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Theorem eflegeo 12714
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 10512 . . 3  |-  NN0  =  ( ZZ>= `  0 )
2 0z 10285 . . . 4  |-  0  e.  ZZ
32a1i 11 . . 3  |-  ( ph  ->  0  e.  ZZ )
4 eqid 2435 . . . . 5  |-  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) )
54eftval 12671 . . . 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 12669 . . . 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 6081 . . . . 5  |-  ( n  =  k  ->  ( A ^ n )  =  ( A ^ k
) )
11 eqid 2435 . . . . 5  |-  ( n  e.  NN0  |->  ( A ^ n ) )  =  ( n  e. 
NN0  |->  ( A ^
n ) )
12 ovex 6098 . . . . 5  |-  ( A ^ k )  e. 
_V
1310, 11, 12fvmpt 5798 . . . 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 11390 . . . 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 11568 . . . . . . 7  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
1817adantl 453 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ! `  k )  e.  NN )
1918nnred 10007 . . . . 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 11533 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  0  <_  ( A ^ k ) )
2518nnge1d 10034 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  1  <_  ( ! `  k ) )
2616, 19, 24, 25lemulge12d 9941 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( A ^ k )  <_ 
( ( ! `  k )  x.  ( A ^ k ) ) )
2718nngt0d 10035 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  0  <  ( ! `  k ) )
28 ledivmul 9875 . . . . 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 9106 . . . 4  |-  ( ph  ->  A  e.  CC )
324efcllem 12672 . . . 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 12217 . . . . . 6  |-  ( ph  ->  ( abs `  A
)  =  A )
35 eflegeo.3 . . . . . 6  |-  ( ph  ->  A  <  1 )
3634, 35eqbrtrd 4224 . . . . 5  |-  ( ph  ->  ( abs `  A
)  <  1 )
3731, 36, 14geolim 12639 . . . 4  |-  ( ph  ->  seq  0 (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  ~~>  ( 1  /  (
1  -  A ) ) )
38 seqex 11317 . . . . 5  |-  seq  0
(  +  ,  ( n  e.  NN0  |->  ( A ^ n ) ) )  e.  _V
39 ovex 6098 . . . . 5  |-  ( 1  /  ( 1  -  A ) )  e. 
_V
4038, 39breldm 5066 . . . 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 12616 . 2  |-  ( ph  -> 
sum_ k  e.  NN0  ( ( A ^
k )  /  ( ! `  k )
)  <_  sum_ k  e. 
NN0  ( A ^
k ) )
43 efval 12674 . . 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 11391 . . . . 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 12533 . . 3  |-  ( ph  -> 
sum_ k  e.  NN0  ( A ^ k )  =  ( 1  / 
( 1  -  A
) ) )
4847eqcomd 2440 . 2  |-  ( ph  ->  ( 1  /  (
1  -  A ) )  =  sum_ k  e.  NN0  ( A ^
k ) )
4942, 44, 483brtr4d 4234 1  |-  ( ph  ->  ( exp `  A
)  <_  ( 1  /  ( 1  -  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4204    e. cmpt 4258   dom cdm 4870   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    < clt 9112    <_ cle 9113    - cmin 9283    / cdiv 9669   NNcn 9992   NN0cn0 10213   ZZcz 10274    seq cseq 11315   ^cexp 11374   !cfa 11558   abscabs 12031    ~~> cli 12270   sum_csu 12471   expce 12656
This theorem is referenced by:  birthdaylem3  20784  logdiflbnd  20825  emcllem2  20827
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-ico 10914  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-fac 11559  df-hash 11611  df-shft 11874  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-limsup 12257  df-clim 12274  df-rlim 12275  df-sum 12472  df-ef 12662
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