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Theorem efgt1p2 12715
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 10520 . . 3  |-  NN0  =  ( ZZ>= `  0 )
2 1nn0 10237 . . 3  |-  1  e.  NN0
3 df-2 10058 . . 3  |-  2  =  ( 1  +  1 )
4 rpcn 10620 . . . 4  |-  ( A  e.  RR+  ->  A  e.  CC )
5 0nn0 10236 . . . . 5  |-  0  e.  NN0
6 1e0p1 10410 . . . . 5  |-  1  =  ( 0  +  1 )
7 0z 10293 . . . . . 6  |-  0  e.  ZZ
8 eqid 2436 . . . . . . . . 9  |-  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) )
98eftval 12679 . . . . . . . 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 12713 . . . . . . 7  |-  ( A  e.  CC  ->  (
( A ^ 0 )  /  ( ! `
 0 ) )  =  1 )
1210, 11syl5eq 2480 . . . . . 6  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  0
)  =  1 )
137, 12seq1i 11337 . . . . 5  |-  ( A  e.  CC  ->  (  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
0 )  =  1 )
148eftval 12679 . . . . . . 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 11570 . . . . . . . 8  |-  ( ! `
 1 )  =  1
1716oveq2i 6092 . . . . . . 7  |-  ( ( A ^ 1 )  /  ( ! ` 
1 ) )  =  ( ( A ^
1 )  /  1
)
18 exp1 11387 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
1918oveq1d 6096 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  1 )  =  ( A  / 
1 ) )
20 div1 9707 . . . . . . . 8  |-  ( A  e.  CC  ->  ( A  /  1 )  =  A )
2119, 20eqtrd 2468 . . . . . . 7  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  1 )  =  A )
2217, 21syl5eq 2480 . . . . . 6  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  ( ! `
 1 ) )  =  A )
2315, 22syl5eq 2480 . . . . 5  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  1
)  =  A )
241, 5, 6, 13, 23seqp1i 11339 . . . 4  |-  ( A  e.  CC  ->  (  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
1 )  =  ( 1  +  A ) )
254, 24syl 16 . . 3  |-  ( A  e.  RR+  ->  (  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
1 )  =  ( 1  +  A ) )
26 2nn0 10238 . . . . . 6  |-  2  e.  NN0
278eftval 12679 . . . . . 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 11572 . . . . . 6  |-  ( ! `
 2 )  =  2
3029oveq2i 6092 . . . . 5  |-  ( ( A ^ 2 )  /  ( ! ` 
2 ) )  =  ( ( A ^
2 )  /  2
)
3128, 30eqtri 2456 . . . 4  |-  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 2 )  =  ( ( A ^
2 )  /  2
)
3231a1i 11 . . 3  |-  ( A  e.  RR+  ->  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 2 )  =  ( ( A ^
2 )  /  2
) )
331, 2, 3, 25, 32seqp1i 11339 . 2  |-  ( A  e.  RR+  ->  (  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
2 )  =  ( ( 1  +  A
)  +  ( ( A ^ 2 )  /  2 ) ) )
34 id 20 . . 3  |-  ( A  e.  RR+  ->  A  e.  RR+ )
3526a1i 11 . . 3  |-  ( A  e.  RR+  ->  2  e. 
NN0 )
368, 34, 35effsumlt 12712 . 2  |-  ( A  e.  RR+  ->  (  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
2 )  <  ( exp `  A ) )
3733, 36eqbrtrrd 4234 1  |-  ( A  e.  RR+  ->  ( ( 1  +  A )  +  ( ( A ^ 2 )  / 
2 ) )  < 
( exp `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   class class class wbr 4212    e. cmpt 4266   ` cfv 5454  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991    + caddc 8993    < clt 9120    / cdiv 9677   2c2 10049   NN0cn0 10221   RR+crp 10612    seq cseq 11323   ^cexp 11382   !cfa 11566   expce 12664
This theorem is referenced by:  cxp2limlem  20814  pntpbnd1a  21279
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-ico 10922  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-fac 11567  df-hash 11619  df-shft 11882  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-limsup 12265  df-clim 12282  df-rlim 12283  df-sum 12480  df-ef 12670
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