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Theorem effsumlt 11325
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 9328 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
2 0zd 9034 . . . . 5  |-  ( ph  ->  0  e.  ZZ )
3 effsumlt.2 . . . . . . . 8  |-  ( ph  ->  A  e.  RR+ )
43rpcnd 9453 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
5 effsumlt.1 . . . . . . . 8  |-  F  =  ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) )
65eftvalcn 11290 . . . . . . 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 9451 . . . . . . 7  |-  ( ph  ->  A  e.  RR )
9 reeftcl 11288 . . . . . . 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 2194 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  e.  RR )
121, 2, 11serfre 10216 . . . 4  |-  ( ph  ->  seq 0 (  +  ,  F ) : NN0 --> RR )
13 effsumlt.3 . . . 4  |-  ( ph  ->  N  e.  NN0 )
1412, 13ffvelrnd 5524 . . 3  |-  ( ph  ->  (  seq 0 (  +  ,  F ) `
 N )  e.  RR )
15 eqid 2117 . . . 4  |-  ( ZZ>= `  ( N  +  1
) )  =  (
ZZ>= `  ( N  + 
1 ) )
16 peano2nn0 8985 . . . . 5  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
1713, 16syl 14 . . . 4  |-  ( ph  ->  ( N  +  1 )  e.  NN0 )
18 eqidd 2118 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  ( F `  k ) )
19 nn0z 9042 . . . . . . 7  |-  ( k  e.  NN0  ->  k  e.  ZZ )
20 rpexpcl 10280 . . . . . . 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 10449 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
2322adantl 275 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ! `  k )  e.  NN )
2423nnrpd 9450 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ! `  k )  e.  RR+ )
2521, 24rpdivcld 9469 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( A ^ k )  / 
( ! `  k
) )  e.  RR+ )
267, 25eqeltrd 2194 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  e.  RR+ )
275efcllem 11292 . . . . 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 11231 . . 3  |-  ( ph  -> 
sum_ k  e.  (
ZZ>= `  ( N  + 
1 ) ) ( F `  k )  e.  RR+ )
3014, 29ltaddrpd 9485 . 2  |-  ( ph  ->  (  seq 0 (  +  ,  F ) `
 N )  < 
( (  seq 0
(  +  ,  F
) `  N )  +  sum_ k  e.  (
ZZ>= `  ( N  + 
1 ) ) ( F `  k ) ) )
315efval2 11298 . . . 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 7762 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  e.  CC )
341, 15, 17, 18, 33, 28isumsplit 11228 . . 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 9000 . . . . . . . 8  |-  ( ph  ->  N  e.  CC )
36 ax-1cn 7681 . . . . . . . 8  |-  1  e.  CC
37 pncan 7936 . . . . . . . 8  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  + 
1 )  -  1 )  =  N )
3835, 36, 37sylancl 409 . . . . . . 7  |-  ( ph  ->  ( ( N  + 
1 )  -  1 )  =  N )
3938oveq2d 5758 . . . . . 6  |-  ( ph  ->  ( 0 ... (
( N  +  1 )  -  1 ) )  =  ( 0 ... N ) )
4039sumeq1d 11103 . . . . 5  |-  ( ph  -> 
sum_ k  e.  ( 0 ... ( ( N  +  1 )  -  1 ) ) ( F `  k
)  =  sum_ k  e.  ( 0 ... N
) ( F `  k ) )
41 eqidd 2118 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( F `  k )  =  ( F `  k ) )
4213, 1eleqtrdi 2210 . . . . . 6  |-  ( ph  ->  N  e.  ( ZZ>= ` 
0 ) )
43 elnn0uz 9331 . . . . . . 7  |-  ( k  e.  NN0  <->  k  e.  (
ZZ>= `  0 ) )
4443, 33sylan2br 286 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( F `  k )  e.  CC )
4541, 42, 44fsum3ser 11134 . . . . 5  |-  ( ph  -> 
sum_ k  e.  ( 0 ... N ) ( F `  k
)  =  (  seq 0 (  +  ,  F ) `  N
) )
4640, 45eqtrd 2150 . . . 4  |-  ( ph  -> 
sum_ k  e.  ( 0 ... ( ( N  +  1 )  -  1 ) ) ( F `  k
)  =  (  seq 0 (  +  ,  F ) `  N
) )
4746oveq1d 5757 . . 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 2154 . 2  |-  ( ph  ->  ( exp `  A
)  =  ( (  seq 0 (  +  ,  F ) `  N )  +  sum_ k  e.  ( ZZ>= `  ( N  +  1
) ) ( F `
 k ) ) )
4930, 48breqtrrd 3926 1  |-  ( ph  ->  (  seq 0 (  +  ,  F ) `
 N )  < 
( exp `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465   class class class wbr 3899    |-> cmpt 3959   dom cdm 4509   ` cfv 5093  (class class class)co 5742   CCcc 7586   RRcr 7587   0cc0 7588   1c1 7589    + caddc 7591    < clt 7768    - cmin 7901    / cdiv 8400   NNcn 8688   NN0cn0 8945   ZZcz 9022   ZZ>=cuz 9294   RR+crp 9409   ...cfz 9758    seqcseq 10186   ^cexp 10260   !cfa 10439    ~~> cli 11015   sum_csu 11090   expce 11275
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707  ax-caucvg 7708
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-isom 5102  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-frec 6256  df-1o 6281  df-oadd 6285  df-er 6397  df-en 6603  df-dom 6604  df-fin 6605  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-2 8747  df-3 8748  df-4 8749  df-n0 8946  df-z 9023  df-uz 9295  df-q 9380  df-rp 9410  df-ico 9645  df-fz 9759  df-fzo 9888  df-seqfrec 10187  df-exp 10261  df-fac 10440  df-ihash 10490  df-cj 10582  df-re 10583  df-im 10584  df-rsqrt 10738  df-abs 10739  df-clim 11016  df-sumdc 11091  df-ef 11281
This theorem is referenced by:  efgt1p2  11328  efgt1p  11329
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