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Theorem efgt1p2 11300
Description: The exponential of a positive real number is greater than the sum of 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 variables  k  n  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1nn0 8944 . . . . . . 7  |-  1  e.  NN0
2 nn0uz 9309 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
31, 2eleqtri 2190 . . . . . 6  |-  1  e.  ( ZZ>= `  0 )
43a1i 9 . . . . 5  |-  ( A  e.  RR+  ->  1  e.  ( ZZ>= `  0 )
)
5 elnn0uz 9312 . . . . . . . . 9  |-  ( k  e.  NN0  <->  k  e.  (
ZZ>= `  0 ) )
65biimpri 132 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  0
)  ->  k  e.  NN0 )
76adantl 273 . . . . . . 7  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  k  e.  NN0 )
8 simpl 108 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  A  e.  RR+ )
9 eluzelz 9284 . . . . . . . . . 10  |-  ( k  e.  ( ZZ>= `  0
)  ->  k  e.  ZZ )
109adantl 273 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  k  e.  ZZ )
118, 10rpexpcld 10388 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( A ^ k )  e.  RR+ )
127faccld 10422 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( ! `  k )  e.  NN )
1312nnrpd 9428 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( ! `  k )  e.  RR+ )
1411, 13rpdivcld 9447 . . . . . . 7  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( ( A ^ k )  / 
( ! `  k
) )  e.  RR+ )
15 oveq2 5748 . . . . . . . . 9  |-  ( n  =  k  ->  ( A ^ n )  =  ( A ^ k
) )
16 fveq2 5387 . . . . . . . . 9  |-  ( n  =  k  ->  ( ! `  n )  =  ( ! `  k ) )
1715, 16oveq12d 5758 . . . . . . . 8  |-  ( n  =  k  ->  (
( A ^ n
)  /  ( ! `
 n ) )  =  ( ( A ^ k )  / 
( ! `  k
) ) )
18 eqid 2115 . . . . . . . 8  |-  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) )
1917, 18fvmptg 5463 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ( A ^
k )  /  ( ! `  k )
)  e.  RR+ )  ->  ( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  k )  =  ( ( A ^ k
)  /  ( ! `
 k ) ) )
207, 14, 19syl2anc 406 . . . . . 6  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( (
n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 k )  =  ( ( A ^
k )  /  ( ! `  k )
) )
2120, 14eqeltrd 2192 . . . . 5  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( (
n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 k )  e.  RR+ )
22 rpaddcl 9413 . . . . . 6  |-  ( ( k  e.  RR+  /\  y  e.  RR+ )  ->  (
k  +  y )  e.  RR+ )
2322adantl 273 . . . . 5  |-  ( ( A  e.  RR+  /\  (
k  e.  RR+  /\  y  e.  RR+ ) )  -> 
( k  +  y )  e.  RR+ )
244, 21, 23seq3p1 10175 . . . 4  |-  ( A  e.  RR+  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) `  ( 1  +  1 ) )  =  ( (  seq 0 (  +  ,  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) ) `  1 )  +  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 ( 1  +  1 ) ) ) )
25 df-2 8736 . . . . 5  |-  2  =  ( 1  +  1 )
2625fveq2i 5390 . . . 4  |-  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
2 )  =  (  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ) `
 ( 1  +  1 ) )
2725fveq2i 5390 . . . . 5  |-  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 2 )  =  ( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  ( 1  +  1 ) )
2827oveq2i 5751 . . . 4  |-  ( (  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ) `
 1 )  +  ( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ` 
2 ) )  =  ( (  seq 0
(  +  ,  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) ) `  1 )  +  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 ( 1  +  1 ) ) )
2924, 26, 283eqtr4g 2173 . . 3  |-  ( A  e.  RR+  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
2 )  =  ( (  seq 0 (  +  ,  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) ) `  1 )  +  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 2 ) ) )
30 0nn0 8943 . . . . . . . . 9  |-  0  e.  NN0
3130, 2eleqtri 2190 . . . . . . . 8  |-  0  e.  ( ZZ>= `  0 )
3231a1i 9 . . . . . . 7  |-  ( A  e.  RR+  ->  0  e.  ( ZZ>= `  0 )
)
3332, 21, 23seq3p1 10175 . . . . . 6  |-  ( A  e.  RR+  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) `  ( 0  +  1 ) )  =  ( (  seq 0 (  +  ,  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) ) `  0 )  +  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 ( 0  +  1 ) ) ) )
34 1e0p1 9174 . . . . . . 7  |-  1  =  ( 0  +  1 )
3534fveq2i 5390 . . . . . 6  |-  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
1 )  =  (  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ) `
 ( 0  +  1 ) )
3634fveq2i 5390 . . . . . . 7  |-  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 1 )  =  ( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  ( 0  +  1 ) )
3736oveq2i 5751 . . . . . 6  |-  ( (  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ) `
 0 )  +  ( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ` 
1 ) )  =  ( (  seq 0
(  +  ,  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) ) `  0 )  +  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 ( 0  +  1 ) ) )
3833, 35, 373eqtr4g 2173 . . . . 5  |-  ( A  e.  RR+  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
