MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  effsumlt Unicode version

Theorem effsumlt 12391
Description: The partial sums of the series expansion of the exponential function of a positive real number are bounded by the value of the function. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 29-Apr-2014.)
Hypotheses
Ref Expression
effsumlt.1  |-  F  =  ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) )
effsumlt.2  |-  ( ph  ->  A  e.  RR+ )
effsumlt.3  |-  ( ph  ->  N  e.  NN0 )
Assertion
Ref Expression
effsumlt  |-  ( ph  ->  (  seq  0 (  +  ,  F ) `
 N )  < 
( exp `  A
) )
Distinct variable group:    A, n
Allowed substitution hints:    ph( n)    F( n)    N( n)

Proof of Theorem effsumlt
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 nn0uz 10262 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
2 0z 10035 . . . . . 6  |-  0  e.  ZZ
32a1i 10 . . . . 5  |-  ( ph  ->  0  e.  ZZ )
4 effsumlt.1 . . . . . . . 8  |-  F  =  ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) )
54eftval 12358 . . . . . . 7  |-  ( k  e.  NN0  ->  ( F `
 k )  =  ( ( A ^
k )  /  ( ! `  k )
) )
65adantl 452 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  ( ( A ^ k
)  /  ( ! `
 k ) ) )
7 effsumlt.2 . . . . . . . 8  |-  ( ph  ->  A  e.  RR+ )
87rpred 10390 . . . . . . 7  |-  ( ph  ->  A  e.  RR )
9 reeftcl 12356 . . . . . . 7  |-  ( ( A  e.  RR  /\  k  e.  NN0 )  -> 
( ( A ^
k )  /  ( ! `  k )
)  e.  RR )
108, 9sylan 457 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( A ^ k )  / 
( ! `  k
) )  e.  RR )
116, 10eqeltrd 2357 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  e.  RR )
121, 3, 11serfre 11075 . . . 4  |-  ( ph  ->  seq  0 (  +  ,  F ) : NN0 --> RR )
13 effsumlt.3 . . . 4  |-  ( ph  ->  N  e.  NN0 )
14 ffvelrn 5663 . . . 4  |-  ( (  seq  0 (  +  ,  F ) : NN0 --> RR  /\  N  e.  NN0 )  ->  (  seq  0 (  +  ,  F ) `  N
)  e.  RR )
1512, 13, 14syl2anc 642 . . 3  |-  ( ph  ->  (  seq  0 (  +  ,  F ) `
 N )  e.  RR )
16 eqid 2283 . . . 4  |-  ( ZZ>= `  ( N  +  1
) )  =  (
ZZ>= `  ( N  + 
1 ) )
17 peano2nn0 10004 . . . . 5  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
1813, 17syl 15 . . . 4  |-  ( ph  ->  ( N  +  1 )  e.  NN0 )
19 eqidd 2284 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  ( F `  k ) )
20 nn0z 10046 . . . . . . 7  |-  ( k  e.  NN0  ->  k  e.  ZZ )
21 rpexpcl 11122 . . . . . . 7  |-  ( ( A  e.  RR+  /\  k  e.  ZZ )  ->  ( A ^ k )  e.  RR+ )
227, 20, 21syl2an 463 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( A ^ k )  e.  RR+ )
23 faccl 11298 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
2423adantl 452 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ! `  k )  e.  NN )
2524nnrpd 10389 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ! `  k )  e.  RR+ )
2622, 25rpdivcld 10407 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( A ^ k )  / 
( ! `  k
) )  e.  RR+ )
276, 26eqeltrd 2357 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  e.  RR+ )
288recnd 8861 . . . . 5  |-  ( ph  ->  A  e.  CC )
294efcllem 12359 . . . . 5  |-  ( A  e.  CC  ->  seq  0 (  +  ,  F )  e.  dom  ~~>  )
3028, 29syl 15 . . . 4  |-  ( ph  ->  seq  0 (  +  ,  F )  e. 
