Theorem geoisumr 11553

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 9587	. 2 |- NN0 = ( ZZ>= ` 0 )
2		0zd 9290	. 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 11217	. . . . . . 7 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> ( abs ` A ) e. RR )
6		0red 7983	. . . . . . . 8 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> 0 e. RR )
7		1red 7997	. . . . . . . 8 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> 1 e. RR )
8		0lt1 8109	. . . . . . . . 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 8108	. . . . . . 7 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> 0 < ( abs ` A ) )
12	5, 11	gt0ap0d 8611	. . . . . 6 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> ( abs ` A ) # 0 )
13		abs00ap 11098	. . . . . . 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 8763	. . . 4 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> ( 1 / A ) e. CC )
17	16, 3	expcld 10680	. . 3 |- ( ( ( A e. CC /\ 1 < ( abs ` A ) ) /\ k e. NN0 ) -> ( ( 1 / A ) ^ k ) e. CC )
18		oveq2 5900	. . . 4 |- ( n = k -> ( ( 1 / A ) ^ n ) = ( ( 1 / A ) ^ k ) )
19		eqid 2189	. . . 4 |- ( n e. NN0 |-> ( ( 1 / A ) ^ n ) ) = ( n e. NN0 |-> ( ( 1 / A ) ^ n ) )
20	18, 19	fvmptg 5609	. . 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 11548	. 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 11456	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 2160   class class class wbr 4018   |-> cmpt 4079   ` cfv 5232  (class class class)co 5892   CCcc 7834   0cc0 7836   1c1 7837   < clt 8017   - cmin 8153   # cap 8563   / cdiv 8654   NN0cn0 9201   ^cexp 10545   abscabs 11033   sum_csu 11388

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7927  ax-resscn 7928  ax-1cn 7929  ax-1re 7930  ax-icn 7931  ax-addcl 7932  ax-addrcl 7933  ax-mulcl 7934  ax-mulrcl 7935  ax-addcom 7936  ax-mulcom 7937  ax-addass 7938  ax-mulass 7939  ax-distr 7940  ax-i2m1 7941  ax-0lt1 7942  ax-1rid 7943  ax-0id 7944  ax-rnegex 7945  ax-precex 7946  ax-cnre 7947  ax-pre-ltirr 7948  ax-pre-ltwlin 7949  ax-pre-lttrn 7950  ax-pre-apti 7951  ax-pre-ltadd 7952  ax-pre-mulgt0 7953  ax-pre-mulext 7954  ax-arch 7955  ax-caucvg 7956

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 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-isom 5241  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-recs 6325  df-irdg 6390  df-frec 6411  df-1o 6436  df-oadd 6440  df-er 6554  df-en 6762  df-dom 6763  df-fin 6764  df-pnf 8019  df-mnf 8020  df-xr 8021  df-ltxr 8022  df-le 8023  df-sub 8155  df-neg 8156  df-reap 8557  df-ap 8564  df-div 8655  df-inn 8945  df-2 9003  df-3 9004  df-4 9005  df-n0 9202  df-z 9279  df-uz 9554  df-q 9645  df-rp 9679  df-fz 10034  df-fzo 10168  df-seqfrec 10472  df-exp 10546  df-ihash 10783  df-cj 10878  df-re 10879  df-im 10880  df-rsqrt 11034  df-abs 11035  df-clim 11314  df-sumdc 11389

This theorem is referenced by: (None)