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Theorem eflegeo 11046
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 9107 . . 3  |-  NN0  =  ( ZZ>= `  0 )
2 0zd 8816 . . 3  |-  ( ph  ->  0  e.  ZZ )
3 eflegeo.1 . . . . 5  |-  ( ph  ->  A  e.  RR )
43recnd 7570 . . . 4  |-  ( ph  ->  A  e.  CC )
5 eqid 2089 . . . . 5  |-  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) )
65eftvalcn 11001 . . . 4  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  k )  =  ( ( A ^ k
)  /  ( ! `
 k ) ) )
74, 6sylan 278 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 k )  =  ( ( A ^
k )  /  ( ! `  k )
) )
8 reeftcl 10999 . . . 4  |-  ( ( A  e.  RR  /\  k  e.  NN0 )  -> 
( ( A ^
k )  /  ( ! `  k )
)  e.  RR )
93, 8sylan 278 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( A ^ k )  / 
( ! `  k
) )  e.  RR )
10 simpr 109 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  k  e.  NN0 )
113adantr 271 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  A  e.  RR )
1211, 10reexpcld 10157 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( A ^ k )  e.  RR )
13 oveq2 5674 . . . . 5  |-  ( n  =  k  ->  ( A ^ n )  =  ( A ^ k
) )
14 eqid 2089 . . . . 5  |-  ( n  e.  NN0  |->  ( A ^ n ) )  =  ( n  e. 
NN0  |->  ( A ^
n ) )
1513, 14fvmptg 5393 . . . 4  |-  ( ( k  e.  NN0  /\  ( A ^ k )  e.  RR )  -> 
( ( n  e. 
NN0  |->  ( A ^
n ) ) `  k )  =  ( A ^ k ) )
1610, 12, 15syl2anc 404 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 k )  =  ( A ^ k
) )
17 reexpcl 10026 . . . 4  |-  ( ( A  e.  RR  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  RR )
183, 17sylan 278 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( A ^ k )  e.  RR )
19 faccl 10197 . . . . . . 7  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
2019adantl 272 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ! `  k )  e.  NN )
2120nnred 8489 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ! `  k )  e.  RR )
22 eflegeo.2 . . . . . . 7  |-  ( ph  ->  0  <_  A )
2322adantr 271 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  0  <_  A )
2411, 10, 23expge0d 10158 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  0  <_  ( A ^ k ) )
2520nnge1d 8519 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  1  <_  ( ! `  k ) )
2618, 21, 24, 25lemulge12d 8453 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( A ^ k )  <_ 
( ( ! `  k )  x.  ( A ^ k ) ) )
2720nngt0d 8520 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  0  <  ( ! `  k ) )
28 ledivmul 8392 . . . . 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 ) ) ) )
2918, 18, 21, 27, 28syl112anc 1179 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( (
( A ^ k
)  /  ( ! `
 k ) )  <_  ( A ^
k )  <->  ( A ^ k )  <_ 
( ( ! `  k )  x.  ( A ^ k ) ) ) )
3026, 29mpbird 166 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( A ^ k )  / 
( ! `  k
) )  <_  ( A ^ k ) )
315efcllem 11003 . . . 4  |-  ( A  e.  CC  ->  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) )  e. 
dom 
~~>  )
324, 31syl 14 . . 3  |-  ( ph  ->  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) )  e.  dom  ~~>  )
33 seqex 9911 . . . 4  |-  seq 0
(  +  ,  ( n  e.  NN0  |->  ( A ^ n ) ) )  e.  _V
34 eflegeo.3 . . . . . 6  |-  ( ph  ->  A  <  1 )
35 1red 7557 . . . . . . 7  |-  ( ph  ->  1  e.  RR )
36 difrp 9224 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  e.  RR )  ->  ( A  <  1  <->  ( 1  -  A )  e.  RR+ ) )
373, 35, 36syl2anc 404 . . . . . 6  |-  ( ph  ->  ( A  <  1  <->  ( 1  -  A )  e.  RR+ ) )
3834, 37mpbid 146 . . . . 5  |-  ( ph  ->  ( 1  -  A
)  e.  RR+ )
3938rpreccld 9238 . . . 4  |-  ( ph  ->  ( 1  /  (
1  -  A ) )  e.  RR+ )
403, 22absidd 10654 . . . . . 6  |-  ( ph  ->  ( abs `  A
)  =  A )
4140, 34eqbrtrd 3871 . . . . 5  |-  ( ph  ->  ( abs `  A
)  <  1 )
424, 41, 16geolim 10959 . . . 4  |-  ( ph  ->  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  ~~>  ( 1  /  (
1  -  A ) ) )
43 breldmg 4655 . . . 4  |-  ( (  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  e.  _V  /\  (
1  /  ( 1  -  A ) )  e.  RR+  /\  seq 0
(  +  ,  ( n  e.  NN0  |->  ( A ^ n ) ) )  ~~>  ( 1  / 
( 1  -  A
) ) )  ->  seq 0 (  +  , 
( n  e.  NN0  |->  ( A ^ n ) ) )  e.  dom  ~~>  )
4433, 39, 42, 43mp3an2i 1279 . . 3  |-  ( ph  ->  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  e.  dom  ~~>  )
451, 2, 7, 9, 16, 18, 30, 32, 44isumle 10943 . 2  |-  ( ph  -> 
sum_ k  e.  NN0  ( ( A ^
k )  /  ( ! `  k )
)  <_  sum_ k  e. 
