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.

Step	Hyp	Ref	Expression
1		rpcn 9619	. . 3 |- ( A e. RR+ -> A e. CC )
2		1e0p1 9384	. . . . 5 |- 1 = ( 0 + 1 )
3	2	fveq2i 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 )
6	4, 5	eleqtri 2245	. . . . . . 7 |- 0 e. ( ZZ>= ` 0 )
7	6	a1i 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 ) ) )
10	9	eftvalcn 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 )
12	10, 11	eqeltrd 2247	. . . . . . 7 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) e. CC )
13	8, 12	sylan2br 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 )
15	14	adantl 275	. . . . . 6 |- ( ( A e. CC /\ ( k e. CC /\ y e. CC ) ) -> ( k + y ) e. CC )
16	7, 13, 15	seq3p1 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 )
18	17, 13, 15	seq3-1 10416	. . . . . . 7 |- ( A e. CC -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 0 ) = ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 0 ) )
19	9	eftvalcn 11620	. . . . . . . . 9 |- ( ( A e. CC /\ 0 e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) ) )
20	4, 19	mpan2 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 )
22	20, 21	eqtrd 2203	. . . . . . 7 |- ( A e. CC -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 0 ) = 1 )
23	18, 22	eqtrd 2203	. . . . . 6 |- ( A e. CC -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 0 ) = 1 )
24	2	fveq2i 5499	. . . . . . 7 |- ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` ( 0 + 1 ) )
25		1nn0 9151	. . . . . . . . 9 |- 1 e. NN0
26	9	eftvalcn 11620	. . . . . . . . 9 |- ( ( A e. CC /\ 1 e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = ( ( A ^ 1 ) / ( ! ` 1 ) ) )
27	25, 26	mpan2 423	. . . . . . . 8 |- ( A e. CC -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = ( ( A ^ 1 ) / ( ! ` 1 ) ) )
28		fac1 10663	. . . . . . . . . 10 |- ( ! ` 1 ) = 1
29	28	oveq2i 5864	. . . . . . . . 9 |- ( ( A ^ 1 ) / ( ! ` 1 ) ) = ( ( A ^ 1 ) / 1 )
30		exp1 10482	. . . . . . . . . . 11 |- ( A e. CC -> ( A ^ 1 ) = A )
31	30	oveq1d 5868	. . . . . . . . . 10 |- ( A e. CC -> ( ( A ^ 1 ) / 1 ) = ( A / 1 ) )
32		div1 8620	. . . . . . . . . 10 |- ( A e. CC -> ( A / 1 ) = A )
33	31, 32	eqtrd 2203	. . . . . . . . 9 |- ( A e. CC -> ( ( A ^ 1 ) / 1 ) = A )
34	29, 33	eqtrid 2215	. . . . . . . 8 |- ( A e. CC -> ( ( A ^ 1 ) / ( ! ` 1 ) ) = A )
35	27, 34	eqtrd 2203	. . . . . . 7 |- ( A e. CC -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = A )
36	24, 35	eqtr3id 2217	. . . . . 6 |- ( A e. CC -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` ( 0 + 1 ) ) = A )
37	23, 36	oveq12d 5871	. . . . 5 |- ( A e. CC -> ( ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 0 ) + ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` ( 0 + 1 ) ) ) = ( 1 + A ) )
38	16, 37	eqtrd 2203	. . . 4 |- ( A e. CC -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` ( 0 + 1 ) ) = ( 1 + A ) )
39	3, 38	eqtrid 2215	. . 3 |- ( A e. CC -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 1 ) = ( 1 + A ) )
40	1, 39	syl 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+ )
42	25	a1i 9	. . 3 |- ( A e. RR+ -> 1 e. NN0 )
43	9, 41, 42	effsumlt 11655	. 2 |- ( A e. RR+ -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 1 ) < ( exp ` A ) )
44	40, 43	eqbrtrrd 4013	1 |- ( A e. RR+ -> ( 1 + A ) < ( exp ` A ) )

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