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Theorem geolim3 23160
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 12215 . . 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 5 . 2  |-  seq A
(  +  ,  F
)  =  seq A
(  +  ,  ( k  e.  ( ZZ>= `  A )  |->  ( C  x.  ( B ^
( k  -  A
) ) ) ) )
4 nn0uz 11193 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
5 0zd 10949 . . . . 5  |-  ( ph  ->  0  e.  ZZ )
6 geolim3.c . . . . 5  |-  ( ph  ->  C  e.  CC )
7 geolim3.b1 . . . . . 6  |-  ( ph  ->  B  e.  CC )
8 geolim3.b2 . . . . . 6  |-  ( ph  ->  ( abs `  B
)  <  1 )
9 oveq2 6313 . . . . . . . 8  |-  ( k  =  a  ->  ( B ^ k )  =  ( B ^ a
) )
10 eqid 2429 . . . . . . . 8  |-  ( k  e.  NN0  |->  ( B ^ k ) )  =  ( k  e. 
NN0  |->  ( B ^
k ) )
11 ovex 6333 . . . . . . . 8  |-  ( B ^ a )  e. 
_V
129, 10, 11fvmpt 5964 . . . . . . 7  |-  ( a  e.  NN0  ->  ( ( k  e.  NN0  |->  ( B ^ k ) ) `
 a )  =  ( B ^ a
) )
1312adantl 467 . . . . . 6  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( (
k  e.  NN0  |->  ( B ^ k ) ) `
 a )  =  ( B ^ a
) )
147, 8, 13geolim 13904 . . . . 5  |-  ( ph  ->  seq 0 (  +  ,  ( k  e. 
NN0  |->  ( B ^
k ) ) )  ~~>  ( 1  /  (
1  -  B ) ) )
15 expcl 12287 . . . . . . 7  |-  ( ( B  e.  CC  /\  a  e.  NN0 )  -> 
( B ^ a
)  e.  CC )
167, 15sylan 473 . . . . . 6  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( B ^ a )  e.  CC )
1713, 16eqeltrd 2517 . . . . 5  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( (
k  e.  NN0  |->  ( B ^ k ) ) `
 a )  e.  CC )
18 geolim3.a . . . . . . . 8  |-  ( ph  ->  A  e.  ZZ )
1918zcnd 11041 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
20 nn0cn 10879 . . . . . . 7  |-  ( a  e.  NN0  ->  a  e.  CC )
21 fvex 5891 . . . . . . . . 9  |-  ( ZZ>= `  A )  e.  _V
2221mptex 6151 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  A
)  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) )  e.  _V
2322shftval4 13119 . . . . . . 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 )
) )
2419, 20, 23syl2an 479 . . . . . 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 ) ) )
25 uzid 11173 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  A  e.  ( ZZ>= `  A )
)
2618, 25syl 17 . . . . . . . 8  |-  ( ph  ->  A  e.  ( ZZ>= `  A ) )
27 uzaddcl 11215 . . . . . . . 8  |-  ( ( A  e.  ( ZZ>= `  A )  /\  a  e.  NN0 )  ->  ( A  +  a )  e.  ( ZZ>= `  A )
)
2826, 27sylan 473 . . . . . . 7  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( A  +  a )  e.  ( ZZ>= `  A )
)
29 oveq1 6312 . . . . . . . . . 10  |-  ( k  =  ( A  +  a )  ->  (
k  -  A )  =  ( ( A  +  a )  -  A ) )
3029oveq2d 6321 . . . . . . . . 9  |-  ( k  =  ( A  +  a )  ->  ( B ^ ( k  -  A ) )  =  ( B ^ (
( A  +  a )  -  A ) ) )
3130oveq2d 6321 . . . . . . . 8  |-  ( k  =  ( A  +  a )  ->  ( C  x.  ( B ^ ( k  -  A ) ) )  =  ( C  x.  ( B ^ ( ( A  +  a )  -  A ) ) ) )
32 eqid 2429 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  A
)  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) )  =  ( k  e.  ( ZZ>= `  A )  |->  ( C  x.  ( B ^
( k  -  A
) ) ) )
33 ovex 6333 . . . . . . . 8  |-  ( C  x.  ( B ^
( ( A  +  a )  -  A
) ) )  e. 
_V
3431, 32, 33fvmpt 5964 . . . . . . 7  |-  ( ( A  +  a )  e.  ( ZZ>= `  A
)  ->  ( (
k  e.  ( ZZ>= `  A )  |->  ( C  x.  ( B ^
( k  -  A
) ) ) ) `
 ( A  +  a ) )  =  ( C  x.  ( B ^ ( ( A  +  a )  -  A ) ) ) )
3528, 34syl 17 . . . . . 6  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( (
k  e.  ( ZZ>= `  A )  |->  ( C  x.  ( B ^
( k  -  A
) ) ) ) `
 ( A  +  a ) )  =  ( C  x.  ( B ^ ( ( A  +  a )  -  A ) ) ) )
36 pncan2 9881 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  a  e.  CC )  ->  ( ( A  +  a )  -  A
)  =  a )
3719, 20, 36syl2an 479 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( ( A  +  a )  -  A )  =  a )
3837oveq2d 6321 . . . . . . . 8  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( B ^ ( ( A  +  a )  -  A ) )  =  ( B ^ a
) )
3938, 13eqtr4d 2473 . . . . . . 7  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( B ^ ( ( A  +  a )  -  A ) )  =  ( ( k  e. 
