Theorem geoisum1c 8507

Description: The infinite sum of A x. (R^1) + A x. (R^2)... is (A x. R) / (1 - R).

Assertion

Ref	Expression
geoisum1c	|- ((A e. CC /\ R e. CC /\ (abs` R) < 1) -> sum_k e. NN (A x. (R^k)) = ((A x. R) / (1 - R)))

Distinct variable groups:   A,k   R,k

Proof of Theorem geoisum1c

Step	Hyp	Ref	Expression
1		divass 6924	. . 3 |- ((A e. CC /\ R e. CC /\ ((1 - R) e. CC /\ (1 - R) =/= 0)) -> ((A x. R) / (1 - R)) = (A x. (R / (1 - R))))
2		ax1cn 6422	. . . . . 6 |- 1 e. CC
3		subcl 6524	. . . . . 6 |- ((1 e. CC /\ R e. CC) -> (1 - R) e. CC)
4	2, 3	mpan 759	. . . . 5 |- (R e. CC -> (1 - R) e. CC)
5	4	3ad2ant2 898	. . . 4 |- ((A e. CC /\ R e. CC /\ (abs` R) < 1) -> (1 - R) e. CC)
6		1re 6598	. . . . . 6 |- 1 e. RR
7		abssubne0 8134	. . . . . 6 |- ((R e. CC /\ 1 e. RR /\ (abs` R) < 1) -> (1 - R) =/= 0)
8	6, 7	mp3an2 1179	. . . . 5 |- ((R e. CC /\ (abs` R) < 1) -> (1 - R) =/= 0)
9	8	3adant1 894	. . . 4 |- ((A e. CC /\ R e. CC /\ (abs` R) < 1) -> (1 - R) =/= 0)
10	5, 9	jca 310	. . 3 |- ((A e. CC /\ R e. CC /\ (abs` R) < 1) -> ((1 - R) e. CC /\ (1 - R) =/= 0))
11	1, 10	syld3an3 1142	. 2 |- ((A e. CC /\ R e. CC /\ (abs` R) < 1) -> ((A x. R) / (1 - R)) = (A x. (R / (1 - R))))
12		geoisum1 8506	. . . . 5 |- ((R e. CC /\ (abs` R) < 1) -> sum_k e. NN (R^k) = (R / (1 - R)))
13		nnuz 7608	. . . . . 6 |- NN = (ZZ>=` 1)
14	13	sumeq1i 8247	. . . . 5 |- sum_k e. NN (R^k) = sum_k e. (ZZ>=` 1)(R^k)
15	12, 14	syl5eqr 1942	. . . 4 |- ((R e. CC /\ (abs` R) < 1) -> sum_k e. (ZZ>=` 1)(R^k) = (R / (1 - R)))
16	15	3adant1 894	. . 3 |- ((A e. CC /\ R e. CC /\ (abs` R) < 1) -> sum_k e. (ZZ>=` 1)(R^k) = (R / (1 - R)))
17	16	opreq2d 4898	. 2 |- ((A e. CC /\ R e. CC /\ (abs` R) < 1) -> (A x. sum_k e. (ZZ>=` 1)(R^k)) = (A x. (R / (1 - R))))
18		1z 7368	. . . . . 6 |- 1 e. ZZ
19		oprex 4907	. . . . . . 7 |- (R^k) e. _V
20		nnex 7116	. . . . . . . 8 |- NN e. _V
21	20	opabex2 4539	. . . . . . 7 |- {<.k, y>. | (k e. NN /\ y = (R^k))} e. _V
22		hbopab1 3562	. . . . . . 7 |- (z e. {<.k, y>. | (k e. NN /\ y = (R^k))} -> A.k z e. {<.k, y>. | (k e. NN /\ y = (R^k))})
23		elnnuz 7609	. . . . . . . 8 |- (k e. NN <-> k e. (ZZ>=` 1))
24		fvopab2 4754	. . . . . . . . 9 |- ((k e. NN /\ (R^k) e. _V) -> ({<.k, y>. | (k e. NN /\ y = (R^k))}` k) = (R^k))
25	19, 24	mpan2 760	. . . . . . . 8 |- (k e. NN -> ({<.k, y>. | (k e. NN /\ y = (R^k))}` k) = (R^k))
26	23, 25	sylbir 218	. . . . . . 7 |- (k e. (ZZ>=` 1) -> ({<.k, y>. | (k e. NN /\ y = (R^k))}` k) = (R^k))
27	19, 21, 22, 26	isummulc1ai 8475	. . . . . 6 |- (((1 e. ZZ /\ A e. CC) /\ (A.k e. (ZZ>=` 1)(R^k) e. CC /\ E.x(<.1, + >. seq {<.k, y>. | (k e. NN /\ y = (R^k))}) ~~> x)) -> (A x. sum_k e. (ZZ>=` 1)(R^k)) = sum_k e. (ZZ>=` 1)(A x. (R^k)))
28	18, 27	mpanl1 770	. . . . 5 |- ((A e. CC /\ (A.k e. (ZZ>=` 1)(R^k) e. CC /\ E.x(<.1, + >. seq {<.k, y>. | (k e. NN /\ y = (R^k))}) ~~> x)) -> (A x. sum_k e. (ZZ>=` 1)(R^k)) = sum_k e. (ZZ>=` 1)(A x. (R^k)))
29		expcl 7824	. . . . . . . . 9 |- ((R e. CC /\ k e. NN0) -> (R^k) e. CC)
30		nnnn0 7315	. . . . . . . . . 10 |- (k e. NN -> k e. NN0)
31	23, 30	sylbir 218	. . . . . . . . 9 |- (k e. (ZZ>=` 1) -> k e. NN0)