NN0  ( A ^
k ) )
46 efval 11005 . . 3  |-  ( A  e.  CC  ->  ( exp `  A )  = 
sum_ k  e.  NN0  ( ( A ^
k )  /  ( ! `  k )
) )
474, 46syl 14 . 2  |-  ( ph  ->  ( exp `  A
)  =  sum_ k  e.  NN0  ( ( A ^ k )  / 
( ! `  k
) ) )
48 expcl 10027 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
494, 48sylan 278 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( A ^ k )  e.  CC )
501, 2, 16, 49, 42isumclim 10869 . . 3  |-  ( ph  -> 
sum_ k  e.  NN0  ( A ^ k )  =  ( 1  / 
( 1  -  A
) ) )
5150eqcomd 2094 . 2  |-  ( ph  ->  ( 1  /  (
1  -  A ) )  =  sum_ k  e.  NN0  ( A ^
k ) )
5245, 47, 513brtr4d 3881 1  |-  ( ph  ->  ( exp `  A
)  <_  ( 1  /  ( 1  -  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1290    e. wcel 1439   _Vcvv 2620   class class class wbr 3851    |-> cmpt 3905   dom cdm 4451   ` cfv 5028  (class class class)co 5666   CCcc 7402   RRcr 7403   0cc0 7404   1c1 7405    + caddc 7407    x. cmul 7409    < clt 7576    <_ cle 7577    - cmin 7707    / cdiv 8193   NNcn 8476   NN0cn0 8727   RR+crp 9188    seqcseq 9906   ^cexp 10008   !cfa 10187   abscabs 10484    ~~> cli 10720   sum_csu 10796   expce 10986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7490  ax-resscn 7491  ax-1cn 7492  ax-1re 7493  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-mulrcl 7498  ax-addcom 7499  ax-mulcom 7500  ax-addass 7501  ax-mulass 7502  ax-distr 7503  ax-i2m1 7504  ax-0lt1 7505  ax-1rid 7506  ax-0id 7507  ax-rnegex 7508  ax-precex 7509  ax-cnre 7510  ax-pre-ltirr 7511  ax-pre-ltwlin 7512  ax-pre-lttrn 7513  ax-pre-apti 7514  ax-pre-ltadd 7515  ax-pre-mulgt0 7516  ax-pre-mulext 7517  ax-arch 7518  ax-caucvg 7519
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-if 3398  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-isom 5037  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-frec 6170  df-1o 6195  df-oadd 6199  df-er 6306  df-en 6512  df-dom 6513  df-fin 6514  df-pnf 7578  df-mnf 7579  df-xr 7580  df-ltxr 7581  df-le 7582  df-sub 7709  df-neg 7710  df-reap 8106  df-ap 8113  df-div 8194  df-inn 8477  df-2 8535  df-3 8536  df-4 8537  df-n0 8728  df-z 8805  df-uz 9074  df-q 9159  df-rp 9189  df-ico 9366  df-fz 9479  df-fzo 9608  df-iseq 9907  df-seq3 9908  df-exp 10009  df-fac 10188  df-ihash 10238  df-cj 10330  df-re 10331  df-im 10332  df-rsqrt 10485  df-abs 10486  df-clim 10721  df-isum 10797  df-ef 10992
This theorem is referenced by: (None)
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