Theorem efgt1p2 12462

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 9579	. . . . . . 7 |- 1 e. NN0
2		nn0uz 9957	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
3	1, 2	eleqtri 2313	. . . . . 6 |- 1 e. ( ZZ>= ` 0 )
4	3	a1i 9	. . . . 5 |- ( A e. RR+ -> 1 e. ( ZZ>= ` 0 ) )
5		elnn0uz 9960	. . . . . . . . 9 |- ( k e. NN0 <-> k e. ( ZZ>= ` 0 ) )
6	5	biimpri 133	. . . . . . . 8 |- ( k e. ( ZZ>= ` 0 ) -> k e. NN0 )
7	6	adantl 277	. . . . . . 7 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> k e. NN0 )
8		simpl 109	. . . . . . . . 9 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> A e. RR+ )
9		eluzelz 9931	. . . . . . . . . 10 |- ( k e. ( ZZ>= ` 0 ) -> k e. ZZ )
10	9	adantl 277	. . . . . . . . 9 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> k e. ZZ )
11	8, 10	rpexpcld 11135	. . . . . . . 8 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( A ^ k ) e. RR+ )
12	7	faccld 11174	. . . . . . . . 9 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( ! ` k ) e. NN )
13	12	nnrpd 10095	. . . . . . . 8 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( ! ` k ) e. RR+ )
14	11, 13	rpdivcld 10115	. . . . . . 7 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( ( A ^ k ) / ( ! ` k ) ) e. RR+ )
15		oveq2 6093	. . . . . . . . 9 |- ( n = k -> ( A ^ n ) = ( A ^ k ) )
16		fveq2 5695	. . . . . . . . 9 |- ( n = k -> ( ! ` n ) = ( ! ` k ) )
17	15, 16	oveq12d 6103	. . . . . . . 8 |- ( n = k -> ( ( A ^ n ) / ( ! ` n ) ) = ( ( A ^ k ) / ( ! ` k ) ) )
18		eqid 2238	. . . . . . . 8 |- ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
19	17, 18	fvmptg 5781	. . . . . . 7 |- ( ( k e. NN0 /\ ( ( A ^ k ) / ( ! ` k ) ) e. RR+ ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
20	7, 14, 19	syl2anc 415	. . . . . 6 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
21	20, 14	eqeltrd 2315	. . . . 5 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) e. RR+ )
22		rpaddcl 10078	. . . . . 6 |- ( ( k e. RR+ /\ y e. RR+ ) -> ( k + y ) e. RR+ )
23	22	adantl 277	. . . . 5 |- ( ( A e. RR+ /\ ( k e. RR+ /\ y e. RR+ ) ) -> ( k + y ) e. RR+ )
24	4, 21, 23	seq3p1 10902	. . . 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 9363	. . . . 5 |- 2 = ( 1 + 1 )
26	25	fveq2i 5698	. . . 4 |- ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 2 ) = ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` ( 1 + 1 ) )
27	25	fveq2i 5698	. . . . 5 |- ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 2 ) = ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` ( 1 + 1 ) )
28	27	oveq2i 6096	. . . 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 2296	. . 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 9578	. . . . . . . . 9 |- 0 e. NN0
31	30, 2	eleqtri 2313	. . . . . . . 8 |- 0 e. ( ZZ>= ` 0 )
32	31	a1i 9	. . . . . . 7 |- ( A e. RR+ -> 0 e. ( ZZ>= ` 0 ) )
33	32, 21, 23	seq3p1 10902	. . . . . 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 9818	. . . . . . 7 |- 1 = ( 0 + 1 )
35	34	fveq2i 5698	. . . . . 6 |- ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 1 ) = ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` ( 0 + 1 ) )
36	34	fveq2i 5698	. . . . . . 7 |- ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` ( 0 + 1 ) )
37	36	oveq2i 6096	. . . . . 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 2296	. . . . 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 9656	. . . . . . . 8 |- ( A e. RR+ -> 0 e. ZZ )
