Theorem eflegeo 12468

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 9957	. . 3 |- NN0 = ( ZZ>= ` 0 )
2		0zd 9656	. . 3 |- ( ph -> 0 e. ZZ )
3		eflegeo.1	. . . . 5 |- ( ph -> A e. RR )
4	3	recnd 8354	. . . 4 |- ( ph -> A e. CC )
5		eqid 2238	. . . . 5 |- ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
6	5	eftvalcn 12424	. . . 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 12422	. . . 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 11128	. . . 4 |- ( ( ph /\ k e. NN0 ) -> ( A ^ k ) e. RR )
13		oveq2 6093	. . . . 5 |- ( n = k -> ( A ^ n ) = ( A ^ k ) )
14		eqid 2238	. . . . 5 |- ( n e. NN0 |-> ( A ^ n ) ) = ( n e. NN0 |-> ( A ^ n ) )
15	13, 14	fvmptg 5781	. . . 4 |- ( ( k e. NN0 /\ ( A ^ k ) e. RR ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( A ^ k ) )
16	10, 12, 15	syl2anc 415	. . 3 |- ( ( ph /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( A ^ k ) )
17		reexpcl 10993	. . . 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 11173	. . . . . . 7 |- ( k e. NN0 -> ( ! ` k ) e. NN )
20	19	adantl 277	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> ( ! ` k ) e. NN )
21	20	nnred 9317	. . . . 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 11129	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> 0 <_ ( A ^ k ) )
25	20	nnge1d 9347	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> 1 <_ ( ! ` k ) )
26	18, 21, 24, 25	lemulge12d 9268	. . . 4 |- ( ( ph /\ k e. NN0 ) -> ( A ^ k ) <_ ( ( ! ` k ) x. ( A ^ k ) ) )
27	20	nngt0d 9348	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> 0 < ( ! ` k ) )
28		ledivmul 9207	. . . . 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 1282	. . . 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 12426	. . . 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 10886	. . . 4 |- seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. _V
34		eflegeo.3	. . . . . 6 |- ( ph -> A < 1 )
35		1red 8341	. . . . . . 7 |- ( ph -> 1 e. RR )
36		difrp 10093	. . . . . . 7 |- ( ( A e. RR /\ 1 e. RR ) -> ( A < 1 <-> ( 1 - A ) e. RR+ ) )
37	3, 35, 36	syl2anc 415	. . . . . 6 |- ( ph -> ( A < 1 <-> ( 1 - A ) e. RR+ ) )
38	34, 37	mpbid 147	. . . . 5 |- ( ph -> ( 1 - A ) e. RR+ )
39	38	rpreccld 10108	. . . 4 |- ( ph -> ( 1 / ( 1 - A ) ) e. RR+ )
40	3, 22	absidd 11933	. . . . . 6 |- ( ph -> ( abs ` A ) = A )
41	40, 34	eqbrtrd 4152	. . . . 5 |- ( ph -> ( abs ` A ) < 1 )
42	4, 41, 16	geolim 12278	. . . 4 |- ( ph -> seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) ~~> ( 1 / ( 1 - A ) ) )
43		breldmg 4987	. . . 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 1383	. . 3 |- ( ph -> seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. dom ~~> )
45	1, 2, 7, 9, 16, 18, 30, 32, 44	isumle 12262	. 2 |- ( ph -> sum_ k e. NN0 ( ( A ^ k ) / ( ! ` k ) ) <_ sum_ k e. NN0 ( A ^ k ) )
46		efval 12428	. . 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 10994	. . . . 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 12188	. . 3 |- ( ph -> sum_ k e. NN0 ( A ^ k ) = ( 1 / ( 1 - A ) ) )
51	50	eqcomd 2244	. 2 |- ( ph -> ( 1 / ( 1 - A ) ) = sum_ k e. NN0 ( A ^ k ) )
52	45, 47, 51	3brtr4d 4162	1 |- ( ph -> ( exp ` A ) <_ ( 1 / ( 1 - A ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   _Vcvv 2821   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   x. cmul 8184   < clt 8360   <_ cle 8361   - cmin 8497   / cdiv 9002   NNcn 9304   NN0cn0 9563   RR+crp 10054   seqcseq 10884   ^cexp 10975   !cfa 11163   abscabs 11763   ~~> 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:  birthdaylem3  16089