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Theorem efgt1p 11409
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 9457 . . 3  |-  ( A  e.  RR+  ->  A  e.  CC )
2 1e0p1 9230 . . . . 5  |-  1  =  ( 0  +  1 )
32fveq2i 5424 . . . 4  |-  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
1 )  =  (  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ) `
 ( 0  +  1 ) )
4 0nn0 8999 . . . . . . . 8  |-  0  e.  NN0
5 nn0uz 9367 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
64, 5eleqtri 2214 . . . . . . 7  |-  0  e.  ( ZZ>= `  0 )
76a1i 9 . . . . . 6  |-  ( A  e.  CC  ->  0  e.  ( ZZ>= `  0 )
)
8 elnn0uz 9370 . . . . . . 7  |-  ( k  e.  NN0  <->  k  e.  (
ZZ>= `  0 ) )
9 eqid 2139 . . . . . . . . 9  |-  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) )
109eftvalcn 11370 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  k )  =  ( ( A ^ k
)  /  ( ! `
 k ) ) )
11 eftcl 11367 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( A ^
k )  /  ( ! `  k )
)  e.  CC )
1210, 11eqeltrd 2216 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  k )  e.  CC )
138, 12sylan2br 286 . . . . . 6  |-  ( ( A  e.  CC  /\  k  e.  ( ZZ>= ` 
0 ) )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  k )  e.  CC )
14 addcl 7752 . . . . . . 7  |-  ( ( k  e.  CC  /\  y  e.  CC )  ->  ( k  +  y )  e.  CC )
1514adantl 275 . . . . . 6  |-  ( ( A  e.  CC  /\  ( k  e.  CC  /\  y  e.  CC ) )  ->  ( k  +  y )  e.  CC )
167, 13, 15seq3p1 10242 . . . . 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 9073 . . . . . . . 8  |-  ( A  e.  CC  ->  0  e.  ZZ )
1817, 13, 15seq3-1 10240 . . . . . . 7  |-  ( A  e.  CC  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
0 )  =  ( ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  0
) )
199eftvalcn 11370 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  0  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ` 
0 )  =  ( ( A ^ 0 )  /  ( ! `
 0 ) ) )
204, 19mpan2 421 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  0
)  =  ( ( A ^ 0 )  /  ( ! ` 
0 ) ) )
21 eft0val 11406 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( A ^ 0 )  /  ( ! `
 0 ) )  =  1 )
2220, 21eqtrd 2172 . . . . . . 7  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  0
)  =  1 )
2318, 22eqtrd 2172 . . . . . 6  |-  ( A  e.  CC  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
0 )  =  1 )
242fveq2i 5424 . . . . . . 7  |-  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 1 )  =  ( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  ( 0  +  1 ) )
25 1nn0 9000 . . . . . . . . 9  |-  1  e.  NN0
269eftvalcn 11370 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  1  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ` 
1 )  =  ( ( A ^ 1 )  /  ( ! `
 1 ) ) )
2725, 26mpan2 421 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  1
)  =  ( ( A ^ 1 )  /  ( ! ` 
1 ) ) )
28 fac1 10482 . . . . . . . . . 10  |-  ( ! `
 1 )  =  1
2928oveq2i 5785 . . . . . . . . 9  |-  ( ( A ^ 1 )  /  ( ! ` 
1 ) )  =  ( ( A ^
1 )  /  1
)
30 exp1 10306 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
3130oveq1d 5789 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  1 )  =  ( A  / 
1 ) )
32 div1 8470 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( A  /  1 )  =  A )
3331, 32eqtrd 2172 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  1 )  =  A )
3429, 33syl5eq 2184 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  ( ! `
 1 ) )  =  A )
3527, 34eqtrd 2172 . . . . . . 7  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  1
)  =  A )
3624, 35syl5eqr 2186 . . . . . 6  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  (
0  +  1 ) )  =  A )
3723, 36oveq12d 5792 . . . . 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 2172 . . . 4  |-  ( A  e.  CC  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) `  ( 0  +  1 ) )  =  ( 1  +  A ) )
393, 38syl5eq 2184 . . 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 11405 . 2  |-  ( A  e.  RR+  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
1 )  <  ( exp `  A ) )
4440, 43eqbrtrrd 3952 1  |-  ( A  e.  RR+  ->  ( 1  +  A )  < 
( exp `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   class class class wbr 3929    |-> cmpt 3989   ` cfv 5123  (class class class)co 5774   CCcc 7625   0cc0 7627   1c1 7628    + caddc 7630    < clt 7807    / cdiv 8439   NN0cn0 8984   ZZ>=cuz 9333   RR+crp 9448    seqcseq 10225   ^cexp 10299   !cfa 10478   expce 11355
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745  ax-arch 7746  ax-caucvg 7747
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440  df-inn 8728  df-2 8786  df-3 8787  df-4 8788  df-n0 8985  df-z 9062  df-uz 9334  df-q 9419  df-rp 9449  df-ico 9684  df-fz 9798  df-fzo 9927  df-seqfrec 10226  df-exp 10300  df-fac 10479  df-ihash 10529  df-cj 10621  df-re 10622  df-im 10623  df-rsqrt 10777  df-abs 10778  df-clim 11055  df-sumdc 11130  df-ef 11361
This theorem is referenced by:  efgt1  11410  reeff1olem  12870
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