Theorem geolim 7172

Description: The partial sums in the infinite series 1 + A^1 + A^2... converge to (1 / (1 - A)).

Hypothesis

Ref	Expression
geolim.1	|- F = {<.k, y>. | (k e. NN0 /\ y = (A^k))}

Assertion

Ref	Expression
geolim	|- ((A e. CC /\ (abs` A) < 1) -> ( + seq0 F) ~~> (1 / (1 - A)))

Distinct variable group:   y,k,A

Proof of Theorem geolim

Step	Hyp	Ref	Expression
1		opreq1 3953	. . . . . . . 8 |- (A = if((A e. CC /\ (abs` A) < 1), A, 0) -> (A^k) = (if((A e. CC /\ (abs` A) < 1), A, 0)^k))
2	1	eqeq2d 1478	. . . . . . 7 |- (A = if((A e. CC /\ (abs` A) < 1), A, 0) -> (y = (A^k) <-> y = (if((A e. CC /\ (abs` A) < 1), A, 0)^k)))
3	2	anbi2d 614	. . . . . 6 |- (A = if((A e. CC /\ (abs` A) < 1), A, 0) -> ((k e. NN0 /\ y = (A^k)) <-> (k e. NN0 /\ y = (if((A e. CC /\ (abs` A) < 1), A, 0)^k))))
4	3	opabbidv 2660	. . . . 5 |- (A = if((A e. CC /\ (abs` A) < 1), A, 0) -> {<.k, y>. | (k e. NN0 /\ y = (A^k))} = {<.k, y>. | (k e. NN0 /\ y = (if((A e. CC /\ (abs` A) < 1), A, 0)^k))})
5	4	opreq2d 3961	. . . 4 |- (A = if((A e. CC /\ (abs` A) < 1), A, 0) -> ( + seq0 {<.k, y>. | (k e. NN0 /\ y = (A^k))}) = ( + seq0 {<.k, y>. | (k e. NN0 /\ y = (if((A e. CC /\ (abs` A) < 1), A, 0)^k))}))
6		opreq2 3954	. . . . 5 |- (A = if((A e. CC /\ (abs` A) < 1), A, 0) -> (1 - A) = (1 - if((A e. CC /\ (abs` A) < 1), A, 0)))
7	6	opreq2d 3961	. . . 4 |- (A = if((A e. CC /\ (abs` A) < 1), A, 0) -> (1 / (1 - A)) = (1 / (1 - if((A e. CC /\ (abs` A) < 1), A, 0))))
8	5, 7	breq12d 2621	. . 3 |- (A = if((A e. CC /\ (abs` A) < 1), A, 0) -> (( + seq0 {<.k, y>. | (k e. NN0 /\ y = (A^k))}) ~~> (1 / (1 - A)) <-> ( + seq0 {<.k, y>. | (k e. NN0 /\ y = (if((A e. CC /\ (abs` A) < 1), A, 0)^k))}) ~~> (1 / (1 - if((A e. CC /\ (abs` A) < 1), A, 0)))))
9		eqid 1468	. . . 4 |- {<.k, y>. | (k e. NN0 /\ y = (if((A e. CC /\ (abs` A) < 1), A, 0)^k))} = {<.k, y>. | (k e. NN0 /\ y = (if((A e. CC /\ (abs` A) < 1), A, 0)^k))}
10		eleq1 1526	. . . . . . 7 |- (A = if((A e. CC /\ (abs` A) < 1), A, 0) -> (A e. CC <-> if((A e. CC /\ (abs` A) < 1), A, 0) e. CC))
11		fveq2 3709	. . . . . . . 8 |- (A = if((A e. CC /\ (abs` A) < 1), A, 0) -> (abs` A) = (abs` if((A e. CC /\ (abs` A) < 1), A, 0)))
12	11	breq1d 2619	. . . . . . 7 |- (A = if((A e. CC /\ (abs` A) < 1), A, 0) -> ((abs` A) < 1 <-> (abs` if((A e. CC /\ (abs` A) < 1), A, 0)) < 1))
13	10, 12	anbi12d 626	. . . . . 6 |- (A = if((A e. CC /\ (abs` A) < 1), A, 0) -> ((A e. CC /\ (abs` A) < 1) <-> (if((A e. CC /\ (abs` A) < 1), A, 0) e. CC /\ (abs` if((A e. CC /\ (abs` A) < 1), A, 0)) < 1)))
