Theorem geolim1 7191

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

Hypothesis

Ref	Expression
geolim1.1	|- F = {<.k, y>. | (k e. NN /\ y = (A^k))}

Assertion

Ref	Expression
geolim1	|- ((A e. CC /\ (abs` A) < 1) -> ( + seq1 F) ~~> (A / (1 - A)))

Distinct variable group:   y,k,A

Proof of Theorem geolim1

Step	Hyp	Ref	Expression
1		opreq1 3963	. . . . . . . 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 1484	. . . . . . 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 615	. . . . . 6 |- (A = if((A e. CC /\ (abs` A) < 1), A, 0) -> ((k e. NN /\ y = (A^k)) <-> (k e. NN /\ y = (if((A e. CC /\ (abs` A) < 1), A, 0)^k))))
4	3	opabbidv 2666	. . . . 5 |- (A = if((A e. CC /\ (abs` A) < 1), A, 0) -> {<.k, y>. | (k e. NN /\ y = (A^k))} = {<.k, y>. | (k e. NN /\ y = (if((A e. CC /\ (abs` A) < 1), A, 0)^k))})
5	4	opreq2d 3971	. . . 4 |- (A = if((A e. CC /\ (abs` A) < 1), A, 0) -> ( + seq1 {<.k, y>. | (k e. NN /\ y = (A^k))}) = ( + seq1 {<.k, y>. | (k e. NN /\ y = (if((A e. CC /\ (abs` A) < 1), A, 0)^k))}))
6		id 59	. . . . 5 |- (A = if((A e. CC /\ (abs` A) < 1), A, 0) -> A = if((A e. CC /\ (abs` A) < 1), A, 0))
7		opreq2 3964	. . . . 5 |- (A = if((A e. CC /\ (abs` A) < 1), A, 0) -> (1 - A) = (1 - if((A e. CC /\ (abs` A) < 1), A, 0)))
8	6, 7	opreq12d 3973	. . . 4 |- (A = if((A e. CC /\ (abs` A) < 1), A, 0) -> (A / (1 - A)) = (if((A e. CC /\ (abs` A) < 1), A, 0) / (1 - if((A e. CC /\ (abs` A) < 1), A, 0))))
9	5, 8	breq12d 2627	. . 3 |- (A = if((A e. CC /\ (abs` A) < 1), A, 0) -> (( + seq1 {<.k, y>. | (k e. NN /\ y = (A^k))}) ~~> (A / (1 - A)) <-> ( + seq1 {<.k, y>. | (k e. NN /\ y = (if((A e. CC /\ (abs` A) < 1), A, 0)^k))}) ~~> (if((A e. CC /\ (abs` A) < 1), A, 0) / (1 - if((A e. CC /\ (abs` A) < 1), A, 0)))))
10		eqid 1474	. . . 4 |- {<.k, y>. | (k e. NN /\ y = (if((A e. CC /\ (abs` A) < 1), A, 0)^k))} = {<.k, y>. | (k e. NN /\ y = (if((A e. CC /\ (abs` A) < 1), A, 0)^k))}
11		eleq1 1532	. . . . . . 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))
12		fveq2 3719	. . . . . . . 8 |- (A = if((A e. CC /\ (abs` A) < 1), A, 0) -> (abs` A) = (abs` if((A e. CC /\ (abs` A) < 1), A, 0)))
13	12	breq1d 2625	. . . . . . 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))
14	11, 13	anbi12d 627	. . . . . 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)))
15		eleq1 1532	. . . . . . 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))
16		fveq2 3719	. . . . . . . 8 |- (0 = if((A e. CC /\ (abs` A) < 1), A, 0) -> (abs` 0) = (abs` if((A e. CC /\ (abs` A) < 1), A, 0)))
17	16	breq1d 2625	. . . . . . 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))
18	15, 17	anbi12d 627	. . . . . 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)))
19		0cn 5311	. . . . . . 7 |- 0 e. CC
20		abs0 6829	. . . . . . . 8 |- (abs` 0) = 0
21		lt01 5663	. . . . . . . 8 |- 0 < 1
22	20, 21	eqbrtr 2630	. . . . . . 7 |- (abs` 0) < 1
23	19, 22	pm3.2i 285	. . . . . 6 |- (0 e. CC /\ (abs` 0) < 1)
24	14, 18, 23	elimhyp 2387	. . . . 5 |- (if((A e. CC /\ (abs` A) < 1), A, 0) e. CC /\ (abs` if((A e. CC /\ (abs` A) < 1), A, 0)) < 1)
25	24	pm3.26i 320	. . . 4 |- if((A e. CC /\ (abs` A) < 1), A, 0) e. CC
26	24	pm3.27i 324	. . . 4 |- (abs` if((A e. CC /\ (abs` A) < 1), A, 0)) < 1
27	10, 25, 26	geolim1i 7190	. . 3 |- ( + seq1 {<.k, y>. | (k e. NN /\ y = (if((A e. CC /\ (abs` A) < 1), A, 0)^k))}) ~~> (if((A e. CC /\ (abs` A) < 1), A, 0) / (1 - if((A e. CC /\ (abs` A) < 1), A, 0)))
28	9, 27	dedth 2380	. 2 |- ((A e. CC /\ (abs` A) < 1) -> ( + seq1 {<.k, y>. | (k e. NN /\ y = (A^k))}) ~~> (A / (1 - A)))
29		geolim1.1	. . 3 |- F = {<.k, y>. | (k e. NN /\ y = (A^k))}
30	29	opreq2i 3967	. 2 |- ( + seq1 F) = ( + seq1 {<.k, y>. | (k e. NN /\ y = (A^k))})
31	28, 30	syl5eqbr 2644	1 |- ((A e. CC /\ (abs` A) < 1) -> ( + seq1 F) ~~> (A / (1 - A)))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 955   e. wcel 957  ifcif 2358   class class class wbr 2615  {copab 2662  ` cfv 3178  (class class class)co 3958  CCcc 5215  0cc0 5217  1c1 5218   + caddc 5220   - cmin 5275   / cdiv 5277  NNcn 5279   < clt 5469   seq1 cseq1 6257  ^cexp 6513  abscabs 6696   ~~> cli 6927

