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Theorem effsumlt 12407
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 10278 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
2 0z 10051 . . . . . 6  |-  0  e.  ZZ
32a1i 10 . . . . 5  |-  ( ph  ->  0  e.  ZZ )
4 effsumlt.1 . . . . . . . 8  |-  F  =  ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) )
54eftval 12374 . . . . . . 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 10406 . . . . . . 7  |-  ( ph  ->  A  e.  RR )
9 reeftcl 12372 . . . . . . 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 2370 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  e.  RR )
121, 3, 11serfre 11091 . . . 4  |-  ( ph  ->  seq  0 (  +  ,  F ) : NN0 --> RR )
13 effsumlt.3 . . . 4  |-  ( ph  ->  N  e.  NN0 )
14 ffvelrn 5679 . . . 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 2296 . . . 4  |-  ( ZZ>= `  ( N  +  1
) )  =  (
ZZ>= `  ( N  + 
1 ) )
17 peano2nn0 10020 . . . . 5  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
1813, 17syl 15 . . . 4  |-  ( ph  ->  ( N  +  1 )  e.  NN0 )
19 eqidd 2297 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  ( F `  k ) )
20 nn0z 10062 . . . . . . 7  |-  ( k  e.  NN0  ->  k  e.  ZZ )
21 rpexpcl 11138 . . . . . . 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 11314 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
2423adantl 452 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ! `  k )  e.  NN )
2524nnrpd 10405 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ! `  k )  e.  RR+ )
2622, 25rpdivcld 10423 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( A ^ k )  / 
( ! `  k
) )  e.  RR+ )
276, 26eqeltrd 2370 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  e.  RR+ )
288recnd 8877 . . . . 5  |-  ( ph  ->  A  e.  CC )
294efcllem 12375 . . . . 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 12318 . . 3  |-  ( ph  -> 
sum_ k  e.  (
ZZ>= `  ( N  + 
1 ) ) ( F `  k )  e.  RR+ )
3215, 31ltaddrpd 10435 . 2  |-  ( ph  ->  (  seq  0 (  +  ,  F ) `
 N )  < 
( (  seq  0
(  +  ,  F
) `  N )  +  sum_ k  e.  (
ZZ>= `  ( N  + 
1 ) ) ( F `  k ) ) )
334efval2 12381 . . . 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 8877 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  e.  CC )
361, 16, 18, 19, 35, 30isumsplit 12315 . . 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 10036 . . . . . . . 8  |-  ( ph  ->  N  e.  CC )
38 ax-1cn 8811 . . . . . . . 8  |-  1  e.  CC
39 pncan 9073 . . . . . . . 8  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  + 
1 )  -  1 )  =  N )
4037, 38, 39sylancl 643 . . . . . . 7  |-  ( ph  ->  ( ( N  + 
1 )  -  1 )  =  N )
4140oveq2d 5890 . . . . . 6  |-  ( ph  ->  ( 0 ... (
( N  +  1 )  -  1 ) )  =  ( 0 ... N ) )
4241sumeq1d 12190 . . . . 5  |-  ( ph  -> 
sum_ k  e.  ( 0 ... ( ( N  +  1 )  -  1 ) ) ( F `  k
)  =  sum_ k  e.  ( 0 ... N
) ( F `  k ) )
43 eqidd 2297 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  ( F `  k )  =  ( F `  k ) )
4413, 1syl6eleq 2386 . . . . . 6  |-  ( ph  ->  N  e.  ( ZZ>= ` 
0 ) )
45 elfznn0 10838 . . . . . . 7  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
4645, 35sylan2 460 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  ( F `  k )  e.  CC )
4743, 44, 46fsumser 12219 . . . . 5  |-  ( ph  -> 
sum_ k  e.  ( 0 ... N ) ( F `  k
)  =  (  seq  0 (  +  ,  F ) `  N
) )
4842, 47eqtrd 2328 . . . 4  |-  ( ph  -> 
sum_ k  e.  ( 0 ... ( ( N  +  1 )  -  1 ) ) ( F `  k
)  =  (  seq  0 (  +  ,  F ) `  N
) )
4948oveq1d 5889 . . 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 2332 . 2  |-  ( ph  ->  ( exp `  A
)  =  ( (  seq  0 (  +  ,  F ) `  N )  +  sum_ k  e.  ( ZZ>= `  ( N  +  1
) ) ( F `
 k ) ) )
5132, 50breqtrrd 4065 1  |-  ( ph  ->  (  seq  0 (  +  ,  F ) `
 N )  < 
( exp `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    < clt 8883    - cmin 9053    / cdiv 9439   NNcn 9762   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   RR+crp 10370   ...cfz 10798    seq cseq 11062   ^cexp 11120   !cfa 11304    ~~> cli 11974   sum_csu 12174   expce 12359
This theorem is referenced by:  efgt1p2  12410  efgt1p  12411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-ico 10678  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-fac 11305  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365
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