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Theorem efgt1p2 12394
Description: The exponential function of a positive real number is greater than the first three terms of the series expansion. (Contributed by Mario Carneiro, 15-Sep-2014.)
Assertion
Ref Expression
efgt1p2  |-  ( A  e.  RR+  ->  ( ( 1  +  A )  +  ( ( A ^ 2 )  / 
2 ) )  < 
( exp `  A
) )

Proof of Theorem efgt1p2
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 nn0uz 10262 . . 3  |-  NN0  =  ( ZZ>= `  0 )
2 1nn0 9981 . . 3  |-  1  e.  NN0
3 df-2 9804 . . 3  |-  2  =  ( 1  +  1 )
4 rpcn 10362 . . . 4  |-  ( A  e.  RR+  ->  A  e.  CC )
5 0nn0 9980 . . . . 5  |-  0  e.  NN0
6 1e0p1 10152 . . . . 5  |-  1  =  ( 0  +  1 )
7 0z 10035 . . . . . 6  |-  0  e.  ZZ
8 eqid 2283 . . . . . . . . 9  |-  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) )
98eftval 12358 . . . . . . . 8  |-  ( 0  e.  NN0  ->  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 0 )  =  ( ( A ^
0 )  /  ( ! `  0 )
) )
105, 9ax-mp 8 . . . . . . 7  |-  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 0 )  =  ( ( A ^
0 )  /  ( ! `  0 )
)
11 eft0val 12392 . . . . . . 7  |-  ( A  e.  CC  ->  (
( A ^ 0 )  /  ( ! `
 0 ) )  =  1 )
1210, 11syl5eq 2327 . . . . . 6  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  0
)  =  1 )
137, 12seq1i 11060 . . . . 5  |-  ( A  e.  CC  ->  (  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
0 )  =  1 )
148eftval 12358 . . . . . . 7  |-  ( 1  e.  NN0  ->  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 1 )  =  ( ( A ^
1 )  /  ( ! `  1 )
) )
152, 14ax-mp 8 . . . . . 6  |-  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 1 )  =  ( ( A ^
1 )  /  ( ! `  1 )
)
16 fac1 11292 . . . . . . . 8  |-  ( ! `
 1 )  =  1
1716oveq2i 5869 . . . . . . 7  |-  ( ( A ^ 1 )  /  ( ! ` 
1 ) )  =  ( ( A ^
1 )  /  1
)
18 exp1 11109 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
1918oveq1d 5873 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  1 )  =  ( A  / 
1 ) )
20 div1 9453 . . . . . . . 8  |-  ( A  e.  CC  ->  ( A  /  1 )  =  A )
2119, 20eqtrd 2315 . . . . . . 7  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  1 )  =  A )
2217, 21syl5eq 2327 . . . . . 6  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  ( ! `
 1 ) )  =  A )
2315, 22syl5eq 2327 . . . . 5  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  1
)  =  A )
241, 5, 6, 13, 23seqp1i 11062 . . . 4  |-  ( A  e.  CC  ->  (  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
1 )  =  ( 1  +  A ) )
254, 24syl 15 . . 3  |-  ( A  e.  RR+  ->  (  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
1 )  =  ( 1  +  A ) )
26 2nn0 9982 . . . . . 6  |-  2  e.  NN0
278eftval 12358 . . . . . 6  |-  ( 2  e.  NN0  ->  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 2 )  =  ( ( A ^
2 )  /  ( ! `  2 )
) )
2826, 27ax-mp 8 . . . . 5  |-  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 2 )  =  ( ( A ^
2 )  /  ( ! `  2 )
)
29 fac2 11294 . . . . . 6  |-  ( ! `
 2 )  =  2
3029oveq2i 5869 . . . . 5  |-  ( ( A ^ 2 )  /  ( ! ` 
2 ) )  =  ( ( A ^
2 )  /  2
)
3128, 30eqtri 2303 . . . 4  |-  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 2 )  =  ( ( A ^
2 )  /  2
)
3231a1i 10 . . 3  |-  ( A  e.  RR+  ->  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 2 )  =  ( ( A ^
2 )  /  2
) )
331, 2, 3, 25, 32seqp1i 11062 . 2  |-  ( A  e.  RR+  ->  (  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
2 )  =  ( ( 1  +  A
)  +  ( ( A ^ 2 )  /  2 ) ) )
34 id 19 . . 3  |-  ( A  e.  RR+  ->  A  e.  RR+ )
3526a1i 10 . . 3  |-  ( A  e.  RR+  ->  2  e. 
NN0 )
368, 34, 35effsumlt 12391 . 2  |-  ( A  e.  RR+  ->  (  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
2 )  <  ( exp `  A ) )
3733, 36eqbrtrrd 4045 1  |-  ( A  e.  RR+  ->  ( ( 1  +  A )  +  ( ( A ^ 2 )  / 
2 ) )  < 
( exp `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    < clt 8867    / cdiv 9423   2c2 9795   NN0cn0 9965   RR+crp 10354    seq cseq 11046   ^cexp 11104   !cfa 11288   expce 12343
This theorem is referenced by:  cxp2limlem  20270  pntpbnd1a  20734
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ico 10662  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-fac 11289  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349
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