Theorem eflegeo 11693

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 9551	. . 3 |- NN0 = ( ZZ>= ` 0 )
2		0zd 9254	. . 3 |- ( ph -> 0 e. ZZ )
3		eflegeo.1	. . . . 5 |- ( ph -> A e. RR )
4	3	recnd 7976	. . . 4 |- ( ph -> A e. CC )
5		eqid 2177	. . . . 5 |- ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
6	5	eftvalcn 11649	. . . 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 11647	. . . 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 10656	. . . 4 |- ( ( ph /\ k e. NN0 ) -> ( A ^ k ) e. RR )
13		oveq2 5877	. . . . 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 5588	. . . 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 10523	. . . 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 10699	. . . . . . 7 |- ( k e. NN0 -> ( ! ` k ) e. NN )
20	19	adantl 277	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> ( ! ` k ) e. NN )
21	20	nnred 8921	. . . . 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 10657	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> 0 <_ ( A ^ k ) )
25	20	nnge1d 8951	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> 1 <_ ( ! ` k ) )
26	18, 21, 24, 25	lemulge12d 8884	. . . 4 |- ( ( ph /\ k e. NN0 ) -> ( A ^ k ) <_ ( ( ! ` k ) x. ( A ^ k ) ) )
27	20	nngt0d 8952	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> 0 < ( ! ` k ) )
28		ledivmul 8823	. . . . 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 11651	. . . 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 10433	. . . 4 |- seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. _V
34		eflegeo.3	. . . . . 6 |- ( ph -> A < 1 )
35		1red 7963	. . . . . . 7 |- ( ph -> 1 e. RR )
36		difrp 9679	. . . . . . 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 9694	. . . 4 |- ( ph -> ( 1 / ( 1 - A ) ) e. RR+ )
40	3, 22	absidd 11160	. . . . . 6 |- ( ph -> ( abs ` A ) = A )
41	40, 34	eqbrtrd 4022	. . . . 5 |- ( ph -> ( abs ` A ) < 1 )
42	4, 41, 16	geolim 11503	. . . 4 |- ( ph -> seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) ~~> ( 1 / ( 1 - A ) ) )
43		breldmg 4829	. . . 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 11487	. 2 |- ( ph -> sum_ k e. NN0 ( ( A ^ k ) / ( ! ` k ) ) <_ sum_ k e. NN0 ( A ^ k ) )
46		efval 11653	. . 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 10524	. . . . 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 11413	. . 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 4032	1 |- ( ph -> ( exp ` A ) <_ ( 1 / ( 1 - A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   _Vcvv 2737   class class class wbr 4000   |-> cmpt 4061   dom cdm 4623   ` cfv 5212  (class class class)co 5869   CCcc 7800   RRcr 7801   0cc0 7802   1c1 7803   + caddc 7805   x. cmul 7807   < clt 7982   <_ cle 7983   - cmin 8118   / cdiv 8618   NNcn 8908   NN0cn0 9165   RR+crp 9640   seqcseq 10431   ^cexp 10505   !cfa 10689   abscabs 10990   ~~> cli 11270   sum_csu 11345   expce 11634

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 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922

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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-oadd 6415  df-er 6529  df-en 6735  df-dom 6736  df-fin 6737  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-ico 9881  df-fz 9996  df-fzo 10129  df-seqfrec 10432  df-exp 10506  df-fac 10690  df-ihash 10740  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-clim 11271  df-sumdc 11346  df-ef 11640

This theorem is referenced by: (None)