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Theorem efgt1p2 11048
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 8752 . . . . . . 7  |-  1  e.  NN0
2 nn0uz 9116 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
31, 2eleqtri 2163 . . . . . 6  |-  1  e.  ( ZZ>= `  0 )
43a1i 9 . . . . 5  |-  ( A  e.  RR+  ->  1  e.  ( ZZ>= `  0 )
)
5 elnn0uz 9119 . . . . . . . . 9  |-  ( k  e.  NN0  <->  k  e.  (
ZZ>= `  0 ) )
65biimpri 132 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  0
)  ->  k  e.  NN0 )
76adantl 272 . . . . . . 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 9091 . . . . . . . . . 10  |-  ( k  e.  ( ZZ>= `  0
)  ->  k  e.  ZZ )
109adantl 272 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  k  e.  ZZ )
118, 10rpexpcld 10173 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( A ^ k )  e.  RR+ )
127faccld 10207 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( ! `  k )  e.  NN )
1312nnrpd 9235 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( ! `  k )  e.  RR+ )
1411, 13rpdivcld 9254 . . . . . . 7  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( ( A ^ k )  / 
( ! `  k
) )  e.  RR+ )
15 oveq2 5676 . . . . . . . . 9  |-  ( n  =  k  ->  ( A ^ n )  =  ( A ^ k
) )
16 fveq2 5320 . . . . . . . . 9  |-  ( n  =  k  ->  ( ! `  n )  =  ( ! `  k ) )
1715, 16oveq12d 5686 . . . . . . . 8  |-  ( n  =  k  ->  (
( A ^ n
)  /  ( ! `
 n ) )  =  ( ( A ^ k )  / 
( ! `  k
) ) )
18 eqid 2089 . . . . . . . 8  |-  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) )
1917, 18fvmptg 5395 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ( A ^
k )  /  ( ! `  k )
)  e.  RR+ )  ->  ( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  k )  =  ( ( A ^ k
)  /  ( ! `
 k ) ) )
207, 14, 19syl2anc 404 . . . . . 6  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( (
n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 k )  =  ( ( A ^
k )  /  ( ! `  k )
) )
2120, 14eqeltrd 2165 . . . . 5  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( (
n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 k )  e.  RR+ )
22 rpaddcl 9220 . . . . . 6  |-  ( ( k  e.  RR+  /\  y  e.  RR+ )  ->  (
k  +  y )  e.  RR+ )
2322adantl 272 . . . . 5  |-  ( ( A  e.  RR+  /\  (
k  e.  RR+  /\  y  e.  RR+ ) )  -> 
( k  +  y )  e.  RR+ )
244, 21, 23seq3p1 9947 . . . 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 8544 . . . . 5  |-  2  =  ( 1  +  1 )
2625fveq2i 5323 . . . 4  |-  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
2 )  =  (  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ) `
 ( 1  +  1 ) )
2725fveq2i 5323 . . . . 5  |-  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 2 )  =  ( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  ( 1  +  1 ) )
2827oveq2i 5679 . . . 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 2146 . . 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 8751 . . . . . . . . 9  |-  0  e.  NN0
3130, 2eleqtri 2163 . . . . . . . 8  |-  0  e.  ( ZZ>= `  0 )
3231a1i 9 . . . . . . 7  |-  ( A  e.  RR+  ->  0  e.  ( ZZ>= `  0 )
)
3332, 21, 23seq3p1 9947 . . . . . 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 8981 . . . . . . 7  |-  1  =  ( 0  +  1 )
3534fveq2i 5323 . . . . . 6  |-  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
1 )  =  (  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ) `
 ( 0  +  1 ) )
3634fveq2i 5323 . . . . . . 7  |-  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 1 )  =  ( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  ( 0  +  1 ) )
3736oveq2i 5679 . . . . . 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 2146 . . . . 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 8825 . . . . . . . 8  |-  ( A  e.  RR+  ->  0  e.  ZZ )
4039, 21, 23seq3-1 9940 . . . . . . 7  |-  ( A  e.  RR+  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
0 )  =  ( ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  0
) )
41 rpcn 9205 . . . . . . . . 9  |-  ( A  e.  RR+  ->  A  e.  CC )
4218eftvalcn 11010 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  0  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ` 
0 )  =  ( ( A ^ 0 )  /  ( ! `
 0 ) ) )
4330, 42mpan2 417 . . . . . . . . 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 11046 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A ^ 0 )  /  ( ! `
 0 ) )  =  1 )
4641, 45syl 14 . . . . . . . 8  |-  ( A  e.  RR+  ->  ( ( A ^ 0 )  /  ( ! ` 
0 ) )  =  1 )
4744, 46eqtrd 2121 . . . . . . 7  |-  ( A  e.  RR+  ->  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 0 )  =  1 )
