Theorem efgt1p 12382

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 9995	. . 3 |- ( A e. RR+ -> A e. CC )
2		1e0p1 9750	. . . . 5 |- 1 = ( 0 + 1 )
3	2	fveq2i 5673	. . . 4 |- ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 1 ) = ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` ( 0 + 1 ) )
4		0nn0 9511	. . . . . . . 8 |- 0 e. NN0
5		nn0uz 9889	. . . . . . . 8 |- NN0 = ( ZZ>= ` 0 )
6	4, 5	eleqtri 2307	. . . . . . 7 |- 0 e. ( ZZ>= ` 0 )
7	6	a1i 9	. . . . . 6 |- ( A e. CC -> 0 e. ( ZZ>= ` 0 ) )
8		elnn0uz 9892	. . . . . . 7 |- ( k e. NN0 <-> k e. ( ZZ>= ` 0 ) )
9		eqid 2232	. . . . . . . . 9 |- ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
10	9	eftvalcn 12343	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
11		eftcl 12340	. . . . . . . 8 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
12	10, 11	eqeltrd 2309	. . . . . . 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 8252	. . . . . . 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 10827	. . . . 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 9589	. . . . . . . 8 |- ( A e. CC -> 0 e. ZZ )
18	17, 13, 15	seq3-1 10824	. . . . . . 7 |- ( A e. CC -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 0 ) = ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 0 ) )
19	9	eftvalcn 12343	. . . . . . . . 9 |- ( ( A e. CC /\ 0 e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) ) )
20	4, 19	mpan2 425	. . . . . . . 8 |- ( A e. CC -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) ) )
21		eft0val 12379	. . . . . . . 8 |- ( A e. CC -> ( ( A ^ 0 ) / ( ! ` 0 ) ) = 1 )
22	20, 21	eqtrd 2265	. . . . . . 7 |- ( A e. CC -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 0 ) = 1 )
23	18, 22	eqtrd 2265	. . . . . 6 |- ( A e. CC -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 0 ) = 1 )
24	2	fveq2i 5673	. . . . . . 7 |- ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` ( 0 + 1 ) )
25		1nn0 9512	. . . . . . . . 9 |- 1 e. NN0
26	9	eftvalcn 12343	. . . . . . . . 9 |- ( ( A e. CC /\ 1 e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = ( ( A ^ 1 ) / ( ! ` 1 ) ) )
27	25, 26	mpan2 425	. . . . . . . 8 |- ( A e. CC -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = ( ( A ^ 1 ) / ( ! ` 1 ) ) )
28		fac1 11091	. . . . . . . . . 10 |- ( ! ` 1 ) = 1
29	28	oveq2i 6061	. . . . . . . . 9 |- ( ( A ^ 1 ) / ( ! ` 1 ) ) = ( ( A ^ 1 ) / 1 )
30		exp1 10907	. . . . . . . . . . 11 |- ( A e. CC -> ( A ^ 1 ) = A )
31	30	oveq1d 6065	. . . . . . . . . 10 |- ( A e. CC -> ( ( A ^ 1 ) / 1 ) = ( A / 1 ) )
32		div1 8977	. . . . . . . . . 10 |- ( A e. CC -> ( A / 1 ) = A )
33	31, 32	eqtrd 2265	. . . . . . . . 9 |- ( A e. CC -> ( ( A ^ 1 ) / 1 ) = A )
34	29, 33	eqtrid 2277	. . . . . . . 8 |- ( A e. CC -> ( ( A ^ 1 ) / ( ! ` 1 ) ) = A )
35	27, 34	eqtrd 2265	. . . . . . 7 |- ( A e. CC -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = A )
36	24, 35	eqtr3id 2279	. . . . . 6 |- ( A e. CC -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` ( 0 + 1 ) ) = A )
37	23, 36	oveq12d 6068	. . . . 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 2265	. . . 4 |- ( A e. CC -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` ( 0 + 1 ) ) = ( 1 + A ) )
39	3, 38	eqtrid 2277	. . 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 12378	. 2 |- ( A e. RR+ -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 1 ) < ( exp ` A ) )
44	40, 43	eqbrtrrd 4133	1 |- ( A e. RR+ -> ( 1 + A ) < ( exp ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   class class class wbr 4109   |-> cmpt 4171   ` cfv 5352  (class class class)co 6050   CCcc 8125   0cc0 8127   1c1 8128   + caddc 8130   < clt 8308   / cdiv 8946   NN0cn0 9496   ZZ>=cuz 9853   RR+crp 9986   seqcseq 10809   ^cexp 10900   !cfa 11087   expce 12328

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-frec 6622  df-1o 6647  df-oadd 6651  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-ico 10227  df-fz 10343  df-fzo 10477  df-seqfrec 10810  df-exp 10901  df-fac 11088  df-ihash 11139  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-clim 11964  df-sumdc 12039  df-ef 12334

This theorem is referenced by:  efgt1  12383  reeff1olem  15636  logdivlti  15746