re 10230 . . . . 5  |-  ( N  e.  NN0  ->  N  e.  RR )
433ad2ant1 978 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  N  e.  RR )
5 prmnn 13082 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  NN )
653ad2ant3 980 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  P  e.  NN )
7 eluznn0 10546 . . . . . . 7  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N ) )  ->  M  e.  NN0 )
873adant3 977 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  M  e.  NN0 )
96, 8nnexpcld 11544 . . . . 5  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( P ^ M )  e.  NN )
109nnred 10015 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( P ^ M )  e.  RR )
119nngt0d 10043 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  0  <  ( P ^ M
) )
12 ge0div 9877 . . . 4  |-  ( ( N  e.  RR  /\  ( P ^ M )  e.  RR  /\  0  <  ( P ^ M
) )  ->  (
0  <_  N  <->  0  <_  ( N  /  ( P ^ M ) ) ) )
134, 10, 11, 12syl3anc 1184 . . 3  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  (
0  <_  N  <->  0  <_  ( N  /  ( P ^ M ) ) ) )
142, 13mpbid 202 . 2  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  0  <_  ( N  /  ( P ^ M ) ) )
158nn0red 10275 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  M  e.  RR )
16 eluzle 10498 . . . . . . 7  |-  ( M  e.  ( ZZ>= `  N
)  ->  N  <_  M )
17163ad2ant2 979 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  N  <_  M )
18 prmuz2 13097 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
19183ad2ant3 980 . . . . . . 7  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  P  e.  ( ZZ>= `  2 )
)
20 bernneq3 11507 . . . . . . 7  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  M  e.  NN0 )  ->  M  <  ( P ^ M
) )
2119, 8, 20syl2anc 643 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  M  <  ( P ^ M
) )
224, 15, 10, 17, 21lelttrd 9228 . . . . 5  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  N  <  ( P ^ M
) )
239nncnd 10016 . . . . . 6  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( P ^ M )  e.  CC )
2423mulid1d 9105 . . . . 5  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  (
( P ^ M
)  x.  1 )  =  ( P ^ M ) )
2522, 24breqtrrd 4238 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  N  <  ( ( P ^ M )  x.  1 ) )
26 1re 9090 . . . . . 6  |-  1  e.  RR
2726a1i 11 . . . . 5  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  1  e.  RR )
28 ltdivmul 9882 . . . . 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 1188 . . . 4  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  (
( N  /  ( P ^ M ) )  <  1  <->  N  <  ( ( P ^ M
)  x.  1 ) ) )
3025, 29mpbird 224 . . 3  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( N  /  ( P ^ M ) )  <  1 )
31 0p1e1 10093 . . 3  |-  ( 0  +  1 )  =  1
3230, 31syl6breqr 4252 . 2  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( N  /  ( P ^ M ) )  < 
( 0  +  1 ) )
334, 9nndivred 10048 . . 3  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  P  e.  Prime )  ->  ( N  /  ( P ^ M ) )  e.  RR )
34 0z 10293 . . 3  |-  0  e.  ZZ
35 flbi 11223 . . 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 644 . 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 889 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    < clt 9120    <_ cle 9121    / cdiv 9677   NNcn 10000   2c2 10049   NN0cn0 10221   ZZcz 10282   ZZ>=cuz 10488   |_cfl 11201   ^cexp 11382   Primecprime 13079
This theorem is referenced by:  pcfac  13268
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-fl 11202  df-seq 11324  df-exp 11383  df-dvds 12853  df-prm 13080
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