4840, 47eqtrd 2121 . . . . . 6  |-  ( A  e.  RR+  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
0 )  =  1 )
4918eftvalcn 11010 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  1  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ` 
1 )  =  ( ( A ^ 1 )  /  ( ! `
 1 ) ) )
501, 49mpan2 417 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  1
)  =  ( ( A ^ 1 )  /  ( ! ` 
1 ) ) )
51 fac1 10200 . . . . . . . . . 10  |-  ( ! `
 1 )  =  1
5251oveq2i 5679 . . . . . . . . 9  |-  ( ( A ^ 1 )  /  ( ! ` 
1 ) )  =  ( ( A ^
1 )  /  1
)
53 exp1 10024 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
5453oveq1d 5683 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  1 )  =  ( A  / 
1 ) )
55 div1 8233 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( A  /  1 )  =  A )
5654, 55eqtrd 2121 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  1 )  =  A )
5752, 56syl5eq 2133 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  ( ! `
 1 ) )  =  A )
5850, 57eqtrd 2121 . . . . . . 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 5686 . . . . 5  |-  ( A  e.  RR+  ->  ( (  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ) `
 0 )  +  ( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ` 
1 ) )  =  ( 1  +  A
) )
6138, 60eqtrd 2121 . . . 4  |-  ( A  e.  RR+  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
1 )  =  ( 1  +  A ) )
62 2nn0 8753 . . . . . . 7  |-  2  e.  NN0
6318eftvalcn 11010 . . . . . . 7  |-  ( ( A  e.  CC  /\  2  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ` 
2 )  =  ( ( A ^ 2 )  /  ( ! `
 2 ) ) )
6462, 63mpan2 417 . . . . . 6  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  2
)  =  ( ( A ^ 2 )  /  ( ! ` 
2 ) ) )
65 fac2 10202 . . . . . . 7  |-  ( ! `
 2 )  =  2
6665oveq2i 5679 . . . . . 6  |-  ( ( A ^ 2 )  /  ( ! ` 
2 ) )  =  ( ( A ^
2 )  /  2
)
6764, 66syl6eq 2137 . . . . 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 5686 . . 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 2121 . 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 11045 . 2  |-  ( A  e.  RR+  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
2 )  <  ( exp `  A ) )
7470, 73eqbrtrrd 3875 1  |-  ( A  e.  RR+  ->  ( ( 1  +  A )  +  ( ( A ^ 2 )  / 
2 ) )  < 
( exp `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1290    e. wcel 1439   class class class wbr 3853    |-> cmpt 3907   ` cfv 5030  (class class class)co 5668   CCcc 7411   0cc0 7413   1c1 7414    + caddc 7416    < clt 7585    / cdiv 8202   2c2 8536   NN0cn0 8736   ZZcz 8813   ZZ>=cuz 9082   RR+crp 9197    seqcseq 9915   ^cexp 10017   !cfa 10196   expce 10995
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3962  ax-sep 3965  ax-nul 3973  ax-pow 4017  ax-pr 4047  ax-un 4271  ax-setind 4368  ax-iinf 4418  ax-cnex 7499  ax-resscn 7500  ax-1cn 7501  ax-1re 7502  ax-icn 7503  ax-addcl 7504  ax-addrcl 7505  ax-mulcl 7506  ax-mulrcl 7507  ax-addcom 7508  ax-mulcom 7509  ax-addass 7510  ax-mulass 7511  ax-distr 7512  ax-i2m1 7513  ax-0lt1 7514  ax-1rid 7515  ax-0id 7516  ax-rnegex 7517  ax-precex 7518  ax-cnre 7519  ax-pre-ltirr 7520  ax-pre-ltwlin 7521  ax-pre-lttrn 7522  ax-pre-apti 7523  ax-pre-ltadd 7524  ax-pre-mulgt0 7525  ax-pre-mulext 7526  ax-arch 7527  ax-caucvg 7528
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2624  df-sbc 2844  df-csb 2937  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-nul 3290  df-if 3400  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-int 3697  df-iun 3740  df-br 3854  df-opab 3908  df-mpt 3909  df-tr 3945  df-id 4131  df-po 4134  df-iso 4135  df-iord 4204  df-on 4206  df-ilim 4207  df-suc 4209  df-iom 4421  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-f1 5035  df-fo 5036  df-f1o 5037  df-fv 5038  df-isom 5039  df-riota 5624  df-ov 5671  df-oprab 5672  df-mpt2 5673  df-1st 5927  df-2nd 5928  df-recs 6086  df-irdg 6151  df-frec 6172  df-1o 6197  df-oadd 6201  df-er 6308  df-en 6514  df-dom 6515  df-fin 6516  df-pnf 7587  df-mnf 7588  df-xr 7589  df-ltxr 7590  df-le 7591  df-sub 7718  df-neg 7719  df-reap 8115  df-ap 8122  df-div 8203  df-inn 8486  df-2 8544  df-3 8545  df-4 8546  df-n0 8737  df-z 8814  df-uz 9083  df-q 9168  df-rp 9198  df-ico 9375  df-fz 9488  df-fzo 9617  df-iseq 9916  df-seq3 9917  df-exp 10018  df-fac 10197  df-ihash 10247  df-cj 10339  df-re 10340  df-im 10341  df-rsqrt 10494  df-abs 10495  df-clim 10730  df-isum 10806  df-ef 11001
This theorem is referenced by: (None)
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