Theorem efgt1p2 11401

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 8993	. . . . . . 7 |- 1 e. NN0
2		nn0uz 9360	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
3	1, 2	eleqtri 2214	. . . . . 6 |- 1 e. ( ZZ>= ` 0 )
4	3	a1i 9	. . . . 5 |- ( A e. RR+ -> 1 e. ( ZZ>= ` 0 ) )
5		elnn0uz 9363	. . . . . . . . 9 |- ( k e. NN0 <-> k e. ( ZZ>= ` 0 ) )
6	5	biimpri 132	. . . . . . . 8 |- ( k e. ( ZZ>= ` 0 ) -> k e. NN0 )
7	6	adantl 275	. . . . . . 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 9335	. . . . . . . . . 10 |- ( k e. ( ZZ>= ` 0 ) -> k e. ZZ )
10	9	adantl 275	. . . . . . . . 9 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> k e. ZZ )
11	8, 10	rpexpcld 10448	. . . . . . . 8 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( A ^ k ) e. RR+ )
12	7	faccld 10482	. . . . . . . . 9 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( ! ` k ) e. NN )
13	12	nnrpd 9482	. . . . . . . 8 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( ! ` k ) e. RR+ )
14	11, 13	rpdivcld 9501	. . . . . . 7 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( ( A ^ k ) / ( ! ` k ) ) e. RR+ )
15		oveq2 5782	. . . . . . . . 9 |- ( n = k -> ( A ^ n ) = ( A ^ k ) )
16		fveq2 5421	. . . . . . . . 9 |- ( n = k -> ( ! ` n ) = ( ! ` k ) )
17	15, 16	oveq12d 5792	. . . . . . . 8 |- ( n = k -> ( ( A ^ n ) / ( ! ` n ) ) = ( ( A ^ k ) / ( ! ` k ) ) )
18		eqid 2139	. . . . . . . 8 |- ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
19	17, 18	fvmptg 5497	. . . . . . 7 |- ( ( k e. NN0 /\ ( ( A ^ k ) / ( ! ` k ) ) e. RR+ ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
20	7, 14, 19	syl2anc 408	. . . . . 6 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
21	20, 14	eqeltrd 2216	. . . . 5 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) e. RR+ )
22		rpaddcl 9465	. . . . . 6 |- ( ( k e. RR+ /\ y e. RR+ ) -> ( k + y ) e. RR+ )
23	22	adantl 275	. . . . 5 |- ( ( A e. RR+ /\ ( k e. RR+ /\ y e. RR+ ) ) -> ( k + y ) e. RR+ )
24	4, 21, 23	seq3p1 10235	. . . 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 8779	. . . . 5 |- 2 = ( 1 + 1 )
26	25	fveq2i 5424	. . . 4 |- ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 2 ) = ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` ( 1 + 1 ) )
27	25	fveq2i 5424	. . . . 5 |- ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 2 ) = ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` ( 1 + 1 ) )
28	27	oveq2i 5785	. . . 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 2197	. . 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 8992	. . . . . . . . 9 |- 0 e. NN0
31	30, 2	eleqtri 2214	. . . . . . . 8 |- 0 e. ( ZZ>= ` 0 )
32	31	a1i 9	. . . . . . 7 |- ( A e. RR+ -> 0 e. ( ZZ>= ` 0 ) )
33	32, 21, 23	seq3p1 10235	. . . . . 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 9223	. . . . . . 7 |- 1 = ( 0 + 1 )
35	34	fveq2i 5424	. . . . . 6 |- ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 1 ) = ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` ( 0 + 1 ) )
36	34	fveq2i 5424	. . . . . . 7 |- ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` ( 0 + 1 ) )
37	36	oveq2i 5785	. . . . . 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 2197	. . . . 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 9066	. . . . . . . 8 |- ( A e. RR+ -> 0 e. ZZ )
