Theorem effsumlt 12036

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 9685	. . . . 5 |- NN0 = ( ZZ>= ` 0 )
2		0zd 9386	. . . . 5 |- ( ph -> 0 e. ZZ )
3		effsumlt.2	. . . . . . . 8 |- ( ph -> A e. RR+ )
4	3	rpcnd 9822	. . . . . . 7 |- ( ph -> A e. CC )
5		effsumlt.1	. . . . . . . 8 |- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
6	5	eftvalcn 12001	. . . . . . 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 9820	. . . . . . 7 |- ( ph -> A e. RR )
9		reeftcl 11999	. . . . . . 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 2282	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) e. RR )
12	1, 2, 11	serfre 10631	. . . 4 |- ( ph -> seq 0 ( + , F ) : NN0 --> RR )
13		effsumlt.3	. . . 4 |- ( ph -> N e. NN0 )
14	12, 13	ffvelcdmd 5718	. . 3 |- ( ph -> ( seq 0 ( + , F ) ` N ) e. RR )
15		eqid 2205	. . . 4 |- ( ZZ>= ` ( N + 1 ) ) = ( ZZ>= ` ( N + 1 ) )
16		peano2nn0 9337	. . . . 5 |- ( N e. NN0 -> ( N + 1 ) e. NN0 )
17	13, 16	syl 14	. . . 4 |- ( ph -> ( N + 1 ) e. NN0 )
18		eqidd 2206	. . . 4 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) = ( F ` k ) )
19		nn0z 9394	. . . . . . 7 |- ( k e. NN0 -> k e. ZZ )
20		rpexpcl 10705	. . . . . . 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 10882	. . . . . . . 8 |- ( k e. NN0 -> ( ! ` k ) e. NN )
23	22	adantl 277	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> ( ! ` k ) e. NN )
24	23	nnrpd 9818	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> ( ! ` k ) e. RR+ )
25	21, 24	rpdivcld 9838	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. RR+ )
26	7, 25	eqeltrd 2282	. . . 4 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) e. RR+ )
27	5	efcllem 12003	. . . . 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 11838	. . 3 |- ( ph -> sum_ k e. ( ZZ>= ` ( N + 1 ) ) ( F ` k ) e. RR+ )
30	14, 29	ltaddrpd 9854	. 2 |- ( ph -> ( seq 0 ( + , F ) ` N ) < ( ( seq 0 ( + , F ) ` N ) + sum_ k e. ( ZZ>= ` ( N + 1 ) ) ( F ` k ) ) )
31	5	efval2 12009	. . . 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 8103	. . . 4 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) e. CC )
34	1, 15, 17, 18, 33, 28	isumsplit 11835	. . 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 9352	. . . . . . . 8 |- ( ph -> N e. CC )
36		ax-1cn 8020	. . . . . . . 8 |- 1 e. CC
37		pncan 8280	. . . . . . . 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 5962	. . . . . 6 |- ( ph -> ( 0 ... ( ( N + 1 ) - 1 ) ) = ( 0 ... N ) )
40	39	sumeq1d 11710	. . . . 5 |- ( ph -> sum_ k e. ( 0 ... ( ( N + 1 ) - 1 ) ) ( F ` k ) = sum_ k e. ( 0 ... N ) ( F ` k ) )
41		eqidd 2206	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` 0 ) ) -> ( F ` k ) = ( F ` k ) )
42	13, 1	eleqtrdi 2298	. . . . . 6 |- ( ph -> N e. ( ZZ>= ` 0 ) )
43		elnn0uz 9688	. . . . . . 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 11741	. . . . 5 |- ( ph -> sum_ k e. ( 0 ... N ) ( F ` k ) = ( seq 0 ( + , F ) ` N ) )
46	40, 45	eqtrd 2238	. . . 4 |- ( ph -> sum_ k e. ( 0 ... ( ( N + 1 ) - 1 ) ) ( F ` k ) = ( seq 0 ( + , F ) ` N ) )
47	46	oveq1d 5961	. . 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 2242	. 2 |- ( ph -> ( exp ` A ) = ( ( seq 0 ( + , F ) ` N ) + sum_ k e. ( ZZ>= ` ( N + 1 ) ) ( F ` k ) ) )
49	30, 48	breqtrrd 4073	1 |- ( ph -> ( seq 0 ( + , F ) ` N ) < ( exp ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   class class class wbr 4045   |-> cmpt 4106   dom cdm 4676   ` cfv 5272  (class class class)co 5946   CCcc 7925   RRcr 7926   0cc0 7927   1c1 7928   + caddc 7930   < clt 8109   - cmin 8245   / cdiv 8747   NNcn 9038   NN0cn0 9297   ZZcz 9374   ZZ>=cuz 9650   RR+crp 9777   ...cfz 10132   seqcseq 10594   ^cexp 10685   !cfa 10872   ~~> cli 11622   sum_csu 11697   expce 11986

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044  ax-pre-mulext 8045  ax-arch 8046  ax-caucvg 8047

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-po 4344  df-iso 4345  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-isom 5281  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-irdg 6458  df-frec 6479  df-1o 6504  df-oadd 6508  df-er 6622  df-en 6830  df-dom 6831  df-fin 6832  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-div 8748  df-inn 9039  df-2 9097  df-3 9098  df-4 9099  df-n0 9298  df-z 9375  df-uz 9651  df-q 9743  df-rp 9778  df-ico 10018  df-fz 10133  df-fzo 10267  df-seqfrec 10595  df-exp 10686  df-fac 10873  df-ihash 10923  df-cj 11186  df-re 11187  df-im 11188  df-rsqrt 11342  df-abs 11343  df-clim 11623  df-sumdc 11698  df-ef 11992

This theorem is referenced by:  efgt1p2  12039  efgt1p  12040