NN0  |->  ( B ^
k ) ) `  a ) )
4039oveq2d 6321 . . . . . 6  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( C  x.  ( B ^ (
( A  +  a )  -  A ) ) )  =  ( C  x.  ( ( k  e.  NN0  |->  ( B ^ k ) ) `
 a ) ) )
4124, 35, 403eqtrd 2474 . . . . 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 ) ) )
424, 5, 6, 14, 17, 41isermulc2 13699 . . . 4  |-  ( ph  ->  seq 0 (  +  ,  ( ( k  e.  ( ZZ>= `  A
)  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) )  shift  -u A
) )  ~~>  ( C  x.  ( 1  / 
( 1  -  B
) ) ) )
4319negidd 9975 . . . . 5  |-  ( ph  ->  ( A  +  -u A )  =  0 )
4443seqeq1d 12216 . . . 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 ) ) )
45 ax-1cn 9596 . . . . . 6  |-  1  e.  CC
46 subcl 9873 . . . . . 6  |-  ( ( 1  e.  CC  /\  B  e.  CC )  ->  ( 1  -  B
)  e.  CC )
4745, 7, 46sylancr 667 . . . . 5  |-  ( ph  ->  ( 1  -  B
)  e.  CC )
48 abs1 13339 . . . . . . . . 9  |-  ( abs `  1 )  =  1
4948a1i 11 . . . . . . . 8  |-  ( ph  ->  ( abs `  1
)  =  1 )
507abscld 13476 . . . . . . . . 9  |-  ( ph  ->  ( abs `  B
)  e.  RR )
5150, 8gtned 9769 . . . . . . . 8  |-  ( ph  ->  1  =/=  ( abs `  B ) )
5249, 51eqnetrd 2724 . . . . . . 7  |-  ( ph  ->  ( abs `  1
)  =/=  ( abs `  B ) )
53 fveq2 5881 . . . . . . . 8  |-  ( 1  =  B  ->  ( abs `  1 )  =  ( abs `  B
) )
5453necon3i 2671 . . . . . . 7  |-  ( ( abs `  1 )  =/=  ( abs `  B
)  ->  1  =/=  B )
5552, 54syl 17 . . . . . 6  |-  ( ph  ->  1  =/=  B )
56 subeq0 9899 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  B  e.  CC )  ->  ( ( 1  -  B )  =  0  <->  1  =  B ) )
5745, 7, 56sylancr 667 . . . . . . 7  |-  ( ph  ->  ( ( 1  -  B )  =  0  <->  1  =  B ) )
5857necon3bid 2689 . . . . . 6  |-  ( ph  ->  ( ( 1  -  B )  =/=  0  <->  1  =/=  B ) )
5955, 58mpbird 235 . . . . 5  |-  ( ph  ->  ( 1  -  B
)  =/=  0 )
606, 47, 59divrecd 10385 . . . 4  |-  ( ph  ->  ( C  /  (
1  -  B ) )  =  ( C  x.  ( 1  / 
( 1  -  B
) ) ) )
6142, 44, 603brtr4d 4456 . . 3  |-  ( ph  ->  seq ( A  +  -u A ) (  +  ,  ( ( k  e.  ( ZZ>= `  A
)  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) )  shift  -u A
) )  ~~>  ( C  /  ( 1  -  B ) ) )
6218znegcld 11042 . . . 4  |-  ( ph  -> 
-u A  e.  ZZ )
6322isershft 13705 . . . 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
) ) ) )
6418, 62, 63syl2anc 665 . . 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
) ) ) )
6561, 64mpbird 235 . 2  |-  ( ph  ->  seq A (  +  ,  ( k  e.  ( ZZ>= `  A )  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) ) )  ~~>  ( C  /  ( 1  -  B ) ) )
663, 65syl5eqbr 4459 1  |-  ( ph  ->  seq A (  +  ,  F )  ~~>  ( C  /  ( 1  -  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   class class class wbr 4426    |-> cmpt 4484   ` cfv 5601  (class class class)co 6305   CCcc 9536   0cc0 9538   1c1 9539    + caddc 9541    x. cmul 9543    < clt 9674    - cmin 9859   -ucneg 9860    / cdiv 10268   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159    seqcseq 12210   ^cexp 12269    shift cshi 13108   abscabs 13276    ~~> cli 13526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-oi 8025  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11783  df-fzo 11914  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-shft 13109  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-rlim 13531  df-sum 13731
This theorem is referenced by:  aaliou3lem3  23165
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