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Theorem geolim3 19719
Description: Geometric series convergence with arbitrary shift, radix, and multiplicative constant. (Contributed by Stefan O'Rear, 16-Nov-2014.)
Hypotheses
Ref Expression
geolim3.a  |-  ( ph  ->  A  e.  ZZ )
geolim3.b1  |-  ( ph  ->  B  e.  CC )
geolim3.b2  |-  ( ph  ->  ( abs `  B
)  <  1 )
geolim3.c  |-  ( ph  ->  C  e.  CC )
geolim3.f  |-  F  =  ( k  e.  (
ZZ>= `  A )  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) )
Assertion
Ref Expression
geolim3  |-  ( ph  ->  seq  A (  +  ,  F )  ~~>  ( C  /  ( 1  -  B ) ) )
Distinct variable groups:    ph, k    A, k    B, k    C, k
Allowed substitution hint:    F( k)

Proof of Theorem geolim3
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 geolim3.f . . 3  |-  F  =  ( k  e.  (
ZZ>= `  A )  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) )
2 seqeq3 11051 . . 3  |-  ( F  =  ( k  e.  ( ZZ>= `  A )  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) )  ->  seq  A (  +  ,  F )  =  seq  A (  +  ,  ( k  e.  ( ZZ>= `  A
)  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) ) ) )
31, 2ax-mp 8 . 2  |-  seq  A
(  +  ,  F
)  =  seq  A
(  +  ,  ( k  e.  ( ZZ>= `  A )  |->  ( C  x.  ( B ^
( k  -  A
) ) ) ) )
4 nn0uz 10262 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
5 0z 10035 . . . . . 6  |-  0  e.  ZZ
65a1i 10 . . . . 5  |-  ( ph  ->  0  e.  ZZ )
7 geolim3.c . . . . 5  |-  ( ph  ->  C  e.  CC )
8 geolim3.b1 . . . . . 6  |-  ( ph  ->  B  e.  CC )
9 geolim3.b2 . . . . . 6  |-  ( ph  ->  ( abs `  B
)  <  1 )
10 oveq2 5866 . . . . . . . 8  |-  ( k  =  a  ->  ( B ^ k )  =  ( B ^ a
) )
11 eqid 2283 . . . . . . . 8  |-  ( k  e.  NN0  |->  ( B ^ k ) )  =  ( k  e. 
NN0  |->  ( B ^
k ) )
12 ovex 5883 . . . . . . . 8  |-  ( B ^ a )  e. 
_V
1310, 11, 12fvmpt 5602 . . . . . . 7  |-  ( a  e.  NN0  ->  ( ( k  e.  NN0  |->  ( B ^ k ) ) `
 a )  =  ( B ^ a
) )
1413adantl 452 . . . . . 6  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( (
k  e.  NN0  |->  ( B ^ k ) ) `
 a )  =  ( B ^ a
) )
158, 9, 14geolim 12326 . . . . 5  |-  ( ph  ->  seq  0 (  +  ,  ( k  e. 
NN0  |->  ( B ^
k ) ) )  ~~>  ( 1  /  (
1  -  B ) ) )
16 expcl 11121 . . . . . . 7  |-  ( ( B  e.  CC  /\  a  e.  NN0 )  -> 
( B ^ a
)  e.  CC )
178, 16sylan 457 . . . . . 6  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( B ^ a )  e.  CC )
1814, 17eqeltrd 2357 . . . . 5  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( (
k  e.  NN0  |->  ( B ^ k ) ) `
 a )  e.  CC )
19 geolim3.a . . . . . . . 8  |-  ( ph  ->  A  e.  ZZ )
2019zcnd 10118 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
21 nn0cn 9975 . . . . . . 7  |-  ( a  e.  NN0  ->  a  e.  CC )
22 fvex 5539 . . . . . . . . 9  |-  ( ZZ>= `  A )  e.  _V
2322mptex 5746 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  A
)  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) )  e.  _V
2423shftval4 11572 . . . . . . 7  |-  ( ( A  e.  CC  /\  a  e.  CC )  ->  ( ( ( k  e.  ( ZZ>= `  A
)  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) )  shift  -u A
) `  a )  =  ( ( k  e.  ( ZZ>= `  A