32	29, 31	sylan2 500	. . . . . . . 8 |- ((R e. CC /\ k e. (ZZ>=` 1)) -> (R^k) e. CC)
33	32	r19.21aiva 2176	. . . . . . 7 |- (R e. CC -> A.k e. (ZZ>=` 1)(R^k) e. CC)
34	33	adantr 425	. . . . . 6 |- ((R e. CC /\ (abs` R) < 1) -> A.k e. (ZZ>=` 1)(R^k) e. CC)
35		eqid 1884	. . . . . . . 8 |- {<.k, y>. | (k e. NN /\ y = (R^k))} = {<.k, y>. | (k e. NN /\ y = (R^k))}
36	35	geolim1 8501	. . . . . . 7 |- ((R e. CC /\ (abs` R) < 1) -> ( + seq1 {<.k, y>. | (k e. NN /\ y = (R^k))}) ~~> (R / (1 - R)))
37		oprex 4907	. . . . . . . 8 |- (R / (1 - R)) e. _V
38		breq2 3342	. . . . . . . . 9 |- (x = (R / (1 - R)) -> (( + seq1 {<.k, y>. | (k e. NN /\ y = (R^k))}) ~~> x <-> ( + seq1 {<.k, y>. | (k e. NN /\ y = (R^k))}) ~~> (R / (1 - R))))
39		addex 6470	. . . . . . . . . . 11 |- + e. _V
40	39, 21	seq1seqz 7784	. . . . . . . . . 10 |- ( + seq1 {<.k, y>. | (k e. NN /\ y = (R^k))}) = (<.1, + >. seq {<.k, y>. | (k e. NN /\ y = (R^k))})
41	40	breq1i 3345	. . . . . . . . 9 |- (( + seq1 {<.k, y>. | (k e. NN /\ y = (R^k))}) ~~> x <-> (<.1, + >. seq {<.k, y>. | (k e. NN /\ y = (R^k))}) ~~> x)
42	38, 41	syl5bbr 593	. . . . . . . 8 |- (x = (R / (1 - R)) -> ((<.1, + >. seq {<.k, y>. | (k e. NN /\ y = (R^k))}) ~~> x <-> ( + seq1 {<.k, y>. | (k e. NN /\ y = (R^k))}) ~~> (R / (1 - R))))
43	37, 42	cla4ev 2371	. . . . . . 7 |- (( + seq1 {<.k, y>. | (k e. NN /\ y = (R^k))}) ~~> (R / (1 - R)) -> E.x(<.1, + >. seq {<.k, y>. | (k e. NN /\ y = (R^k))}) ~~> x)
44	36, 43	syl 12	. . . . . 6 |- ((R e. CC /\ (abs` R) < 1) -> E.x(<.1, + >. seq {<.k, y>. | (k e. NN /\ y = (R^k))}) ~~> x)
45	34, 44	jca 310	. . . . 5 |- ((R e. CC /\ (abs` R) < 1) -> (A.k e. (ZZ>=` 1)(R^k) e. CC /\ E.x(<.1, + >. seq {<.k, y>. | (k e. NN /\ y = (R^k))}) ~~> x))
46	28, 45	sylan2 500	. . . 4 |- ((A e. CC /\ (R e. CC /\ (abs` R) < 1)) -> (A x. sum_k e. (ZZ>=` 1)(R^k)) = sum_k e. (ZZ>=` 1)(A x. (R^k)))
47	46	3impb 1063	. . 3 |- ((A e. CC /\ R e. CC /\ (abs` R) < 1) -> (A x. sum_k e. (ZZ>=` 1)(R^k)) = sum_k e. (ZZ>=` 1)(A x. (R^k)))
48	13	sumeq1i 8247	. . 3 |- sum_k e. NN (A x. (R^k)) = sum_k e. (ZZ>=` 1)(A x. (R^k))
49	47, 48	syl6eqr 1946	. 2 |- ((A e. CC /\ R e. CC /\ (abs` R) < 1) -> (A x. sum_k e. (ZZ>=` 1)(R^k)) = sum_k e. NN (A x. (R^k)))
50	11, 17, 49	3eqtr2rd 1933	1 |- ((A e. CC /\ R e. CC /\ (abs` R) < 1) -> sum_k e. NN (A x. (R^k)) = ((A x. R) / (1 - R)))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  A.wral 2105  _Vcvv 2292  <.cop 3046   class class class wbr 3338  {copab 3395  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   x. cmul 6391   - cmin 6445   / cdiv 6447  NNcn 6449  NN0cn0 6450  ZZcz 6451   < clt 6653  ZZ>=cuz 7586   seq1 cseq1 7720   seq cseqz 7774  ^cexp 7811  abscabs 8000   ~~> cli 8234  sum_csu 8239

This theorem is referenced by:  0.999... 8508  cntrsetlem 14999

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  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-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-seq0 7777  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-clim 8235  df-sum 8240

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