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Theorem eflegeo 11632
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 9492 . . 3  |-  NN0  =  ( ZZ>= `  0 )
2 0zd 9195 . . 3  |-  ( ph  ->  0  e.  ZZ )
3 eflegeo.1 . . . . 5  |-  ( ph  ->  A  e.  RR )
43recnd 7919 . . . 4  |-  ( ph  ->  A  e.  CC )
5 eqid 2164 . . . . 5  |-  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) )
65eftvalcn 11588 . . . 4  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  k )  =  ( ( A ^ k
)  /  ( ! `
 k ) ) )
74, 6sylan 281 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 k )  =  ( ( A ^
k )  /  ( ! `  k )
) )
8 reeftcl 11586 . . . 4  |-  ( ( A  e.  RR  /\  k  e.  NN0 )  -> 
( ( A ^
k )  /  ( ! `  k )
)  e.  RR )
93, 8sylan 281 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( A ^ k )  / 
( ! `  k
) )  e.  RR )
10 simpr 109 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  k  e.  NN0 )
113adantr 274 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  A  e.  RR )
1211, 10reexpcld 10595 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( A ^ k )  e.  RR )
13 oveq2 5845 . . . . 5  |-  ( n  =  k  ->  ( A ^ n )  =  ( A ^ k
) )
14 eqid 2164 . . . . 5  |-  ( n  e.  NN0  |->  ( A ^ n ) )  =  ( n  e. 
NN0  |->  ( A ^
n ) )
1513, 14fvmptg 5557 . . . 4  |-  ( ( k  e.  NN0  /\  ( A ^ k )  e.  RR )  -> 
( ( n  e. 
NN0  |->  ( A ^
n ) ) `  k )  =  ( A ^ k ) )
1610, 12, 15syl2anc 409 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 k )  =  ( A ^ k
) )
17 reexpcl 10463 . . . 4  |-  ( ( A  e.  RR  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  RR )
183, 17sylan 281 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( A ^ k )  e.  RR )
19 faccl 10638 . . . . . . 7  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
2019adantl 275 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ! `  k )  e.  NN )
2120nnred 8862 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ! `  k )  e.  RR )
22 eflegeo.2 . . . . . . 7  |-  ( ph  ->  0  <_  A )
2322adantr 274 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  0  <_  A )
2411, 10, 23expge0d 10596 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  0  <_  ( A ^ k ) )
2520nnge1d 8892 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  1  <_  ( ! `  k ) )
2618, 21, 24, 25lemulge12d 8825 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( A ^ k )  <_ 
( ( ! `  k )  x.  ( A ^ k ) ) )
2720nngt0d 8893 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  0  <  ( ! `  k ) )
28 ledivmul 8764 . . . . 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 1231 . . . 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 11590 . . . 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 10373 . . . 4  |-  seq 0
(  +  ,  ( n  e.  NN0  |->  ( A ^ n ) ) )  e.  _V
34 eflegeo.3 . . . . . 6  |-  ( ph  ->  A  <  1 )
35 1red 7906 . . . . . . 7  |-  ( ph  ->  1  e.  RR )
36 difrp 9620 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  e.  RR )  ->  ( A  <  1  <->  ( 1  -  A )  e.  RR+ ) )
373, 35, 36syl2anc 409 . . . . . 6  |-  ( ph  ->  ( A  <  1  <->  ( 1  -  A )  e.  RR+ ) )
3834, 37mpbid 146 . . . . 5  |-  ( ph  ->  ( 1  -  A
)  e.  RR+ )
3938rpreccld 9635 . . . 4  |-  ( ph  ->  ( 1  /  (
1  -  A ) )  e.  RR+ )
403, 22absidd 11099 . . . . . 6  |-  ( ph  ->  ( abs `  A
)  =  A )
4140, 34eqbrtrd 3999 . . . . 5  |-  ( ph  ->  ( abs `  A
)  <  1 )
424, 41, 16geolim 11442 . . . 4  |-  ( ph  ->  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  ~~>  ( 1  /  (
1  -  A ) ) )
43 breldmg 4805 . . . 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 1331 . . 3  |-  ( ph  ->  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  e.  dom  ~~>  )
451, 2, 7, 9, 16, 18, 30, 32, 44isumle 11426 . 2  |-  ( ph  -> 
sum_ k  e.  NN0  ( ( A ^
k )  /  ( ! `  k )
)  <_  sum_ k  e. 
