Theorem effsumlt 12459

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 9957	. . . . 5 |- NN0 = ( ZZ>= ` 0 )
2		0zd 9656	. . . . 5 |- ( ph -> 0 e. ZZ )
3		effsumlt.2	. . . . . . . 8 |- ( ph -> A e. RR+ )
4	3	rpcnd 10099	. . . . . . 7 |- ( ph -> A e. CC )
5		effsumlt.1	. . . . . . . 8 |- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
6	5	eftvalcn 12424	. . . . . . 7 |- ( ( A e. CC /\ k e. NN0 ) -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
7	4, 6	sylan 283	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
8	3	rpred 10097	. . . . . . 7 |- ( ph -> A e. RR )
9		reeftcl 12422	. . . . . . 7 |- ( ( A e. RR /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. RR )
10	8, 9	sylan 283	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. RR )
11	7, 10	eqeltrd 2315	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) e. RR )
12	1, 2, 11	serfre 10921	. . . 4 |- ( ph -> seq 0 ( + , F ) : NN0 --> RR )
13		effsumlt.3	. . . 4 |- ( ph -> N e. NN0 )
14	12, 13	ffvelcdmd 5844	. . 3 |- ( ph -> ( seq 0 ( + , F ) ` N ) e. RR )
15		eqid 2238	. . . 4 |- ( ZZ>= ` ( N + 1 ) ) = ( ZZ>= ` ( N + 1 ) )
16		peano2nn0 9603	. . . . 5 |- ( N e. NN0 -> ( N + 1 ) e. NN0 )
17	13, 16	syl 14	. . . 4 |- ( ph -> ( N + 1 ) e. NN0 )
18		eqidd 2239	. . . 4 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) = ( F ` k ) )
19		nn0z 9664	. . . . . . 7 |- ( k e. NN0 -> k e. ZZ )
20		rpexpcl 10995	. . . . . . 7 |- ( ( A e. RR+ /\ k e. ZZ ) -> ( A ^ k ) e. RR+ )
21	3, 19, 20	syl2an 289	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> ( A ^ k ) e. RR+ )
22		faccl 11173	. . . . . . . 8 |- ( k e. NN0 -> ( ! ` k ) e. NN )
23	22	adantl 277	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> ( ! ` k ) e. NN )
24	23	nnrpd 10095	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> ( ! ` k ) e. RR+ )
25	21, 24	rpdivcld 10115	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. RR+ )
26	7, 25	eqeltrd 2315	. . . 4 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) e. RR+ )
27	5	efcllem 12426	. . . . 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 12261	. . 3 |- ( ph -> sum_ k e. ( ZZ>= ` ( N + 1 ) ) ( F ` k ) e. RR+ )
30	14, 29	ltaddrpd 10131	. 2 |- ( ph -> ( seq 0 ( + , F ) ` N ) < ( ( seq 0 ( + , F ) ` N ) + sum_ k e. ( ZZ>= ` ( N + 1 ) ) ( F ` k ) ) )
31	5	efval2 12432	. . . 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 8354	. . . 4 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) e. CC )
34	1, 15, 17, 18, 33, 28	isumsplit 12258	. . 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 9622	. . . . . . . 8 |- ( ph -> N e. CC )
36		ax-1cn 8272	. . . . . . . 8 |- 1 e. CC
37		pncan 8532	. . . . . . . 8 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N + 1 ) - 1 ) = N )
38	35, 36, 37	sylancl 417	. . . . . . 7 |- ( ph -> ( ( N + 1 ) - 1 ) = N )
39	38	oveq2d 6101	. . . . . 6 |- ( ph -> ( 0 ... ( ( N + 1 ) - 1 ) ) = ( 0 ... N ) )
40	39	sumeq1d 12132	. . . . 5 |- ( ph -> sum_ k e. ( 0 ... ( ( N + 1 ) - 1 ) ) ( F ` k ) = sum_ k e. ( 0 ... N ) ( F ` k ) )
41		eqidd 2239	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` 0 ) ) -> ( F ` k ) = ( F ` k ) )
42	13, 1	eleqtrdi 2331	. . . . . 6 |- ( ph -> N e. ( ZZ>= ` 0 ) )
43		elnn0uz 9960	. . . . . . 7 |- ( k e. NN0 <-> k e. ( ZZ>= ` 0 ) )
44	43, 33	sylan2br 288	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` 0 ) ) -> ( F ` k ) e. CC )
45	41, 42, 44	fsum3ser 12164	. . . . 5 |- ( ph -> sum_ k e. ( 0 ... N ) ( F ` k ) = ( seq 0 ( + , F ) ` N ) )
46	40, 45	eqtrd 2271	. . . 4 |- ( ph -> sum_ k e. ( 0 ... ( ( N + 1 ) - 1 ) ) ( F ` k ) = ( seq 0 ( + , F ) ` N ) )
47	46	oveq1d 6100	. . 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 2275	. 2 |- ( ph -> ( exp ` A ) = ( ( seq 0 ( + , F ) ` N ) + sum_ k e. ( ZZ>= ` ( N + 1 ) ) ( F ` k ) ) )
49	30, 48	breqtrrd 4158	1 |- ( ph -> ( seq 0 ( + , F ) ` N ) < ( exp ` A ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   class class class wbr 4130   |-> cmpt 4192   dom cdm 4774   ` cfv 5377  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   1c1 8180   + caddc 8182   < clt 8360   - cmin 8497   / cdiv 9002   NNcn 9304   NN0cn0 9563   ZZcz 9644   ZZ>=cuz 9921   RR+crp 10054   ...cfz 10411   seqcseq 10884   ^cexp 10975   !cfa 11163   ~~> cli 12044   sum_csu 12119   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:  efgt1p2  12462  efgt1p  12463