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Theorem eflegeo 12677
Description: The exponential function on the reals between 0 and 1 lies below the comparable geometric series sum. (Contributed by Paul Chapman, 11-Sep-2007.)
Hypotheses
Ref Expression
eflegeo.1  |-  ( ph  ->  A  e.  RR )
eflegeo.2  |-  ( ph  ->  0  <_  A )
eflegeo.3  |-  ( ph  ->  A  <  1 )
Assertion
Ref Expression
eflegeo  |-  ( ph  ->  ( exp `  A
)  <_  ( 1  /  ( 1  -  A ) ) )

Proof of Theorem eflegeo
Dummy variables  k  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nn0uz 10476 . . 3  |-  NN0  =  ( ZZ>= `  0 )
2 0z 10249 . . . 4  |-  0  e.  ZZ
32a1i 11 . . 3  |-  ( ph  ->  0  e.  ZZ )
4 eqid 2404 . . . . 5  |-  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) )
54eftval 12634 . . . 4  |-  ( k  e.  NN0  ->  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 k )  =  ( ( A ^
k )  /  ( ! `  k )
) )
65adantl 453 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 k )  =  ( ( A ^
k )  /  ( ! `  k )
) )
7 eflegeo.1 . . . 4  |-  ( ph  ->  A  e.  RR )
8 reeftcl 12632 . . . 4  |-  ( ( A  e.  RR  /\  k  e.  NN0 )  -> 
( ( A ^
k )  /  ( ! `  k )
)  e.  RR )
97, 8sylan 458 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( A ^ k )  / 
( ! `  k
) )  e.  RR )
10 oveq2 6048 . . . . 5  |-  ( n  =  k  ->  ( A ^ n )  =  ( A ^ k
) )
11 eqid 2404 . . . . 5  |-  ( n  e.  NN0  |->  ( A ^ n ) )  =  ( n  e. 
NN0  |->  ( A ^
n ) )
12 ovex 6065 . . . . 5  |-  ( A ^ k )  e. 
_V
1310, 11, 12fvmpt 5765 . . . 4  |-  ( k  e.  NN0  ->  ( ( n  e.  NN0  |->  ( A ^ n ) ) `
 k )  =  ( A ^ k
) )
1413adantl 453 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 k )  =  ( A ^ k
) )
15 reexpcl 11353 . . . 4  |-  ( ( A  e.  RR  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  RR )
167, 15sylan 458 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( A ^ k )  e.  RR )
17 faccl 11531 . . . . . . 7  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
1817adantl 453 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ! `  k )  e.  NN )
1918nnred 9971 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ! `  k )  e.  RR )
207adantr 452 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  A  e.  RR )
21 simpr 448 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  k  e.  NN0 )
22 eflegeo.2 . . . . . . 7  |-  ( ph  ->  0  <_  A )
2322adantr 452 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  0  <_  A )
2420, 21, 23expge0d 11496 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  0  <_  ( A ^ k ) )
2518nnge1d 9998 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  1  <_  ( ! `  k ) )
2616, 19, 24, 25lemulge12d 9905 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( A ^ k )  <_ 
( ( ! `  k )  x.  ( A ^ k ) ) )
2718nngt0d 9999 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  0  <  ( ! `  k ) )
28 ledivmul 9839 . . . . 5  |-  ( ( ( A ^ k
)  e.  RR  /\  ( A ^ k )  e.  RR  /\  (
( ! `  k
)  e.  RR  /\  0  <  ( ! `  k ) ) )  ->  ( ( ( A ^ k )  /  ( ! `  k ) )  <_ 
( A ^ k
)  <->  ( A ^
k )  <_  (
( ! `  k
)  x.  ( A ^ k ) ) ) )
2916, 16, 19, 27, 28syl112anc 1188 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( (
( A ^ k
)  /  ( ! `
 k ) )  <_  ( A ^
k )  <->  ( A ^ k )  <_ 
( ( ! `  k )  x.  ( A ^ k ) ) ) )
3026, 29mpbird 224 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( A ^ k )  / 
( ! `  k
) )  <_  ( A ^ k ) )
317recnd 9070 . . . 4  |-  ( ph  ->  A  e.  CC )
324efcllem 12635 . . . 4  |-  ( A  e.  CC  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) )  e. 
dom 
~~>  )
3331, 32syl 16 . . 3  |-  ( ph  ->  seq  0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) )  e.  dom  ~~>  )
347, 22absidd 12180 . . . . . 6  |-  ( ph  ->  ( abs `  A
)  =  A )
35 eflegeo.3 . . . . . 6  |-  ( ph  ->  A  <  1 )
3634, 35eqbrtrd 4192 . . . . 5  |-  ( ph  ->  ( abs `  A
)  <  1 )
3731, 36, 14geolim 12602 . . . 4  |-  ( ph  ->  seq  0 (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  ~~>  ( 1  /  (
1  -  A ) ) )
38 seqex 11280 . . . . 5  |-  seq  0
(  +  ,  ( n  e.  NN0  |->  ( A ^ n ) ) )  e.  _V
39 ovex 6065 . . . . 5  |-  ( 1  /  ( 1  -  A ) )  e. 
_V
4038, 39breldm 5033 . . . 4  |-  (  seq  0 (  +  , 
( n  e.  NN0  |->  ( A ^ n ) ) )  ~~>  ( 1  /  ( 1  -  A ) )  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( A ^ n ) ) )  e.  dom  ~~>  )
4137, 40syl 16 . . 3  |-  ( ph  ->  seq  0 (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  e.  dom  ~~>  )
421, 3, 6, 9, 14, 16, 30, 33, 41isumle 12579 . 2  |-  ( ph  -> 
sum_ k  e.  NN0  ( ( A ^
k )  /  ( ! `  k )
)  <_  sum_ k  e. 
NN0  ( A ^
k ) )
43 efval 12637 . . 3  |-  ( A  e.  CC  ->  ( exp `  A )  = 
sum_ k  e.  NN0  ( ( A ^
k )  /  ( ! `  k )
) )
4431, 43syl 16 . 2  |-  ( ph  ->  ( exp `  A
)  =  sum_ k  e.  NN0  ( ( A ^ k )  / 
( ! `  k
) ) )
45 expcl 11354 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
4631, 45sylan 458 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( A ^ k )  e.  CC )
471, 3, 14, 46, 37isumclim 12496 . . 3  |-  ( ph  -> 
sum_ k  e.  NN0  ( A ^ k )  =  ( 1  / 
( 1  -  A
) ) )
4847eqcomd 2409 . 2  |-  ( ph  ->  ( 1  /  (
1  -  A ) )  =  sum_ k  e.  NN0  ( A ^
k ) )
4942, 44, 483brtr4d 4202 1  |-  ( ph  ->  ( exp `  A
)  <_  ( 1  /  ( 1  -  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   class class class wbr 4172    e. cmpt 4226   dom cdm 4837   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    < clt 9076    <_ cle 9077    - cmin 9247    / cdiv 9633   NNcn 9956   NN0cn0 10177   ZZcz 10238    seq cseq 11278   ^cexp 11337   !cfa 11521   abscabs 11994    ~~> cli 12233   sum_csu 12434   expce 12619
This theorem is referenced by:  birthdaylem3  20745  logdiflbnd  20786  emcllem2  20788
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-ico 10878  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-fac 11522  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625
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