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Theorem efgt1p2 11632
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 9126 . . . . . . 7  |-  1  e.  NN0
2 nn0uz 9496 . . . . . . 7  |-  NN0  =  ( ZZ>= `  0 )
31, 2eleqtri 2240 . . . . . 6  |-  1  e.  ( ZZ>= `  0 )
43a1i 9 . . . . 5  |-  ( A  e.  RR+  ->  1  e.  ( ZZ>= `  0 )
)
5 elnn0uz 9499 . . . . . . . . 9  |-  ( k  e.  NN0  <->  k  e.  (
ZZ>= `  0 ) )
65biimpri 132 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  0
)  ->  k  e.  NN0 )
76adantl 275 . . . . . . 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 9471 . . . . . . . . . 10  |-  ( k  e.  ( ZZ>= `  0
)  ->  k  e.  ZZ )
109adantl 275 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  k  e.  ZZ )
118, 10rpexpcld 10608 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( A ^ k )  e.  RR+ )
127faccld 10645 . . . . . . . . 9  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( ! `  k )  e.  NN )
1312nnrpd 9626 . . . . . . . 8  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( ! `  k )  e.  RR+ )
1411, 13rpdivcld 9646 . . . . . . 7  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( ( A ^ k )  / 
( ! `  k
) )  e.  RR+ )
15 oveq2 5849 . . . . . . . . 9  |-  ( n  =  k  ->  ( A ^ n )  =  ( A ^ k
) )
16 fveq2 5485 . . . . . . . . 9  |-  ( n  =  k  ->  ( ! `  n )  =  ( ! `  k ) )
1715, 16oveq12d 5859 . . . . . . . 8  |-  ( n  =  k  ->  (
( A ^ n
)  /  ( ! `
 n ) )  =  ( ( A ^ k )  / 
( ! `  k
) ) )
18 eqid 2165 . . . . . . . 8  |-  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) )
1917, 18fvmptg 5561 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ( A ^
k )  /  ( ! `  k )
)  e.  RR+ )  ->  ( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  k )  =  ( ( A ^ k
)  /  ( ! `
 k ) ) )
207, 14, 19syl2anc 409 . . . . . 6  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( (
n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 k )  =  ( ( A ^
k )  /  ( ! `  k )
) )
2120, 14eqeltrd 2242 . . . . 5  |-  ( ( A  e.  RR+  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( (
n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 k )  e.  RR+ )
22 rpaddcl 9609 . . . . . 6  |-  ( ( k  e.  RR+  /\  y  e.  RR+ )  ->  (
k  +  y )  e.  RR+ )
2322adantl 275 . . . . 5  |-  ( ( A  e.  RR+  /\  (
k  e.  RR+  /\  y  e.  RR+ ) )  -> 
( k  +  y )  e.  RR+ )
244, 21, 23seq3p1 10393 . . . 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 8912 . . . . 5  |-  2  =  ( 1  +  1 )
2625fveq2i 5488 . . . 4  |-  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
2 )  =  (  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ) `
 ( 1  +  1 ) )
2725fveq2i 5488 . . . . 5  |-  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 2 )  =  ( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  ( 1  +  1 ) )
2827oveq2i 5852 . . . 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 2223 . . 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 9125 . . . . . . . . 9  |-  0  e.  NN0
3130, 2eleqtri 2240 . . . . . . . 8  |-  0  e.  ( ZZ>= `  0 )
3231a1i 9 . . . . . . 7  |-  ( A  e.  RR+  ->  0  e.  ( ZZ>= `  0 )
)
3332, 21, 23seq3p1 10393 . . . . . 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 9359 . . . . . . 7  |-  1  =  ( 0  +  1 )
3534fveq2i 5488 . . . . . 6  |-  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
1 )  =  (  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ) `
 ( 0  +  1 ) )
3634fveq2i 5488 . . . . . . 7  |-  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 1 )  =  ( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  ( 0  +  1 ) )
3736oveq2i 5852 . . . . . 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 2223 . . . . 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 9199 . . . . . . . 8  |-  ( A  e.  RR+  ->  0  e.  ZZ )
4039, 21, 23seq3-1 10391 . . . . . . 7  |-  ( A  e.  RR+  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
0 )  =  ( ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  0
) )
41 rpcn 9594 . . . . . . . . 9  |-  ( A  e.  RR+  ->  A  e.  CC )
4218eftvalcn 11594 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  0  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ` 
0 )  =  ( ( A ^ 0 )  /  ( ! `
 0 ) ) )
4330, 42mpan2 422 . . . . . . . . 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 11630 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A ^ 0 )  /  ( ! `
 0 ) )  =  1 )
4641, 45syl 14 . . . . . . . 8  |-  ( A  e.  RR+  ->  ( ( A ^ 0 )  /  ( ! ` 
0 ) )  =  1 )
4744, 46eqtrd 2198 . . . . . . 7  |-  ( A  e.  RR+  ->  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 0 )  =  1 )
4840, 47eqtrd 2198 . . . . . 6  |-  ( A  e.  RR+  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
