Theorem effsumlt 11655

Description: The partial sums of the series expansion of the exponential function at a positive real number are bounded by the value of the function. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 29-Apr-2014.)

Hypotheses

Ref	Expression
effsumlt.1	|- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
effsumlt.2	|- ( ph -> A e. RR+ )
effsumlt.3	|- ( ph -> N e. NN0 )

Assertion

Ref	Expression
effsumlt	|- ( ph -> ( seq 0 ( + , F ) ` N ) < ( exp ` A ) )

Distinct variable group:   A, n

Allowed substitution hints:   ph( n)   F( n)   N( n)

Proof of Theorem effsumlt

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0uz 9521	. . . . 5 |- NN0 = ( ZZ>= ` 0 )
2		0zd 9224	. . . . 5 |- ( ph -> 0 e. ZZ )
3		effsumlt.2	. . . . . . . 8 |- ( ph -> A e. RR+ )
4	3	rpcnd 9655	. . . . . . 7 |- ( ph -> A e. CC )
5		effsumlt.1	. . . . . . . 8 |- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
6	5	eftvalcn 11620	. . . . . . 7 |- ( ( A e. CC /\ k e. NN0 ) -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
7	4, 6	sylan 281	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
8	3	rpred 9653	. . . . . . 7 |- ( ph -> A e. RR )
9		reeftcl 11618	. . . . . . 7 |- ( ( A e. RR /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. RR )
10	8, 9	sylan 281	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. RR )
11	7, 10	eqeltrd 2247	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) e. RR )
12	1, 2, 11	serfre 10431	. . . 4 |- ( ph -> seq 0 ( + , F ) : NN0 --> RR )
13		effsumlt.3	. . . 4 |- ( ph -> N e. NN0 )
14	12, 13	ffvelrnd 5632	. . 3 |- ( ph -> ( seq 0 ( + , F ) ` N ) e. RR )
15		eqid 2170	. . . 4 |- ( ZZ>= ` ( N + 1 ) ) = ( ZZ>= ` ( N + 1 ) )
16		peano2nn0 9175	. . . . 5 |- ( N e. NN0 -> ( N + 1 ) e. NN0 )
17	13, 16	syl 14	. . . 4 |- ( ph -> ( N + 1 ) e. NN0 )
18		eqidd 2171	. . . 4 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) = ( F ` k ) )
19		nn0z 9232	. . . . . . 7 |- ( k e. NN0 -> k e. ZZ )
20		rpexpcl 10495	. . . . . . 7 |- ( ( A e. RR+ /\ k e. ZZ ) -> ( A ^ k ) e. RR+ )
21	3, 19, 20	syl2an 287	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> ( A ^ k ) e. RR+ )
22		faccl 10669	. . . . . . . 8 |- ( k e. NN0 -> ( ! ` k ) e. NN )
23	22	adantl 275	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> ( ! ` k ) e. NN )
24	23	nnrpd 9651	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> ( ! ` k ) e. RR+ )
25	21, 24	rpdivcld 9671	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. RR+ )
26	7, 25	eqeltrd 2247	. . . 4 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) e. RR+ )
27	5	efcllem 11622	. . . . 5 |- ( A e. CC -> seq 0 ( + , F ) e. dom ~~> )
28	4, 27	syl 14	. . . 4 |- ( ph -> seq 0 ( + , F ) e. dom ~~> )
29	1, 15, 17, 18, 26, 28	isumrpcl 11457	. . 3 |- ( ph -> sum_ k e. ( ZZ>= ` ( N + 1 ) ) ( F ` k ) e. RR+ )
