Theorem geoisumr 11702

Description: The infinite sum of reciprocals 1 + ( 1 / A ) ^ 1 + ( 1 / A ) ^ 2... is A / ( A - 1 ). (Contributed by rpenner, 3-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)

Assertion

Ref	Expression
geoisumr	|- ( ( A e. CC /\ 1 < ( abs ` A ) ) -> sum_ k e. NN0 ( ( 1 / A ) ^ k ) = ( A / ( A - 1 ) ) )

Distinct variable group:   A, k

Proof of Theorem geoisumr

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0uz 9655	. 2 |- NN0 = ( ZZ>= ` 0 )
2		0zd 9357	. 2 |- ( ( A e. CC /\ 1 < ( abs ` A ) ) -> 0 e. ZZ )
3		simpr 110	. . 3 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> k e. NN0 )
4		simpll 527	. . . . 5 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> A e. CC )
5	4	abscld 11365	. . . . . . 7 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> ( abs ` A ) e. RR )
6		0red 8046	. . . . . . . 8 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> 0 e. RR )
7		1red 8060	. . . . . . . 8 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> 1 e. RR )
8		0lt1 8172	. . . . . . . . 9 |- 0 < 1
9	8	a1i 9	. . . . . . . 8 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> 0 < 1 )
10		simplr 528	. . . . . . . 8 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> 1 < ( abs ` A ) )
11	6, 7, 5, 9, 10	lttrd 8171	. . . . . . 7 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> 0 < ( abs ` A ) )
12	5, 11	gt0ap0d 8675	. . . . . 6 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> ( abs ` A ) # 0 )
13		abs00ap 11246	. . . . . . 7 |- ( A e. CC -> ( ( abs ` A ) # 0 <-> A # 0 ) )
14	13	ad2antrr 488	. . . . . 6 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> ( ( abs ` A ) # 0 <-> A # 0 ) )
15	12, 14	mpbid 147	. . . . 5 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> A # 0 )
16	4, 15	recclapd 8827	. . . 4 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> ( 1 / A ) e. CC )
17	16, 3	expcld 10784	. . 3 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> ( ( 1 / A ) ^ k ) e. CC )
18		oveq2 5933	. . . 4 |- ( n = k -> ( ( 1 / A ) ^ n ) = ( ( 1 / A ) ^ k ) )
19		eqid 2196	. . . 4 |- ( n e. NN0 |-> ( ( 1 / A ) ^ n ) ) = ( n e. NN0 |-> ( ( 1 / A ) ^ n ) )
20	18, 19	fvmptg 5640	. . 3 |- ( ( k e. NN0 /\ ( ( 1 / A ) ^ k ) e. CC ) -> ( ( n e. NN0 |-> ( ( 1 / A ) ^ n ) ) ` k ) = ( ( 1 / A ) ^ k ) )
21	3, 17, 20	syl2anc 411	. 2 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( ( 1 / A ) ^ n ) ) ` k ) = ( ( 1 / A ) ^ k ) )
22		simpl 109	. . 3 |- ( ( A e. CC /\ 1 < ( abs ` A ) ) -> A e. CC )
23		simpr 110	. . 3 |- ( ( A e. CC /\ 1 < ( abs ` A ) ) -> 1 < ( abs ` A ) )
24	22, 23, 21	georeclim 11697	. 2 |- ( ( A e. CC /\ 1 < ( abs ` A ) ) -> seq 0 ( + , ( n e. NN0 |-> ( ( 1 / A ) ^ n ) ) ) ~~> ( A / ( A - 1 ) ) )
25	1, 2, 21, 17, 24	isumclim 11605	1 |- ( ( A e. CC /\ 1 < ( abs ` A ) ) -> sum_ k e. NN0 ( ( 1 / A ) ^ k ) = ( A / ( A - 1 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   class class class wbr 4034   |-> cmpt 4095   ` cfv 5259  (class class class)co 5925   CCcc 7896   0cc0 7898   1c1 7899   < clt 8080   - cmin 8216   # cap 8627   / cdiv 8718   NN0cn0 9268   ^cexp 10649   abscabs 11181   sum_csu 11537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7989  ax-resscn 7990  ax-1cn 7991  ax-1re 7992  ax-icn 7993  ax-addcl 7994  ax-addrcl 7995  ax-mulcl 7996  ax-mulrcl 7997  ax-addcom 7998  ax-mulcom 7999  ax-addass 8000  ax-mulass 8001  ax-distr 8002  ax-i2m1 8003  ax-0lt1 8004  ax-1rid 8005  ax-0id 8006  ax-rnegex 8007  ax-precex 8008  ax-cnre 8009  ax-pre-ltirr 8010  ax-pre-ltwlin 8011  ax-pre-lttrn 8012  ax-pre-apti 8013  ax-pre-ltadd 8014  ax-pre-mulgt0 8015  ax-pre-mulext 8016  ax-arch 8017  ax-caucvg 8018

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-frec 6458  df-1o 6483  df-oadd 6487  df-er 6601  df-en 6809  df-dom 6810  df-fin 6811  df-pnf 8082  df-mnf 8083  df-xr 8084  df-ltxr 8085  df-le 8086  df-sub 8218  df-neg 8219  df-reap 8621  df-ap 8628  df-div 8719  df-inn 9010  df-2 9068  df-3 9069  df-4 9070  df-n0 9269  df-z 9346  df-uz 9621  df-q 9713  df-rp 9748  df-fz 10103  df-fzo 10237  df-seqfrec 10559  df-exp 10650  df-ihash 10887  df-cj 11026  df-re 11027  df-im 11028  df-rsqrt 11182  df-abs 11183  df-clim 11463  df-sumdc 11538

This theorem is referenced by: (None)