Theorem eflegeo 12252

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 9781	. . 3 |- NN0 = ( ZZ>= ` 0 )
2		0zd 9481	. . 3 |- ( ph -> 0 e. ZZ )
3		eflegeo.1	. . . . 5 |- ( ph -> A e. RR )
4	3	recnd 8198	. . . 4 |- ( ph -> A e. CC )
5		eqid 2229	. . . . 5 |- ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
6	5	eftvalcn 12208	. . . 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 12206	. . . 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 10942	. . . 4 |- ( ( ph /\ k e. NN0 ) -> ( A ^ k ) e. RR )
13		oveq2 6021	. . . . 5 |- ( n = k -> ( A ^ n ) = ( A ^ k ) )
14		eqid 2229	. . . . 5 |- ( n e. NN0 |-> ( A ^ n ) ) = ( n e. NN0 |-> ( A ^ n ) )
15	13, 14	fvmptg 5718	. . . 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 10808	. . . 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 10987	. . . . . . 7 |- ( k e. NN0 -> ( ! ` k ) e. NN )
20	19	adantl 277	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> ( ! ` k ) e. NN )
21	20	nnred 9146	. . . . 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 10943	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> 0 <_ ( A ^ k ) )
25	20	nnge1d 9176	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> 1 <_ ( ! ` k ) )
26	18, 21, 24, 25	lemulge12d 9108	. . . 4 |- ( ( ph /\ k e. NN0 ) -> ( A ^ k ) <_ ( ( ! ` k ) x. ( A ^ k ) ) )
27	20	nngt0d 9177	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> 0 < ( ! ` k ) )
28		ledivmul 9047	. . . . 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 1275	. . . 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 12210	. . . 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 10701	. . . 4 |- seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. _V
34		eflegeo.3	. . . . . 6 |- ( ph -> A < 1 )
35		1red 8184	. . . . . . 7 |- ( ph -> 1 e. RR )
36		difrp 9917	. . . . . . 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 9932	. . . 4 |- ( ph -> ( 1 / ( 1 - A ) ) e. RR+ )
40	3, 22	absidd 11718	. . . . . 6 |- ( ph -> ( abs ` A ) = A )
41	40, 34	eqbrtrd 4108	. . . . 5 |- ( ph -> ( abs ` A ) < 1 )
42	4, 41, 16	geolim 12062	. . . 4 |- ( ph -> seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) ~~> ( 1 / ( 1 - A ) ) )
43		breldmg 4935	. . . 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 1376	. . 3 |- ( ph -> seq 0 ( + , ( n e. NN0 |-> ( A ^ n ) ) ) e. dom ~~> )
45	1, 2, 7, 9, 16, 18, 30, 32, 44	isumle 12046	. 2 |- ( ph -> sum_ k e. NN0 ( ( A ^ k ) / ( ! ` k ) ) <_ sum_ k e. NN0 ( A ^ k ) )
46		efval 12212	. . 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 10809	. . . . 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 11972	. . 3 |- ( ph -> sum_ k e. NN0 ( A ^ k ) = ( 1 / ( 1 - A ) ) )
51	50	eqcomd 2235	. 2 |- ( ph -> ( 1 / ( 1 - A ) ) = sum_ k e. NN0 ( A ^ k ) )
52	45, 47, 51	3brtr4d 4118	1 |- ( ph -> ( exp ` A ) <_ ( 1 / ( 1 - A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   _Vcvv 2800   class class class wbr 4086   |-> cmpt 4148   dom cdm 4723   ` cfv 5324  (class class class)co 6013   CCcc 8020   RRcr 8021   0cc0 8022   1c1 8023   + caddc 8025   x. cmul 8027   < clt 8204   <_ cle 8205   - cmin 8340   / cdiv 8842   NNcn 9133   NN0cn0 9392   RR+crp 9878   seqcseq 10699   ^cexp 10790   !cfa 10977   abscabs 11548   ~~> cli 11829   sum_csu 11904   expce 12193

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141  ax-caucvg 8142

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-frec 6552  df-1o 6577  df-oadd 6581  df-er 6697  df-en 6905  df-dom 6906  df-fin 6907  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-n0 9393  df-z 9470  df-uz 9746  df-q 9844  df-rp 9879  df-ico 10119  df-fz 10234  df-fzo 10368  df-seqfrec 10700  df-exp 10791  df-fac 10978  df-ihash 11028  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550  df-clim 11830  df-sumdc 11905  df-ef 12199

This theorem is referenced by: (None)