Theorem efgt1p2 11048

Description: The exponential of a positive real number is greater than the sum of the first three terms of the series expansion. (Contributed by Mario Carneiro, 15-Sep-2014.)

Assertion

Ref	Expression
efgt1p2	|- ( A e. RR+ -> ( ( 1 + A ) + ( ( A ^ 2 ) / 2 ) ) < ( exp ` A ) )

Proof of Theorem efgt1p2

Dummy variables k n y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1nn0 8752	. . . . . . 7 |- 1 e. NN0
2		nn0uz 9116	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
3	1, 2	eleqtri 2163	. . . . . 6 |- 1 e. ( ZZ>= ` 0 )
4	3	a1i 9	. . . . 5 |- ( A e. RR+ -> 1 e. ( ZZ>= ` 0 ) )
5		elnn0uz 9119	. . . . . . . . 9 |- ( k e. NN0 <-> k e. ( ZZ>= ` 0 ) )
6	5	biimpri 132	. . . . . . . 8 |- ( k e. ( ZZ>= ` 0 ) -> k e. NN0 )
7	6	adantl 272	. . . . . . 7 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> k e. NN0 )
8		simpl 108	. . . . . . . . 9 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> A e. RR+ )
9		eluzelz 9091	. . . . . . . . . 10 |- ( k e. ( ZZ>= ` 0 ) -> k e. ZZ )
10	9	adantl 272	. . . . . . . . 9 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> k e. ZZ )
11	8, 10	rpexpcld 10173	. . . . . . . 8 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( A ^ k ) e. RR+ )
12	7	faccld 10207	. . . . . . . . 9 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( ! ` k ) e. NN )
13	12	nnrpd 9235	. . . . . . . 8 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( ! ` k ) e. RR+ )
14	11, 13	rpdivcld 9254	. . . . . . 7 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( ( A ^ k ) / ( ! ` k ) ) e. RR+ )
15		oveq2 5676	. . . . . . . . 9 |- ( n = k -> ( A ^ n ) = ( A ^ k ) )
16		fveq2 5320	. . . . . . . . 9 |- ( n = k -> ( ! ` n ) = ( ! ` k ) )
17	15, 16	oveq12d 5686	. . . . . . . 8 |- ( n = k -> ( ( A ^ n ) / ( ! ` n ) ) = ( ( A ^ k ) / ( ! ` k ) ) )
18		eqid 2089	. . . . . . . 8 |- ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
19	17, 18	fvmptg 5395	. . . . . . 7 |- ( ( k e. NN0 /\ ( ( A ^ k ) / ( ! ` k ) ) e. RR+ ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
20	7, 14, 19	syl2anc 404	. . . . . 6 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
21	20, 14	eqeltrd 2165	. . . . 5 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) e. RR+ )
22		rpaddcl 9220	. . . . . 6 |- ( ( k e. RR+ /\ y e. RR+ ) -> ( k + y ) e. RR+ )
23	22	adantl 272	. . . . 5 |- ( ( A e. RR+ /\ ( k e. RR+ /\ y e. RR+ ) ) -> ( k + y ) e. RR+ )
24	4, 21, 23	seq3p1 9947	. . . 4 |- ( A e. RR+ -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` ( 1 + 1 ) ) = ( ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 1 ) + ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` ( 1 + 1 ) ) ) )
25		df-2 8544	. . . . 5 |- 2 = ( 1 + 1 )
26	25	fveq2i 5323	. . . 4 |- ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 2 ) = ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` ( 1 + 1 ) )
27	25	fveq2i 5323	. . . . 5 |- ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 2 ) = ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` ( 1 + 1 ) )
28	27	oveq2i 5679	. . . 4 |- ( ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 1 ) + ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 2 ) ) = ( ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 1 ) + ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` ( 1 + 1 ) ) )
29	24, 26, 28	3eqtr4g 2146	. . 3 |- ( A e. RR+ -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 2 ) = ( ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 1 ) + ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 2 ) ) )
30		0nn0 8751	. . . . . . . . 9 |- 0 e. NN0
31	30, 2	eleqtri 2163	. . . . . . . 8 |- 0 e. ( ZZ>= ` 0 )
32	31	a1i 9	. . . . . . 7 |- ( A e. RR+ -> 0 e. ( ZZ>= ` 0 ) )
33	32, 21, 23	seq3p1 9947	. . . . . 6 |- ( A e. RR+ -> ( 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 ) ) ) )
34		1e0p1 8981	. . . . . . 7 |- 1 = ( 0 + 1 )
35	34	fveq2i 5323	. . . . . 6 |- ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 1 ) = ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` ( 0 + 1 ) )
36	34	fveq2i 5323	. . . . . . 7 |- ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` ( 0 + 1 ) )
37	36	oveq2i 5679	. . . . . 6 |- ( ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 0 ) + ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) ) = ( ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 0 ) + ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` ( 0 + 1 ) ) )
