Theorem efgt1p2 11438

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 9017	. . . . . . 7 |- 1 e. NN0
2		nn0uz 9384	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
3	1, 2	eleqtri 2215	. . . . . 6 |- 1 e. ( ZZ>= ` 0 )
4	3	a1i 9	. . . . 5 |- ( A e. RR+ -> 1 e. ( ZZ>= ` 0 ) )
5		elnn0uz 9387	. . . . . . . . 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 9359	. . . . . . . . . 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 10479	. . . . . . . 8 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( A ^ k ) e. RR+ )
12	7	faccld 10514	. . . . . . . . 9 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( ! ` k ) e. NN )
13	12	nnrpd 9511	. . . . . . . 8 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( ! ` k ) e. RR+ )
14	11, 13	rpdivcld 9531	. . . . . . 7 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( ( A ^ k ) / ( ! ` k ) ) e. RR+ )
15		oveq2 5790	. . . . . . . . 9 |- ( n = k -> ( A ^ n ) = ( A ^ k ) )
16		fveq2 5429	. . . . . . . . 9 |- ( n = k -> ( ! ` n ) = ( ! ` k ) )
17	15, 16	oveq12d 5800	. . . . . . . 8 |- ( n = k -> ( ( A ^ n ) / ( ! ` n ) ) = ( ( A ^ k ) / ( ! ` k ) ) )
18		eqid 2140	. . . . . . . 8 |- ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
19	17, 18	fvmptg 5505	. . . . . . 7 |- ( ( k e. NN0 /\ ( ( A ^ k ) / ( ! ` k ) ) e. RR+ ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
20	7, 14, 19	syl2anc 409	. . . . . 6 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
21	20, 14	eqeltrd 2217	. . . . 5 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) e. RR+ )
22		rpaddcl 9494	. . . . . 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 10266	. . . 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 8803	. . . . 5 |- 2 = ( 1 + 1 )
26	25	fveq2i 5432	. . . 4 |- ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 2 ) = ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` ( 1 + 1 ) )
27	25	fveq2i 5432	. . . . 5 |- ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 2 ) = ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` ( 1 + 1 ) )
28	27	oveq2i 5793	. . . 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 2198	. . 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 9016	. . . . . . . . 9 |- 0 e. NN0
31	30, 2	eleqtri 2215	. . . . . . . 8 |- 0 e. ( ZZ>= ` 0 )
32	31	a1i 9	. . . . . . 7 |- ( A e. RR+ -> 0 e. ( ZZ>= ` 0 ) )
33	32, 21, 23	seq3p1 10266	. . . . . 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 9247	. . . . . . 7 |- 1 = ( 0 + 1 )
35	34	fveq2i 5432	. . . . . 6 |- ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 1 ) = ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` ( 0 + 1 ) )
36	34	fveq2i 5432	. . . . . . 7 |- ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` ( 0 + 1 ) )
37	36	oveq2i 5793	. . . . . 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 2198	. . . . 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 9090	. . . . . . . 8 |- ( A e. RR+ -> 0 e. ZZ )
40	39, 21, 23	seq3-1 10264	. . . . . . 7 |- ( A e. RR+ -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 0 ) = ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 0 ) )
41		rpcn 9479	. . . . . . . . 9 |- ( A e. RR+ -> A e. CC )
42	18	eftvalcn 11400	. . . . . . . . . 10 |- ( ( A e. CC /\ 0 e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) ) )
43	30, 42	mpan2 422	. . . . . . . . 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 11436	. . . . . . . . 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 2173	. . . . . . 7 |- ( A e. RR+ -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 0 ) = 1 )
48	40, 47	eqtrd 2173	. . . . . 6 |- ( A e. RR+ -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 0 ) = 1 )
49	18	eftvalcn 11400	. . . . . . . . 9 |- ( ( A e. CC /\ 1 e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = ( ( A ^ 1 ) / ( ! ` 1 ) ) )
50	1, 49	mpan2 422	. . . . . . . 8 |- ( A e. CC -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = ( ( A ^ 1 ) / ( ! ` 1 ) ) )
51		fac1 10507	. . . . . . . . . 10 |- ( ! ` 1 ) = 1
52	51	oveq2i 5793	. . . . . . . . 9 |- ( ( A ^ 1 ) / ( ! ` 1 ) ) = ( ( A ^ 1 ) / 1 )
53		exp1 10330	. . . . . . . . . . 11 |- ( A e. CC -> ( A ^ 1 ) = A )
54	53	oveq1d 5797	. . . . . . . . . 10 |- ( A e. CC -> ( ( A ^ 1 ) / 1 ) = ( A / 1 ) )
55		div1 8487	. . . . . . . . . 10 |- ( A e. CC -> ( A / 1 ) = A )
56	54, 55	eqtrd 2173	. . . . . . . . 9 |- ( A e. CC -> ( ( A ^ 1 ) / 1 ) = A )
57	52, 56	syl5eq 2185	. . . . . . . 8 |- ( A e. CC -> ( ( A ^ 1 ) / ( ! ` 1 ) ) = A )
58	50, 57	eqtrd 2173	. . . . . . 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 5800	. . . . 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 2173	. . . 4 |- ( A e. RR+ -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 1 ) = ( 1 + A ) )
62		2nn0 9018	. . . . . . 7 |- 2 e. NN0
63	18	eftvalcn 11400	. . . . . . 7 |- ( ( A e. CC /\ 2 e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 2 ) = ( ( A ^ 2 ) / ( ! ` 2 ) ) )
64	62, 63	mpan2 422	. . . . . 6 |- ( A e. CC -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 2 ) = ( ( A ^ 2 ) / ( ! ` 2 ) ) )
65		fac2 10509	. . . . . . 7 |- ( ! ` 2 ) = 2
66	65	oveq2i 5793	. . . . . 6 |- ( ( A ^ 2 ) / ( ! ` 2 ) ) = ( ( A ^ 2 ) / 2 )
67	64, 66	eqtrdi 2189	. . . . 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 5800	. . 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 2173	. 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 11435	. 2 |- ( A e. RR+ -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 2 ) < ( exp ` A ) )
74	70, 73	eqbrtrrd 3960	1 |- ( A e. RR+ -> ( ( 1 + A ) + ( ( A ^ 2 ) / 2 ) ) < ( exp ` A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   class class class wbr 3937   |-> cmpt 3997   ` cfv 5131  (class class class)co 5782   CCcc 7642   0cc0 7644   1c1 7645   + caddc 7647   < clt 7824   / cdiv 8456   2c2 8795   NN0cn0 9001   ZZcz 9078   ZZ>=cuz 9350   RR+crp 9470   seqcseq 10249   ^cexp 10323   !cfa 10503   expce 11385

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-isom 5140  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-frec 6296  df-1o 6321  df-oadd 6325  df-er 6437  df-en 6643  df-dom 6644  df-fin 6645  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-n0 9002  df-z 9079  df-uz 9351  df-q 9439  df-rp 9471  df-ico 9707  df-fz 9822  df-fzo 9951  df-seqfrec 10250  df-exp 10324  df-fac 10504  df-ihash 10554  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803  df-clim 11080  df-sumdc 11155  df-ef 11391

This theorem is referenced by: (None)