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Theorem efgt1p 11047
Description: The exponential of a positive real number is greater than 1 plus that number. (Contributed by Mario Carneiro, 14-Mar-2014.) (Revised by Mario Carneiro, 30-Apr-2014.)
Assertion
Ref Expression
efgt1p  |-  ( A  e.  RR+  ->  ( 1  +  A )  < 
( exp `  A
) )

Proof of Theorem efgt1p
Dummy variables  k  n  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rpcn 9203 . . 3  |-  ( A  e.  RR+  ->  A  e.  CC )
2 1e0p1 8979 . . . . 5  |-  1  =  ( 0  +  1 )
32fveq2i 5321 . . . 4  |-  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
1 )  =  (  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ) `
 ( 0  +  1 ) )
4 0nn0 8749 . . . . . . . 8  |-  0  e.  NN0
5 nn0uz 9114 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
64, 5eleqtri 2163 . . . . . . 7  |-  0  e.  ( ZZ>= `  0 )
76a1i 9 . . . . . 6  |-  ( A  e.  CC  ->  0  e.  ( ZZ>= `  0 )
)
8 elnn0uz 9117 . . . . . . 7  |-  ( k  e.  NN0  <->  k  e.  (
ZZ>= `  0 ) )
9 eqid 2089 . . . . . . . . 9  |-  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) )
109eftvalcn 11008 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  k )  =  ( ( A ^ k
)  /  ( ! `
 k ) ) )
11 eftcl 11005 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( A ^
k )  /  ( ! `  k )
)  e.  CC )
1210, 11eqeltrd 2165 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  k )  e.  CC )
138, 12sylan2br 283 . . . . . 6  |-  ( ( A  e.  CC  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  k )  e.  CC )
14 addcl 7528 . . . . . . 7  |-  ( ( k  e.  CC  /\  y  e.  CC )  ->  ( k  +  y )  e.  CC )
1514adantl 272 . . . . . 6  |-  ( ( A  e.  CC  /\  ( k  e.  CC  /\  y  e.  CC ) )  ->  ( k  +  y )  e.  CC )
167, 13, 15seq3p1 9945 . . . . 5  |-  ( A  e.  CC  ->  (  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 ) ) ) )
17 0zd 8823 . . . . . . . 8  |-  ( A  e.  CC  ->  0  e.  ZZ )
1817, 13, 15seq3-1 9938 . . . . . . 7  |-  ( A  e.  CC  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
0 )  =  ( ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  0
) )
199eftvalcn 11008 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  0  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ` 
0 )  =  ( ( A ^ 0 )  /  ( ! `
 0 ) ) )
204, 19mpan2 417 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  0
)  =  ( ( A ^ 0 )  /  ( ! ` 
0 ) ) )
21 eft0val 11044 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( A ^ 0 )  /  ( ! `
 0 ) )  =  1 )
2220, 21eqtrd 2121 . . . . . . 7  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  0
)  =  1 )
2318, 22eqtrd 2121 . . . . . 6  |-  ( A  e.  CC  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
0 )  =  1 )
242fveq2i 5321 . . . . . . 7  |-  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 1 )  =  ( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  ( 0  +  1 ) )
25 1nn0 8750 . . . . . . . . 9  |-  1  e.  NN0
269eftvalcn 11008 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  1  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ` 
1 )  =  ( ( A ^ 1 )  /  ( ! `
 1 ) ) )
2725, 26mpan2 417 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  1
)  =  ( ( A ^ 1 )  /  ( ! ` 
1 ) ) )
28 fac1 10198 . . . . . . . . . 10  |-  ( ! `
 1 )  =  1
2928oveq2i 5677 . . . . . . . . 9  |-  ( ( A ^ 1 )  /  ( ! ` 
1 ) )  =  ( ( A ^
1 )  /  1
)
30 exp1 10022 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
3130oveq1d 5681 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  1 )  =  ( A  / 
1 ) )
32 div1 8231 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( A  /  1 )  =  A )
3331, 32eqtrd 2121 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  1 )  =  A )
3429, 33syl5eq 2133 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  ( ! `
 1 ) )  =  A )
3527, 34eqtrd 2121 . . . . . . 7  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  1
)  =  A )
3624, 35syl5eqr 2135 . . . . . 6  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  (
0  +  1 ) )  =  A )
3723, 36oveq12d 5684 . . . . 5  |-  ( A  e.  CC  ->  (
(  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ) `
 0 )  +  ( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  ( 0  +  1 ) ) )  =  ( 1  +  A
) )
3816, 37eqtrd 2121 . . . 4  |-  ( A  e.  CC  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) `  ( 0  +  1 ) )  =  ( 1  +  A ) )
393, 38syl5eq 2133 . . 3  |-  ( A  e.  CC  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
1 )  =  ( 1  +  A ) )
401, 39syl 14 . 2  |-  ( A  e.  RR+  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
1 )  =  ( 1  +  A ) )
41 id 19 . . 3  |-  ( A  e.  RR+  ->  A  e.  RR+ )
4225a1i 9 . . 3  |-  ( A  e.  RR+  ->  1  e. 
NN0 )
439, 41, 42effsumlt 11043 . 2  |-  ( A  e.  RR+  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
1 )  <  ( exp `  A ) )
4440, 43eqbrtrrd 3873 1  |-  ( A  e.  RR+  ->  ( 1  +  A )  < 
( exp `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1290    e. wcel 1439   class class class wbr 3851    |-> cmpt 3905   ` cfv 5028  (class class class)co 5666   CCcc 7409   0cc0 7411   1c1 7412    + caddc 7414    < clt 7583    / cdiv 8200   NN0cn0 8734   ZZ>=cuz 9080   RR+crp 9195    seqcseq 9913   ^cexp 10015   !cfa 10194   expce 10993
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 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-mulrcl 7505  ax-addcom 7506  ax-mulcom 7507  ax-addass 7508  ax-mulass 7509  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-1rid 7513  ax-0id 7514  ax-rnegex 7515  ax-precex 7516  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-apti 7521  ax-pre-ltadd 7522  ax-pre-mulgt0 7523  ax-pre-mulext 7524  ax-arch 7525  ax-caucvg 7526
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 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-if 3398  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-isom 5037  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-frec 6170  df-1o 6195  df-oadd 6199  df-er 6306  df-en 6512  df-dom 6513  df-fin 6514  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-reap 8113  df-ap 8120  df-div 8201  df-inn 8484  df-2 8542  df-3 8543  df-4 8544  df-n0 8735  df-z 8812  df-uz 9081  df-q 9166  df-rp 9196  df-ico 9373  df-fz 9486  df-fzo 9615  df-iseq 9914  df-seq3 9915  df-exp 10016  df-fac 10195  df-ihash 10245  df-cj 10337  df-re 10338  df-im 10339  df-rsqrt 10492  df-abs 10493  df-clim 10728  df-isum 10804  df-ef 10999
This theorem is referenced by:  efgt1  11048
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