)  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) ) `  ( A  +  a )
) )
2520, 21, 24syl2an 463 . . . . . 6  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( (
( k  e.  (
ZZ>= `  A )  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) )  shift  -u A ) `
 a )  =  ( ( k  e.  ( ZZ>= `  A )  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) ) `  ( A  +  a ) ) )
26 uzid 10242 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  A  e.  ( ZZ>= `  A )
)
2719, 26syl 15 . . . . . . . 8  |-  ( ph  ->  A  e.  ( ZZ>= `  A ) )
28 uzaddcl 10275 . . . . . . . 8  |-  ( ( A  e.  ( ZZ>= `  A )  /\  a  e.  NN0 )  ->  ( A  +  a )  e.  ( ZZ>= `  A )
)
2927, 28sylan 457 . . . . . . 7  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( A  +  a )  e.  ( ZZ>= `  A )
)
30 oveq1 5865 . . . . . . . . . 10  |-  ( k  =  ( A  +  a )  ->  (
k  -  A )  =  ( ( A  +  a )  -  A ) )
3130oveq2d 5874 . . . . . . . . 9  |-  ( k  =  ( A  +  a )  ->  ( B ^ ( k  -  A ) )  =  ( B ^ (
( A  +  a )  -  A ) ) )
3231oveq2d 5874 . . . . . . . 8  |-  ( k  =  ( A  +  a )  ->  ( C  x.  ( B ^ ( k  -  A ) ) )  =  ( C  x.  ( B ^ ( ( A  +  a )  -  A ) ) ) )
33 eqid 2283 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  A
)  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) )  =  ( k  e.  ( ZZ>= `  A )  |->  ( C  x.  ( B ^
( k  -  A
) ) ) )
34 ovex 5883 . . . . . . . 8  |-  ( C  x.  ( B ^
( ( A  +  a )  -  A
) ) )  e. 
_V
3532, 33, 34fvmpt 5602 . . . . . . 7  |-  ( ( A  +  a )  e.  ( ZZ>= `  A
)  ->  ( (
k  e.  ( ZZ>= `  A )  |->  ( C  x.  ( B ^
( k  -  A
) ) ) ) `
 ( A  +  a ) )  =  ( C  x.  ( B ^ ( ( A  +  a )  -  A ) ) ) )
3629, 35syl 15 . . . . . 6  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( (
k  e.  ( ZZ>= `  A )  |->  ( C  x.  ( B ^
( k  -  A
) ) ) ) `
 ( A  +  a ) )  =  ( C  x.  ( B ^ ( ( A  +  a )  -  A ) ) ) )
37 pncan2 9058 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  a  e.  CC )  ->  ( ( A  +  a )  -  A
)  =  a )
3820, 21, 37syl2an 463 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( ( A  +  a )  -  A )  =  a )
3938oveq2d 5874 . . . . . . . 8  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( B ^ ( ( A  +  a )  -  A ) )  =  ( B ^ a
) )
4039, 14eqtr4d 2318 . . . . . . 7  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( B ^ ( ( A  +  a )  -  A ) )  =  ( ( k  e. 
NN0  |->  ( B ^
k ) ) `  a ) )
4140oveq2d 5874 . . . . . 6  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( C  x.  ( B ^ (
( A  +  a )  -  A ) ) )  =  ( C  x.  ( ( k  e.  NN0  |->  ( B ^ k ) ) `
 a ) ) )
4225, 36, 413eqtrd 2319 . . . . 5  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( (
( k  e.  (
ZZ>= `  A )  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) )  shift  -u A ) `
 a )  =  ( C  x.  (
( k  e.  NN0  |->  ( B ^ k ) ) `  a ) ) )
434, 6, 7, 15, 18, 42isermulc2 12131 . . . 4  |-  ( ph  ->  seq  0 (  +  ,  ( ( k  e.  ( ZZ>= `  A
)  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) )  shift  -u A
) )  ~~>  ( C  x.  ( 1  / 
( 1  -  B
) ) ) )
4420negidd 9147 . . . . 5  |-  ( ph  ->  ( A  +  -u A )  =  0 )