NN0  ( A ^
k ) )
46 efval 11592 . . 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 10464 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
494, 48sylan 281 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( A ^ k )  e.  CC )
501, 2, 16, 49, 42isumclim 11352 . . 3  |-  ( ph  -> 
sum_ k  e.  NN0  ( A ^ k )  =  ( 1  / 
( 1  -  A
) ) )
5150eqcomd 2170 . 2  |-  ( ph  ->  ( 1  /  (
1  -  A ) )  =  sum_ k  e.  NN0  ( A ^
k ) )
5245, 47, 513brtr4d 4009 1  |-  ( ph  ->  ( exp `  A
)  <_  ( 1  /  ( 1  -  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1342    e. wcel 2135   _Vcvv 2722   class class class wbr 3977    |-> cmpt 4038   dom cdm 4599   ` cfv 5183  (class class class)co 5837   CCcc 7743   RRcr 7744   0cc0 7745   1c1 7746    + caddc 7748    x. cmul 7750    < clt 7925    <_ cle 7926    - cmin 8061    / cdiv 8560   NNcn 8849   NN0cn0 9106   RR+crp 9581    seqcseq 10371   ^cexp 10445   !cfa 10628   abscabs 10929    ~~> cli 11209   sum_csu 11284   expce 11573
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4092  ax-sep 4095  ax-nul 4103  ax-pow 4148  ax-pr 4182  ax-un 4406  ax-setind 4509  ax-iinf 4560  ax-cnex 7836  ax-resscn 7837  ax-1cn 7838  ax-1re 7839  ax-icn 7840  ax-addcl 7841  ax-addrcl 7842  ax-mulcl 7843  ax-mulrcl 7844  ax-addcom 7845  ax-mulcom 7846  ax-addass 7847  ax-mulass 7848  ax-distr 7849  ax-i2m1 7850  ax-0lt1 7851  ax-1rid 7852  ax-0id 7853  ax-rnegex 7854  ax-precex 7855  ax-cnre 7856  ax-pre-ltirr 7857  ax-pre-ltwlin 7858  ax-pre-lttrn 7859  ax-pre-apti 7860  ax-pre-ltadd 7861  ax-pre-mulgt0 7862  ax-pre-mulext 7863  ax-arch 7864  ax-caucvg 7865
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2724  df-sbc 2948  df-csb 3042  df-dif 3114  df-un 3116  df-in 3118  df-ss 3125  df-nul 3406  df-if 3517  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-int 3820  df-iun 3863  df-br 3978  df-opab 4039  df-mpt 4040  df-tr 4076  df-id 4266  df-po 4269  df-iso 4270  df-iord 4339  df-on 4341  df-ilim 4342  df-suc 4344  df-iom 4563  df-xp 4605  df-rel 4606  df-cnv 4607  df-co 4608  df-dm 4609  df-rn 4610  df-res 4611  df-ima 4612  df-iota 5148  df-fun 5185  df-fn 5186  df-f 5187  df-f1 5188  df-fo 5189  df-f1o 5190  df-fv 5191  df-isom 5192  df-riota 5793  df-ov 5840  df-oprab 5841  df-mpo 5842  df-1st 6101  df-2nd 6102  df-recs 6265  df-irdg 6330  df-frec 6351  df-1o 6376  df-oadd 6380  df-er 6493  df-en 6699  df-dom 6700  df-fin 6701  df-pnf 7927  df-mnf 7928  df-xr 7929  df-ltxr 7930  df-le 7931  df-sub 8063  df-neg 8064  df-reap 8465  df-ap 8472  df-div 8561  df-inn 8850  df-2 8908  df-3 8909  df-4 8910  df-n0 9107  df-z 9184  df-uz 9459  df-q 9550  df-rp 9582  df-ico 9822  df-fz 9937  df-fzo 10069  df-seqfrec 10372  df-exp 10446  df-fac 10629  df-ihash 10679  df-cj 10774  df-re 10775  df-im 10776  df-rsqrt 10930  df-abs 10931  df-clim 11210  df-sumdc 11285  df-ef 11579
This theorem is referenced by: (None)
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