Theorem georeclim 12224

Description: The limit of a geometric series of reciprocals. (Contributed by Paul Chapman, 28-Dec-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)

Hypotheses

Ref	Expression
georeclim.1	|- ( ph -> A e. CC )
georeclim.2	|- ( ph -> 1 < ( abs ` A ) )
georeclim.3	|- ( ( ph /\ k e. NN0 ) -> ( F ` k ) = ( ( 1 / A ) ^ k ) )

Assertion

Ref	Expression
georeclim	|- ( ph -> seq 0 ( + , F ) ~~> ( A / ( A - 1 ) ) )

Distinct variable groups:   A, k   k, F   ph, k

Proof of Theorem georeclim

Step	Hyp	Ref	Expression
1		georeclim.1	. . . 4 |- ( ph -> A e. CC )
2	1	abscld 11891	. . . . . 6 |- ( ph -> ( abs ` A ) e. RR )
3		0red 8291	. . . . . . 7 |- ( ph -> 0 e. RR )
4		1red 8305	. . . . . . 7 |- ( ph -> 1 e. RR )
5		0lt1 8416	. . . . . . . 8 |- 0 < 1
6	5	a1i 9	. . . . . . 7 |- ( ph -> 0 < 1 )
7		georeclim.2	. . . . . . 7 |- ( ph -> 1 < ( abs ` A ) )
8	3, 4, 2, 6, 7	lttrd 8415	. . . . . 6 |- ( ph -> 0 < ( abs ` A ) )
9	2, 8	gt0ap0d 8920	. . . . 5 |- ( ph -> ( abs ` A ) # 0 )
10		abs00ap 11772	. . . . . 6 |- ( A e. CC -> ( ( abs ` A ) # 0 <-> A # 0 ) )
11	1, 10	syl 14	. . . . 5 |- ( ph -> ( ( abs ` A ) # 0 <-> A # 0 ) )
12	9, 11	mpbid 147	. . . 4 |- ( ph -> A # 0 )
13	1, 12	recclapd 9072	. . 3 |- ( ph -> ( 1 / A ) e. CC )
14		1cnd 8306	. . . . . 6 |- ( ph -> 1 e. CC )
15	14, 1, 12	absdivapd 11905	. . . . 5 |- ( ph -> ( abs ` ( 1 / A ) ) = ( ( abs ` 1 ) / ( abs ` A ) ) )
16		abs1 11782	. . . . . 6 |- ( abs ` 1 ) = 1
17	16	oveq1i 6068	. . . . 5 |- ( ( abs ` 1 ) / ( abs ` A ) ) = ( 1 / ( abs ` A ) )
18	15, 17	eqtrdi 2283	. . . 4 |- ( ph -> ( abs ` ( 1 / A ) ) = ( 1 / ( abs ` A ) ) )
19	2, 8	elrpd 10044	. . . . . 6 |- ( ph -> ( abs ` A ) e. RR+ )
20	19	recgt1d 10062	. . . . 5 |- ( ph -> ( 1 < ( abs ` A ) <-> ( 1 / ( abs ` A ) ) < 1 ) )
21	7, 20	mpbid 147	. . . 4 |- ( ph -> ( 1 / ( abs ` A ) ) < 1 )
22	18, 21	eqbrtrd 4136	. . 3 |- ( ph -> ( abs ` ( 1 / A ) ) < 1 )
23		georeclim.3	. . 3 |- ( ( ph /\ k e. NN0 ) -> ( F ` k ) = ( ( 1 / A ) ^ k ) )
24	13, 22, 23	geolim 12222	. 2 |- ( ph -> seq 0 ( + , F ) ~~> ( 1 / ( 1 - ( 1 / A ) ) ) )
25	1, 14, 1, 12	divsubdirapd 9121	. . . . 5 |- ( ph -> ( ( A - 1 ) / A ) = ( ( A / A ) - ( 1 / A ) ) )
26	1, 12	dividapd 9077	. . . . . 6 |- ( ph -> ( A / A ) = 1 )
27	26	oveq1d 6073	. . . . 5 |- ( ph -> ( ( A / A ) - ( 1 / A ) ) = ( 1 - ( 1 / A ) ) )
28	25, 27	eqtrd 2267	. . . 4 |- ( ph -> ( ( A - 1 ) / A ) = ( 1 - ( 1 / A ) ) )
29	28	oveq2d 6074	. . 3 |- ( ph -> ( 1 / ( ( A - 1 ) / A ) ) = ( 1 / ( 1 - ( 1 / A ) ) ) )
30		ax-1cn 8236	. . . . 5 |- 1 e. CC
31		subcl 8488	. . . . 5 |- ( ( A e. CC /\ 1 e. CC ) -> ( A - 1 ) e. CC )
32	1, 30, 31	sylancl 413	. . . 4 |- ( ph -> ( A - 1 ) e. CC )
33	4, 6	elrpd 10044	. . . . . 6 |- ( ph -> 1 e. RR+ )
34	1, 33, 7	absgtap 12221	. . . . 5 |- ( ph -> A # 1 )
35	1, 14, 34	subap0d 8935	. . . 4 |- ( ph -> ( A - 1 ) # 0 )
36	32, 1, 35, 12	recdivapd 9098	. . 3 |- ( ph -> ( 1 / ( ( A - 1 ) / A ) ) = ( A / ( A - 1 ) ) )
37	29, 36	eqtr3d 2269	. 2 |- ( ph -> ( 1 / ( 1 - ( 1 / A ) ) ) = ( A / ( A - 1 ) ) )
38	24, 37	breqtrd 4140	1 |- ( ph -> seq 0 ( + , F ) ~~> ( A / ( A - 1 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   CCcc 8141   0cc0 8143   1c1 8144   + caddc 8146   < clt 8324   - cmin 8460   # cap 8872   / cdiv 8963   NN0cn0 9513   seqcseq 10833   ^cexp 10924   abscabs 11707   ~~> cli 11988

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-seqfrec 10834  df-exp 10925  df-ihash 11164  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-clim 11989  df-sumdc 12064

This theorem is referenced by:  geoisumr  12229  ege2le3  12382  eftlub  12401