Theorem geoisum1c 7245

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		divasst 5748	. . 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		pm3.26 319	. . . . 5 |- ((A e. CC /\ R e. CC) -> A e. CC)
3		pm3.27 323	. . . . 5 |- ((A e. CC /\ R e. CC) -> R e. CC)
4		ax1cn 5281	. . . . . . 7 |- 1 e. CC
5		subclt 5379	. . . . . . 7 |- ((1 e. CC /\ R e. CC) -> (1 - R) e. CC)
6	4, 5	mpan 697	. . . . . 6 |- (R e. CC -> (1 - R) e. CC)
7	6	adantl 390	. . . . 5 |- ((A e. CC /\ R e. CC) -> (1 - R) e. CC)
8	2, 3, 7	3jca 821	. . . 4 |- ((A e. CC /\ R e. CC) -> (A e. CC /\ R e. CC /\ (1 - R) e. CC))
9	8	3adant3 801	. . 3 |- ((A e. CC /\ R e. CC /\ (abs` R) < 1) -> (A e. CC /\ R e. CC /\ (1 - R) e. CC))
10		1re 5447	. . . . 5 |- 1 e. RR
11		abssubne0t 6882	. . . . 5 |- ((R e. CC /\ 1 e. RR /\ (abs` R) < 1) -> (1 - R) =/= 0)
12	10, 11	mp3an2 906	. . . 4 |- ((R e. CC /\ (abs` R) < 1) -> (1 - R) =/= 0)
13	12	3adant1 799	. . 3 |- ((A e. CC /\ R e. CC /\ (abs` R) < 1) -> (1 - R) =/= 0)
14	1, 9, 13	sylanc 473	. 2 |- ((A e. CC /\ R e. CC /\ (abs` R) < 1) -> ((A x. R) / (1 - R)) = (A x. (R / (1 - R))))
15		geoisum1 7244	. . . . 5 |- ((R e. CC /\ (abs` R) < 1) -> sum_k e. NN (R^k) = (R / (1 - R)))
16		nnuz 6440	. . . . . 6 |- NN = (ZZ>` 1)
17	16	sumeq1i 6987	. . . . 5 |- sum_k e. NN (R^k) = sum_k e. (ZZ>` 1)(R^k)
18	15, 17	syl5eqr 1524	. . . 4 |- ((R e. CC /\ (abs` R) < 1) -> sum_k e. (ZZ>` 1)(R^k) = (R / (1 - R)))
19	18	3adant1 799	. . 3 |- ((A e. CC /\ R e. CC /\ (abs` R) < 1) -> sum_k e. (ZZ>` 1)(R^k) = (R / (1 - R)))
20	19	opreq2d 3982	. 2 |- ((A e. CC /\ R e. CC /\ (abs` R) < 1) -> (A x. sum_k e. (ZZ>` 1)(R^k)) = (A x. (R / (1 - R))))
21		1z 6161	. . . . . 6 |- 1 e. ZZ
22		oprex 3989	. . . . . . 7 |- (R^k) e. V
23		nnex 5935	. . . . . . . 8 |- NN e. V
24	23	opabex2 3616	. . . . . . 7 |- {<.k, y>. | (k e. NN /\ y = (R^k))} e. V
25		hbopab1 2819	. . . . . . 7 |- (z e. {<.k, y>. | (k e. NN /\ y = (R^k))} -> A.k z e. {<.k, y>. | (k e. NN /\ y = (R^k))})
26		elnnuz 6441	. . . . . . . 8 |- (k e. NN <-> k e. (ZZ>` 1))
27		fvopab2 3797	. . . . . . . . 9 |- ((k e. NN /\ (R^k) e. V) -> ({<.k, y>. | (k e. NN /\ y = (R^k))}` k) = (R^k))
28	22, 27	mpan2 698	. . . . . . . 8 |- (k e. NN -> ({<.k, y>. | (k e. NN /\ y = (R^k))}` k) = (R^k))
29	26, 28	sylbir 201	. . . . . . 7 |- (k e. (ZZ>` 1) -> ({<.k, y>. | (k e. NN /\ y = (R^k))}` k) = (R^k))
30	22, 24, 25, 29	isummulc1a 7214	. . . . . 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)))
31	21, 30	mpanl1 708	. . . . 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)))
32		expclt 6582	. . . . . . . . 9 |- ((R e. CC /\ k e. NN0) -> (R^k) e. CC)
33		nnnn0t 6108	. . . . . . . . . 10 |- (k e. NN -> k e. NN0)
34	26, 33	sylbir 201	. . . . . . . . 9 |- (k e. (ZZ>` 1) -> k e. NN0)
35	32, 34	sylan2 453	. . . . . . . 8 |- ((R e. CC /\ k e. (ZZ>` 1)) -> (R^k) e. CC)
36	35	r19.21aiva 1717	. . . . . . 7 |- (R e. CC -> A.k e. (ZZ>` 1)(R^k) e. CC)
37	36	adantr 391	. . . . . 6 |- ((R e. CC /\ (abs` R) < 1) -> A.k e. (ZZ>` 1)(R^k) e. CC)
38		eqid 1478	. . . . . . . 8 |- {<.k, y>. | (k e. NN /\ y = (R^k))} = {<.k, y>. | (k e. NN /\ y = (R^k))}
39	38	geolim1 7239	. . . . . . 7 |- ((R e. CC /\ (abs` R) < 1) -> ( + seq1 {<.k, y>. | (k e. NN /\ y = (R^k))}) ~~> (R / (1 - R)))
40		oprex 3989	. . . . . . . 8 |- (R / (1 - R)) e. V
41		breq2 2628	. . . . . . . . 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))))
42		addex 5329	. . . . . . . . . . 11 |- + e. V
43	42, 24	seq1seqz 6542	. . . . . . . . . 10 |- ( + seq1 {<.k, y>. | (k e. NN /\ y = (R^k))}) = (<.1, + >. seq {<.k, y>. | (k e. NN /\ y = (R^k))})
44	43	breq1i 2631	. . . . . . . . 9 |- (( + seq1 {<.k, y>. | (k e. NN /\ y = (R^k))}) ~~> x <-> (<.1, + >. seq {<.k, y>. | (k e. NN /\ y = (R^k))}) ~~> x)
45	41, 44	syl5bbr 536	. . . . . . . 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))))
46	40, 45	cla4ev 1872	. . . . . . 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)
47	39, 46	syl 10	. . . . . 6 |- ((R e. CC /\ (abs` R) < 1) -> E.x(<.1, + >. seq {<.k, y>. | (k e. NN /\ y = (R^k))}) ~~> x)
48	37, 47	jca 288	. . . . 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))
49	31, 48	sylan2 453	. . . 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)))
50	49	3impb 831	. . 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)))
51	16	sumeq1i 6987	. . 3 |- sum_k e. NN (A x. (R^k)) = sum_k e. (ZZ>` 1)(A x. (R^k))
52	50, 51	syl6eqr 1528	. 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)))
53	14, 20, 52	3eqtr2rd 1517	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 223   /\ w3a 777   = wceq 958   e. wcel 960  E.wex 982   =/= wne