38	33, 35, 37	3eqtr4g 2146	. . . . 5 |- ( A e. RR+ -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 1 ) = ( ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 0 ) + ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) ) )
39		0zd 8825	. . . . . . . 8 |- ( A e. RR+ -> 0 e. ZZ )
40	39, 21, 23	seq3-1 9940	. . . . . . 7 |- ( A e. RR+ -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 0 ) = ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 0 ) )
41		rpcn 9205	. . . . . . . . 9 |- ( A e. RR+ -> A e. CC )
42	18	eftvalcn 11010	. . . . . . . . . 10 |- ( ( A e. CC /\ 0 e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) ) )
43	30, 42	mpan2 417	. . . . . . . . 9 |- ( A e. CC -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) ) )
44	41, 43	syl 14	. . . . . . . 8 |- ( A e. RR+ -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) ) )
45		eft0val 11046	. . . . . . . . 9 |- ( A e. CC -> ( ( A ^ 0 ) / ( ! ` 0 ) ) = 1 )
46	41, 45	syl 14	. . . . . . . 8 |- ( A e. RR+ -> ( ( A ^ 0 ) / ( ! ` 0 ) ) = 1 )
47	44, 46	eqtrd 2121	. . . . . . 7 |- ( A e. RR+ -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 0 ) = 1 )
48	40, 47	eqtrd 2121	. . . . . 6 |- ( A e. RR+ -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 0 ) = 1 )
49	18	eftvalcn 11010	. . . . . . . . 9 |- ( ( A e. CC /\ 1 e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = ( ( A ^ 1 ) / ( ! ` 1 ) ) )
50	1, 49	mpan2 417	. . . . . . . 8 |- ( A e. CC -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = ( ( A ^ 1 ) / ( ! ` 1 ) ) )
51		fac1 10200	. . . . . . . . . 10 |- ( ! ` 1 ) = 1
52	51	oveq2i 5679	. . . . . . . . 9 |- ( ( A ^ 1 ) / ( ! ` 1 ) ) = ( ( A ^ 1 ) / 1 )
53		exp1 10024	. . . . . . . . . . 11 |- ( A e. CC -> ( A ^ 1 ) = A )
54	53	oveq1d 5683	. . . . . . . . . 10 |- ( A e. CC -> ( ( A ^ 1 ) / 1 ) = ( A / 1 ) )
55		div1 8233	. . . . . . . . . 10 |- ( A e. CC -> ( A / 1 ) = A )
56	54, 55	eqtrd 2121	. . . . . . . . 9 |- ( A e. CC -> ( ( A ^ 1 ) / 1 ) = A )
57	52, 56	syl5eq 2133	. . . . . . . 8 |- ( A e. CC -> ( ( A ^ 1 ) / ( ! ` 1 ) ) = A )
58	50, 57	eqtrd 2121	. . . . . . 7 |- ( A e. CC -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = A )
59	41, 58	syl 14	. . . . . 6 |- ( A e. RR+ -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = A )
60	48, 59	oveq12d 5686	. . . . 5 |- ( A e. RR+ -> ( ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 0 ) + ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) ) = ( 1 + A ) )
61	38, 60	eqtrd 2121	. . . 4 |- ( A e. RR+ -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 1 ) = ( 1 + A ) )
62		2nn0 8753	. . . . . . 7 |- 2 e. NN0
63	18	eftvalcn 11010	. . . . . . 7 |- ( ( A e. CC /\ 2 e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 2 ) = ( ( A ^ 2 ) / ( ! ` 2 ) ) )
64	62, 63	mpan2 417	. . . . . 6 |- ( A e. CC -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 2 ) = ( ( A ^ 2 ) / ( ! ` 2 ) ) )
65		fac2 10202	. . . . . . 7 |- ( ! ` 2 ) = 2
66	65	oveq2i 5679	. . . . . 6 |- ( ( A ^ 2 ) / ( ! ` 2 ) ) = ( ( A ^ 2 ) / 2 )
67	64, 66	syl6eq 2137	. . . . 5 |- ( A e. CC -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 2 ) = ( ( A ^ 2 ) / 2 ) )
68	41, 67	syl 14	. . . 4 |- ( A e. RR+ -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 2 ) = ( ( A ^ 2 ) / 2 ) )
69	61, 68	oveq12d 5686	. . 3 |- ( A e. RR+ -> ( ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 1 ) + ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 2 ) ) = ( ( 1 + A ) + ( ( A ^ 2 ) / 2 ) ) )
70	29, 69	eqtrd 2121	. 2 |- ( A e. RR+ -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 2 ) = ( ( 1 + A ) + ( ( A ^ 2 ) / 2 ) ) )
71		id 19	. . 3 |- ( A e. RR+ -> A e. RR+ )
72	62	a1i 9	. . 3 |- ( A e. RR+ -> 2 e. NN0 )
73	18, 71, 72	effsumlt 11045	. 2 |- ( A e. RR+ -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 2 ) < ( exp ` A ) )
74	70, 73	eqbrtrrd 3875	1 |- ( A e. RR+ -> ( ( 1 + A ) + ( ( A ^ 2 ) / 2 ) ) < ( exp ` A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1290   e. wcel 1439   class class class wbr 3853   |-> cmpt 3907   ` cfv 5030  (class class class)co 5668   CCcc 7411   0cc0 7413   1c1 7414   + caddc 7416   < clt 7585   / cdiv 8202   2c2 8536   NN0cn0 8736   ZZcz 8813   ZZ>=cuz 9082   RR+crp 9197   seqcseq 9915   ^cexp 10017   !cfa 10196   expce 10995