30	14, 29	ltaddrpd 9687	. 2 |- ( ph -> ( seq 0 ( + , F ) ` N ) < ( ( seq 0 ( + , F ) ` N ) + sum_ k e. ( ZZ>= ` ( N + 1 ) ) ( F ` k ) ) )
31	5	efval2 11628	. . . 4 |- ( A e. CC -> ( exp ` A ) = sum_ k e. NN0 ( F ` k ) )
32	4, 31	syl 14	. . 3 |- ( ph -> ( exp ` A ) = sum_ k e. NN0 ( F ` k ) )
33	11	recnd 7948	. . . 4 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) e. CC )
34	1, 15, 17, 18, 33, 28	isumsplit 11454	. . 3 |- ( ph -> sum_ k e. NN0 ( F ` k ) = ( sum_ k e. ( 0 ... ( ( N + 1 ) - 1 ) ) ( F ` k ) + sum_ k e. ( ZZ>= ` ( N + 1 ) ) ( F ` k ) ) )
35	13	nn0cnd 9190	. . . . . . . 8 |- ( ph -> N e. CC )
36		ax-1cn 7867	. . . . . . . 8 |- 1 e. CC
37		pncan 8125	. . . . . . . 8 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N + 1 ) - 1 ) = N )
38	35, 36, 37	sylancl 411	. . . . . . 7 |- ( ph -> ( ( N + 1 ) - 1 ) = N )
39	38	oveq2d 5869	. . . . . 6 |- ( ph -> ( 0 ... ( ( N + 1 ) - 1 ) ) = ( 0 ... N ) )
40	39	sumeq1d 11329	. . . . 5 |- ( ph -> sum_ k e. ( 0 ... ( ( N + 1 ) - 1 ) ) ( F ` k ) = sum_ k e. ( 0 ... N ) ( F ` k ) )
41		eqidd 2171	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` 0 ) ) -> ( F ` k ) = ( F ` k ) )
42	13, 1	eleqtrdi 2263	. . . . . 6 |- ( ph -> N e. ( ZZ>= ` 0 ) )
43		elnn0uz 9524	. . . . . . 7 |- ( k e. NN0 <-> k e. ( ZZ>= ` 0 ) )
44	43, 33	sylan2br 286	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` 0 ) ) -> ( F ` k ) e. CC )
45	41, 42, 44	fsum3ser 11360	. . . . 5 |- ( ph -> sum_ k e. ( 0 ... N ) ( F ` k ) = ( seq 0 ( + , F ) ` N ) )
46	40, 45	eqtrd 2203	. . . 4 |- ( ph -> sum_ k e. ( 0 ... ( ( N + 1 ) - 1 ) ) ( F ` k ) = ( seq 0 ( + , F ) ` N ) )
47	46	oveq1d 5868	. . 3 |- ( ph -> ( sum_ k e. ( 0 ... ( ( N + 1 ) - 1 ) ) ( F ` k ) + sum_ k e. ( ZZ>= ` ( N + 1 ) ) ( F ` k ) ) = ( ( seq 0 ( + , F ) ` N ) + sum_ k e. ( ZZ>= ` ( N + 1 ) ) ( F ` k ) ) )
48	32, 34, 47	3eqtrd 2207	. 2 |- ( ph -> ( exp ` A ) = ( ( seq 0 ( + , F ) ` N ) + sum_ k e. ( ZZ>= ` ( N + 1 ) ) ( F ` k ) ) )
49	30, 48	breqtrrd 4017	1 |- ( ph -> ( seq 0 ( + , F ) ` N ) < ( exp ` A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   class class class wbr 3989   |-> cmpt 4050   dom cdm 4611   ` cfv 5198  (class class class)co 5853   CCcc 7772   RRcr 7773   0cc0 7774   1c1 7775   + caddc 7777   < clt 7954   - cmin 8090   / cdiv 8589   NNcn 8878   NN0cn0 9135   ZZcz 9212   ZZ>=cuz 9487   RR+crp 9610   ...cfz 9965   seqcseq 10401   ^cexp 10475   !cfa 10659   ~~> cli 11241   sum_csu 11316   expce 11605

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-ico 9851  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-fac 10660  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317  df-ef 11611

This theorem is referenced by:  efgt1p2  11658  efgt1p  11659