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Theorem geoisum1c 12659
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 958 . . 3  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  A  e.  CC )
2 simp2 959 . . 3  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  R  e.  CC )
3 ax-1cn 9050 . . . 4  |-  1  e.  CC
4 subcl 9307 . . . 4  |-  ( ( 1  e.  CC  /\  R  e.  CC )  ->  ( 1  -  R
)  e.  CC )
53, 2, 4sylancr 646 . . 3  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  (
1  -  R )  e.  CC )
6 simp3 960 . . . . 5  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  ( abs `  R )  <  1 )
7 1re 9092 . . . . . . . 8  |-  1  e.  RR
87ltnri 9184 . . . . . . 7  |-  -.  1  <  1
9 abs1 12104 . . . . . . . . 9  |-  ( abs `  1 )  =  1
10 fveq2 5730 . . . . . . . . 9  |-  ( 1  =  R  ->  ( abs `  1 )  =  ( abs `  R
) )
119, 10syl5eqr 2484 . . . . . . . 8  |-  ( 1  =  R  ->  1  =  ( abs `  R
) )
1211breq1d 4224 . . . . . . 7  |-  ( 1  =  R  ->  (
1  <  1  <->  ( abs `  R )  <  1
) )
138, 12mtbii 295 . . . . . 6  |-  ( 1  =  R  ->  -.  ( abs `  R )  <  1 )
1413necon2ai 2651 . . . . 5  |-  ( ( abs `  R )  <  1  ->  1  =/=  R )
156, 14syl 16 . . . 4  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  1  =/=  R )
16 subeq0 9329 . . . . . 6  |-  ( ( 1  e.  CC  /\  R  e.  CC )  ->  ( ( 1  -  R )  =  0  <->  1  =  R ) )
1716necon3bid 2638 . . . . 5  |-  ( ( 1  e.  CC  /\  R  e.  CC )  ->  ( ( 1  -  R )  =/=  0  <->  1  =/=  R ) )
183, 2, 17sylancr 646 . . . 4  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  (
( 1  -  R
)  =/=  0  <->  1  =/=  R ) )
1915, 18mpbird 225 . . 3  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  (
1  -  R )  =/=  0 )
201, 2, 5, 19divassd 9827 . 2  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  (
( A  x.  R
)  /  ( 1  -  R ) )  =  ( A  x.  ( R  /  (
1  -  R ) ) ) )
21 geoisum1 12658 . . . 4  |-  ( ( R  e.  CC  /\  ( abs `  R )  <  1 )  ->  sum_ k  e.  NN  ( R ^ k )  =  ( R  /  (
1  -  R ) ) )
22213adant1 976 . . 3  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  sum_ k  e.  NN  ( R ^
k )  =  ( R  /  ( 1  -  R ) ) )
2322oveq2d 6099 . 2  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  ( A  x.  sum_ k  e.  NN  ( R ^
k ) )  =  ( A  x.  ( R  /  ( 1  -  R ) ) ) )
24 nnuz 10523 . . 3  |-  NN  =  ( ZZ>= `  1 )
25 1z 10313 . . . 4  |-  1  e.  ZZ
2625a1i 11 . . 3  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  1  e.  ZZ )
27 oveq2 6091 . . . . 5  |-  ( n  =  k  ->  ( R ^ n )  =  ( R ^ k
) )
28 eqid 2438 . . . . 5  |-  ( n  e.  NN  |->  ( R ^ n ) )  =  ( n  e.  NN  |->  ( R ^
n ) )
29 ovex 6108 . . . . 5  |-  ( R ^ k )  e. 
_V
3027, 28, 29fvmpt 5808 . . . 4  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  ( R ^ n ) ) `  k )  =  ( R ^
k ) )
3130adantl 454 . . 3  |-  ( ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  ( R ^
n ) ) `  k )  =  ( R ^ k ) )
32 nnnn0 10230 . . . 4  |-  ( k  e.  NN  ->  k  e.  NN0 )
33 expcl 11401 . . . 4  |-  ( ( R  e.  CC  /\  k  e.  NN0 )  -> 
( R ^ k
)  e.  CC )
342, 32, 33syl2an 465 . . 3  |-  ( ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  /\  k  e.  NN )  ->  ( R ^ k
)  e.  CC )
35 1nn0 10239 . . . . . 6  |-  1  e.  NN0
3635a1i 11 . . . . 5  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  1  e.  NN0 )
37 elnnuz 10524 . . . . . 6  |-  ( k  e.  NN  <->  k  e.  ( ZZ>= `  1 )
)
3837, 31sylan2br 464 . . . . 5  |-  ( ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  /\  k  e.  ( ZZ>= ` 
1 ) )  -> 
( ( n  e.  NN  |->  ( R ^
n ) ) `  k )  =  ( R ^ k ) )
392, 6, 36, 38geolim2 12650 . . . 4  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  seq  1 (  +  , 
( n  e.  NN  |->  ( R ^ n ) ) )  ~~>  ( ( R ^ 1 )  /  ( 1  -  R ) ) )
40 seqex 11327 . . . . 5  |-  seq  1
(  +  ,  ( n  e.  NN  |->  ( R ^ n ) ) )  e.  _V
41 ovex 6108 . . . . 5  |-  ( ( R ^ 1 )  /  ( 1  -  R ) )  e. 
_V
4240, 41breldm 5076 . . . 4  |-  (  seq  1 (  +  , 
( n  e.  NN  |->  ( R ^ n ) ) )  ~~>  ( ( R ^ 1 )  /  ( 1  -  R ) )  ->  seq  1 (  +  , 
( n  e.  NN  |->  ( R ^ n ) ) )  e.  dom  ~~>  )
4339, 42syl 16 . . 3  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  seq  1 (  +  , 
( n  e.  NN  |->  ( R ^ n ) ) )  e.  dom  ~~>  )
4424, 26, 31, 34, 43, 1isummulc2 12548 . 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 ) ) )
4520, 23, 443eqtr2rd 2477 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    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4214    e. cmpt 4268   dom cdm 4880   ` cfv 5456  (class class class)co 6083   CCcc 8990   0cc0 8992   1c1 8993    + caddc 8995    x. cmul 8997    < clt 9122    - cmin 9293    / cdiv 9679   NNcn 10002   NN0cn0 10223   ZZcz 10284   ZZ>=cuz 10490    seq cseq 11325   ^cexp 11384   abscabs 12041    ~~> cli 12280   sum_csu 12481
This theorem is referenced by:  0.999...  12660
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-fz 11046  df-fzo 11138  df-fl 11204  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-rlim 12285  df-sum 12482
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