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Theorem effsumlt 11732
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 9592 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
2 0zd 9295 . . . . 5  |-  ( ph  ->  0  e.  ZZ )
3 effsumlt.2 . . . . . . . 8  |-  ( ph  ->  A  e.  RR+ )
43rpcnd 9728 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
5 effsumlt.1 . . . . . . . 8  |-  F  =  ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) )
65eftvalcn 11697 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( F `  k
)  =  ( ( A ^ k )  /  ( ! `  k ) ) )
74, 6sylan 283 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  ( ( A ^ k
)  /  ( ! `
 k ) ) )
83rpred 9726 . . . . . . 7  |-  ( ph  ->  A  e.  RR )
9 reeftcl 11695 . . . . . . 7  |-  ( ( A  e.  RR  /\  k  e.  NN0 )  -> 
( ( A ^
k )  /  ( ! `  k )
)  e.  RR )
108, 9sylan 283 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( A ^ k )  / 
( ! `  k
) )  e.  RR )
117, 10eqeltrd 2266 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  e.  RR )
121, 2, 11serfre 10506 . . . 4  |-  ( ph  ->  seq 0 (  +  ,  F ) : NN0 --> RR )
13 effsumlt.3 . . . 4  |-  ( ph  ->  N  e.  NN0 )
1412, 13ffvelcdmd 5673 . . 3  |-  ( ph  ->  (  seq 0 (  +  ,  F ) `
 N )  e.  RR )
15 eqid 2189 . . . 4  |-  ( ZZ>= `  ( N  +  1
) )  =  (
ZZ>= `  ( N  + 
1 ) )
16 peano2nn0 9246 . . . . 5  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
1713, 16syl 14 . . . 4  |-  ( ph  ->  ( N  +  1 )  e.  NN0 )
18 eqidd 2190 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  ( F `  k ) )
19 nn0z 9303 . . . . . . 7  |-  ( k  e.  NN0  ->  k  e.  ZZ )
20 rpexpcl 10570 . . . . . . 7  |-  ( ( A  e.  RR+  /\  k  e.  ZZ )  ->  ( A ^ k )  e.  RR+ )
213, 19, 20syl2an 289 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( A ^ k )  e.  RR+ )
22 faccl 10747 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
2322adantl 277 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ! `  k )  e.  NN )
2423nnrpd 9724 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ! `  k )  e.  RR+ )
2521, 24rpdivcld 9744 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( A ^ k )  / 
( ! `  k
) )  e.  RR+ )
267, 25eqeltrd 2266 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  e.  RR+ )
275efcllem 11699 . . . . 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 11534 . . 3  |-  ( ph  -> 
sum_ k  e.  (
ZZ>= `  ( N  + 
1 ) ) ( F `  k )  e.  RR+ )
3014, 29ltaddrpd 9760 . 2  |-  ( ph  ->  (  seq 0 (  +  ,  F ) `
 N )  < 
( (  seq 0
(  +  ,  F
) `  N )  +  sum_ k  e.  (
ZZ>= `  ( N  + 
1 ) ) ( F `  k ) ) )
315efval2 11705 . . . 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 8016 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  e.  CC )
341, 15, 17, 18, 33, 28isumsplit 11531 . . 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 9261 . . . . . . . 8  |-  ( ph  ->  N  e.  CC )
36 ax-1cn 7934 . . . . . . . 8  |-  1  e.  CC
37 pncan 8193 . . . . . . . 8  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  + 
1 )  -  1 )  =  N )
3835, 36, 37sylancl 413 . . . . . . 7  |-  ( ph  ->  ( ( N  + 
1 )  -  1 )  =  N )
3938oveq2d 5912 . . . . . 6  |-  ( ph  ->  ( 0 ... (
( N  +  1 )  -  1 ) )  =  ( 0 ... N ) )
4039sumeq1d 11406 . . . . 5  |-  ( ph  -> 
sum_ k  e.  ( 0 ... ( ( N  +  1 )  -  1 ) ) ( F `  k
)  =  sum_ k  e.  ( 0 ... N
) ( F `  k ) )
41 eqidd 2190 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( F `  k )  =  ( F `  k ) )
4213, 1eleqtrdi 2282 . . . . . 6  |-  ( ph  ->  N  e.  ( ZZ>= ` 
0 ) )
43 elnn0uz 9595 . . . . . . 7  |-  ( k  e.  NN0  <->  k  e.  (
ZZ>= `  0 ) )
4443, 33sylan2br 288 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( F `  k )  e.  CC )
4541, 42, 44fsum3ser 11437 . . . . 5  |-  ( ph  -> 
sum_ k  e.  ( 0 ... N ) ( F `  k
)  =  (  seq 0 (  +  ,  F ) `  N
) )
4640, 45eqtrd 2222 . . . 4  |-  ( ph  -> 
sum_ k  e.  ( 0 ... ( ( N  +  1 )  -  1 ) ) ( F `  k
)  =  (  seq 0 (  +  ,  F ) `  N
) )
4746oveq1d 5911 . . 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 2226 . 2  |-  ( ph  ->  ( exp `  A
)  =  ( (  seq 0 (  +  ,  F ) `  N )  +  sum_ k  e.  ( ZZ>= `  ( N  +  1
) ) ( F `
 k ) ) )
4930, 48breqtrrd 4046 1  |-  ( ph  ->  (  seq 0 (  +  ,  F ) `
 N )  < 
( exp `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   class class class wbr 4018    |-> cmpt 4079   dom cdm 4644   ` cfv 5235  (class class class)co 5896   CCcc 7839   RRcr 7840   0cc0 7841   1c1 7842    + caddc 7844    < clt 8022    - cmin 8158    / cdiv 8659   NNcn 8949   NN0cn0 9206   ZZcz 9283   ZZ>=cuz 9558   RR+crp 9683   ...cfz 10038    seqcseq 10476   ^cexp 10550   !cfa 10737    ~~> cli 11318   sum_csu 11393   expce 11682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958  ax-pre-mulext 7959  ax-arch 7960  ax-caucvg 7961
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-isom 5244  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-irdg 6395  df-frec 6416  df-1o 6441  df-oadd 6445  df-er 6559  df-en 6767  df-dom 6768  df-fin 6769  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569  df-div 8660  df-inn 8950  df-2 9008  df-3 9009  df-4 9010  df-n0 9207  df-z 9284  df-uz 9559  df-q 9650  df-rp 9684  df-ico 9924  df-fz 10039  df-fzo 10173  df-seqfrec 10477  df-exp 10551  df-fac 10738  df-ihash 10788  df-cj 10883  df-re 10884  df-im 10885  df-rsqrt 11039  df-abs 11040  df-clim 11319  df-sumdc 11394  df-ef 11688
This theorem is referenced by:  efgt1p2  11735  efgt1p  11736
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