40	39, 21, 23	seq3-1 10233	. . . . . . 7 |- ( A e. RR+ -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 0 ) = ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 0 ) )
41		rpcn 9450	. . . . . . . . 9 |- ( A e. RR+ -> A e. CC )
42	18	eftvalcn 11363	. . . . . . . . . 10 |- ( ( A e. CC /\ 0 e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) ) )
43	30, 42	mpan2 421	. . . . . . . . 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 11399	. . . . . . . . 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 2172	. . . . . . 7 |- ( A e. RR+ -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 0 ) = 1 )
48	40, 47	eqtrd 2172	. . . . . 6 |- ( A e. RR+ -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 0 ) = 1 )
49	18	eftvalcn 11363	. . . . . . . . 9 |- ( ( A e. CC /\ 1 e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = ( ( A ^ 1 ) / ( ! ` 1 ) ) )
50	1, 49	mpan2 421	. . . . . . . 8 |- ( A e. CC -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = ( ( A ^ 1 ) / ( ! ` 1 ) ) )
51		fac1 10475	. . . . . . . . . 10 |- ( ! ` 1 ) = 1
52	51	oveq2i 5785	. . . . . . . . 9 |- ( ( A ^ 1 ) / ( ! ` 1 ) ) = ( ( A ^ 1 ) / 1 )
53		exp1 10299	. . . . . . . . . . 11 |- ( A e. CC -> ( A ^ 1 ) = A )
54	53	oveq1d 5789	. . . . . . . . . 10 |- ( A e. CC -> ( ( A ^ 1 ) / 1 ) = ( A / 1 ) )
55		div1 8463	. . . . . . . . . 10 |- ( A e. CC -> ( A / 1 ) = A )
56	54, 55	eqtrd 2172	. . . . . . . . 9 |- ( A e. CC -> ( ( A ^ 1 ) / 1 ) = A )
57	52, 56	syl5eq 2184	. . . . . . . 8 |- ( A e. CC -> ( ( A ^ 1 ) / ( ! ` 1 ) ) = A )
58	50, 57	eqtrd 2172	. . . . . . 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 5792	. . . . 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 2172	. . . 4 |- ( A e. RR+ -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 1 ) = ( 1 + A ) )
62		2nn0 8994	. . . . . . 7 |- 2 e. NN0
63	18	eftvalcn 11363	. . . . . . 7 |- ( ( A e. CC /\ 2 e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 2 ) = ( ( A ^ 2 ) / ( ! ` 2 ) ) )
64	62, 63	mpan2 421	. . . . . 6 |- ( A e. CC -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 2 ) = ( ( A ^ 2 ) / ( ! ` 2 ) ) )
65		fac2 10477	. . . . . . 7 |- ( ! ` 2 ) = 2
66	65	oveq2i 5785	. . . . . 6 |- ( ( A ^ 2 ) / ( ! ` 2 ) ) = ( ( A ^ 2 ) / 2 )
67	64, 66	syl6eq 2188	. . . . 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 5792	. . 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 2172	. 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 11398	. 2 |- ( A e. RR+ -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 2 ) < ( exp ` A ) )
74	70, 73	eqbrtrrd 3952	1 |- ( A e. RR+ -> ( ( 1 + A ) + ( ( A ^ 2 ) / 2 ) ) < ( exp ` A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   class class class wbr 3929   |-> cmpt 3989   ` cfv 5123  (class class class)co 5774   CCcc 7618   0cc0 7620   1c1 7621   + caddc 7623   < clt 7800   / cdiv 8432   2c2 8771   NN0cn0 8977   ZZcz 9054   ZZ>=cuz 9326   RR+crp 9441   seqcseq 10218   ^cexp 10292   !cfa 10471   expce 11348

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-ico 9677  df-fz 9791  df-fzo 9920  df-seqfrec 10219  df-exp 10293  df-fac 10472  df-ihash 10522  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-clim 11048  df-sumdc 11123  df-ef 11354

This theorem is referenced by: (None)