Theorem effsumlt 12203

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 9757	. . . . 5 |- NN0 = ( ZZ>= ` 0 )
2		0zd 9458	. . . . 5 |- ( ph -> 0 e. ZZ )
3		effsumlt.2	. . . . . . . 8 |- ( ph -> A e. RR+ )
4	3	rpcnd 9894	. . . . . . 7 |- ( ph -> A e. CC )
5		effsumlt.1	. . . . . . . 8 |- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
6	5	eftvalcn 12168	. . . . . . 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 9892	. . . . . . 7 |- ( ph -> A e. RR )
9		reeftcl 12166	. . . . . . 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 2306	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) e. RR )
12	1, 2, 11	serfre 10706	. . . 4 |- ( ph -> seq 0 ( + , F ) : NN0 --> RR )
13		effsumlt.3	. . . 4 |- ( ph -> N e. NN0 )
14	12, 13	ffvelcdmd 5771	. . 3 |- ( ph -> ( seq 0 ( + , F ) ` N ) e. RR )
15		eqid 2229	. . . 4 |- ( ZZ>= ` ( N + 1 ) ) = ( ZZ>= ` ( N + 1 ) )
16		peano2nn0 9409	. . . . 5 |- ( N e. NN0 -> ( N + 1 ) e. NN0 )
17	13, 16	syl 14	. . . 4 |- ( ph -> ( N + 1 ) e. NN0 )
18		eqidd 2230	. . . 4 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) = ( F ` k ) )
19		nn0z 9466	. . . . . . 7 |- ( k e. NN0 -> k e. ZZ )
20		rpexpcl 10780	. . . . . . 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 10957	. . . . . . . 8 |- ( k e. NN0 -> ( ! ` k ) e. NN )
23	22	adantl 277	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> ( ! ` k ) e. NN )
24	23	nnrpd 9890	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> ( ! ` k ) e. RR+ )
25	21, 24	rpdivcld 9910	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. RR+ )
26	7, 25	eqeltrd 2306	. . . 4 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) e. RR+ )
27	5	efcllem 12170	. . . . 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 12005	. . 3 |- ( ph -> sum_ k e. ( ZZ>= ` ( N + 1 ) ) ( F ` k ) e. RR+ )
30	14, 29	ltaddrpd 9926	. 2 |- ( ph -> ( seq 0 ( + , F ) ` N ) < ( ( seq 0 ( + , F ) ` N ) + sum_ k e. ( ZZ>= ` ( N + 1 ) ) ( F ` k ) ) )
31	5	efval2 12176	. . . 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 8175	. . . 4 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) e. CC )
34	1, 15, 17, 18, 33, 28	isumsplit 12002	. . 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 9424	. . . . . . . 8 |- ( ph -> N e. CC )
36		ax-1cn 8092	. . . . . . . 8 |- 1 e. CC
37		pncan 8352	. . . . . . . 8 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N + 1 ) - 1 ) = N )
38	35, 36, 37	sylancl 413	. . . . . . 7 |- ( ph -> ( ( N + 1 ) - 1 ) = N )
39	38	oveq2d 6017	. . . . . 6 |- ( ph -> ( 0 ... ( ( N + 1 ) - 1 ) ) = ( 0 ... N ) )
40	39	sumeq1d 11877	. . . . 5 |- ( ph -> sum_ k e. ( 0 ... ( ( N + 1 ) - 1 ) ) ( F ` k ) = sum_ k e. ( 0 ... N ) ( F ` k ) )
41		eqidd 2230	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` 0 ) ) -> ( F ` k ) = ( F ` k ) )
42	13, 1	eleqtrdi 2322	. . . . . 6 |- ( ph -> N e. ( ZZ>= ` 0 ) )
43		elnn0uz 9760	. . . . . . 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 11908	. . . . 5 |- ( ph -> sum_ k e. ( 0 ... N ) ( F ` k ) = ( seq 0 ( + , F ) ` N ) )
46	40, 45	eqtrd 2262	. . . 4 |- ( ph -> sum_ k e. ( 0 ... ( ( N + 1 ) - 1 ) ) ( F ` k ) = ( seq 0 ( + , F ) ` N ) )
47	46	oveq1d 6016	. . 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 2266	. 2 |- ( ph -> ( exp ` A ) = ( ( seq 0 ( + , F ) ` N ) + sum_ k e. ( ZZ>= ` ( N + 1 ) ) ( F ` k ) ) )
49	30, 48	breqtrrd 4111	1 |- ( ph -> ( seq 0 ( + , F ) ` N ) < ( exp ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   class class class wbr 4083   |-> cmpt 4145   dom cdm 4719   ` cfv 5318  (class class class)co 6001   CCcc 7997   RRcr 7998   0cc0 7999   1c1 8000   + caddc 8002   < clt 8181   - cmin 8317   / cdiv 8819   NNcn 9110   NN0cn0 9369   ZZcz 9446   ZZ>=cuz 9722   RR+crp 9849   ...cfz 10204   seqcseq 10669   ^cexp 10760   !cfa 10947   ~~> cli 11789   sum_csu 11864   expce 12153

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-frec 6537  df-1o 6562  df-oadd 6566  df-er 6680  df-en 6888  df-dom 6889  df-fin 6890  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-ico 10090  df-fz 10205  df-fzo 10339  df-seqfrec 10670  df-exp 10761  df-fac 10948  df-ihash 10998  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-clim 11790  df-sumdc 11865  df-ef 12159

This theorem is referenced by:  efgt1p2  12206  efgt1p  12207