1 )  =  ( (  seq 0 (  +  ,  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) ) `  0 )  +  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 1 ) ) )
39 0zd 9017 . . . . . . . 8  |-  ( A  e.  RR+  ->  0  e.  ZZ )
4039, 21, 23seq3-1 10173 . . . . . . 7  |-  ( A  e.  RR+  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
0 )  =  ( ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  0
) )
41 rpcn 9398 . . . . . . . . 9  |-  ( A  e.  RR+  ->  A  e.  CC )
4218eftvalcn 11262 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  0  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ` 
0 )  =  ( ( A ^ 0 )  /  ( ! `
 0 ) ) )
4330, 42mpan2 419 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  0
)  =  ( ( A ^ 0 )  /  ( ! ` 
0 ) ) )
4441, 43syl 14 . . . . . . . 8  |-  ( A  e.  RR+  ->  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 0 )  =  ( ( A ^
0 )  /  ( ! `  0 )
) )
45 eft0val 11298 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A ^ 0 )  /  ( ! `
 0 ) )  =  1 )
4641, 45syl 14 . . . . . . . 8  |-  ( A  e.  RR+  ->  ( ( A ^ 0 )  /  ( ! ` 
0 ) )  =  1 )
4744, 46eqtrd 2148 . . . . . . 7  |-  ( A  e.  RR+  ->  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 0 )  =  1 )
4840, 47eqtrd 2148 . . . . . 6  |-  ( A  e.  RR+  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
0 )  =  1 )
4918eftvalcn 11262 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  1  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ` 
1 )  =  ( ( A ^ 1 )  /  ( ! `
 1 ) ) )
501, 49mpan2 419 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  1
)  =  ( ( A ^ 1 )  /  ( ! ` 
1 ) ) )
51 fac1 10415 . . . . . . . . . 10  |-  ( ! `
 1 )  =  1
5251oveq2i 5751 . . . . . . . . 9  |-  ( ( A ^ 1 )  /  ( ! ` 
1 ) )  =  ( ( A ^
1 )  /  1
)
53 exp1 10239 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
5453oveq1d 5755 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  1 )  =  ( A  / 
1 ) )
55 div1 8423 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( A  /  1 )  =  A )
5654, 55eqtrd 2148 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  1 )  =  A )
5752, 56syl5eq 2160 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  ( ! `
 1 ) )  =  A )
5850, 57eqtrd 2148 . . . . . . 7  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  1
)  =  A )
5941, 58syl 14 . . . . . 6  |-  ( A  e.  RR+  ->  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 1 )  =  A )
6048, 59oveq12d 5758 . . . . 5  |-  ( A  e.  RR+  ->  ( (  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ) `
 0 )  +  ( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ` 
1 ) )  =  ( 1  +  A
) )
6138, 60eqtrd 2148 . . . 4  |-  ( A  e.  RR+  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
1 )  =  ( 1  +  A ) )
62 2nn0 8945 . . . . . . 7  |-  2  e.  NN0
6318eftvalcn 11262 . . . . . . 7  |-  ( ( A  e.  CC  /\  2  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ` 
2 )  =  ( ( A ^ 2 )  /  ( ! `
 2 ) ) )
6462, 63mpan2 419 . . . . . 6  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  2
)  =  ( ( A ^ 2 )  /  ( ! ` 
2 ) ) )
65 fac2 10417 . . . . . . 7  |-  ( ! `
 2 )  =  2
6665oveq2i 5751 . . . . . 6  |-  ( ( A ^ 2 )  /  ( ! ` 
2 ) )  =  ( ( A ^
2 )  /  2
)
6764, 66syl6eq 2164 . . . . 5  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  2
)  =  ( ( A ^ 2 )  /  2 ) )
6841, 67syl 14 . . . 4  |-  ( A  e.  RR+  ->  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 2 )  =  ( ( A ^
2 )  /  2
) )
6961, 68oveq12d 5758 . . 3  |-  ( A  e.  RR+  ->  ( (  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ) `
 1 )  +  ( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ` 
2 ) )  =  ( ( 1  +  A )  +  ( ( A ^ 2 )  /  2 ) ) )
7029, 69eqtrd 2148 . 2  |-  ( A  e.  RR+  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
2 )  =  ( ( 1  +  A
)  +  ( ( A ^ 2 )  /  2 ) ) )
71 id 19 . . 3  |-  ( A  e.  RR+  ->  A  e.  RR+ )
7262a1i 9 . . 3  |-  ( A  e.  RR+  ->  2  e. 
NN0 )
7318, 71, 72effsumlt 11297 . 2  |-  ( A  e.  RR+  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
2 )  <  ( exp `  A ) )
7470, 73eqbrtrrd 3920 1  |-  ( A  e.  RR+  ->  ( ( 1  +  A )  +  ( ( A ^ 2 )  / 
2 ) )  < 
( exp `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463   class class class wbr 3897    |-> cmpt 3957   ` cfv 5091  (class class class)co 5740   CCcc 7582   0cc0 7584   1c1 7585    + caddc 7587    < clt 7764    / cdiv 8392   2c2 8728   NN0cn0 8928   ZZcz 9005   ZZ>=cuz 9275   RR+crp 9390    seqcseq 10158   ^cexp 10232   !cfa 10411   expce 11247
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703  ax-caucvg 7704
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-isom 5100  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-frec 6254  df-1o 6279  df-oadd 6283  df-er 6395  df-en 6601  df-dom 6602  df-fin 6603  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8393  df-inn 8678  df-2 8736  df-3 8737  df-4 8738  df-n0 8929  df-z 9006  df-uz 9276  df-q 9361  df-rp 9391  df-ico 9617  df-fz 9731  df-fzo 9860  df-seqfrec 10159  df-exp 10233  df-fac 10412  df-ihash 10462  df-cj 10554  df-re 10555  df-im 10556  df-rsqrt 10710  df-abs 10711  df-clim 10988  df-sumdc 11063  df-ef 11253
This theorem is referenced by: (None)
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