Theorem eflegeo 11844

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 9627	. . 3 |- NN0 = ( ZZ>= ` 0 )
2		0zd 9329	. . 3 |- ( ph -> 0 e. ZZ )
3		eflegeo.1	. . . . 5 |- ( ph -> A e. RR )
4	3	recnd 8048	. . . 4 |- ( ph -> A e. CC )
5		eqid 2193	. . . . 5 |- ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
6	5	eftvalcn 11800	. . . 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 11798	. . . 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 10761	. . . 4 |- ( ( ph /\ k e. NN0 ) -> ( A ^ k ) e. RR )
13		oveq2 5926	. . . . 5 |- ( n = k -> ( A ^ n ) = ( A ^ k ) )
14		eqid 2193	. . . . 5 |- ( n e. NN0 |-> ( A ^ n ) ) = ( n e. NN0 |-> ( A ^ n ) )
15	13, 14	fvmptg 5633	. . . 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 10627	. . . 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 10806	. . . . . . 7 |- ( k e. NN0 -> ( ! ` k ) e. NN )
20	19	adantl 277	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> ( ! ` k ) e. NN )
21	20	nnred 8995	. . . . 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 10762	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> 0 <_ ( A ^ k ) )
25	20	nnge1d 9025	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> 1 <_ ( ! ` k ) )
26	18, 21, 24, 25	lemulge12d 8957	. . . 4 |- ( ( ph /\ k e. NN0 ) -> ( A ^ k ) <_ ( ( ! ` k ) x. ( A ^ k ) ) )
27	20	nngt0d 9026	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> 0 < ( ! ` k ) )
28		ledivmul 8896	. . . . 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 1253	. . . 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 11802	. . . 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 10520	. . . 4 |- seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. _V
34		eflegeo.3	. . . . . 6 |- ( ph -> A < 1 )
35		1red 8034	. . . . . . 7 |- ( ph -> 1 e. RR )
36		difrp 9758	. . . . . . 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 9773	. . . 4 |- ( ph -> ( 1 / ( 1 - A ) ) e. RR+ )
40	3, 22	absidd 11311	. . . . . 6 |- ( ph -> ( abs ` A ) = A )
41	40, 34	eqbrtrd 4051	. . . . 5 |- ( ph -> ( abs ` A ) < 1 )
42	4, 41, 16	geolim 11654	. . . 4 |- ( ph -> seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) ~~> ( 1 / ( 1 - A ) ) )
43		breldmg 4868	. . . 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 1353	. . 3 |- ( ph -> seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. dom ~~> )
45	1, 2, 7, 9, 16, 18, 30, 32, 44	isumle 11638	. 2 |- ( ph -> sum_ k e. NN0 ( ( A ^ k ) / ( ! ` k ) ) <_ sum_ k e. NN0 ( A ^ k ) )
46		efval 11804	. . 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 10628	. . . . 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 11564	. . 3 |- ( ph -> sum_ k e. NN0 ( A ^ k ) = ( 1 / ( 1 - A ) ) )
51	50	eqcomd 2199	. 2 |- ( ph -> ( 1 / ( 1 - A ) ) = sum_ k e. NN0 ( A ^ k ) )
52	45, 47, 51	3brtr4d 4061	1 |- ( ph -> ( exp ` A ) <_ ( 1 / ( 1 - A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   _Vcvv 2760   class class class wbr 4029   |-> cmpt 4090   dom cdm 4659   ` cfv 5254  (class class class)co 5918   CCcc 7870   RRcr 7871   0cc0 7872   1c1 7873   + caddc 7875   x. cmul 7877   < clt 8054   <_ cle 8055   - cmin 8190   / cdiv 8691   NNcn 8982   NN0cn0 9240   RR+crp 9719   seqcseq 10518   ^cexp 10609   !cfa 10796   abscabs 11141   ~~> cli 11421   sum_csu 11496   expce 11785

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-frec 6444  df-1o 6469  df-oadd 6473  df-er 6587  df-en 6795  df-dom 6796  df-fin 6797  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-ico 9960  df-fz 10075  df-fzo 10209  df-seqfrec 10519  df-exp 10610  df-fac 10797  df-ihash 10847  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-clim 11422  df-sumdc 11497  df-ef 11791

This theorem is referenced by: (None)