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Theorem geolim3 19735
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 11067 . . 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 10278 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
5 0z 10051 . . . . . 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 5882 . . . . . . . 8  |-  ( k  =  a  ->  ( B ^ k )  =  ( B ^ a
) )
11 eqid 2296 . . . . . . . 8  |-  ( k  e.  NN0  |->  ( B ^ k ) )  =  ( k  e. 
NN0  |->  ( B ^
k ) )
12 ovex 5899 . . . . . . . 8  |-  ( B ^ a )  e. 
_V
1310, 11, 12fvmpt 5618 . . . . . . 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 12342 . . . . 5  |-  ( ph  ->  seq  0 (  +  ,  ( k  e. 
NN0  |->  ( B ^
k ) ) )  ~~>  ( 1  /  (
1  -  B ) ) )
16 expcl 11137 . . . . . . 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 2370 . . . . 5  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( (
k  e.  NN0  |->  ( B ^ k ) ) `
 a )  e.  CC )
19 geolim3.a . . . . . . . 8  |-  ( ph  ->  A  e.  ZZ )
2019zcnd 10134 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
21 nn0cn 9991 . . . . . . 7  |-  ( a  e.  NN0  ->  a  e.  CC )
22 fvex 5555 . . . . . . . . 9  |-  ( ZZ>= `  A )  e.  _V
2322mptex 5762 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  A
)  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) )  e.  _V
2423shftval4 11588 . . . . . . 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 10258 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  A  e.  ( ZZ>= `  A )
)
2719, 26syl 15 . . . . . . . 8  |-  ( ph  ->  A  e.  ( ZZ>= `  A ) )
28 uzaddcl 10291 . . . . . . . 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 5881 . . . . . . . . . 10  |-  ( k  =  ( A  +  a )  ->  (
k  -  A )  =  ( ( A  +  a )  -  A ) )
3130oveq2d 5890 . . . . . . . . 9  |-  ( k  =  ( A  +  a )  ->  ( B ^ ( k  -  A ) )  =  ( B ^ (
( A  +  a )  -  A ) ) )
3231oveq2d 5890 . . . . . . . 8  |-  ( k  =  ( A  +  a )  ->  ( C  x.  ( B ^ ( k  -  A ) ) )  =  ( C  x.  ( B ^ ( ( A  +  a )  -  A ) ) ) )
33 eqid 2296 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  A
)  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) )  =  ( k  e.  ( ZZ>= `  A )  |->  ( C  x.  ( B ^
( k  -  A
) ) ) )
34 ovex 5899 . . . . . . . 8  |-  ( C  x.  ( B ^
( ( A  +  a )  -  A
) ) )  e. 
_V
3532, 33, 34fvmpt 5618 . . . . . . 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 9074 . . . . . . . . . 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 5890 . . . . . . . 8  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( B ^ ( ( A  +  a )  -  A ) )  =  ( B ^ a
) )
4039, 14eqtr4d 2331 . . . . . . 7  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( B ^ ( ( A  +  a )  -  A ) )  =  ( ( k  e. 
NN0  |->  ( B ^
k ) ) `  a ) )
4140oveq2d 5890 . . . . . 6  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( C  x.  ( B ^ (
( A  +  a )  -  A ) ) )  =  ( C  x.  ( ( k  e.  NN0  |->  ( B ^ k ) ) `
 a ) ) )
4225, 36, 413eqtrd 2332 . . . . 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 12147 . . . 4  |-  ( ph  ->  seq  0 (  +  ,  ( ( k  e.  ( ZZ>= `  A
)  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) )  shift  -u A
) )  ~~>  ( C  x.  ( 1  / 
( 1  -  B
) ) ) )
4420negidd 9163 . . . . 5  |-  ( ph  ->  ( A  +  -u A )  =  0 )
4544seqeq1d 11068 . . . 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 8811 . . . . . 6  |-  1  e.  CC
47 subcl 9067 . . . . . 6  |-  ( ( 1  e.  CC  /\  B  e.  CC )  ->  ( 1  -  B
)  e.  CC )
4846, 8, 47sylancr 644 . . . . 5  |-  ( ph  ->  ( 1  -  B
)  e.  CC )
49 abs1 11798 . . . . . . . . 9  |-  ( abs `  1 )  =  1
5049a1i 10 . . . . . . . 8  |-  ( ph  ->  ( abs `  1
)  =  1 )
518abscld 11934 . . . . . . . . 9  |-  ( ph  ->  ( abs `  B
)  e.  RR )
5251, 9gtned 8970 . . . . . . . 8  |-  ( ph  ->  1  =/=  ( abs `  B ) )
5350, 52eqnetrd 2477 . . . . . . 7  |-  ( ph  ->  ( abs `  1
)  =/=  ( abs `  B ) )
54 fveq2 5541 . . . . . . . 8  |-  ( 1  =  B  ->  ( abs `  1 )  =  ( abs `  B
) )
5554necon3i 2498 . . . . . . 7  |-  ( ( abs `  1 )  =/=  ( abs `  B
)  ->  1  =/=  B )
5653, 55syl 15 . . . . . 6  |-  ( ph  ->  1  =/=  B )
57 subeq0 9089 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  B  e.  CC )  ->  ( ( 1  -  B )  =  0  <->  1  =  B ) )
5846, 8, 57sylancr 644 . . . . . . 7  |-  ( ph  ->  ( ( 1  -  B )  =  0  <->  1  =  B ) )
5958necon3bid 2494 . . . . . 6  |-  ( ph  ->  ( ( 1  -  B )  =/=  0  <->  1  =/=  B ) )
6056, 59mpbird 223 . . . . 5  |-  ( ph  ->  ( 1  -  B
)  =/=  0 )
617, 48, 60divrecd 9555 . . . 4  |-  ( ph  ->  ( C  /  (
1  -  B ) )  =  ( C  x.  ( 1  / 
( 1  -  B
) ) ) )
6243, 45, 613brtr4d 4069 . . 3  |-  ( ph  ->  seq  ( A  +  -u A ) (  +  ,  ( ( k  e.  ( ZZ>= `  A
)  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) )  shift  -u A
) )  ~~>  ( C  /  ( 1  -  B ) ) )
6319znegcld 10135 . . . 4  |-  ( ph  -> 
-u A  e.  ZZ )
6423isershft 12153 . . . 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 4072 1  |-  ( ph  ->  seq  A (  +  ,  F )  ~~>  ( C  /  ( 1  -  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    - cmin 9053   -ucneg 9054    / cdiv 9439   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246    seq cseq 11062   ^cexp 11120    shift cshi 11577   abscabs 11735    ~~> cli 11974
This theorem is referenced by:  aaliou3lem3  19740
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175
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