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Theorem geoisum1c 11301
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.
StepHypRef 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 7794 . . . 4  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  1  e.  CC )
43, 2subcld 8085 . . 3  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  (
1  -  R )  e.  CC )
5 1red 7793 . . . . . 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 )
72, 5, 6absltap 11290 . . . . 5  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  R #  1 )
8 apsym 8380 . . . . . 6  |-  ( ( R  e.  CC  /\  1  e.  CC )  ->  ( R #  1  <->  1 #  R ) )
92, 3, 8syl2anc 408 . . . . 5  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  ( R #  1  <->  1 #  R )
)
107, 9mpbid 146 . . . 4  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  1 #  R )
113, 2, 10subap0d 8418 . . 3  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  (
1  -  R ) #  0 )
121, 2, 4, 11divassapd 8598 . 2  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  (
( A  x.  R
)  /  ( 1  -  R ) )  =  ( A  x.  ( R  /  (
1  -  R ) ) ) )
13 geoisum1 11300 . . . 4  |-  ( ( R  e.  CC  /\  ( abs `  R )  <  1 )  ->  sum_ k  e.  NN  ( R ^ k )  =  ( R  /  (
1  -  R ) ) )
14133adant1 999 . . 3  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  sum_ k  e.  NN  ( R ^
k )  =  ( R  /  ( 1  -  R ) ) )
1514oveq2d 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 9373 . . 3  |-  NN  =  ( ZZ>= `  1 )
17 1zzd 9093 . . 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 )
2018nnnn0d 9042 . . . . 5  |-  ( ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  /\  k  e.  NN )  ->  k  e.  NN0 )
2119, 20expcld 10436 . . . 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 ) )
2422, 23fvmptg 5497 . . . 4  |-  ( ( k  e.  NN  /\  ( R ^ k )  e.  CC )  -> 
( ( n  e.  NN  |->  ( R ^
n ) ) `  k )  =  ( R ^ k ) )
2518, 21, 24syl2anc 408 . . 3  |-  ( ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  ( R ^
n ) ) `  k )  =  ( R ^ k ) )
26 nnnn0 8996 . . . . 5  |-  ( k  e.  NN  ->  k  e.  NN0 )
2726adantl 275 . . . 4  |-  ( ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  /\  k  e.  NN )  ->  k  e.  NN0 )
2819, 27expcld 10436 . . 3  |-  ( ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  /\  k  e.  NN )  ->  ( R ^ k
)  e.  CC )
29 seqex 10232 . . . 4  |-  seq 1
(  +  ,  ( n  e.  NN  |->  ( R ^ n ) ) )  e.  _V
30 1nn0 9005 . . . . . . 7  |-  1  e.  NN0
3130a1i 9 . . . . . 6  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  1  e.  NN0 )
32 elnnuz 9374 . . . . . . 7  |-  ( k  e.  NN  <->  k  e.  ( ZZ>= `  1 )
)
3332, 25sylan2br 286 . . . . . 6  |-  ( ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  /\  k  e.  ( ZZ>= ` 
1 ) )  -> 
( ( n  e.  NN  |->  ( R ^
n ) ) `  k )  =  ( R ^ k ) )
342, 6, 31, 33geolim2 11293 . . . . 5  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  seq 1 (  +  , 
( n  e.  NN  |->  ( R ^ n ) ) )  ~~>  ( ( R ^ 1 )  /  ( 1  -  R ) ) )
35 climcl 11063 . . . . 5  |-  (  seq 1 (  +  , 
( n  e.  NN  |->  ( R ^ n ) ) )  ~~>  ( ( R ^ 1 )  /  ( 1  -  R ) )  -> 
( ( R ^
1 )  /  (
1  -  R ) )  e.  CC )
3634, 35syl 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  ~~>  )
3829, 36, 34, 37mp3an2i 1320 . . 3  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  seq 1 (  +  , 
( n  e.  NN  |->  ( R ^ n ) ) )  e.  dom  ~~>  )
3916, 17, 25, 28, 38, 1isummulc2 11207 . 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 ) ) )
4012, 15, 393eqtr2rd 2179 1  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  sum_ k  e.  NN  ( A  x.  ( R ^ k ) )  =  ( ( A  x.  R )  /  ( 1  -  R ) ) )
Colors of variables: wff set class
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 7630   1c1 7633    + caddc 7635    x. cmul 7637    < clt 7812    - cmin 7945   # cap 8355    / cdiv 8444   NNcn 8732   NN0cn0 8989   ZZ>=cuz 9338    seqcseq 10230   ^cexp 10304   abscabs 10781    ~~> cli 11059   sum_csu 11134
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 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750  ax-arch 7751  ax-caucvg 7752
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 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445  df-inn 8733  df-2 8791  df-3 8792  df-4 8793  df-n0 8990  df-z 9067  df-uz 9339  df-q 9424  df-rp 9454  df-fz 9803  df-fzo 9932  df-seqfrec 10231  df-exp 10305  df-ihash 10534  df-cj 10626  df-re 10627  df-im 10628  df-rsqrt 10782  df-abs 10783  df-clim 11060  df-sumdc 11135
This theorem is referenced by:  0.999...  11302
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