4544seqeq1d 11052 . . . 4  |-  ( ph  ->  seq  ( A  +  -u A ) (  +  ,  ( ( k  e.  ( ZZ>= `  A
)  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) )  shift  -u A
) )  =  seq  0 (  +  , 
( ( k  e.  ( ZZ>= `  A )  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) )  shift  -u A ) ) )
46 ax-1cn 8795 . . . . . 6  |-  1  e.  CC
47 subcl 9051 . . . . . 6  |-  ( ( 1  e.  CC  /\  B  e.  CC )  ->  ( 1  -  B
)  e.  CC )
4846, 8, 47sylancr 644 . . . . 5  |-  ( ph  ->  ( 1  -  B
)  e.  CC )
49 abs1 11782 . . . . . . . . 9  |-  ( abs `  1 )  =  1
5049a1i 10 . . . . . . . 8  |-  ( ph  ->  ( abs `  1
)  =  1 )
518abscld 11918 . . . . . . . . 9  |-  ( ph  ->  ( abs `  B
)  e.  RR )
5251, 9gtned 8954 . . . . . . . 8  |-  ( ph  ->  1  =/=  ( abs `  B ) )
5350, 52eqnetrd 2464 . . . . . . 7  |-  ( ph  ->  ( abs `  1
)  =/=  ( abs `  B ) )
54 fveq2 5525 . . . . . . . 8  |-  ( 1  =  B  ->  ( abs `  1 )  =  ( abs `  B
) )
5554necon3i 2485 . . . . . . 7  |-  ( ( abs `  1 )  =/=  ( abs `  B
)  ->  1  =/=  B )
5653, 55syl 15 . . . . . 6  |-  ( ph  ->  1  =/=  B )
57 subeq0 9073 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  B  e.  CC )  ->  ( ( 1  -  B )  =  0  <->  1  =  B ) )
5846, 8, 57sylancr 644 . . . . . . 7  |-  ( ph  ->  ( ( 1  -  B )  =  0  <->  1  =  B ) )
5958necon3bid 2481 . . . . . 6  |-  ( ph  ->  ( ( 1  -  B )  =/=  0  <->  1  =/=  B ) )
6056, 59mpbird 223 . . . . 5  |-  ( ph  ->  ( 1  -  B
)  =/=  0 )
617, 48, 60divrecd 9539 . . . 4  |-  ( ph  ->  ( C  /  (
1  -  B ) )  =  ( C  x.  ( 1  / 
( 1  -  B
) ) ) )
6243, 45, 613brtr4d 4053 . . 3  |-  ( ph  ->  seq  ( A  +  -u A ) (  +  ,  ( ( k  e.  ( ZZ>= `  A
)  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) )  shift  -u A
) )  ~~>  ( C  /  ( 1  -  B ) ) )
6319znegcld 10119 . . . 4  |-  ( ph  -> 
-u A  e.  ZZ )
6423isershft 12137 . . . 4  |-  ( ( A  e.  ZZ  /\  -u A  e.  ZZ )  ->  (  seq  A
(  +  ,  ( k  e.  ( ZZ>= `  A )  |->  ( C  x.  ( B ^
( k  -  A
) ) ) ) )  ~~>  ( C  / 
( 1  -  B
) )  <->  seq  ( A  +  -u A ) (  +  ,  ( ( k  e.  ( ZZ>= `  A )  |->  ( C  x.  ( B ^
( k  -  A
) ) ) ) 
shift  -u A ) )  ~~>  ( C  /  (
1  -  B ) ) ) )
6519, 63, 64syl2anc 642 . . 3  |-  ( ph  ->  (  seq  A (  +  ,  ( k  e.  ( ZZ>= `  A
)  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) ) )  ~~>  ( C  /  ( 1  -  B ) )  <->  seq  ( A  +  -u A ) (  +  ,  ( ( k  e.  ( ZZ>= `  A )  |->  ( C  x.  ( B ^
( k  -  A
) ) ) ) 
shift  -u A ) )  ~~>  ( C  /  (
1  -  B ) ) ) )
6662, 65mpbird 223 . 2  |-  ( ph  ->  seq  A (  +  ,  ( k  e.  ( ZZ>= `  A )  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) ) )  ~~>  ( C  /  ( 1  -  B ) ) )
673, 66syl5eqbr 4056 1  |-  ( ph  ->  seq  A (  +  ,  F )  ~~>  ( C  /  ( 1  -  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    - cmin 9037   -ucneg 9038    / cdiv 9423   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230    seq cseq 11046   ^cexp 11104    shift cshi 11561   abscabs 11719    ~~> cli 11958
This theorem is referenced by:  aaliou3lem3  19724
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-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159
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