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Theorem georeclim 12569
Description: The limit of a geometric series of reciprocals. (Contributed by Paul Chapman, 28-Dec-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)
Hypotheses
Ref Expression
georeclim.1  |-  ( ph  ->  A  e.  CC )
georeclim.2  |-  ( ph  ->  1  <  ( abs `  A ) )
georeclim.3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  ( ( 1  /  A
) ^ k ) )
Assertion
Ref Expression
georeclim  |-  ( ph  ->  seq  0 (  +  ,  F )  ~~>  ( A  /  ( A  - 
1 ) ) )
Distinct variable groups:    A, k    k, F    ph, k

Proof of Theorem georeclim
StepHypRef Expression
1 georeclim.1 . . . 4  |-  ( ph  ->  A  e.  CC )
2 georeclim.2 . . . . 5  |-  ( ph  ->  1  <  ( abs `  A ) )
3 0le1 9476 . . . . . . . 8  |-  0  <_  1
4 0re 9017 . . . . . . . . 9  |-  0  e.  RR
5 1re 9016 . . . . . . . . 9  |-  1  e.  RR
64, 5lenlti 9117 . . . . . . . 8  |-  ( 0  <_  1  <->  -.  1  <  0 )
73, 6mpbi 200 . . . . . . 7  |-  -.  1  <  0
8 fveq2 5661 . . . . . . . . 9  |-  ( A  =  0  ->  ( abs `  A )  =  ( abs `  0
) )
9 abs0 12010 . . . . . . . . 9  |-  ( abs `  0 )  =  0
108, 9syl6eq 2428 . . . . . . . 8  |-  ( A  =  0  ->  ( abs `  A )  =  0 )
1110breq2d 4158 . . . . . . 7  |-  ( A  =  0  ->  (
1  <  ( abs `  A )  <->  1  <  0 ) )
127, 11mtbiri 295 . . . . . 6  |-  ( A  =  0  ->  -.  1  <  ( abs `  A
) )
1312necon2ai 2588 . . . . 5  |-  ( 1  <  ( abs `  A
)  ->  A  =/=  0 )
142, 13syl 16 . . . 4  |-  ( ph  ->  A  =/=  0 )
151, 14reccld 9708 . . 3  |-  ( ph  ->  ( 1  /  A
)  e.  CC )
16 ax-1cn 8974 . . . . . . 7  |-  1  e.  CC
1716a1i 11 . . . . . 6  |-  ( ph  ->  1  e.  CC )
1817, 1, 14absdivd 12177 . . . . 5  |-  ( ph  ->  ( abs `  (
1  /  A ) )  =  ( ( abs `  1 )  /  ( abs `  A
) ) )
19 abs1 12022 . . . . . 6  |-  ( abs `  1 )  =  1
2019oveq1i 6023 . . . . 5  |-  ( ( abs `  1 )  /  ( abs `  A
) )  =  ( 1  /  ( abs `  A ) )
2118, 20syl6eq 2428 . . . 4  |-  ( ph  ->  ( abs `  (
1  /  A ) )  =  ( 1  /  ( abs `  A
) ) )
221, 14absrpcld 12170 . . . . . 6  |-  ( ph  ->  ( abs `  A
)  e.  RR+ )
2322recgt1d 10587 . . . . 5  |-  ( ph  ->  ( 1  <  ( abs `  A )  <->  ( 1  /  ( abs `  A
) )  <  1
) )
242, 23mpbid 202 . . . 4  |-  ( ph  ->  ( 1  /  ( abs `  A ) )  <  1 )
2521, 24eqbrtrd 4166 . . 3  |-  ( ph  ->  ( abs `  (
1  /  A ) )  <  1 )
26 georeclim.3 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  ( ( 1  /  A
) ^ k ) )
2715, 25, 26geolim 12567 . 2  |-  ( ph  ->  seq  0 (  +  ,  F )  ~~>  ( 1  /  ( 1  -  ( 1  /  A
) ) ) )
281, 17, 1, 14divsubdird 9754 . . . . 5  |-  ( ph  ->  ( ( A  - 
1 )  /  A
)  =  ( ( A  /  A )  -  ( 1  /  A ) ) )
291, 14dividd 9713 . . . . . 6  |-  ( ph  ->  ( A  /  A
)  =  1 )
3029oveq1d 6028 . . . . 5  |-  ( ph  ->  ( ( A  /  A )  -  (
1  /  A ) )  =  ( 1  -  ( 1  /  A ) ) )
3128, 30eqtrd 2412 . . . 4  |-  ( ph  ->  ( ( A  - 
1 )  /  A
)  =  ( 1  -  ( 1  /  A ) ) )
3231oveq2d 6029 . . 3  |-  ( ph  ->  ( 1  /  (
( A  -  1 )  /  A ) )  =  ( 1  /  ( 1  -  ( 1  /  A
) ) ) )
33 subcl 9230 . . . . 5  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  -  1 )  e.  CC )
341, 16, 33sylancl 644 . . . 4  |-  ( ph  ->  ( A  -  1 )  e.  CC )
355ltnri 9108 . . . . . . . 8  |-  -.  1  <  1
36 fveq2 5661 . . . . . . . . . 10  |-  ( A  =  1  ->  ( abs `  A )  =  ( abs `  1
) )
3736, 19syl6eq 2428 . . . . . . . . 9  |-  ( A  =  1  ->  ( abs `  A )  =  1 )
3837breq2d 4158 . . . . . . . 8  |-  ( A  =  1  ->  (
1  <  ( abs `  A )  <->  1  <  1 ) )
3935, 38mtbiri 295 . . . . . . 7  |-  ( A  =  1  ->  -.  1  <  ( abs `  A
) )
4039necon2ai 2588 . . . . . 6  |-  ( 1  <  ( abs `  A
)  ->  A  =/=  1 )
412, 40syl 16 . . . . 5  |-  ( ph  ->  A  =/=  1 )
42 subeq0 9252 . . . . . . 7  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  - 
1 )  =  0  <-> 
A  =  1 ) )
431, 16, 42sylancl 644 . . . . . 6  |-  ( ph  ->  ( ( A  - 
1 )  =  0  <-> 
A  =  1 ) )
4443necon3bid 2578 . . . . 5  |-  ( ph  ->  ( ( A  - 
1 )  =/=  0  <->  A  =/=  1 ) )
4541, 44mpbird 224 . . . 4  |-  ( ph  ->  ( A  -  1 )  =/=  0 )
4634, 1, 45, 14recdivd 9732 . . 3  |-  ( ph  ->  ( 1  /  (
( A  -  1 )  /  A ) )  =  ( A  /  ( A  - 
1 ) ) )
4732, 46eqtr3d 2414 . 2  |-  ( ph  ->  ( 1  /  (
1  -  ( 1  /  A ) ) )  =  ( A  /  ( A  - 
1 ) ) )
4827, 47breqtrd 4170 1  |-  ( ph  ->  seq  0 (  +  ,  F )  ~~>  ( A  /  ( A  - 
1 ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2543   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   CCcc 8914   0cc0 8916   1c1 8917    + caddc 8919    < clt 9046    <_ cle 9047    - cmin 9216    / cdiv 9602   NN0cn0 10146    seq cseq 11243   ^cexp 11302   abscabs 11959    ~~> cli 12198
This theorem is referenced by:  geoisumr  12575  ege2le3  12612  eftlub  12630
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-pm 6950  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-oi 7405  df-card 7752  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-fz 10969  df-fzo 11059  df-fl 11122  df-seq 11244  df-exp 11303  df-hash 11539  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-clim 12202  df-rlim 12203  df-sum 12400
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