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Theorem geoisum1c 12336
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 955 . . 3  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  A  e.  CC )
2 simp2 956 . . 3  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  R  e.  CC )
3 ax-1cn 8795 . . . 4  |-  1  e.  CC
4 subcl 9051 . . . 4  |-  ( ( 1  e.  CC  /\  R  e.  CC )  ->  ( 1  -  R
)  e.  CC )
53, 2, 4sylancr 644 . . 3  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  (
1  -  R )  e.  CC )
6 simp3 957 . . . . 5  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  ( abs `  R )  <  1 )
7 1re 8837 . . . . . . . 8  |-  1  e.  RR
87ltnri 8929 . . . . . . 7  |-  -.  1  <  1
9 abs1 11782 . . . . . . . . 9  |-  ( abs `  1 )  =  1
10 fveq2 5525 . . . . . . . . 9  |-  ( 1  =  R  ->  ( abs `  1 )  =  ( abs `  R
) )
119, 10syl5eqr 2329 . . . . . . . 8  |-  ( 1  =  R  ->  1  =  ( abs `  R
) )
1211breq1d 4033 . . . . . . 7  |-  ( 1  =  R  ->  (
1  <  1  <->  ( abs `  R )  <  1
) )
138, 12mtbii 293 . . . . . 6  |-  ( 1  =  R  ->  -.  ( abs `  R )  <  1 )
1413necon2ai 2491 . . . . 5  |-  ( ( abs `  R )  <  1  ->  1  =/=  R )
156, 14syl 15 . . . 4  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  1  =/=  R )
16 subeq0 9073 . . . . . 6  |-  ( ( 1  e.  CC  /\  R  e.  CC )  ->  ( ( 1  -  R )  =  0  <->  1  =  R ) )
1716necon3bid 2481 . . . . 5  |-  ( ( 1  e.  CC  /\  R  e.  CC )  ->  ( ( 1  -  R )  =/=  0  <->  1  =/=  R ) )
183, 2, 17sylancr 644 . . . 4  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  (
( 1  -  R
)  =/=  0  <->  1  =/=  R ) )
1915, 18mpbird 223 . . 3  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  (
1  -  R )  =/=  0 )
201, 2, 5, 19divassd 9571 . 2  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  (
( A  x.  R
)  /  ( 1  -  R ) )  =  ( A  x.  ( R  /  (
1  -  R ) ) ) )
21 geoisum1 12335 . . . 4  |-  ( ( R  e.  CC  /\  ( abs `  R )  <  1 )  ->  sum_ k  e.  NN  ( R ^ k )  =  ( R  /  (
1  -  R ) ) )
22213adant1 973 . . 3  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  sum_ k  e.  NN  ( R ^
k )  =  ( R  /  ( 1  -  R ) ) )
2322oveq2d 5874 . 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 10263 . . 3  |-  NN  =  ( ZZ>= `  1 )
25 1z 10053 . . . 4  |-  1  e.  ZZ
2625a1i 10 . . 3  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  1  e.  ZZ )
27 oveq2 5866 . . . . 5  |-  ( n  =  k  ->  ( R ^ n )  =  ( R ^ k
) )
28 eqid 2283 . . . . 5  |-  ( n  e.  NN  |->  ( R ^ n ) )  =  ( n  e.  NN  |->  ( R ^
n ) )
29 ovex 5883 . . . . 5  |-  ( R ^ k )  e. 
_V
3027, 28, 29fvmpt 5602 . . . 4  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  ( R ^ n ) ) `  k )  =  ( R ^
k ) )
3130adantl 452 . . 3  |-  ( ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  ( R ^
n ) ) `  k )  =  ( R ^ k ) )
32 nnnn0 9972 . . . 4  |-  ( k  e.  NN  ->  k  e.  NN0 )
33 expcl 11121 . . . 4  |-  ( ( R  e.  CC  /\  k  e.  NN0 )  -> 
( R ^ k
)  e.  CC )
342, 32, 33syl2an 463 . . 3  |-  ( ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  /\  k  e.  NN )  ->  ( R ^ k
)  e.  CC )
35 1nn0 9981 . . . . . 6  |-  1  e.  NN0
3635a1i 10 . . . . 5  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  1  e.  NN0 )
37 elnnuz 10264 . . . . . 6  |-  ( k  e.  NN  <->  k  e.  ( ZZ>= `  1 )
)
3837, 31sylan2br 462 . . . . 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 12327 . . . 4  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  seq  1 (  +  , 
( n  e.  NN  |->  ( R ^ n ) ) )  ~~>  ( ( R ^ 1 )  /  ( 1  -  R ) ) )
40 seqex 11048 . . . . 5  |-  seq  1
(  +  ,  ( n  e.  NN  |->  ( R ^ n ) ) )  e.  _V
41 ovex 5883 . . . . 5  |-  ( ( R ^ 1 )  /  ( 1  -  R ) )  e. 
_V
4240, 41breldm 4883 . . . 4  |-  (  seq  1 (  +  , 
( n  e.  NN  |->  ( R ^ n ) ) )  ~~>  ( ( R ^ 1 )  /  ( 1  -  R ) )  ->  seq  1 (  +  , 
( n  e.  NN  |->  ( R ^ n ) ) )  e.  dom  ~~>  )
4339, 42syl 15 . . 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 12225 . 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 2322 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023    e. cmpt 4077   dom cdm 4689   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    - cmin 9037    / cdiv 9423   NNcn 9746   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230    seq cseq 11046   ^cexp 11104   abscabs 11719    ~~> cli 11958   sum_csu 12158
This theorem is referenced by:  0.999...  12337
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159
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