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Theorem pcpre1 12895
Description: Value of the prime power pre-function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 26-Apr-2016.)
Hypotheses
Ref Expression
pclem.1  |-  A  =  { n  e.  NN0  |  ( P ^ n
)  ||  N }
pclem.2  |-  S  =  sup ( A ,  RR ,  <  )
Assertion
Ref Expression
pcpre1  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  N  =  1 )  ->  S  =  0 )
Distinct variable groups:    n, N    P, n
Allowed substitution hints:    A( n)    S( n)

Proof of Theorem pcpre1
StepHypRef Expression
1 1z 10053 . . . . . . . . . 10  |-  1  e.  ZZ
2 eleq1 2343 . . . . . . . . . 10  |-  ( N  =  1  ->  ( N  e.  ZZ  <->  1  e.  ZZ ) )
31, 2mpbiri 224 . . . . . . . . 9  |-  ( N  =  1  ->  N  e.  ZZ )
4 ax-1ne0 8806 . . . . . . . . . 10  |-  1  =/=  0
5 neeq1 2454 . . . . . . . . . 10  |-  ( N  =  1  ->  ( N  =/=  0  <->  1  =/=  0 ) )
64, 5mpbiri 224 . . . . . . . . 9  |-  ( N  =  1  ->  N  =/=  0 )
73, 6jca 518 . . . . . . . 8  |-  ( N  =  1  ->  ( N  e.  ZZ  /\  N  =/=  0 ) )
8 pclem.1 . . . . . . . . 9  |-  A  =  { n  e.  NN0  |  ( P ^ n
)  ||  N }
9 pclem.2 . . . . . . . . 9  |-  S  =  sup ( A ,  RR ,  <  )
108, 9pcprecl 12892 . . . . . . . 8  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  e.  NN0  /\  ( P ^ S
)  ||  N )
)
117, 10sylan2 460 . . . . . . 7  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  N  =  1 )  -> 
( S  e.  NN0  /\  ( P ^ S
)  ||  N )
)
1211simprd 449 . . . . . 6  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  N  =  1 )  -> 
( P ^ S
)  ||  N )
13 simpr 447 . . . . . 6  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  N  =  1 )  ->  N  =  1 )
1412, 13breqtrd 4047 . . . . 5  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  N  =  1 )  -> 
( P ^ S
)  ||  1 )
15 eluz2b2 10290 . . . . . . . . . 10  |-  ( P  e.  ( ZZ>= `  2
)  <->  ( P  e.  NN  /\  1  < 
P ) )
1615simplbi 446 . . . . . . . . 9  |-  ( P  e.  ( ZZ>= `  2
)  ->  P  e.  NN )
1716adantr 451 . . . . . . . 8  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  N  =  1 )  ->  P  e.  NN )
1811simpld 445 . . . . . . . 8  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  N  =  1 )  ->  S  e.  NN0 )
1917, 18nnexpcld 11266 . . . . . . 7  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  N  =  1 )  -> 
( P ^ S
)  e.  NN )
2019nnzd 10116 . . . . . 6  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  N  =  1 )  -> 
( P ^ S
)  e.  ZZ )
21 1nn 9757 . . . . . 6  |-  1  e.  NN
22 dvdsle 12574 . . . . . 6  |-  ( ( ( P ^ S
)  e.  ZZ  /\  1  e.  NN )  ->  ( ( P ^ S )  ||  1  ->  ( P ^ S
)  <_  1 ) )
2320, 21, 22sylancl 643 . . . . 5  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  N  =  1 )  -> 
( ( P ^ S )  ||  1  ->  ( P ^ S
)  <_  1 ) )
2414, 23mpd 14 . . . 4  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  N  =  1 )  -> 
( P ^ S
)  <_  1 )
2517nncnd 9762 . . . . 5  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  N  =  1 )  ->  P  e.  CC )
2625exp0d 11239 . . . 4  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  N  =  1 )  -> 
( P ^ 0 )  =  1 )
2724, 26breqtrrd 4049 . . 3  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  N  =  1 )  -> 
( P ^ S
)  <_  ( P ^ 0 ) )
2817nnred 9761 . . . 4  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  N  =  1 )  ->  P  e.  RR )
2918nn0zd 10115 . . . 4  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  N  =  1 )  ->  S  e.  ZZ )
30 0z 10035 . . . . 5  |-  0  e.  ZZ
3130a1i 10 . . . 4  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  N  =  1 )  -> 
0  e.  ZZ )
3215simprbi 450 . . . . 5  |-  ( P  e.  ( ZZ>= `  2
)  ->  1  <  P )
3332adantr 451 . . . 4  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  N  =  1 )  -> 
1  <  P )
3428, 29, 31, 33leexp2d 11275 . . 3  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  N  =  1 )  -> 
( S  <_  0  <->  ( P ^ S )  <_  ( P ^
0 ) ) )
3527, 34mpbird 223 . 2  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  N  =  1 )  ->  S  <_  0 )
3610simpld 445 . . . 4  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  S  e.  NN0 )
377, 36sylan2 460 . . 3  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  N  =  1 )  ->  S  e.  NN0 )
38 nn0le0eq0 9994 . . 3  |-  ( S  e.  NN0  ->  ( S  <_  0  <->  S  = 
0 ) )
3937, 38syl 15 . 2  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  N  =  1 )  -> 
( S  <_  0  <->  S  =  0 ) )
4035, 39mpbid 201 1  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  N  =  1 )  ->  S  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   supcsup 7193   RRcr 8736   0cc0 8737   1c1 8738    < clt 8867    <_ cle 8868   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   ^cexp 11104    || cdivides 12531
This theorem is referenced by:  pczpre  12900  pc1  12908
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fl 10925  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532
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