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Theorem efgt1p2 11735
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 9222 . . . . . . 7  |-  1  e.  NN0
2 nn0uz 9592 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
31, 2eleqtri 2264 . . . . . 6  |-  1  e.  ( ZZ>= `  0 )
43a1i 9 . . . . 5  |-  ( A  e.  RR+  ->  1  e.  ( ZZ>= `  0 )
)
5 elnn0uz 9595 . . . . . . . . 9  |-  ( k  e.  NN0  <->  k  e.  (
ZZ>= `  0 ) )
65biimpri 133 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  0
)  ->  k  e.  NN0 )
76adantl 277 . . . . . . 7  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  k  e.  NN0 )
8 simpl 109 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  A  e.  RR+ )
9 eluzelz 9567 . . . . . . . . . 10  |-  ( k  e.  ( ZZ>= `  0
)  ->  k  e.  ZZ )
109adantl 277 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  k  e.  ZZ )
118, 10rpexpcld 10709 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( A ^ k )  e.  RR+ )
127faccld 10748 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( ! `  k )  e.  NN )
1312nnrpd 9724 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( ! `  k )  e.  RR+ )
1411, 13rpdivcld 9744 . . . . . . 7  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( ( A ^ k )  / 
( ! `  k
) )  e.  RR+ )
15 oveq2 5904 . . . . . . . . 9  |-  ( n  =  k  ->  ( A ^ n )  =  ( A ^ k
) )
16 fveq2 5534 . . . . . . . . 9  |-  ( n  =  k  ->  ( ! `  n )  =  ( ! `  k ) )
1715, 16oveq12d 5914 . . . . . . . 8  |-  ( n  =  k  ->  (
( A ^ n
)  /  ( ! `
 n ) )  =  ( ( A ^ k )  / 
( ! `  k
) ) )
18 eqid 2189 . . . . . . . 8  |-  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) )
1917, 18fvmptg 5613 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ( A ^
k )  /  ( ! `  k )
)  e.  RR+ )  ->  ( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  k )  =  ( ( A ^ k
)  /  ( ! `
 k ) ) )
207, 14, 19syl2anc 411 . . . . . 6  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( (
n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 k )  =  ( ( A ^
k )  /  ( ! `  k )
) )
2120, 14eqeltrd 2266 . . . . 5  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( (
n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 k )  e.  RR+ )
22 rpaddcl 9707 . . . . . 6  |-  ( ( k  e.  RR+  /\  y  e.  RR+ )  ->  (
k  +  y )  e.  RR+ )
2322adantl 277 . . . . 5  |-  ( ( A  e.  RR+  /\  (
k  e.  RR+  /\  y  e.  RR+ ) )  -> 
( k  +  y )  e.  RR+ )
244, 21, 23seq3p1 10493 . . . 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 9008 . . . . 5  |-  2  =  ( 1  +  1 )
2625fveq2i 5537 . . . 4  |-  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
2 )  =  (  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ) `
 ( 1  +  1 ) )
2725fveq2i 5537 . . . . 5  |-  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 2 )  =  ( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  ( 1  +  1 ) )
2827oveq2i 5907 . . . 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 2247 . . 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 9221 . . . . . . . . 9  |-  0  e.  NN0
3130, 2eleqtri 2264 . . . . . . . 8  |-  0  e.  ( ZZ>= `  0 )
3231a1i 9 . . . . . . 7  |-  ( A  e.  RR+  ->  0  e.  ( ZZ>= `  0 )
)
3332, 21, 23seq3p1 10493 . . . . . 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 9455 . . . . . . 7  |-  1  =  ( 0  +  1 )
3534fveq2i 5537 . . . . . 6  |-  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
1 )  =  (  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ) `
 ( 0  +  1 ) )
3634fveq2i 5537 . . . . . . 7  |-  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 1 )  =  ( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  ( 0  +  1 ) )
3736oveq2i 5907 . . . . . 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 2247 . . . . 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 9295 . . . . . . . 8  |-  ( A  e.  RR+  ->  0  e.  ZZ )
4039, 21, 23seq3-1 10491 . . . . . . 7  |-  ( A  e.  RR+  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
0 )  =  ( ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  0
) )
