Theorem geolim 7360

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 4044	. . . . . . . 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 1523	. . . . . . 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 618	. . . . . 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 2721	. . . . 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 4052	. . . 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 4045	. . . . 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 4052	. . . 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 2681	. . 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 1512	. . . 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 1571	. . . . . . 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 3800	. . . . . . . 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 2679	. . . . . . 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 630	. . . . . 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 1571	. . . . . . 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 3800	. . . . . . . 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 2679	. . . . . . 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 630	. . . . . 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 5417	. . . . . . 7 |- 0 e. CC
19		abs0 7000	. . . . . . . 8 |- (abs` 0) = 0
20		lt01 5769	. . . . . . . 8 |- 0 < 1
21	19, 20	eqbrtri 2684	. . . . . . 7 |- (abs` 0) < 1
22	18, 21	pm3.2i 283	. . . . . 6 |- (0 e. CC /\ (abs` 0) < 1)
23	13, 17, 22	elimhyp 2435	. . . . 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 318	. . . 4 |- if((A e. CC /\ (abs` A) < 1), A, 0) e. CC
25	23	pm3.27i 322	. . . 4 |- (abs` if((A e. CC /\ (abs` A) < 1), A, 0)) < 1
26	9, 24, 25	geolimi 7359	. . 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 2428	. 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 4048	. 2 |- ( + seq0 F) = ( + seq0 {<.k, y>. | (k e. NN0 /\ y = (A^k))})
30	27, 29	syl5eqbr 2698	1 |- ((A e. CC /\ (abs` A) < 1) -> ( + seq0 F) ~~> (1 / (1 - A)))

Syntax hints:   -> wi 3   /\ wa 221   = wceq 988   e. wcel 990  ifcif 2406   class class class wbr 2669  {copab 2717  ` cfv 3237  (class class class)co 4039  CCcc 5321  0cc0 5323  1c1 5324   + caddc 5326   - cmin 5381   / cdiv 5383  NN0cn0 5386   < clt 5575   seq0 cseq0 6655  ^cexp 6691  abscabs 6873   ~~> cli 7097

This theorem is referenced by:  georeclim 7363  geoisum 7365

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-9 997  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-rep 2744  ax-sep 2754  ax-nul 2761  ax-pow 2794  ax-pr 2832  ax-un 2920  ax-inf2 4711

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 779  df-3an 780  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-nel 1625  df-ral 1687  df-rex 1688  df-reu 1689  df-rab 1690  df-v 1850  df-sbc 1979  df-csb 2044  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-pss 2099  df-nul 2325  df-if 2407  df-pw 2447  df-sn 2457  df-pr 2458  df-tp 2460  df-op 2461  df-uni 2552  df-int 2582  df-iun 2616  df-br 2670  df-opab 2718  df-tr 2732  df-eprel 2886  df-id 2889  df-po 2894  df-so 2904  df-fr 2972  df-we 2989  df-ord 3006  df-on 3007  df-lim 3008  df-suc 3009  df-om 3193  df-xp 3239  df-rel 3240  df-cnv 3241  df-co 3242  df-dm 3243  df-rn 3244  df-res 3245  df-ima 3246  df-fun 3247  df-fn 3248  df-f 3249  df-f1 3250  df-fo 3251  df-f1o 3252  df-fv 3253  df-rdg 4008  df-opr 4041  df-oprab 4042  df-1st 4157  df-2nd 4158  df-1o 4217  df-oadd 4219  df-omul 4220  df-er 4345  df-ec 4347  df-qs 4350  df-en 4455  df-dom 4456  df-sdom 4457  df-sup 4658  df-ni 5089  df-pli 5090  df-mi 5091  df-lti 5092  df-plpq 5124  df-mpq 5125  df-enq 5126  df-nq 5127  df-plq 5128  df-mq 5129  df-rq 5130  df-ltq 5131  df-1q 5132  df-np 5175  df-1p 5176  df-plp 5177  df-mp 5178  df-ltp 5179  df-plpr 5253  df-mpr 5254  df-enr 5255  df-nr 5256  df-plr 5257  df-mr 5258  df-ltr 5259  df-0r 5260  df-1r 5261  df-m1r 5262  df-c 5329  df-0 5330  df-1 5331  df-i 5332  df-r 5333  df-plus 5334  df-mul 5335  df-lt 5336  df-sub 5445  df-neg 5447  df-pnf 5576  df-mnf 5577  df-xr 5578  df-ltxr 5579  df-le 5580  df-div 5789  df-n 6012  df-2 6058  df-n0 6210  df-z 6246  df-fl 6363  df-uz 6478  df-seq1 6601  df-shft 6634  df-seq0 6657  df-exp 6692  df-sqr 6793  df-re 6874  df-im 6875  df-cj 6876  df-abs 6877  df-clim 7098

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