dom 
~~>  )
311, 16, 18, 19, 27, 30isumrpcl 12302 . . 3  |-  ( ph  -> 
sum_ k  e.  (
ZZ>= `  ( N  + 
1 ) ) ( F `  k )  e.  RR+ )
3215, 31ltaddrpd 10419 . 2  |-  ( ph  ->  (  seq  0 (  +  ,  F ) `
 N )  < 
( (  seq  0
(  +  ,  F
) `  N )  +  sum_ k  e.  (
ZZ>= `  ( N  + 
1 ) ) ( F `  k ) ) )
334efval2 12365 . . . 4  |-  ( A  e.  CC  ->  ( exp `  A )  = 
sum_ k  e.  NN0  ( F `  k ) )
3428, 33syl 15 . . 3  |-  ( ph  ->  ( exp `  A
)  =  sum_ k  e.  NN0  ( F `  k ) )
3511recnd 8861 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  e.  CC )
361, 16, 18, 19, 35, 30isumsplit 12299 . . 3  |-  ( ph  -> 
sum_ k  e.  NN0  ( F `  k )  =  ( sum_ k  e.  ( 0 ... (
( N  +  1 )  -  1 ) ) ( F `  k )  +  sum_ k  e.  ( ZZ>= `  ( N  +  1
) ) ( F `
 k ) ) )
3713nn0cnd 10020 . . . . . . . 8  |-  ( ph  ->  N  e.  CC )
38 ax-1cn 8795 . . . . . . . 8  |-  1  e.  CC
39 pncan 9057 . . . . . . . 8  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  + 
1 )  -  1 )  =  N )
4037, 38, 39sylancl 643 . . . . . . 7  |-  ( ph  ->  ( ( N  + 
1 )  -  1 )  =  N )
4140oveq2d 5874 . . . . . 6  |-  ( ph  ->  ( 0 ... (
( N  +  1 )  -  1 ) )  =  ( 0 ... N ) )
4241sumeq1d 12174 . . . . 5  |-  ( ph  -> 
sum_ k  e.  ( 0 ... ( ( N  +  1 )  -  1 ) ) ( F `  k
)  =  sum_ k  e.  ( 0 ... N
) ( F `  k ) )
43 eqidd 2284 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  ( F `  k )  =  ( F `  k ) )
4413, 1syl6eleq 2373 . . . . . 6  |-  ( ph  ->  N  e.  ( ZZ>= ` 
0 ) )
45 elfznn0 10822 . . . . . . 7  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
4645, 35sylan2 460 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  ( F `  k )  e.  CC )
4743, 44, 46fsumser 12203 . . . . 5  |-  ( ph  -> 
sum_ k  e.  ( 0 ... N ) ( F `  k
)  =  (  seq  0 (  +  ,  F ) `  N
) )
4842, 47eqtrd 2315 . . . 4  |-  ( ph  -> 
sum_ k  e.  ( 0 ... ( ( N  +  1 )  -  1 ) ) ( F `  k
)  =  (  seq  0 (  +  ,  F ) `  N
) )
4948oveq1d 5873 . . 3  |-  ( ph  ->  ( sum_ k  e.  ( 0 ... ( ( N  +  1 )  -  1 ) ) ( F `  k
)  +  sum_ k  e.  ( ZZ>= `  ( N  +  1 ) ) ( F `  k
) )  =  ( (  seq  0 (  +  ,  F ) `
 N )  + 
sum_ k  e.  (
ZZ>= `  ( N  + 
1 ) ) ( F `  k ) ) )
5034, 36, 493eqtrd 2319 . 2  |-  ( ph  ->  ( exp `  A
)  =  ( (  seq  0 (  +  ,  F ) `  N )  +  sum_ k  e.  ( ZZ>= `  ( N  +  1
) ) ( F `
 k ) ) )
5132, 50breqtrrd 4049 1  |-  ( ph  ->  (  seq  0 (  +  ,  F ) `
 N )  < 
( exp `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   class class class wbr 4023    e. cmpt 4077   dom cdm 4689   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    < clt 8867    - cmin 9037    / cdiv 9423   NNcn 9746   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   RR+crp 10354   ...cfz 10782    seq cseq 11046   ^cexp 11104   !cfa 11288    ~~> cli 11958   sum_csu 12158   expce 12343
This theorem is referenced by:  efgt1p2  12394  efgt1p  12395
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  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  ax-addf 8816  ax-mulf 8817
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-int 3863  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-se 4353  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-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  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-ico 10662  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-fac 11289  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349
  Copyright terms: Public domain W3C validator