Theorem geoisumr 11481

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 9521	. 2 |- NN0 = ( ZZ>= ` 0 )
2		0zd 9224	. 2 |- ( ( A e. CC /\ 1 < ( abs ` A ) ) -> 0 e. ZZ )
3		simpr 109	. . 3 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> k e. NN0 )
4		simpll 524	. . . . 5 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> A e. CC )
5	4	abscld 11145	. . . . . . 7 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> ( abs ` A ) e. RR )
6		0red 7921	. . . . . . . 8 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> 0 e. RR )
7		1red 7935	. . . . . . . 8 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> 1 e. RR )
8		0lt1 8046	. . . . . . . . 9 |- 0 < 1
9	8	a1i 9	. . . . . . . 8 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> 0 < 1 )
10		simplr 525	. . . . . . . 8 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> 1 < ( abs ` A ) )
11	6, 7, 5, 9, 10	lttrd 8045	. . . . . . 7 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> 0 < ( abs ` A ) )
12	5, 11	gt0ap0d 8548	. . . . . 6 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> ( abs ` A ) # 0 )
13		abs00ap 11026	. . . . . . 7 |- ( A e. CC -> ( ( abs ` A ) # 0 <-> A # 0 ) )
14	13	ad2antrr 485	. . . . . 6 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> ( ( abs ` A ) # 0 <-> A # 0 ) )
15	12, 14	mpbid 146	. . . . 5 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> A # 0 )
16	4, 15	recclapd 8698	. . . 4 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> ( 1 / A ) e. CC )
17	16, 3	expcld 10609	. . 3 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> ( ( 1 / A ) ^ k ) e. CC )
18		oveq2 5861	. . . 4 |- ( n = k -> ( ( 1 / A ) ^ n ) = ( ( 1 / A ) ^ k ) )
19		eqid 2170	. . . 4 |- ( n e. NN0 |-> ( ( 1 / A ) ^ n ) ) = ( n e. NN0 |-> ( ( 1 / A ) ^ n ) )
20	18, 19	fvmptg 5572	. . 3 |- ( ( k e. NN0 /\ ( ( 1 / A ) ^ k ) e. CC ) -> ( ( n e. NN0 |-> ( ( 1 / A ) ^ n ) ) ` k ) = ( ( 1 / A ) ^ k ) )
21	3, 17, 20	syl2anc 409	. 2 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( ( 1 / A ) ^ n ) ) ` k ) = ( ( 1 / A ) ^ k ) )
22		simpl 108	. . 3 |- ( ( A e. CC /\ 1 < ( abs ` A ) ) -> A e. CC )
23		simpr 109	. . 3 |- ( ( A e. CC /\ 1 < ( abs ` A ) ) -> 1 < ( abs ` A ) )
24	22, 23, 21	georeclim 11476	. 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 11384	1 |- ( ( A e. CC /\ 1 < ( abs ` A ) ) -> sum_ k e. NN0 ( ( 1 / A ) ^ k ) = ( A / ( A - 1 ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   class class class wbr 3989   |-> cmpt 4050   ` cfv 5198  (class class class)co 5853   CCcc 7772   0cc0 7774   1c1 7775   < clt 7954   - cmin 8090   # cap 8500   / cdiv 8589   NN0cn0 9135   ^cexp 10475   abscabs 10961   sum_csu 11316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317

This theorem is referenced by: (None)