This theorem is referenced by:  geoisum1 7196  geoisum1c 7197  cvgratlem5 7206

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-inf2 4608

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-nel 1586  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-pss 2052  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-rdg 3927  df-opr 3960  df-oprab 3961  df-1st 4072  df-2nd 4073  df-1o 4126  df-oadd 4128  df-omul 4129  df-er 4254  df-ec 4256  df-qs 4259  df-en 4360  df-dom 4361  df-sdom 4362  df-sup 4557  df-ni 4983  df-pli 4984  df-mi 4985  df-lti 4986  df-plpq 5018  df-mpq 5019  df-enq 5020  df-nq 5021  df-plq 5022  df-mq 5023  df-rq 5024  df-ltq 5025  df-1q 5026  df-np 5069  df-1p 5070  df-plp 5071  df-mp 5072  df-ltp 5073  df-plpr 5147  df-mpr 5148  df-enr 5149  df-nr 5150  df-plr 5151  df-mr 5152  df-ltr 5153  df-0r 5154  df-1r 5155  df-m1r 5156  df-c 5223  df-0 5224  df-1 5225  df-i 5226  df-r 5227  df-plus 5228  df-mul 5229  df-lt 5230  df-sub 5339  df-neg 5341  df-pnf 5470  df-mnf 5471  df-xr 5472  df-ltxr 5473  df-le 5474  df-div 5682  df-n 5883  df-2 5927  df-n0 6057  df-z 6093  df-fl 6182  df-seq1 6258  df-shft 6291  df-uz 6363  df-fz 6413  df-seqz 6478  df-seq0 6479  df-exp 6514  df-sqr 6615  df-re 6697  df-im 6698  df-cj 6699  df-abs 6700  df-clim 6928  df-sum 6933

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