40	39, 21, 23	seq3-1 10899	. . . . . . 7 |- ( A e. RR+ -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 0 ) = ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 0 ) )
41		rpcn 10063	. . . . . . . . 9 |- ( A e. RR+ -> A e. CC )
42	18	eftvalcn 12424	. . . . . . . . . 10 |- ( ( A e. CC /\ 0 e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) ) )
43	30, 42	mpan2 429	. . . . . . . . 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 12460	. . . . . . . . 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 2271	. . . . . . 7 |- ( A e. RR+ -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 0 ) = 1 )
48	40, 47	eqtrd 2271	. . . . . 6 |- ( A e. RR+ -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 0 ) = 1 )
49	18	eftvalcn 12424	. . . . . . . . 9 |- ( ( A e. CC /\ 1 e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = ( ( A ^ 1 ) / ( ! ` 1 ) ) )
50	1, 49	mpan2 429	. . . . . . . 8 |- ( A e. CC -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = ( ( A ^ 1 ) / ( ! ` 1 ) ) )
51		fac1 11167	. . . . . . . . . 10 |- ( ! ` 1 ) = 1
52	51	oveq2i 6096	. . . . . . . . 9 |- ( ( A ^ 1 ) / ( ! ` 1 ) ) = ( ( A ^ 1 ) / 1 )
53		exp1 10982	. . . . . . . . . . 11 |- ( A e. CC -> ( A ^ 1 ) = A )
54	53	oveq1d 6100	. . . . . . . . . 10 |- ( A e. CC -> ( ( A ^ 1 ) / 1 ) = ( A / 1 ) )
55		div1 9033	. . . . . . . . . 10 |- ( A e. CC -> ( A / 1 ) = A )
56	54, 55	eqtrd 2271	. . . . . . . . 9 |- ( A e. CC -> ( ( A ^ 1 ) / 1 ) = A )
57	52, 56	eqtrid 2283	. . . . . . . 8 |- ( A e. CC -> ( ( A ^ 1 ) / ( ! ` 1 ) ) = A )
58	50, 57	eqtrd 2271	. . . . . . 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 6103	. . . . 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 2271	. . . 4 |- ( A e. RR+ -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 1 ) = ( 1 + A ) )
62		2nn0 9580	. . . . . . 7 |- 2 e. NN0
63	18	eftvalcn 12424	. . . . . . 7 |- ( ( A e. CC /\ 2 e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 2 ) = ( ( A ^ 2 ) / ( ! ` 2 ) ) )
64	62, 63	mpan2 429	. . . . . 6 |- ( A e. CC -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 2 ) = ( ( A ^ 2 ) / ( ! ` 2 ) ) )
65		fac2 11169	. . . . . . 7 |- ( ! ` 2 ) = 2
66	65	oveq2i 6096	. . . . . 6 |- ( ( A ^ 2 ) / ( ! ` 2 ) ) = ( ( A ^ 2 ) / 2 )
67	64, 66	eqtrdi 2287	. . . . 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 6103	. . 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 2271	. 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 12459	. 2 |- ( A e. RR+ -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 2 ) < ( exp ` A ) )
74	70, 73	eqbrtrrd 4154	1 |- ( A e. RR+ -> ( ( 1 + A ) + ( ( A ^ 2 ) / 2 ) ) < ( exp ` A ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   class class class wbr 4130   |-> cmpt 4192   ` cfv 5377  (class class class)co 6085   CCcc 8177   0cc0 8179   1c1 8180   + caddc 8182   < clt 8360   / cdiv 9002   2c2 9355   NN0cn0 9563   ZZcz 9644   ZZ>=cuz 9921   RR+crp 10054   seqcseq 10884   ^cexp 10975   !cfa 11163   expce 12409

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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

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 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-ico 10296  df-fz 10412  df-fzo 10550  df-seqfrec 10885  df-exp 10976  df-fac 11164  df-ihash 11215  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-clim 12045  df-sumdc 12120  df-ef 12415

This theorem is used by: (None)