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Theorem geoisum1c 12352
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 8811 . . . 4  |-  1  e.  CC
4 subcl 9067 . . . 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 8853 . . . . . . . 8  |-  1  e.  RR
87ltnri 8945 . . . . . . 7  |-  -.  1  <  1
9 abs1 11798 . . . . . . . . 9  |-  ( abs `  1 )  =  1
10 fveq2 5541 . . . . . . . . 9  |-  ( 1  =  R  ->  ( abs `  1 )  =  ( abs `  R
) )
119, 10syl5eqr 2342 . . . . . . . 8  |-  ( 1  =  R  ->  1  =  ( abs `  R
) )
1211breq1d 4049 . . . . . . 7  |-  ( 1  =  R  ->  (
1  <  1  <->  ( abs `  R )  <  1
) )
138, 12mtbii 293 . . . . . 6  |-  ( 1  =  R  ->  -.  ( abs `  R )  <  1 )
1413necon2ai 2504 . . . . 5  |-  ( ( abs `  R )  <  1  ->  1  =/=  R )
156, 14syl 15 . . . 4  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  1  =/=  R )
16 subeq0 9089 . . . . . 6  |-  ( ( 1  e.  CC  /\  R  e.  CC )  ->  ( ( 1  -  R )  =  0  <->  1  =  R ) )
1716necon3bid 2494 . . . . 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 9587 . 2  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  (
( A  x.  R
)  /  ( 1  -  R ) )  =  ( A  x.  ( R  /  (
1  -  R ) ) ) )
21 geoisum1 12351 . . . 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 5890 . 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 10279 . . 3  |-  NN  =  ( ZZ>= `  1 )
25 1z 10069 . . . 4  |-  1  e.  ZZ
2625a1i 10 . . 3  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  1  e.  ZZ )
27 oveq2 5882 . . . . 5  |-  ( n  =  k  ->  ( R ^ n )  =  ( R ^ k
) )
28 eqid 2296 . . . . 5  |-  ( n  e.  NN  |->  ( R ^ n ) )  =  ( n  e.  NN  |->  ( R ^
n ) )
29 ovex 5899 . . . . 5  |-  ( R ^ k )  e. 
_V
3027, 28, 29fvmpt 5618 . . . 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 9988 . . . 4  |-  ( k  e.  NN  ->  k  e.  NN0 )
33 expcl 11137 . . . 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 9997 . . . . . 6  |-  1  e.  NN0
3635a1i 10 . . . . 5  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  1  e.  NN0 )
37 elnnuz 10280 . . . . . 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 12343 . . . 4  |-  ( ( A  e.  CC  /\  R  e.  CC  /\  ( abs `  R )  <  1 )  ->  seq  1 (  +  , 
( n  e.  NN  |->  ( R ^ n ) ) )  ~~>  ( ( R ^ 1 )  /  ( 1  -  R ) ) )
40 seqex 11064 . . . . 5  |-  seq  1
(  +  ,  ( n  e.  NN  |->  ( R ^ n ) ) )  e.  _V
41 ovex 5899 . . . . 5  |-  ( ( R ^ 1 )  /  ( 1  -  R ) )  e. 
_V
4240, 41breldm 4899 . . . 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 12241 . 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 2335 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 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    - cmin 9053    / cdiv 9439   NNcn 9762   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246    seq cseq 11062   ^cexp 11120   abscabs 11735    ~~> cli 11974   sum_csu 12174
This theorem is referenced by:  0.999...  12353
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175
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