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3962  ax-sep 3965  ax-nul 3973  ax-pow 4017  ax-pr 4047  ax-un 4271  ax-setind 4368  ax-iinf 4418  ax-cnex 7499  ax-resscn 7500  ax-1cn 7501  ax-1re 7502  ax-icn 7503  ax-addcl 7504  ax-addrcl 7505  ax-mulcl 7506  ax-mulrcl 7507  ax-addcom 7508  ax-mulcom 7509  ax-addass 7510  ax-mulass 7511  ax-distr 7512  ax-i2m1 7513  ax-0lt1 7514  ax-1rid 7515  ax-0id 7516  ax-rnegex 7517  ax-precex 7518  ax-cnre 7519  ax-pre-ltirr 7520  ax-pre-ltwlin 7521  ax-pre-lttrn 7522  ax-pre-apti 7523  ax-pre-ltadd 7524  ax-pre-mulgt0 7525  ax-pre-mulext 7526  ax-arch 7527  ax-caucvg 7528

This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2624  df-sbc 2844  df-csb 2937  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-nul 3290  df-if 3400  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-int 3697  df-iun 3740  df-br 3854  df-opab 3908  df-mpt 3909  df-tr 3945  df-id 4131  df-po 4134  df-iso 4135  df-iord 4204  df-on 4206  df-ilim 4207  df-suc 4209  df-iom 4421  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-f1 5035  df-fo 5036  df-f1o 5037  df-fv 5038  df-isom 5039  df-riota 5624  df-ov 5671  df-oprab 5672  df-mpt2 5673  df-1st 5927  df-2nd 5928  df-recs 6086  df-irdg 6151  df-frec 6172  df-1o 6197  df-oadd 6201  df-er 6308  df-en 6514  df-dom 6515  df-fin 6516  df-pnf 7587  df-mnf 7588  df-xr 7589  df-ltxr 7590  df-le 7591  df-sub 7718  df-neg 7719  df-reap 8115  df-ap 8122  df-div 8203  df-inn 8486  df-2 8544  df-3 8545  df-4 8546  df-n0 8737  df-z 8814  df-uz 9083  df-q 9168  df-rp 9198  df-ico 9375  df-fz 9488  df-fzo 9617  df-iseq 9916  df-seq3 9917  df-exp 10018  df-fac 10197  df-ihash 10247  df-cj 10339  df-re 10340  df-im 10341  df-rsqrt 10494  df-abs 10495  df-clim 10730  df-isum 10806  df-ef 11001

This theorem is referenced by: (None)