Theorem efgt1p2 12336

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 9477	. . . . . . 7 |- 1 e. NN0
2		nn0uz 9852	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
3	1, 2	eleqtri 2306	. . . . . 6 |- 1 e. ( ZZ>= ` 0 )
4	3	a1i 9	. . . . 5 |- ( A e. RR+ -> 1 e. ( ZZ>= ` 0 ) )
5		elnn0uz 9855	. . . . . . . . 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 9826	. . . . . . . . . 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 11022	. . . . . . . 8 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( A ^ k ) e. RR+ )
12	7	faccld 11061	. . . . . . . . 9 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( ! ` k ) e. NN )
13	12	nnrpd 9990	. . . . . . . 8 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( ! ` k ) e. RR+ )
14	11, 13	rpdivcld 10010	. . . . . . 7 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( ( A ^ k ) / ( ! ` k ) ) e. RR+ )
15		oveq2 6036	. . . . . . . . 9 |- ( n = k -> ( A ^ n ) = ( A ^ k ) )
16		fveq2 5648	. . . . . . . . 9 |- ( n = k -> ( ! ` n ) = ( ! ` k ) )
17	15, 16	oveq12d 6046	. . . . . . . 8 |- ( n = k -> ( ( A ^ n ) / ( ! ` n ) ) = ( ( A ^ k ) / ( ! ` k ) ) )
18		eqid 2231	. . . . . . . 8 |- ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
19	17, 18	fvmptg 5731	. . . . . . 7 |- ( ( k e. NN0 /\ ( ( A ^ k ) / ( ! ` k ) ) e. RR+ ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
20	7, 14, 19	syl2anc 411	. . . . . 6 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
21	20, 14	eqeltrd 2308	. . . . 5 |- ( ( A e. RR+ /\ k e. ( ZZ>= ` 0 ) ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) e. RR+ )
22		rpaddcl 9973	. . . . . 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 10790	. . . 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 9261	. . . . 5 |- 2 = ( 1 + 1 )
26	25	fveq2i 5651	. . . 4 |- ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 2 ) = ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` ( 1 + 1 ) )
27	25	fveq2i 5651	. . . . 5 |- ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 2 ) = ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` ( 1 + 1 ) )
28	27	oveq2i 6039	. . . 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 2289	. . 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 9476	. . . . . . . . 9 |- 0 e. NN0
31	30, 2	eleqtri 2306	. . . . . . . 8 |- 0 e. ( ZZ>= ` 0 )
32	31	a1i 9	. . . . . . 7 |- ( A e. RR+ -> 0 e. ( ZZ>= ` 0 ) )
33	32, 21, 23	seq3p1 10790	. . . . . 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 9713	. . . . . . 7 |- 1 = ( 0 + 1 )
35	34	fveq2i 5651	. . . . . 6 |- ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 1 ) = ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` ( 0 + 1 ) )
36	34	fveq2i 5651	. . . . . . 7 |- ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` ( 0 + 1 ) )
37	36	oveq2i 6039	. . . . . 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 2289	. . . . 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 9552	. . . . . . . 8 |- ( A e. RR+ -> 0 e. ZZ )
40	39, 21, 23	seq3-1 10787	. . . . . . 7 |- ( A e. RR+ -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 0 ) = ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 0 ) )
41		rpcn 9958	. . . . . . . . 9 |- ( A e. RR+ -> A e. CC )
42	18	eftvalcn 12298	. . . . . . . . . 10 |- ( ( A e. CC /\ 0 e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) ) )
43	30, 42	mpan2 425	. . . . . . . . 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 12334	. . . . . . . . 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 2264	. . . . . . 7 |- ( A e. RR+ -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 0 ) = 1 )
48	40, 47	eqtrd 2264	. . . . . 6 |- ( A e. RR+ -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 0 ) = 1 )
49	18	eftvalcn 12298	. . . . . . . . 9 |- ( ( A e. CC /\ 1 e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = ( ( A ^ 1 ) / ( ! ` 1 ) ) )
50	1, 49	mpan2 425	. . . . . . . 8 |- ( A e. CC -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 1 ) = ( ( A ^ 1 ) / ( ! ` 1 ) ) )
51		fac1 11054	. . . . . . . . . 10 |- ( ! ` 1 ) = 1
52	51	oveq2i 6039	. . . . . . . . 9 |- ( ( A ^ 1 ) / ( ! ` 1 ) ) = ( ( A ^ 1 ) / 1 )
53		exp1 10870	. . . . . . . . . . 11 |- ( A e. CC -> ( A ^ 1 ) = A )
54	53	oveq1d 6043	. . . . . . . . . 10 |- ( A e. CC -> ( ( A ^ 1 ) / 1 ) = ( A / 1 ) )
55		div1 8942	. . . . . . . . . 10 |- ( A e. CC -> ( A / 1 ) = A )
56	54, 55	eqtrd 2264	. . . . . . . . 9 |- ( A e. CC -> ( ( A ^ 1 ) / 1 ) = A )
57	52, 56	eqtrid 2276	. . . . . . . 8 |- ( A e. CC -> ( ( A ^ 1 ) / ( ! ` 1 ) ) = A )
58	50, 57	eqtrd 2264	. . . . . . 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 6046	. . . . 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 2264	. . . 4 |- ( A e. RR+ -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 1 ) = ( 1 + A ) )
62		2nn0 9478	. . . . . . 7 |- 2 e. NN0
63	18	eftvalcn 12298	. . . . . . 7 |- ( ( A e. CC /\ 2 e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 2 ) = ( ( A ^ 2 ) / ( ! ` 2 ) ) )
64	62, 63	mpan2 425	. . . . . 6 |- ( A e. CC -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` 2 ) = ( ( A ^ 2 ) / ( ! ` 2 ) ) )
65		fac2 11056	. . . . . . 7 |- ( ! ` 2 ) = 2
66	65	oveq2i 6039	. . . . . 6 |- ( ( A ^ 2 ) / ( ! ` 2 ) ) = ( ( A ^ 2 ) / 2 )
67	64, 66	eqtrdi 2280	. . . . 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 6046	. . 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 2264	. 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 12333	. 2 |- ( A e. RR+ -> ( seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) ` 2 ) < ( exp ` A ) )
74	70, 73	eqbrtrrd 4117	1 |- ( A e. RR+ -> ( ( 1 + A ) + ( ( A ^ 2 ) / 2 ) ) < ( exp ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   class class class wbr 4093   |-> cmpt 4155   ` cfv 5333  (class class class)co 6028   CCcc 8090   0cc0 8092   1c1 8093   + caddc 8095   < clt 8273   / cdiv 8911   2c2 9253   NN0cn0 9461   ZZcz 9540   ZZ>=cuz 9816   RR+crp 9949   seqcseq 10772   ^cexp 10863   !cfa 11050   expce 12283

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-frec 6600  df-1o 6625  df-oadd 6629  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-n0 9462  df-z 9541  df-uz 9817  df-q 9915  df-rp 9950  df-ico 10190  df-fz 10306  df-fzo 10440  df-seqfrec 10773  df-exp 10864  df-fac 11051  df-ihash 11101  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639  df-clim 11919  df-sumdc 11994  df-ef 12289

This theorem is referenced by: (None)