14		eleq1 1526	. . . . . . 7 |- (0 = if((A e. CC /\ (abs` A) < 1), A, 0) -> (0 e. CC <-> if((A e. CC /\ (abs` A) < 1), A, 0) e. CC))
15		fveq2 3709	. . . . . . . 8 |- (0 = if((A e. CC /\ (abs` A) < 1), A, 0) -> (abs` 0) = (abs` if((A e. CC /\ (abs` A) < 1), A, 0)))
16	15	breq1d 2619	. . . . . . 7 |- (0 = if((A e. CC /\ (abs` A) < 1), A, 0) -> ((abs` 0) < 1 <-> (abs` if((A e. CC /\ (abs` A) < 1), A, 0)) < 1))
17	14, 16	anbi12d 626	. . . . . 6 |- (0 = if((A e. CC /\ (abs` A) < 1), A, 0) -> ((0 e. CC /\ (abs` 0) < 1) <-> (if((A e. CC /\ (abs` A) < 1), A, 0) e. CC /\ (abs` if((A e. CC /\ (abs` A) < 1), A, 0)) < 1)))
18		0cn 5300	. . . . . . 7 |- 0 e. CC
19		abs0 6814	. . . . . . . 8 |- (abs` 0) = 0
20		lt01 5653	. . . . . . . 8 |- 0 < 1
21	19, 20	eqbrtr 2624	. . . . . . 7 |- (abs` 0) < 1
22	18, 21	pm3.2i 285	. . . . . 6 |- (0 e. CC /\ (abs` 0) < 1)
23	13, 17, 22	elimhyp 2380	. . . . 5 |- (if((A e. CC /\ (abs` A) < 1), A, 0) e. CC /\ (abs` if((A e. CC /\ (abs` A) < 1), A, 0)) < 1)
24	23	pm3.26i 320	. . . 4 |- if((A e. CC /\ (abs` A) < 1), A, 0) e. CC
25	23	pm3.27i 324	. . . 4 |- (abs` if((A e. CC /\ (abs` A) < 1), A, 0)) < 1
26	9, 24, 25	geolimi 7171	. . 3 |- ( + seq0 {<.k, y>. | (k e. NN0 /\ y = (if((A e. CC /\ (abs` A) < 1), A, 0)^k))}) ~~> (1 / (1 - if((A e. CC /\ (abs` A) < 1), A, 0)))
27	8, 26	dedth 2373	. 2 |- ((A e. CC /\ (abs` A) < 1) -> ( + seq0 {<.k, y>. | (k e. NN0 /\ y = (A^k))}) ~~> (1 / (1 - A)))
28		geolim.1	. . 3 |- F = {<.k, y>. | (k e. NN0 /\ y = (A^k))}
29	28	opreq2i 3957	. 2 |- ( + seq0 F) = ( + seq0 {<.k, y>. | (k e. NN0 /\ y = (A^k))})
30	27, 29	syl5eqbr 2638	1 |- ((A e. CC /\ (abs` A) < 1) -> ( + seq0 F) ~~> (1 / (1 - A)))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  ifcif 2351   class class class wbr 2609  {copab 2656  ` cfv 3172  (class class class)co 3948  CCcc 5204  0cc0 5206  1c1 5207   + caddc 5209   - cmin 5264   / cdiv 5266  NN0cn0 5269   < clt 5458   seq0 cseq0 6464  ^cexp 6500  abscabs 6681   ~~> cli 6912

This theorem is referenced by:  georeclim 7175  geoisum 7177

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-sup 4548  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463  df-div 5672  df-n 5873  df-2 5917  df-n0 6047  df-z 6083  df-fl 6172  df-seq1 6245  df-shft 6278  df-uz 6350  df-seq0 6466  df-exp 6501  df-sqr 6600  df-re 6682  df-im 6683  df-cj 6684  df-abs 6685  df-clim 6913

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