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Theorem effsumlt 11387
Description: The partial sums of the series expansion of the exponential function at 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 9353 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
2 0zd 9059 . . . . 5  |-  ( ph  ->  0  e.  ZZ )
3 effsumlt.2 . . . . . . . 8  |-  ( ph  ->  A  e.  RR+ )
43rpcnd 9478 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
5 effsumlt.1 . . . . . . . 8  |-  F  =  ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) )
65eftvalcn 11352 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( F `  k
)  =  ( ( A ^ k )  /  ( ! `  k ) ) )
74, 6sylan 281 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  ( ( A ^ k
)  /  ( ! `
 k ) ) )
83rpred 9476 . . . . . . 7  |-  ( ph  ->  A  e.  RR )
9 reeftcl 11350 . . . . . . 7  |-  ( ( A  e.  RR  /\  k  e.  NN0 )  -> 
( ( A ^
k )  /  ( ! `  k )
)  e.  RR )
108, 9sylan 281 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( A ^ k )  / 
( ! `  k
) )  e.  RR )
117, 10eqeltrd 2214 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  e.  RR )
121, 2, 11serfre 10241 . . . 4  |-  ( ph  ->  seq 0 (  +  ,  F ) : NN0 --> RR )
13 effsumlt.3 . . . 4  |-  ( ph  ->  N  e.  NN0 )
1412, 13ffvelrnd 5549 . . 3  |-  ( ph  ->  (  seq 0 (  +  ,  F ) `
 N )  e.  RR )
15 eqid 2137 . . . 4  |-  ( ZZ>= `  ( N  +  1
) )  =  (
ZZ>= `  ( N  + 
1 ) )
16 peano2nn0 9010 . . . . 5  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
1713, 16syl 14 . . . 4  |-  ( ph  ->  ( N  +  1 )  e.  NN0 )
18 eqidd 2138 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  ( F `  k ) )
19 nn0z 9067 . . . . . . 7  |-  ( k  e.  NN0  ->  k  e.  ZZ )
20 rpexpcl 10305 . . . . . . 7  |-  ( ( A  e.  RR+  /\  k  e.  ZZ )  ->  ( A ^ k )  e.  RR+ )
213, 19, 20syl2an 287 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( A ^ k )  e.  RR+ )
22 faccl 10474 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
2322adantl 275 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ! `  k )  e.  NN )
2423nnrpd 9475 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ! `  k )  e.  RR+ )
2521, 24rpdivcld 9494 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( A ^ k )  / 
( ! `  k
) )  e.  RR+ )
267, 25eqeltrd 2214 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  e.  RR+ )
275efcllem 11354 . . . . 5  |-  ( A  e.  CC  ->  seq 0 (  +  ,  F )  e.  dom  ~~>  )
284, 27syl 14 . . . 4  |-  ( ph  ->  seq 0 (  +  ,  F )  e. 
dom 
~~>  )
291, 15, 17, 18, 26, 28isumrpcl 11256 . . 3  |-  ( ph  -> 
sum_ k  e.  (
ZZ>= `  ( N  + 
1 ) ) ( F `  k )  e.  RR+ )
3014, 29ltaddrpd 9510 . 2  |-  ( ph  ->  (  seq 0 (  +  ,  F ) `
 N )  < 
( (  seq 0
(  +  ,  F
) `  N )  +  sum_ k  e.  (
ZZ>= `  ( N  + 
1 ) ) ( F `  k ) ) )
315efval2 11360 . . . 4  |-  ( A  e.  CC  ->  ( exp `  A )  = 
sum_ k  e.  NN0  ( F `  k ) )
324, 31syl 14 . . 3  |-  ( ph  ->  ( exp `  A
)  =  sum_ k  e.  NN0  ( F `  k ) )
3311recnd 7787 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  e.  CC )
341, 15, 17, 18, 33, 28isumsplit 11253 . . 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 ) ) )
3513nn0cnd 9025 . . . . . . . 8  |-  ( ph  ->  N  e.  CC )
36 ax-1cn 7706 . . . . . . . 8  |-  1  e.  CC
37 pncan 7961 . . . . . . . 8  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  + 
1 )  -  1 )  =  N )
3835, 36, 37sylancl 409 . . . . . . 7  |-  ( ph  ->  ( ( N  + 
1 )  -  1 )  =  N )
3938oveq2d 5783 . . . . . 6  |-  ( ph  ->  ( 0 ... (
( N  +  1 )  -  1 ) )  =  ( 0 ... N ) )
4039sumeq1d 11128 . . . . 5  |-  ( ph  -> 
sum_ k  e.  ( 0 ... ( ( N  +  1 )  -  1 ) ) ( F `  k
)  =  sum_ k  e.  ( 0 ... N
) ( F `  k ) )
41 eqidd 2138 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( F `  k )  =  ( F `  k ) )
4213, 1eleqtrdi 2230 . . . . . 6  |-  ( ph  ->  N  e.  ( ZZ>= ` 
0 ) )
43 elnn0uz 9356 . . . . . . 7  |-  ( k  e.  NN0  <->  k  e.  (
ZZ>= `  0 ) )
4443, 33sylan2br 286 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( F `  k )  e.  CC )
4541, 42, 44fsum3ser 11159 . . . . 5  |-  ( ph  -> 
sum_ k  e.  ( 0 ... N ) ( F `  k
)  =  (  seq 0 (  +  ,  F ) `  N
) )
4640, 45eqtrd 2170 . . . 4  |-  ( ph  -> 
sum_ k  e.  ( 0 ... ( ( N  +  1 )  -  1 ) ) ( F `  k
)  =  (  seq 0 (  +  ,  F ) `  N
) )
4746oveq1d 5782 . . 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 ) ) )
4832, 34, 473eqtrd 2174 . 2  |-  ( ph  ->  ( exp `  A
)  =  ( (  seq 0 (  +  ,  F ) `  N )  +  sum_ k  e.  ( ZZ>= `  ( N  +  1
) ) ( F `
 k ) ) )
4930, 48breqtrrd 3951 1  |-  ( ph  ->  (  seq 0 (  +  ,  F ) `
 N )  < 
( exp `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   class class class wbr 3924    |-> cmpt 3984   dom cdm 4534   ` cfv 5118  (class class class)co 5767   CCcc 7611   RRcr 7612   0cc0 7613   1c1 7614    + caddc 7616    < clt 7793    - cmin 7926    / cdiv 8425   NNcn 8713   NN0cn0 8970   ZZcz 9047   ZZ>=cuz 9319   RR+crp 9434   ...cfz 9783    seqcseq 10211   ^cexp 10285   !cfa 10464    ~~> cli 11040   sum_csu 11115   expce 11337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732  ax-caucvg 7733
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-isom 5127  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-frec 6281  df-1o 6306  df-oadd 6310  df-er 6422  df-en 6628  df-dom 6629  df-fin 6630  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-3 8773  df-4 8774  df-n0 8971  df-z 9048  df-uz 9320  df-q 9405  df-rp 9435  df-ico 9670  df-fz 9784  df-fzo 9913  df-seqfrec 10212  df-exp 10286  df-fac 10465  df-ihash 10515  df-cj 10607  df-re 10608  df-im 10609  df-rsqrt 10763  df-abs 10764  df-clim 11041  df-sumdc 11116  df-ef 11343
This theorem is referenced by:  efgt1p2  11390  efgt1p  11391
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