Theorem geoisum1c 11289

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

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

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 981	. . 3 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> A e. CC )
2		simp2 982	. . 3 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> R e. CC )
3		1cnd 7782	. . . 4 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> 1 e. CC )
4	3, 2	subcld 8073	. . 3 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> ( 1 - R ) e. CC )
5		1red 7781	. . . . . 6 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> 1 e. RR )
6		simp3 983	. . . . . 6 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> ( abs ` R ) < 1 )
7	2, 5, 6	absltap 11278	. . . . 5 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> R # 1 )
8		apsym 8368	. . . . . 6 |- ( ( R e. CC /\ 1 e. CC ) -> ( R # 1 <-> 1 # R ) )
9	2, 3, 8	syl2anc 408	. . . . 5 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> ( R # 1 <-> 1 # R ) )
10	7, 9	mpbid 146	. . . 4 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> 1 # R )
11	3, 2, 10	subap0d 8406	. . 3 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> ( 1 - R ) # 0 )
12	1, 2, 4, 11	divassapd 8586	. 2 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> ( ( A x. R ) / ( 1 - R ) ) = ( A x. ( R / ( 1 - R ) ) ) )
13		geoisum1 11288	. . . 4 |- ( ( R e. CC /\ ( abs ` R ) < 1 ) -> sum_ k e. NN ( R ^ k ) = ( R / ( 1 - R ) ) )
14	13	3adant1 999	. . 3 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> sum_ k e. NN ( R ^ k ) = ( R / ( 1 - R ) ) )
15	14	oveq2d 5790	. 2 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> ( A x. sum_ k e. NN ( R ^ k ) ) = ( A x. ( R / ( 1 - R ) ) ) )
16		nnuz 9361	. . 3 |- NN = ( ZZ>= ` 1 )
17		1zzd 9081	. . 3 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> 1 e. ZZ )
18		simpr 109	. . . 4 |- ( ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) /\ k e. NN ) -> k e. NN )
19		simpl2 985	. . . . 5 |- ( ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) /\ k e. NN ) -> R e. CC )
20	18	nnnn0d 9030	. . . . 5 |- ( ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) /\ k e. NN ) -> k e. NN0 )
21	19, 20	expcld 10424	. . . 4 |- ( ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) /\ k e. NN ) -> ( R ^ k ) e. CC )
22		oveq2 5782	. . . . 5 |- ( n = k -> ( R ^ n ) = ( R ^ k ) )
23		eqid 2139	. . . . 5 |- ( n e. NN |-> ( R ^ n ) ) = ( n e. NN |-> ( R ^ n ) )
24	22, 23	fvmptg 5497	. . . 4 |- ( ( k e. NN /\ ( R ^ k ) e. CC ) -> ( ( n e. NN |-> ( R ^ n ) ) ` k ) = ( R ^ k ) )
25	18, 21, 24	syl2anc 408	. . 3 |- ( ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) /\ k e. NN ) -> ( ( n e. NN |-> ( R ^ n ) ) ` k ) = ( R ^ k ) )
26		nnnn0 8984	. . . . 5 |- ( k e. NN -> k e. NN0 )
27	26	adantl 275	. . . 4 |- ( ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) /\ k e. NN ) -> k e. NN0 )
28	19, 27	expcld 10424	. . 3 |- ( ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) /\ k e. NN ) -> ( R ^ k ) e. CC )
29		seqex 10220	. . . 4 |- seq 1 ( + , ( n e. NN |-> ( R ^ n ) ) ) e. _V
30		1nn0 8993	. . . . . . 7 |- 1 e. NN0
31	30	a1i 9	. . . . . 6 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> 1 e. NN0 )
32		elnnuz 9362	. . . . . . 7 |- ( k e. NN <-> k e. ( ZZ>= ` 1 ) )
33	32, 25	sylan2br 286	. . . . . 6 |- ( ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) /\ k e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> ( R ^ n ) ) ` k ) = ( R ^ k ) )
34	2, 6, 31, 33	geolim2 11281	. . . . 5 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> seq 1 ( + , ( n e. NN |-> ( R ^ n ) ) ) ~~> ( ( R ^ 1 ) / ( 1 - R ) ) )
35		climcl 11051	. . . . 5 |- ( seq 1 ( + , ( n e. NN |-> ( R ^ n ) ) ) ~~> ( ( R ^ 1 ) / ( 1 - R ) ) -> ( ( R ^ 1 ) / ( 1 - R ) ) e. CC )
36	34, 35	syl 14	. . . 4 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> ( ( R ^ 1 ) / ( 1 - R ) ) e. CC )
37		breldmg 4745	. . . 4 |- ( ( seq 1 ( + , ( n e. NN |-> ( R ^ n ) ) ) e. _V /\ ( ( R ^ 1 ) / ( 1 - R ) ) e. CC /\ seq 1 ( + , ( n e. NN |-> ( R ^ n ) ) ) ~~> ( ( R ^ 1 ) / ( 1 - R ) ) ) -> seq 1 ( + , ( n e. NN |-> ( R ^ n ) ) ) e. dom ~~> )
38	29, 36, 34, 37	mp3an2i 1320	. . 3 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> seq 1 ( + , ( n e. NN |-> ( R ^ n ) ) ) e. dom ~~> )
39	16, 17, 25, 28, 38, 1	isummulc2 11195	. 2 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> ( A x. sum_ k e. NN ( R ^ k ) ) = sum_ k e. NN ( A x. ( R ^ k ) ) )
40	12, 15, 39	3eqtr2rd 2179	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 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   _Vcvv 2686   class class class wbr 3929   |-> cmpt 3989   dom cdm 4539   ` cfv 5123  (class class class)co 5774   CCcc 7618   1c1 7621   + caddc 7623   x. cmul 7625   < clt 7800   - cmin 7933   # cap 8343   / cdiv 8432   NNcn 8720   NN0cn0 8977   ZZ>=cuz 9326   seqcseq 10218   ^cexp 10292   abscabs 10769   ~~> cli 11047   sum_csu 11122

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-fz 9791  df-fzo 9920  df-seqfrec 10219  df-exp 10293  df-ihash 10522  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-clim 11048  df-sumdc 11123

This theorem is referenced by:  0.999...  11290