MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  georeclim Unicode version

Theorem georeclim 12328
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 9297 . . . . . . . 8  |-  0  <_  1
4 0re 8838 . . . . . . . . 9  |-  0  e.  RR
5 1re 8837 . . . . . . . . 9  |-  1  e.  RR
64, 5lenlti 8938 . . . . . . . 8  |-  ( 0  <_  1  <->  -.  1  <  0 )
73, 6mpbi 199 . . . . . . 7  |-  -.  1  <  0
8 fveq2 5525 . . . . . . . . 9  |-  ( A  =  0  ->  ( abs `  A )  =  ( abs `  0
) )
9 abs0 11770 . . . . . . . . 9  |-  ( abs `  0 )  =  0
108, 9syl6eq 2331 . . . . . . . 8  |-  ( A  =  0  ->  ( abs `  A )  =  0 )
1110breq2d 4035 . . . . . . 7  |-  ( A  =  0  ->  (
1  <  ( abs `  A )  <->  1  <  0 ) )
127, 11mtbiri 294 . . . . . 6  |-  ( A  =  0  ->  -.  1  <  ( abs `  A
) )
1312necon2ai 2491 . . . . 5  |-  ( 1  <  ( abs `  A
)  ->  A  =/=  0 )
142, 13syl 15 . . . 4  |-  ( ph  ->  A  =/=  0 )
151, 14reccld 9529 . . 3  |-  ( ph  ->  ( 1  /  A
)  e.  CC )
16 ax-1cn 8795 . . . . . . 7  |-  1  e.  CC
1716a1i 10 . . . . . 6  |-  ( ph  ->  1  e.  CC )
1817, 1, 14absdivd 11937 . . . . 5  |-  ( ph  ->  ( abs `  (
1  /  A ) )  =  ( ( abs `  1 )  /  ( abs `  A
) ) )
19 abs1 11782 . . . . . 6  |-  ( abs `  1 )  =  1
2019oveq1i 5868 . . . . 5  |-  ( ( abs `  1 )  /  ( abs `  A
) )  =  ( 1  /  ( abs `  A ) )
2118, 20syl6eq 2331 . . . 4  |-  ( ph  ->  ( abs `  (
1  /  A ) )  =  ( 1  /  ( abs `  A
) ) )
221, 14absrpcld 11930 . . . . . 6  |-  ( ph  ->  ( abs `  A
)  e.  RR+ )
2322recgt1d 10404 . . . . 5  |-  ( ph  ->  ( 1  <  ( abs `  A )  <->  ( 1  /  ( abs `  A
) )  <  1
) )
242, 23mpbid 201 . . . 4  |-  ( ph  ->  ( 1  /  ( abs `  A ) )  <  1 )
2521, 24eqbrtrd 4043 . . 3  |-  ( ph  ->  ( abs `  (
1  /  A ) )  <  1 )
26 georeclim.3 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  ( ( 1  /  A
) ^ k ) )
2715, 25, 26geolim 12326 . 2  |-  ( ph  ->  seq  0 (  +  ,  F )  ~~>  ( 1  /  ( 1  -  ( 1  /  A
) ) ) )
281, 17, 1, 14divsubdird 9575 . . . . 5  |-  ( ph  ->  ( ( A  - 
1 )  /  A
)  =  ( ( A  /  A )  -  ( 1  /  A ) ) )
291, 14dividd 9534 . . . . . 6  |-  ( ph  ->  ( A  /  A
)  =  1 )
3029oveq1d 5873 . . . . 5  |-  ( ph  ->  ( ( A  /  A )  -  (
1  /  A ) )  =  ( 1  -  ( 1  /  A ) ) )
3128, 30eqtrd 2315 . . . 4  |-  ( ph  ->  ( ( A  - 
1 )  /  A
)  =  ( 1  -  ( 1  /  A ) ) )
3231oveq2d 5874 . . 3  |-  ( ph  ->  ( 1  /  (
( A  -  1 )  /  A ) )  =  ( 1  /  ( 1  -  ( 1  /  A
) ) ) )
33 subcl 9051 . . . . 5  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  -  1 )  e.  CC )
341, 16, 33sylancl 643 . . . 4  |-  ( ph  ->  ( A  -  1 )  e.  CC )
355ltnri 8929 . . . . . . . 8  |-  -.  1  <  1
36 fveq2 5525 . . . . . . . . . 10  |-  ( A  =  1  ->  ( abs `  A )  =  ( abs `  1
) )
3736, 19syl6eq 2331 . . . . . . . . 9  |-  ( A  =  1  ->  ( abs `  A )  =  1 )
3837breq2d 4035 . . . . . . . 8  |-  ( A  =  1  ->  (
1  <  ( abs `  A )  <->  1  <  1 ) )
3935, 38mtbiri 294 . . . . . . 7  |-  ( A  =  1  ->  -.  1  <  ( abs `  A
) )
4039necon2ai 2491 . . . . . 6  |-  ( 1  <  ( abs `  A
)  ->  A  =/=  1 )
412, 40syl 15 . . . . 5  |-  ( ph  ->  A  =/=  1 )
42 subeq0 9073 . . . . . . 7  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  - 
1 )  =  0  <-> 
A  =  1 ) )
431, 16, 42sylancl 643 . . . . . 6  |-  ( ph  ->  ( ( A  - 
1 )  =  0  <-> 
A  =  1 ) )
4443necon3bid 2481 . . . . 5  |-  ( ph  ->  ( ( A  - 
1 )  =/=  0  <->  A  =/=  1 ) )
4541, 44mpbird 223 . . . 4  |-  ( ph  ->  ( A  -  1 )  =/=  0 )
4634, 1, 45, 14recdivd 9553 . . 3  |-  ( ph  ->  ( 1  /  (
( A  -  1 )  /  A ) )  =  ( A  /  ( A  - 
1 ) ) )
4732, 46eqtr3d 2317 . 2  |-  ( ph  ->  ( 1  /  (
1  -  ( 1  /  A ) ) )  =  ( A  /  ( A  - 
1 ) ) )
4827, 47breqtrd 4047 1  |-  ( ph  ->  seq  0 (  +  ,  F )  ~~>  ( A  /  ( A  - 
1 ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   NN0cn0 9965    seq cseq 11046   ^cexp 11104   abscabs 11719    ~~> cli 11958
This theorem is referenced by:  geoisumr  12334  ege2le3  12371  eftlub  12389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159
  Copyright terms: Public domain W3C validator