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

Theorem georeclim 13928
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 10137 . . . . . . . 8  |-  0  <_  1
4 0re 9643 . . . . . . . . 9  |-  0  e.  RR
5 1re 9642 . . . . . . . . 9  |-  1  e.  RR
64, 5lenlti 9754 . . . . . . . 8  |-  ( 0  <_  1  <->  -.  1  <  0 )
73, 6mpbi 212 . . . . . . 7  |-  -.  1  <  0
8 fveq2 5865 . . . . . . . . 9  |-  ( A  =  0  ->  ( abs `  A )  =  ( abs `  0
) )
9 abs0 13348 . . . . . . . . 9  |-  ( abs `  0 )  =  0
108, 9syl6eq 2501 . . . . . . . 8  |-  ( A  =  0  ->  ( abs `  A )  =  0 )
1110breq2d 4414 . . . . . . 7  |-  ( A  =  0  ->  (
1  <  ( abs `  A )  <->  1  <  0 ) )
127, 11mtbiri 305 . . . . . 6  |-  ( A  =  0  ->  -.  1  <  ( abs `  A
) )
1312necon2ai 2653 . . . . 5  |-  ( 1  <  ( abs `  A
)  ->  A  =/=  0 )
142, 13syl 17 . . . 4  |-  ( ph  ->  A  =/=  0 )
151, 14reccld 10376 . . 3  |-  ( ph  ->  ( 1  /  A
)  e.  CC )
16 1cnd 9659 . . . . . 6  |-  ( ph  ->  1  e.  CC )
1716, 1, 14absdivd 13517 . . . . 5  |-  ( ph  ->  ( abs `  (
1  /  A ) )  =  ( ( abs `  1 )  /  ( abs `  A
) ) )
18 abs1 13360 . . . . . 6  |-  ( abs `  1 )  =  1
1918oveq1i 6300 . . . . 5  |-  ( ( abs `  1 )  /  ( abs `  A
) )  =  ( 1  /  ( abs `  A ) )
2017, 19syl6eq 2501 . . . 4  |-  ( ph  ->  ( abs `  (
1  /  A ) )  =  ( 1  /  ( abs `  A
) ) )
211, 14absrpcld 13510 . . . . . 6  |-  ( ph  ->  ( abs `  A
)  e.  RR+ )
2221recgt1d 11355 . . . . 5  |-  ( ph  ->  ( 1  <  ( abs `  A )  <->  ( 1  /  ( abs `  A
) )  <  1
) )
232, 22mpbid 214 . . . 4  |-  ( ph  ->  ( 1  /  ( abs `  A ) )  <  1 )
2420, 23eqbrtrd 4423 . . 3  |-  ( ph  ->  ( abs `  (
1  /  A ) )  <  1 )
25 georeclim.3 . . 3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  ( ( 1  /  A
) ^ k ) )
2615, 24, 25geolim 13926 . 2  |-  ( ph  ->  seq 0 (  +  ,  F )  ~~>  ( 1  /  ( 1  -  ( 1  /  A
) ) ) )
271, 16, 1, 14divsubdird 10422 . . . . 5  |-  ( ph  ->  ( ( A  - 
1 )  /  A
)  =  ( ( A  /  A )  -  ( 1  /  A ) ) )
281, 14dividd 10381 . . . . . 6  |-  ( ph  ->  ( A  /  A
)  =  1 )
2928oveq1d 6305 . . . . 5  |-  ( ph  ->  ( ( A  /  A )  -  (
1  /  A ) )  =  ( 1  -  ( 1  /  A ) ) )
3027, 29eqtrd 2485 . . . 4  |-  ( ph  ->  ( ( A  - 
1 )  /  A
)  =  ( 1  -  ( 1  /  A ) ) )
3130oveq2d 6306 . . 3  |-  ( ph  ->  ( 1  /  (
( A  -  1 )  /  A ) )  =  ( 1  /  ( 1  -  ( 1  /  A
) ) ) )
32 ax-1cn 9597 . . . . 5  |-  1  e.  CC
33 subcl 9874 . . . . 5  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A  -  1 )  e.  CC )
341, 32, 33sylancl 668 . . . 4  |-  ( ph  ->  ( A  -  1 )  e.  CC )
355ltnri 9743 . . . . . . . 8  |-  -.  1  <  1
36 fveq2 5865 . . . . . . . . . 10  |-  ( A  =  1  ->  ( abs `  A )  =  ( abs `  1
) )
3736, 18syl6eq 2501 . . . . . . . . 9  |-  ( A  =  1  ->  ( abs `  A )  =  1 )
3837breq2d 4414 . . . . . . . 8  |-  ( A  =  1  ->  (
1  <  ( abs `  A )  <->  1  <  1 ) )
3935, 38mtbiri 305 . . . . . . 7  |-  ( A  =  1  ->  -.  1  <  ( abs `  A
) )
4039necon2ai 2653 . . . . . 6  |-  ( 1  <  ( abs `  A
)  ->  A  =/=  1 )
412, 40syl 17 . . . . 5  |-  ( ph  ->  A  =/=  1 )
42 subeq0 9900 . . . . . . 7  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  - 
1 )  =  0  <-> 
A  =  1 ) )
431, 32, 42sylancl 668 . . . . . 6  |-  ( ph  ->  ( ( A  - 
1 )  =  0  <-> 
A  =  1 ) )
4443necon3bid 2668 . . . . 5  |-  ( ph  ->  ( ( A  - 
1 )  =/=  0  <->  A  =/=  1 ) )
4541, 44mpbird 236 . . . 4  |-  ( ph  ->  ( A  -  1 )  =/=  0 )
4634, 1, 45, 14recdivd 10400 . . 3  |-  ( ph  ->  ( 1  /  (
( A  -  1 )  /  A ) )  =  ( A  /  ( A  - 
1 ) ) )
4731, 46eqtr3d 2487 . 2  |-  ( ph  ->  ( 1  /  (
1  -  ( 1  /  A ) ) )  =  ( A  /  ( A  - 
1 ) ) )
4826, 47breqtrd 4427 1  |-  ( ph  ->  seq 0 (  +  ,  F )  ~~>  ( A  /  ( A  - 
1 ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   CCcc 9537   0cc0 9539   1c1 9540    + caddc 9542    < clt 9675    <_ cle 9676    - cmin 9860    / cdiv 10269   NN0cn0 10869    seqcseq 12213   ^cexp 12272   abscabs 13297    ~~> cli 13548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-fzo 11916  df-fl 12028  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-rlim 13553  df-sum 13753
This theorem is referenced by:  geoisumr  13934  ege2le3  14144  eftlub  14163
  Copyright terms: Public domain W3C validator