0 )  =  1 )
4918eftvalcn 11594 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  1  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ` 
1 )  =  ( ( A ^ 1 )  /  ( ! `
 1 ) ) )
501, 49mpan2 422 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  1
)  =  ( ( A ^ 1 )  /  ( ! ` 
1 ) ) )
51 fac1 10638 . . . . . . . . . 10  |-  ( ! `
 1 )  =  1
5251oveq2i 5852 . . . . . . . . 9  |-  ( ( A ^ 1 )  /  ( ! ` 
1 ) )  =  ( ( A ^
1 )  /  1
)
53 exp1 10457 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
5453oveq1d 5856 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  1 )  =  ( A  / 
1 ) )
55 div1 8595 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( A  /  1 )  =  A )
5654, 55eqtrd 2198 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  1 )  =  A )
5752, 56syl5eq 2210 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  ( ! `
 1 ) )  =  A )
5850, 57eqtrd 2198 . . . . . . 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 5859 . . . . 5  |-  ( A  e.  RR+  ->  ( (  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ) `
 0 )  +  ( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ` 
1 ) )  =  ( 1  +  A
) )
6138, 60eqtrd 2198 . . . 4  |-  ( A  e.  RR+  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
1 )  =  ( 1  +  A ) )
62 2nn0 9127 . . . . . . 7  |-  2  e.  NN0
6318eftvalcn 11594 . . . . . . 7  |-  ( ( A  e.  CC  /\  2  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ` 
2 )  =  ( ( A ^ 2 )  /  ( ! `
 2 ) ) )
6462, 63mpan2 422 . . . . . 6  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  2
)  =  ( ( A ^ 2 )  /  ( ! ` 
2 ) ) )
65 fac2 10640 . . . . . . 7  |-  ( ! `
 2 )  =  2
6665oveq2i 5852 . . . . . 6  |-  ( ( A ^ 2 )  /  ( ! ` 
2 ) )  =  ( ( A ^
2 )  /  2
)
6764, 66eqtrdi 2214 . . . . 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 5859 . . 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 2198 . 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 11629 . 2  |-  ( A  e.  RR+  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
2 )  <  ( exp `  A ) )
7470, 73eqbrtrrd 4005 1  |-  ( A  e.  RR+  ->  ( ( 1  +  A )  +  ( ( A ^ 2 )  / 
2 ) )  < 
( exp `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1343    e. wcel 2136   class class class wbr 3981    |-> cmpt 4042   ` cfv 5187  (class class class)co 5841   CCcc 7747   0cc0 7749   1c1 7750    + caddc 7752    < clt 7929    / cdiv 8564   2c2 8904   NN0cn0 9110   ZZcz 9187   ZZ>=cuz 9462   RR+crp 9585    seqcseq 10376   ^cexp 10450   !cfa 10634   expce 11579
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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4096  ax-sep 4099  ax-nul 4107  ax-pow 4152  ax-pr 4186  ax-un 4410  ax-setind 4513  ax-iinf 4564  ax-cnex 7840  ax-resscn 7841  ax-1cn 7842  ax-1re 7843  ax-icn 7844  ax-addcl 7845  ax-addrcl 7846  ax-mulcl 7847  ax-mulrcl 7848  ax-addcom 7849  ax-mulcom 7850  ax-addass 7851  ax-mulass 7852  ax-distr 7853  ax-i2m1 7854  ax-0lt1 7855  ax-1rid 7856  ax-0id 7857  ax-rnegex 7858  ax-precex 7859  ax-cnre 7860  ax-pre-ltirr 7861  ax-pre-ltwlin 7862  ax-pre-lttrn 7863  ax-pre-apti 7864  ax-pre-ltadd 7865  ax-pre-mulgt0 7866  ax-pre-mulext 7867  ax-arch 7868  ax-caucvg 7869
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-nel 2431  df-ral 2448  df-rex 2449  df-reu 2450  df-rmo 2451  df-rab 2452  df-v 2727  df-sbc 2951  df-csb 3045  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-nul 3409  df-if 3520  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-int 3824  df-iun 3867  df-br 3982  df-opab 4043  df-mpt 4044  df-tr 4080  df-id 4270  df-po 4273  df-iso 4274  df-iord 4343  df-on 4345  df-ilim 4346  df-suc 4348  df-iom 4567  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-rn 4614  df-res 4615  df-ima 4616  df-iota 5152  df-fun 5189  df-fn 5190  df-f 5191  df-f1 5192  df-fo 5193  df-f1o 5194  df-fv 5195  df-isom 5196  df-riota 5797  df-ov 5844  df-oprab 5845  df-mpo 5846  df-1st 6105  df-2nd 6106  df-recs 6269  df-irdg 6334  df-frec 6355  df-1o 6380  df-oadd 6384  df-er 6497  df-en 6703  df-dom 6704  df-fin 6705  df-pnf 7931  df-mnf 7932  df-xr 7933  df-ltxr 7934  df-le 7935  df-sub 8067  df-neg 8068  df-reap 8469  df-ap 8476  df-div 8565  df-inn 8854  df-2 8912  df-3 8913  df-4 8914  df-n0 9111  df-z 9188  df-uz 9463  df-q 9554  df-rp 9586  df-ico 9826  df-fz 9941  df-fzo 10074  df-seqfrec 10377  df-exp 10451  df-fac 10635  df-ihash 10685  df-cj 10780  df-re 10781  df-im 10782  df-rsqrt 10936  df-abs 10937  df-clim 11216  df-sumdc 11291  df-ef 11585
This theorem is referenced by: (None)
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