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Theorem georeclim 12641
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 9543 . . . . . . . 8  |-  0  <_  1
4 0re 9083 . . . . . . . . 9  |-  0  e.  RR
5 1re 9082 . . . . . . . . 9  |-  1  e.  RR
64, 5lenlti 9185 . . . . . . . 8  |-  ( 0  <_  1  <->  -.  1  <  0 )
73, 6mpbi 200 . . . . . . 7  |-  -.  1  <  0
8 fveq2 5720 . . . . . . . . 9  |-  ( A  =  0  ->  ( abs `  A )  =  ( abs `  0
) )
9 abs0 12082 . . . . . . . . 9  |-  ( abs `  0 )  =  0
108, 9syl6eq 2483 . . . . . . . 8  |-  ( A  =  0  ->  ( abs `  A )  =  0 )
1110breq2d 4216 . . . . . . 7  |-  ( A  =  0  ->  (
1  <  ( abs `  A )  <->  1  <  0 ) )
127, 11mtbiri 295 . . . . . 6  |-  ( A  =  0  ->  -.  1  <  ( abs `  A
) )
1312necon2ai 2643 . . . . 5  |-  ( 1  <  ( abs `  A
)  ->  A  =/=  0 )
142, 13syl 16 . . . 4  |-  ( ph  ->  A  =/=  0 )
151, 14reccld 9775 . . 3  |-  ( ph  ->  ( 1  /  A
)  e.  CC )
16 ax-1cn 9040 . . . . . . 7  |-  1  e.  CC
1716a1i 11 . . . . . 6  |-  ( ph  ->  1  e.  CC )
1817, 1, 14absdivd 12249 . . . . 5  |-  ( ph  ->  ( abs `  (
1  /  A ) )  =  ( ( abs `  1 )  /  ( abs `  A
) ) )
19 abs1 12094 . . . . . 6  |-  ( abs `  1 )  =  1
2019oveq1i 6083 . . . . 5  |-  ( ( abs `  1 )  /  ( abs `  A
) )  =  ( 1  /  ( abs `  A ) )
2118, 20syl6eq 2483 . . . 4  |-  ( ph  ->  ( abs `  (
1  /  A ) )  =  ( 1  /  ( abs `  A
) ) )
221, 14absrpcld 12242 . . . . . 6  |-  ( ph  ->  ( abs `  A
)  e.  RR+ )
2322recgt1d 10654 . . . . 5  |-  ( ph  ->  ( 1  <  ( abs `  A )  <->  ( 1  /  ( abs `  A
) )  <  1
) )
242, 23mpbid 202 . . . 4  |-  ( ph  ->  ( 1  /  ( abs `  A ) )  <  1 )
2521, 24eqbrtrd 4224 . . 3  |-  ( ph  ->  ( abs `  (
1  /  A ) )  <  1 )
26 georeclim.3 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  ( ( 1  /  A
) ^ k ) )
2715, 25, 26geolim 12639 . 2  |-  ( ph  ->  seq  0 (  +  ,  F )  ~~>  ( 1  /  ( 1  -  ( 1  /  A
) ) ) )
281, 17, 1, 14divsubdird 9821 . . . . 5  |-  ( ph  ->  ( ( A  - 
1 )  /  A
)  =  ( ( A  /  A )  -  ( 1  /  A ) ) )
291, 14dividd 9780 . . . . . 6  |-  ( ph  ->  ( A  /  A
)  =  1 )
3029oveq1d 6088 . . . . 5  |-  ( ph  ->  ( ( A  /  A )  -  (
1  /  A ) )  =  ( 1  -  ( 1  /  A ) ) )
3128, 30eqtrd 2467 . . . 4  |-  ( ph  ->  ( ( A  - 
1 )  /  A
)  =  ( 1  -  ( 1  /  A ) ) )
3231oveq2d 6089 . . 3  |-  ( ph  ->  ( 1  /  (
( A  -  1 )  /  A ) )  =  ( 1  /  ( 1  -  ( 1  /  A
) ) ) )
33 subcl 9297 . . . . 5  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  -  1 )  e.  CC )
341, 16, 33sylancl 644 . . . 4  |-  ( ph  ->  ( A  -  1 )  e.  CC )
355ltnri 9174 . . . . . . . 8  |-  -.  1  <  1
36 fveq2 5720 . . . . . . . . . 10  |-  ( A  =  1  ->  ( abs `  A )  =  ( abs `  1
) )
3736, 19syl6eq 2483 . . . . . . . . 9  |-  ( A  =  1  ->  ( abs `  A )  =  1 )
3837breq2d 4216 . . . . . . . 8  |-  ( A  =  1  ->  (
1  <  ( abs `  A )  <->  1  <  1 ) )
3935, 38mtbiri 295 . . . . . . 7  |-  ( A  =  1  ->  -.  1  <  ( abs `  A
) )
4039necon2ai 2643 . . . . . 6  |-  ( 1  <  ( abs `  A
)  ->  A  =/=  1 )
412, 40syl 16 . . . . 5  |-  ( ph  ->  A  =/=  1 )
42 subeq0 9319 . . . . . . 7  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  - 
1 )  =  0  <-> 
A  =  1 ) )
431, 16, 42sylancl 644 . . . . . 6  |-  ( ph  ->  ( ( A  - 
1 )  =  0  <-> 
A  =  1 ) )
4443necon3bid 2633 . . . . 5  |-  ( ph  ->  ( ( A  - 
1 )  =/=  0  <->  A  =/=  1 ) )
4541, 44mpbird 224 . . . 4  |-  ( ph  ->  ( A  -  1 )  =/=  0 )
4634, 1, 45, 14recdivd 9799 . . 3  |-  ( ph  ->  ( 1  /  (
( A  -  1 )  /  A ) )  =  ( A  /  ( A  - 
1 ) ) )
4732, 46eqtr3d 2469 . 2  |-  ( ph  ->  ( 1  /  (
1  -  ( 1  /  A ) ) )  =  ( A  /  ( A  - 
1 ) ) )
4827, 47breqtrd 4228 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 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982   1c1 8983    + caddc 8985    < clt 9112    <_ cle 9113    - cmin 9283    / cdiv 9669   NN0cn0 10213    seq cseq 11315   ^cexp 11374   abscabs 12031    ~~> cli 12270
This theorem is referenced by:  geoisumr  12647  ege2le3  12684  eftlub  12702
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-rlim 12275  df-sum 12472
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