Theorem efgt1p 12488

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 10080	. . 3 |- ( A e. RR+ -> A e. CC )
2		1e0p1 9834	. . . . 5 |- 1 = ( 0 + 1 )
3	2	fveq2i 5701	. . . 4 |- ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 1 ) = ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` ( 0 + 1 ) )
4		0nn0 9589	. . . . . . . 8 |- 0 e. NN0
5		nn0uz 9973	. . . . . . . 8 |- NN0 = ( ZZ>= ` 0 )
6	4, 5	eleqtri 2313	. . . . . . 7 |- 0 e. ( ZZ>= ` 0 )
7	6	a1i 9	. . . . . 6 |- ( A e. CC -> 0 e. ( ZZ>= ` 0 ) )
8		elnn0uz 9976	. . . . . . 7 |- ( k e. NN0 <-> k e. ( ZZ>= ` 0 ) )
9		eqid 2238	. . . . . . . . 9 |- ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
10	9	eftvalcn 12449	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
11		eftcl 12446	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
12	10, 11	eqeltrd 2315	. . . . . . 7 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) e. CC )
13	8, 12	sylan2br 288	. . . . . 6 |- ( ( A e. CC /\ k e. ( ZZ>= ` 0 ) ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) e. CC )
14		addcl 8310	. . . . . . 7 |- ( ( k e. CC /\ y e. CC ) -> ( k + y ) e. CC )
15	14	adantl 277	. . . . . 6 |- ( ( A e. CC /\ ( k e. CC /\ y e. CC ) ) -> ( k + y ) e. CC )
16	7, 13, 15	seq3p1 10923	. . . . 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 9667	. . . . . . . 8 |- ( A e. CC -> 0 e. ZZ )
18	17, 13, 15	seq3-1 10920	. . . . . . 7 |- ( A e. CC -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 0 ) = ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 0 ) )
19	9	eftvalcn 12449	. . . . . . . . 9 |- ( ( A e. CC /\ 0 e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) ) )
20	4, 19	mpan2 429	. . . . . . . 8 |- ( A e. CC -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) ) )
21		eft0val 12485	. . . . . . . 8 |- ( A e. CC -> ( ( A ^ 0 ) / ( ! ` 0 ) ) = 1 )
22	20, 21	eqtrd 2271	. . . . . . 7 |- ( A e. CC -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 0 ) = 1 )
23	18, 22	eqtrd 2271	. . . . . 6 |- ( A e. CC -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 0 ) = 1 )
24	2	fveq2i 5701	. . . . . . 7 |- ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` ( 0 + 1 ) )
25		1nn0 9590	. . . . . . . . 9 |- 1 e. NN0
26	9	eftvalcn 12449	. . . . . . . . 9 |- ( ( A e. CC /\ 1 e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = ( ( A ^ 1 ) / ( ! ` 1 ) ) )
27	25, 26	mpan2 429	. . . . . . . 8 |- ( A e. CC -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = ( ( A ^ 1 ) / ( ! ` 1 ) ) )
28		fac1 11189	. . . . . . . . . 10 |- ( ! ` 1 ) = 1
29	28	oveq2i 6099	. . . . . . . . 9 |- ( ( A ^ 1 ) / ( ! ` 1 ) ) = ( ( A ^ 1 ) / 1 )
30		exp1 11003	. . . . . . . . . . 11 |- ( A e. CC -> ( A ^ 1 ) = A )
31	30	oveq1d 6103	. . . . . . . . . 10 |- ( A e. CC -> ( ( A ^ 1 ) / 1 ) = ( A / 1 ) )
32		div1 9041	. . . . . . . . . 10 |- ( A e. CC -> ( A / 1 ) = A )
33	31, 32	eqtrd 2271	. . . . . . . . 9 |- ( A e. CC -> ( ( A ^ 1 ) / 1 ) = A )
34	29, 33	eqtrid 2283	. . . . . . . 8 |- ( A e. CC -> ( ( A ^ 1 ) / ( ! ` 1 ) ) = A )
35	27, 34	eqtrd 2271	. . . . . . 7 |- ( A e. CC -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = A )
36	24, 35	eqtr3id 2285	. . . . . 6 |- ( A e. CC -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` ( 0 + 1 ) ) = A )
37	23, 36	oveq12d 6106	. . . . 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 2271	. . . 4 |- ( A e. CC -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` ( 0 + 1 ) ) = ( 1 + A ) )
39	3, 38	eqtrid 2283	. . 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 12484	. 2 |- ( A e. RR+ -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 1 ) < ( exp ` A ) )
44	40, 43	eqbrtrrd 4155	1 |- ( A e. RR+ -> ( 1 + A ) < ( exp ` A ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   class class class wbr 4131   |-> cmpt 4193   ` cfv 5380  (class class class)co 6088   CCcc 8183   0cc0 8185   1c1 8186   + caddc 8188   < clt 8366   / cdiv 9010   NN0cn0 9574   ZZ>=cuz 9937   RR+crp 10071   seqcseq 10905   ^cexp 10996   !cfa 11185   expce 12434

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-oadd 6695  df-er 6811  df-en 7027  df-dom 7028  df-fin 7029  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-ico 10313  df-fz 10429  df-fzo 10567  df-seqfrec 10906  df-exp 10997  df-fac 11186  df-ihash 11237  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-clim 12070  df-sumdc 12145  df-ef 12440

This theorem is used by:  efgt1  12489  reeff1olem  15997  logdivlti  16111