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Theorem eflegeo 11415
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 9367 . . 3  |-  NN0  =  ( ZZ>= `  0 )
2 0zd 9073 . . 3  |-  ( ph  ->  0  e.  ZZ )
3 eflegeo.1 . . . . 5  |-  ( ph  ->  A  e.  RR )
43recnd 7801 . . . 4  |-  ( ph  ->  A  e.  CC )
5 eqid 2139 . . . . 5  |-  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) )
65eftvalcn 11370 . . . 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 11368 . . . 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 10448 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( A ^ k )  e.  RR )
13 oveq2 5782 . . . . 5  |-  ( n  =  k  ->  ( A ^ n )  =  ( A ^ k
) )
14 eqid 2139 . . . . 5  |-  ( n  e.  NN0  |->  ( A ^ n ) )  =  ( n  e. 
NN0  |->  ( A ^
n ) )
1513, 14fvmptg 5497 . . . 4  |-  ( ( k  e.  NN0  /\  ( A ^ k )  e.  RR )  -> 
( ( n  e. 
NN0  |->  ( A ^
n ) ) `  k )  =  ( A ^ k ) )
1610, 12, 15syl2anc 408 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 k )  =  ( A ^ k
) )
17 reexpcl 10317 . . . 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 10488 . . . . . . 7  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
2019adantl 275 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ! `  k )  e.  NN )
2120nnred 8740 . . . . 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 10449 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  0  <_  ( A ^ k ) )
2520nnge1d 8770 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  1  <_  ( ! `  k ) )
2618, 21, 24, 25lemulge12d 8703 . . . 4  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( A ^ k )  <_ 
( ( ! `  k )  x.  ( A ^ k ) ) )
2720nngt0d 8771 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  0  <  ( ! `  k ) )
28 ledivmul 8642 . . . . 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 1220 . . . 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 11372 . . . 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 10227 . . . 4  |-  seq 0
(  +  ,  ( n  e.  NN0  |->  ( A ^ n ) ) )  e.  _V
34 eflegeo.3 . . . . . 6  |-  ( ph  ->  A  <  1 )
35 1red 7788 . . . . . . 7  |-  ( ph  ->  1  e.  RR )
36 difrp 9487 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  e.  RR )  ->  ( A  <  1  <->  ( 1  -  A )  e.  RR+ ) )
373, 35, 36syl2anc 408 . . . . . 6  |-  ( ph  ->  ( A  <  1  <->  ( 1  -  A )  e.  RR+ ) )
3834, 37mpbid 146 . . . . 5  |-  ( ph  ->  ( 1  -  A
)  e.  RR+ )
3938rpreccld 9501 . . . 4  |-  ( ph  ->  ( 1  /  (
1  -  A ) )  e.  RR+ )
403, 22absidd 10946 . . . . . 6  |-  ( ph  ->  ( abs `  A
)  =  A )
4140, 34eqbrtrd 3950 . . . . 5  |-  ( ph  ->  ( abs `  A
)  <  1 )
424, 41, 16geolim 11287 . . . 4  |-  ( ph  ->  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  ~~>  ( 1  /  (
1  -  A ) ) )
43 breldmg 4745 . . . 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 1320 . . 3  |-  ( ph  ->  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  e.  dom  ~~>  )
451, 2, 7, 9, 16, 18, 30, 32, 44isumle 11271 . 2  |-  ( ph  -> 
sum_ k  e.  NN0  ( ( A ^
k )  /  ( ! `  k )
)  <_  sum_ k  e. 
NN0  ( A ^
k ) )
46 efval 11374 . . 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 10318 . . . . 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 11197 . . 3  |-  ( ph  -> 
sum_ k  e.  NN0  ( A ^ k )  =  ( 1  / 
( 1  -  A
) ) )
5150eqcomd 2145 . 2  |-  ( ph  ->  ( 1  /  (
1  -  A ) )  =  sum_ k  e.  NN0  ( A ^
k ) )
5245, 47, 513brtr4d 3960 1  |-  ( ph  ->  ( exp `  A
)  <_  ( 1  /  ( 1  -  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   _Vcvv 2686   class class class wbr 3929    |-> cmpt 3989   dom cdm 4539   ` cfv 5123  (class class class)co 5774   CCcc 7625   RRcr 7626   0cc0 7627   1c1 7628    + caddc 7630    x. cmul 7632    < clt 7807    <_ cle 7808    - cmin 7940    / cdiv 8439   NNcn 8727   NN0cn0 8984   RR+crp 9448    seqcseq 10225   ^cexp 10299   !cfa 10478   abscabs 10776    ~~> cli 11054   sum_csu 11129   expce 11355
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745  ax-arch 7746  ax-caucvg 7747
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440  df-inn 8728  df-2 8786  df-3 8787  df-4 8788  df-n0 8985  df-z 9062  df-uz 9334  df-q 9419  df-rp 9449  df-ico 9684  df-fz 9798  df-fzo 9927  df-seqfrec 10226  df-exp 10300  df-fac 10479  df-ihash 10529  df-cj 10621  df-re 10622  df-im 10623  df-rsqrt 10777  df-abs 10778  df-clim 11055  df-sumdc 11130  df-ef 11361
This theorem is referenced by: (None)
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