Theorem eflegeo 11711

Description: The exponential function on the reals between 0 and 1 lies below the comparable geometric series sum. (Contributed by Paul Chapman, 11-Sep-2007.)

Hypotheses

Ref	Expression
eflegeo.1	|- ( ph -> A e. RR )
eflegeo.2	|- ( ph -> 0 <_ A )
eflegeo.3	|- ( ph -> A < 1 )

Assertion

Ref	Expression
eflegeo	|- ( ph -> ( exp ` A ) <_ ( 1 / ( 1 - A ) ) )

Proof of Theorem eflegeo

Dummy variables k n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0uz 9564	. . 3 |- NN0 = ( ZZ>= ` 0 )
2		0zd 9267	. . 3 |- ( ph -> 0 e. ZZ )
3		eflegeo.1	. . . . 5 |- ( ph -> A e. RR )
4	3	recnd 7988	. . . 4 |- ( ph -> A e. CC )
5		eqid 2177	. . . . 5 |- ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
6	5	eftvalcn 11667	. . . 4 |- ( ( A e. CC /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
7	4, 6	sylan 283	. . 3 |- ( ( ph /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
8		reeftcl 11665	. . . 4 |- ( ( A e. RR /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. RR )
9	3, 8	sylan 283	. . 3 |- ( ( ph /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. RR )
10		simpr 110	. . . 4 |- ( ( ph /\ k e. NN0 ) -> k e. NN0 )
11	3	adantr 276	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> A e. RR )
12	11, 10	reexpcld 10673	. . . 4 |- ( ( ph /\ k e. NN0 ) -> ( A ^ k ) e. RR )
13		oveq2 5885	. . . . 5 |- ( n = k -> ( A ^ n ) = ( A ^ k ) )
14		eqid 2177	. . . . 5 |- ( n e. NN0 |-> ( A ^ n ) ) = ( n e. NN0 |-> ( A ^ n ) )
15	13, 14	fvmptg 5594	. . . 4 |- ( ( k e. NN0 /\ ( A ^ k ) e. RR ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( A ^ k ) )
16	10, 12, 15	syl2anc 411	. . 3 |- ( ( ph /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( A ^ k ) )
17		reexpcl 10539	. . . 4 |- ( ( A e. RR /\ k e. NN0 ) -> ( A ^ k ) e. RR )
18	3, 17	sylan 283	. . 3 |- ( ( ph /\ k e. NN0 ) -> ( A ^ k ) e. RR )
19		faccl 10717	. . . . . . 7 |- ( k e. NN0 -> ( ! ` k ) e. NN )
20	19	adantl 277	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> ( ! ` k ) e. NN )
21	20	nnred 8934	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> ( ! ` k ) e. RR )
22		eflegeo.2	. . . . . . 7 |- ( ph -> 0 <_ A )
23	22	adantr 276	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> 0 <_ A )
24	11, 10, 23	expge0d 10674	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> 0 <_ ( A ^ k ) )
25	20	nnge1d 8964	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> 1 <_ ( ! ` k ) )
26	18, 21, 24, 25	lemulge12d 8897	. . . 4 |- ( ( ph /\ k e. NN0 ) -> ( A ^ k ) <_ ( ( ! ` k ) x. ( A ^ k ) ) )
27	20	nngt0d 8965	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> 0 < ( ! ` k ) )
28		ledivmul 8836	. . . . 5 |- ( ( ( A ^ k ) e. RR /\ ( A ^ k ) e. RR /\ ( ( ! ` k ) e. RR /\ 0 < ( ! ` k ) ) ) -> ( ( ( A ^ k ) / ( ! ` k ) ) <_ ( A ^ k ) <-> ( A ^ k ) <_ ( ( ! ` k ) x. ( A ^ k ) ) ) )
29	18, 18, 21, 27, 28	syl112anc 1242	. . . 4 |- ( ( ph /\ k e. NN0 ) -> ( ( ( A ^ k ) / ( ! ` k ) ) <_ ( A ^ k ) <-> ( A ^ k ) <_ ( ( ! ` k ) x. ( A ^ k ) ) ) )