41 rpcn 9692 . . . . . . . . 9  |-  ( A  e.  RR+  ->  A  e.  CC )
4218eftvalcn 11697 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  0  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ` 
0 )  =  ( ( A ^ 0 )  /  ( ! `
 0 ) ) )
4330, 42mpan2 425 . . . . . . . . 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 11733 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A ^ 0 )  /  ( ! `
 0 ) )  =  1 )
4641, 45syl 14 . . . . . . . 8  |-  ( A  e.  RR+  ->  ( ( A ^ 0 )  /  ( ! ` 
0 ) )  =  1 )
4744, 46eqtrd 2222 . . . . . . 7  |-  ( A  e.  RR+  ->  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 0 )  =  1 )
4840, 47eqtrd 2222 . . . . . 6  |-  ( A  e.  RR+  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
0 )  =  1 )
4918eftvalcn 11697 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  1  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ` 
1 )  =  ( ( A ^ 1 )  /  ( ! `
 1 ) ) )
501, 49mpan2 425 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  1
)  =  ( ( A ^ 1 )  /  ( ! ` 
1 ) ) )
51 fac1 10741 . . . . . . . . . 10  |-  ( ! `
 1 )  =  1
5251oveq2i 5907 . . . . . . . . 9  |-  ( ( A ^ 1 )  /  ( ! ` 
1 ) )  =  ( ( A ^
1 )  /  1
)
53 exp1 10557 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
5453oveq1d 5911 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  1 )  =  ( A  / 
1 ) )
55 div1 8690 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( A  /  1 )  =  A )
5654, 55eqtrd 2222 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  1 )  =  A )
5752, 56eqtrid 2234 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  ( ! `
 1 ) )  =  A )
5850, 57eqtrd 2222 . . . . . . 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 5914 . . . . 5  |-  ( A  e.  RR+  ->  ( (  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ) `
 0 )  +  ( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ` 
1 ) )  =  ( 1  +  A
) )
6138, 60eqtrd 2222 . . . 4  |-  ( A  e.  RR+  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
1 )  =  ( 1  +  A ) )
62 2nn0 9223 . . . . . . 7  |-  2  e.  NN0
6318eftvalcn 11697 . . . . . . 7  |-  ( ( A  e.  CC  /\  2  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ` 
2 )  =  ( ( A ^ 2 )  /  ( ! `
 2 ) ) )
6462, 63mpan2 425 . . . . . 6  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  2
)  =  ( ( A ^ 2 )  /  ( ! ` 
2 ) ) )
65 fac2 10743 . . . . . . 7  |-  ( ! `
 2 )  =  2
6665oveq2i 5907 . . . . . 6  |-  ( ( A ^ 2 )  /  ( ! ` 
2 ) )  =  ( ( A ^
2 )  /  2
)
6764, 66eqtrdi 2238 . . . . 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 5914 . . 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 2222 . 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 11732 . 2  |-  ( A  e.  RR+  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
2 )  <  ( exp `  A ) )
7470, 73eqbrtrrd 4042 1  |-  ( A  e.  RR+  ->  ( ( 1  +  A )  +  ( ( A ^ 2 )  / 
2 ) )  < 
( exp `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   class class class wbr 4018    |-> cmpt 4079   ` cfv 5235  (class class class)co 5896   CCcc 7839   0cc0 7841   1c1 7842    + caddc 7844    < clt 8022    / cdiv 8659   2c2 9000   NN0cn0 9206   ZZcz 9283   ZZ>=cuz 9558   RR+crp 9683    seqcseq 10476   ^cexp 10550   !cfa 10737   expce 11682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958  ax-pre-mulext 7959  ax-arch 7960  ax-caucvg 7961
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-isom 5244  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-irdg 6395  df-frec 6416  df-1o 6441  df-oadd 6445  df-er 6559  df-en 6767  df-dom 6768  df-fin 6769  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569  df-div 8660  df-inn 8950  df-2 9008  df-3 9009  df-4 9010  df-n0 9207  df-z 9284  df-uz 9559  df-q 9650  df-rp 9684  df-ico 9924  df-fz 10039  df-fzo 10173  df-seqfrec 10477  df-exp 10551  df-fac 10738  df-ihash 10788  df-cj 10883  df-re 10884  df-im 10885  df-rsqrt 11039  df-abs 11040  df-clim 11319  df-sumdc 11394  df-ef 11688
This theorem is referenced by: (None)
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