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Theorem geoisum1c 11472
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 992 . . 3  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  A  e.  CC )
2 simp2 993 . . 3  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  R  e.  CC )
3 1cnd 7925 . . . 4  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  1  e.  CC )
43, 2subcld 8219 . . 3  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  (
1  -  R )  e.  CC )
5 1red 7924 . . . . . 6  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  1  e.  RR )
6 simp3 994 . . . . . 6  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  ( abs `  R )  <  1 )
72, 5, 6absltap 11461 . . . . 5  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  R #  1 )
8 apsym 8514 . . . . . 6  |-  ( ( R  e.  CC  /\  1  e.  CC )  ->  ( R #  1  <->  1 #  R ) )
92, 3, 8syl2anc 409 . . . . 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 8552 . . 3  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  (
1  -  R ) #  0 )
121, 2, 4, 11divassapd 8732 . 2  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  (
( A  x.  R
)  /  ( 1  -  R ) )  =  ( A  x.  ( R  /  (
1  -  R ) ) ) )
13 geoisum1 11471 . . . 4  |-  ( ( R  e.  CC  /\  ( abs `  R )  <  1 )  ->  sum_ k  e.  NN  ( R ^ k )  =  ( R  /  (
1  -  R ) ) )
14133adant1 1010 . . 3  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  sum_ k  e.  NN  ( R ^
k )  =  ( R  /  ( 1  -  R ) ) )
1514oveq2d 5867 . 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 9511 . . 3  |-  NN  =  ( ZZ>= `  1 )
17 1zzd 9228 . . 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 996 . . . . 5  |-  ( ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  /\  k  e.  NN )  ->  R  e.  CC )
2018nnnn0d 9177 . . . . 5  |-  ( ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  /\  k  e.  NN )  ->  k  e.  NN0 )
2119, 20expcld 10598 . . . 4  |-  ( ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  /\  k  e.  NN )  ->  ( R ^ k
)  e.  CC )
22 oveq2 5859 . . . . 5  |-  ( n  =  k  ->  ( R ^ n )  =  ( R ^ k
) )
23 eqid 2170 . . . . 5  |-  ( n  e.  NN  |->  ( R ^ n ) )  =  ( n  e.  NN  |->  ( R ^
n ) )
2422, 23fvmptg 5570 . . . 4  |-  ( ( k  e.  NN  /\  ( R ^ k )  e.  CC )  -> 
( ( n  e.  NN  |->  ( R ^
n ) ) `  k )  =  ( R ^ k ) )
2518, 21, 24syl2anc 409 . . 3  |-  ( ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  ( R ^
n ) ) `  k )  =  ( R ^ k ) )
26 nnnn0 9131 . . . . 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 10598 . . 3  |-  ( ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  /\  k  e.  NN )  ->  ( R ^ k
)  e.  CC )
29 seqex 10392 . . . 4  |-  seq 1
(  +  ,  ( n  e.  NN  |->  ( R ^ n ) ) )  e.  _V
30 1nn0 9140 . . . . . . 7  |-  1  e.  NN0
3130a1i 9 . . . . . 6  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  1  e.  NN0 )
32 elnnuz 9512 . . . . . . 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 11464 . . . . 5  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  seq 1 (  +  , 
( n  e.  NN  |->  ( R ^ n ) ) )  ~~>  ( ( R ^ 1 )  /  ( 1  -  R ) ) )
35 climcl 11234 . . . . 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 4815 . . . 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 1337 . . 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 11378 . 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 2210 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 973    = wceq 1348    e. wcel 2141   _Vcvv 2730   class class class wbr 3987    |-> cmpt 4048   dom cdm 4609   ` cfv 5196  (class class class)co 5851   CCcc 7761   1c1 7764    + caddc 7766    x. cmul 7768    < clt 7943    - cmin 8079   # cap 8489    / cdiv 8578   NNcn 8867   NN0cn0 9124   ZZ>=cuz 9476    seqcseq 10390   ^cexp 10464   abscabs 10950    ~~> cli 11230   sum_csu 11305
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7854  ax-resscn 7855  ax-1cn 7856  ax-1re 7857  ax-icn 7858  ax-addcl 7859  ax-addrcl 7860  ax-mulcl 7861  ax-mulrcl 7862  ax-addcom 7863  ax-mulcom 7864  ax-addass 7865  ax-mulass 7866  ax-distr 7867  ax-i2m1 7868  ax-0lt1 7869  ax-1rid 7870  ax-0id 7871  ax-rnegex 7872  ax-precex 7873  ax-cnre 7874  ax-pre-ltirr 7875  ax-pre-ltwlin 7876  ax-pre-lttrn 7877  ax-pre-apti 7878  ax-pre-ltadd 7879  ax-pre-mulgt0 7880  ax-pre-mulext 7881  ax-arch 7882  ax-caucvg 7883
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-isom 5205  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-recs 6282  df-irdg 6347  df-frec 6368  df-1o 6393  df-oadd 6397  df-er 6510  df-en 6716  df-dom 6717  df-fin 6718  df-pnf 7945  df-mnf 7946  df-xr 7947  df-ltxr 7948  df-le 7949  df-sub 8081  df-neg 8082  df-reap 8483  df-ap 8490  df-div 8579  df-inn 8868  df-2 8926  df-3 8927  df-4 8928  df-n0 9125  df-z 9202  df-uz 9477  df-q 9568  df-rp 9600  df-fz 9955  df-fzo 10088  df-seqfrec 10391  df-exp 10465  df-ihash 10699  df-cj 10795  df-re 10796  df-im 10797  df-rsqrt 10951  df-abs 10952  df-clim 11231  df-sumdc 11306
This theorem is referenced by:  0.999...  11473
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