30	26, 29	mpbird 167	. . 3 |- ( ( ph /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) <_ ( A ^ k ) )
31	5	efcllem 11669	. . . 4 |- ( A e. CC -> seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) e. dom ~~> )
32	4, 31	syl 14	. . 3 |- ( ph -> seq 0 ( + , ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) ) e. dom ~~> )
33		seqex 10449	. . . 4 |- seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. _V
34		eflegeo.3	. . . . . 6 |- ( ph -> A < 1 )
35		1red 7974	. . . . . . 7 |- ( ph -> 1 e. RR )
36		difrp 9694	. . . . . . 7 |- ( ( A e. RR /\ 1 e. RR ) -> ( A < 1 <-> ( 1 - A ) e. RR+ ) )
37	3, 35, 36	syl2anc 411	. . . . . 6 |- ( ph -> ( A < 1 <-> ( 1 - A ) e. RR+ ) )
38	34, 37	mpbid 147	. . . . 5 |- ( ph -> ( 1 - A ) e. RR+ )
39	38	rpreccld 9709	. . . 4 |- ( ph -> ( 1 / ( 1 - A ) ) e. RR+ )
40	3, 22	absidd 11178	. . . . . 6 |- ( ph -> ( abs ` A ) = A )
41	40, 34	eqbrtrd 4027	. . . . 5 |- ( ph -> ( abs ` A ) < 1 )
42	4, 41, 16	geolim 11521	. . . 4 |- ( ph -> seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) ~~> ( 1 / ( 1 - A ) ) )
43		breldmg 4835	. . . 4 |- ( ( seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. _V /\ ( 1 / ( 1 - A ) ) e. RR+ /\ seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) ~~> ( 1 / ( 1 - A ) ) ) -> seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. dom ~~> )
44	33, 39, 42, 43	mp3an2i 1342	. . 3 |- ( ph -> seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. dom ~~> )
45	1, 2, 7, 9, 16, 18, 30, 32, 44	isumle 11505	. 2 |- ( ph -> sum_ k e. NN0 ( ( A ^ k ) / ( ! ` k ) ) <_ sum_ k e. NN0 ( A ^ k ) )
46		efval 11671	. . 3 |- ( A e. CC -> ( exp ` A ) = sum_ k e. NN0 ( ( A ^ k ) / ( ! ` k ) ) )
47	4, 46	syl 14	. 2 |- ( ph -> ( exp ` A ) = sum_ k e. NN0 ( ( A ^ k ) / ( ! ` k ) ) )
48		expcl 10540	. . . . 5 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ k ) e. CC )
49	4, 48	sylan 283	. . . 4 |- ( ( ph /\ k e. NN0 ) -> ( A ^ k ) e. CC )
50	1, 2, 16, 49, 42	isumclim 11431	. . 3 |- ( ph -> sum_ k e. NN0 ( A ^ k ) = ( 1 / ( 1 - A ) ) )
51	50	eqcomd 2183	. 2 |- ( ph -> ( 1 / ( 1 - A ) ) = sum_ k e. NN0 ( A ^ k ) )
52	45, 47, 51	3brtr4d 4037	1 |- ( ph -> ( exp ` A ) <_ ( 1 / ( 1 - A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   _Vcvv 2739   class class class wbr 4005   |-> cmpt 4066   dom cdm 4628   ` cfv 5218  (class class class)co 5877   CCcc 7811   RRcr 7812   0cc0 7813   1c1 7814   + caddc 7816   x. cmul 7818   < clt 7994   <_ cle 7995   - cmin 8130   / cdiv 8631   NNcn 8921   NN0cn0 9178   RR+crp 9655   seqcseq 10447   ^cexp 10521   !cfa 10707   abscabs 11008   ~~> cli 11288   sum_csu 11363   expce 11652

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-frec 6394  df-1o 6419  df-oadd 6423  df-er 6537  df-en 6743  df-dom 6744  df-fin 6745  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-ico 9896  df-fz 10011  df-fzo 10145  df-seqfrec 10448  df-exp 10522  df-fac 10708  df-ihash 10758  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-clim 11289  df-sumdc 11364  df-ef 11658

This theorem is referenced by: (None)