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Theorem georeclim 13895
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 10126 . . . . . . . 8  |-  0  <_  1
4 0re 9632 . . . . . . . . 9  |-  0  e.  RR
5 1re 9631 . . . . . . . . 9  |-  1  e.  RR
64, 5lenlti 9743 . . . . . . . 8  |-  ( 0  <_  1  <->  -.  1  <  0 )
73, 6mpbi 211 . . . . . . 7  |-  -.  1  <  0
8 fveq2 5872 . . . . . . . . 9  |-  ( A  =  0  ->  ( abs `  A )  =  ( abs `  0
) )
9 abs0 13316 . . . . . . . . 9  |-  ( abs `  0 )  =  0
108, 9syl6eq 2477 . . . . . . . 8  |-  ( A  =  0  ->  ( abs `  A )  =  0 )
1110breq2d 4429 . . . . . . 7  |-  ( A  =  0  ->  (
1  <  ( abs `  A )  <->  1  <  0 ) )
127, 11mtbiri 304 . . . . . 6  |-  ( A  =  0  ->  -.  1  <  ( abs `  A
) )
1312necon2ai 2657 . . . . 5  |-  ( 1  <  ( abs `  A
)  ->  A  =/=  0 )
142, 13syl 17 . . . 4  |-  ( ph  ->  A  =/=  0 )
151, 14reccld 10365 . . 3  |-  ( ph  ->  ( 1  /  A
)  e.  CC )
16 1cnd 9648 . . . . . 6  |-  ( ph  ->  1  e.  CC )
1716, 1, 14absdivd 13484 . . . . 5  |-  ( ph  ->  ( abs `  (
1  /  A ) )  =  ( ( abs `  1 )  /  ( abs `  A
) ) )
18 abs1 13328 . . . . . 6  |-  ( abs `  1 )  =  1
1918oveq1i 6306 . . . . 5  |-  ( ( abs `  1 )  /  ( abs `  A
) )  =  ( 1  /  ( abs `  A ) )
2017, 19syl6eq 2477 . . . 4  |-  ( ph  ->  ( abs `  (
1  /  A ) )  =  ( 1  /  ( abs `  A
) ) )
211, 14absrpcld 13477 . . . . . 6  |-  ( ph  ->  ( abs `  A
)  e.  RR+ )
2221recgt1d 11344 . . . . 5  |-  ( ph  ->  ( 1  <  ( abs `  A )  <->  ( 1  /  ( abs `  A
) )  <  1
) )
232, 22mpbid 213 . . . 4  |-  ( ph  ->  ( 1  /  ( abs `  A ) )  <  1 )
2420, 23eqbrtrd 4437 . . 3  |-  ( ph  ->  ( abs `  (
1  /  A ) )  <  1 )
25 georeclim.3 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  ( ( 1  /  A
) ^ k ) )
2615, 24, 25geolim 13893 . 2  |-  ( ph  ->  seq 0 (  +  ,  F )  ~~>  ( 1  /  ( 1  -  ( 1  /  A
) ) ) )
271, 16, 1, 14divsubdird 10411 . . . . 5  |-  ( ph  ->  ( ( A  - 
1 )  /  A
)  =  ( ( A  /  A )  -  ( 1  /  A ) ) )
281, 14dividd 10370 . . . . . 6  |-  ( ph  ->  ( A  /  A
)  =  1 )
2928oveq1d 6311 . . . . 5  |-  ( ph  ->  ( ( A  /  A )  -  (
1  /  A ) )  =  ( 1  -  ( 1  /  A ) ) )
3027, 29eqtrd 2461 . . . 4  |-  ( ph  ->  ( ( A  - 
1 )  /  A
)  =  ( 1  -  ( 1  /  A ) ) )
3130oveq2d 6312 . . 3  |-  ( ph  ->  ( 1  /  (
( A  -  1 )  /  A ) )  =  ( 1  /  ( 1  -  ( 1  /  A
) ) ) )
32 ax-1cn 9586 . . . . 5  |-  1  e.  CC
33 subcl 9863 . . . . 5  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  -  1 )  e.  CC )
341, 32, 33sylancl 666 . . . 4  |-  ( ph  ->  ( A  -  1 )  e.  CC )
355ltnri 9732 . . . . . . . 8  |-  -.  1  <  1
36 fveq2 5872 . . . . . . . . . 10  |-  ( A  =  1  ->  ( abs `  A )  =  ( abs `  1
) )
3736, 18syl6eq 2477 . . . . . . . . 9  |-  ( A  =  1  ->  ( abs `  A )  =  1 )
3837breq2d 4429 . . . . . . . 8  |-  ( A  =  1  ->  (
1  <  ( abs `  A )  <->  1  <  1 ) )
3935, 38mtbiri 304 . . . . . . 7  |-  ( A  =  1  ->  -.  1  <  ( abs `  A
) )
4039necon2ai 2657 . . . . . 6  |-  ( 1  <  ( abs `  A
)  ->  A  =/=  1 )
412, 40syl 17 . . . . 5  |-  ( ph  ->  A  =/=  1 )
42 subeq0 9889 . . . . . . 7  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  - 
1 )  =  0  <-> 
A  =  1 ) )
431, 32, 42sylancl 666 . . . . . 6  |-  ( ph  ->  ( ( A  - 
1 )  =  0  <-> 
A  =  1 ) )
4443necon3bid 2680 . . . . 5  |-  ( ph  ->  ( ( A  - 
1 )  =/=  0  <->  A  =/=  1 ) )
4541, 44mpbird 235 . . . 4  |-  ( ph  ->  ( A  -  1 )  =/=  0 )
4634, 1, 45, 14recdivd 10389 . . 3  |-  ( ph  ->  ( 1  /  (
( A  -  1 )  /  A ) )  =  ( A  /  ( A  - 
1 ) ) )
4731, 46eqtr3d 2463 . 2  |-  ( ph  ->  ( 1  /  (
1  -  ( 1  /  A ) ) )  =  ( A  /  ( A  - 
1 ) ) )
4826, 47breqtrd 4441 1  |-  ( ph  ->  seq 0 (  +  ,  F )  ~~>  ( A  /  ( A  - 
1 ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1867    =/= wne 2616   class class class wbr 4417   ` cfv 5592  (class class class)co 6296   CCcc 9526   0cc0 9528   1c1 9529    + caddc 9531    < clt 9664    <_ cle 9665    - cmin 9849    / cdiv 10258   NN0cn0 10858    seqcseq 12199   ^cexp 12258   abscabs 13265    ~~> cli 13515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-pm 7474  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-sup 7953  df-oi 8016  df-card 8363  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-n0 10859  df-z 10927  df-uz 11149  df-rp 11292  df-fz 11772  df-fzo 11903  df-fl 12014  df-seq 12200  df-exp 12259  df-hash 12502  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-clim 13519  df-rlim 13520  df-sum 13720
This theorem is referenced by:  geoisumr  13901  ege2le3  14111  eftlub  14130
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