Theorem effsumlt 11633

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 9500	. . . . 5 |- NN0 = ( ZZ>= ` 0 )
2		0zd 9203	. . . . 5 |- ( ph -> 0 e. ZZ )
3		effsumlt.2	. . . . . . . 8 |- ( ph -> A e. RR+ )
4	3	rpcnd 9634	. . . . . . 7 |- ( ph -> A e. CC )
5		effsumlt.1	. . . . . . . 8 |- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
6	5	eftvalcn 11598	. . . . . . 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 9632	. . . . . . 7 |- ( ph -> A e. RR )
9		reeftcl 11596	. . . . . . 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 2243	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) e. RR )
12	1, 2, 11	serfre 10410	. . . 4 |- ( ph -> seq 0 ( + , F ) : NN0 --> RR )
13		effsumlt.3	. . . 4 |- ( ph -> N e. NN0 )
14	12, 13	ffvelrnd 5621	. . 3 |- ( ph -> ( seq 0 ( + , F ) ` N ) e. RR )
15		eqid 2165	. . . 4 |- ( ZZ>= ` ( N + 1 ) ) = ( ZZ>= ` ( N + 1 ) )
16		peano2nn0 9154	. . . . 5 |- ( N e. NN0 -> ( N + 1 ) e. NN0 )
17	13, 16	syl 14	. . . 4 |- ( ph -> ( N + 1 ) e. NN0 )
18		eqidd 2166	. . . 4 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) = ( F ` k ) )
19		nn0z 9211	. . . . . . 7 |- ( k e. NN0 -> k e. ZZ )
20		rpexpcl 10474	. . . . . . 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 10648	. . . . . . . 8 |- ( k e. NN0 -> ( ! ` k ) e. NN )
23	22	adantl 275	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> ( ! ` k ) e. NN )
24	23	nnrpd 9630	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> ( ! ` k ) e. RR+ )
25	21, 24	rpdivcld 9650	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. RR+ )
26	7, 25	eqeltrd 2243	. . . 4 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) e. RR+ )
27	5	efcllem 11600	. . . . 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 11435	. . 3 |- ( ph -> sum_ k e. ( ZZ>= ` ( N + 1 ) ) ( F ` k ) e. RR+ )
30	14, 29	ltaddrpd 9666	. 2 |- ( ph -> ( seq 0 ( + , F ) ` N ) < ( ( seq 0 ( + , F ) ` N ) + sum_ k e. ( ZZ>= ` ( N + 1 ) ) ( F ` k ) ) )
31	5	efval2 11606	. . . 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 7927	. . . 4 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) e. CC )
34	1, 15, 17, 18, 33, 28	isumsplit 11432	. . 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 9169	. . . . . . . 8 |- ( ph -> N e. CC )
36		ax-1cn 7846	. . . . . . . 8 |- 1 e. CC
37		pncan 8104	. . . . . . . 8 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N + 1 ) - 1 ) = N )
38	35, 36, 37	sylancl 410	. . . . . . 7 |- ( ph -> ( ( N + 1 ) - 1 ) = N )
39	38	oveq2d 5858	. . . . . 6 |- ( ph -> ( 0 ... ( ( N + 1 ) - 1 ) ) = ( 0 ... N ) )
40	39	sumeq1d 11307	. . . . 5 |- ( ph -> sum_ k e. ( 0 ... ( ( N + 1 ) - 1 ) ) ( F ` k ) = sum_ k e. ( 0 ... N ) ( F ` k ) )
41		eqidd 2166	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` 0 ) ) -> ( F ` k ) = ( F ` k ) )
42	13, 1	eleqtrdi 2259	. . . . . 6 |- ( ph -> N e. ( ZZ>= ` 0 ) )
43		elnn0uz 9503	. . . . . . 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 11338	. . . . 5 |- ( ph -> sum_ k e. ( 0 ... N ) ( F ` k ) = ( seq 0 ( + , F ) ` N ) )
46	40, 45	eqtrd 2198	. . . 4 |- ( ph -> sum_ k e. ( 0 ... ( ( N + 1 ) - 1 ) ) ( F ` k ) = ( seq 0 ( + , F ) ` N ) )
47	46	oveq1d 5857	. . 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 2202	. 2 |- ( ph -> ( exp ` A ) = ( ( seq 0 ( + , F ) ` N ) + sum_ k e. ( ZZ>= ` ( N + 1 ) ) ( F ` k ) ) )
49	30, 48	breqtrrd 4010	1 |- ( ph -> ( seq 0 ( + , F ) ` N ) < ( exp ` A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   class class class wbr 3982   |-> cmpt 4043   dom cdm 4604   ` cfv 5188  (class class class)co 5842   CCcc 7751   RRcr 7752   0cc0 7753   1c1 7754   + caddc 7756   < clt 7933   - cmin 8069   / cdiv 8568   NNcn 8857   NN0cn0 9114   ZZcz 9191   ZZ>=cuz 9466   RR+crp 9589   ...cfz 9944   seqcseq 10380   ^cexp 10454   !cfa 10638   ~~> cli 11219   sum_csu 11294   expce 11583

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-frec 6359  df-1o 6384  df-oadd 6388  df-er 6501  df-en 6707  df-dom 6708  df-fin 6709  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-ico 9830  df-fz 9945  df-fzo 10078  df-seqfrec 10381  df-exp 10455  df-fac 10639  df-ihash 10689  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220  df-sumdc 11295  df-ef 11589

This theorem is referenced by:  efgt1p2  11636  efgt1p  11637