Theorem geolim 8499

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 4889	. . . . . . . 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 1895	. . . . . . 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 678	. . . . . 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 3401	. . . . 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 4898	. . . 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 4890	. . . . 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 4898	. . . 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 3351	. . 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 1884	. . . 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 1957	. . . . . . 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 4681	. . . . . . . 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 3348	. . . . . . 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 690	. . . . . 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 1957	. . . . . . 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 4681	. . . . . . . 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 3348	. . . . . . 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 690	. . . . . 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 6481	. . . . . . 7 |- 0 e. CC
19		abs0 8129	. . . . . . . 8 |- (abs` 0) = 0
20		lt01 6871	. . . . . . . 8 |- 0 < 1
21	19, 20	eqbrtri 3356	. . . . . . 7 |- (abs` 0) < 1
22	18, 21	pm3.2i 307	. . . . . 6 |- (0 e. CC /\ (abs` 0) < 1)
23	13, 17, 22	elimhyp 3021	. . . . 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	simpli 347	. . . 4 |- if((A e. CC /\ (abs` A) < 1), A, 0) e. CC
25	23	simpri 351	. . . 4 |- (abs` if((A e. CC /\ (abs` A) < 1), A, 0)) < 1
26	9, 24, 25	geolimi 8498	. . 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 3011	. 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 4893	. 2 |- ( + seq0 F) = ( + seq0 {<.k, y>. | (k e. NN0 /\ y = (A^k))})
30	27, 29	syl5eqbr 3370	1 |- ((A e. CC /\ (abs` A) < 1) -> ( + seq0 F) ~~> (1 / (1 - A)))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  ifcif 2982   class class class wbr 3338  {copab 3395  ` cfv 3998  (class class class)co 4884  CCcc 6384  0cc0 6386  1c1 6387   + caddc 6389   - cmin 6445   / cdiv 6447  NN0cn0 6450   < clt 6653   seq0 cseq0 7775  ^cexp 7811  abscabs 8000   ~~> cli 8234

This theorem is referenced by:  georeclim 8502  geoisum 8504  geomcau 15849

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-n0 7309  df-z 7345  df-fl 7463  df-uz 7587  df-seq1 7721  df-shft 7754  df-seq0 7777  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-clim 8235

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