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Theorem efgt1p 11659
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 9619 . . 3  |-  ( A  e.  RR+  ->  A  e.  CC )
2 1e0p1 9384 . . . . 5  |-  1  =  ( 0  +  1 )
32fveq2i 5499 . . . 4  |-  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
1 )  =  (  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ) `
 ( 0  +  1 ) )
4 0nn0 9150 . . . . . . . 8  |-  0  e.  NN0
5 nn0uz 9521 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  0 )
64, 5eleqtri 2245 . . . . . . 7  |-  0  e.  ( ZZ>= `  0 )
76a1i 9 . . . . . 6  |-  ( A  e.  CC  ->  0  e.  ( ZZ>= `  0 )
)
8 elnn0uz 9524 . . . . . . 7  |-  ( k  e.  NN0  <->  k  e.  (
ZZ>= `  0 ) )
9 eqid 2170 . . . . . . . . 9  |-  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) )
109eftvalcn 11620 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  k )  =  ( ( A ^ k
)  /  ( ! `
 k ) ) )
11 eftcl 11617 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( A ^
k )  /  ( ! `  k )
)  e.  CC )
1210, 11eqeltrd 2247 . . . . . . 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 7899 . . . . . . 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 10418 . . . . 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 9224 . . . . . . . 8  |-  ( A  e.  CC  ->  0  e.  ZZ )
1817, 13, 15seq3-1 10416 . . . . . . 7  |-  ( A  e.  CC  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
0 )  =  ( ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  0
) )
199eftvalcn 11620 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  0  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ` 
0 )  =  ( ( A ^ 0 )  /  ( ! `
 0 ) ) )
204, 19mpan2 423 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  0
)  =  ( ( A ^ 0 )  /  ( ! ` 
0 ) ) )
21 eft0val 11656 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( A ^ 0 )  /  ( ! `
 0 ) )  =  1 )
2220, 21eqtrd 2203 . . . . . . 7  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  0
)  =  1 )
2318, 22eqtrd 2203 . . . . . 6  |-  ( A  e.  CC  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
0 )  =  1 )
242fveq2i 5499 . . . . . . 7  |-  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 1 )  =  ( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  ( 0  +  1 ) )
25 1nn0 9151 . . . . . . . . 9  |-  1  e.  NN0
269eftvalcn 11620 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  1  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ` 
1 )  =  ( ( A ^ 1 )  /  ( ! `
 1 ) ) )
2725, 26mpan2 423 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  1
)  =  ( ( A ^ 1 )  /  ( ! ` 
1 ) ) )
28 fac1 10663 . . . . . . . . . 10  |-  ( ! `
 1 )  =  1
2928oveq2i 5864 . . . . . . . . 9  |-  ( ( A ^ 1 )  /  ( ! ` 
1 ) )  =  ( ( A ^
1 )  /  1
)
30 exp1 10482 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
3130oveq1d 5868 . . . . . . . . . 10  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  1 )  =  ( A  / 
1 ) )
32 div1 8620 . . . . . . . . . 10  |-  ( A  e.  CC  ->  ( A  /  1 )  =  A )
3331, 32eqtrd 2203 . . . . . . . . 9  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  1 )  =  A )
3429, 33eqtrid 2215 . . . . . . . 8  |-  ( A  e.  CC  ->  (
( A ^ 1 )  /  ( ! `
 1 ) )  =  A )
3527, 34eqtrd 2203 . . . . . . 7  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  1
)  =  A )
3624, 35eqtr3id 2217 . . . . . 6  |-  ( A  e.  CC  ->  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  (
0  +  1 ) )  =  A )
3723, 36oveq12d 5871 . . . . 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 2203 . . . 4  |-  ( A  e.  CC  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) `  ( 0  +  1 ) )  =  ( 1  +  A ) )
393, 38eqtrid 2215 . . 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 11655 . 2  |-  ( A  e.  RR+  ->  (  seq 0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) ` 
1 )  <  ( exp `  A ) )
4440, 43eqbrtrrd 4013 1  |-  ( A  e.  RR+  ->  ( 1  +  A )  < 
( exp `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    e. wcel 2141   class class class wbr 3989    |-> cmpt 4050   ` cfv 5198  (class class class)co 5853   CCcc 7772   0cc0 7774   1c1 7775    + caddc 7777    < clt 7954    / cdiv 8589   NN0cn0 9135   ZZ>=cuz 9487   RR+crp 9610    seqcseq 10401   ^cexp 10475   !cfa 10659   expce 11605
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-ico 9851  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-fac 10660  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317  df-ef 11611
This theorem is referenced by:  efgt1  11660  reeff1olem  13486  logdivlti  13596
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