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Theorem geolim3 21803
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 11809 . . 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 10893 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
5 0zd 10656 . . . . 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 6097 . . . . . . . 8  |-  ( k  =  a  ->  ( B ^ k )  =  ( B ^ a
) )
10 eqid 2441 . . . . . . . 8  |-  ( k  e.  NN0  |->  ( B ^ k ) )  =  ( k  e. 
NN0  |->  ( B ^
k ) )
11 ovex 6114 . . . . . . . 8  |-  ( B ^ a )  e. 
_V
129, 10, 11fvmpt 5772 . . . . . . 7  |-  ( a  e.  NN0  ->  ( ( k  e.  NN0  |->  ( B ^ k ) ) `
 a )  =  ( B ^ a
) )
1312adantl 466 . . . . . 6  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( (
k  e.  NN0  |->  ( B ^ k ) ) `
 a )  =  ( B ^ a
) )
147, 8, 13geolim 13328 . . . . 5  |-  ( ph  ->  seq 0 (  +  ,  ( k  e. 
NN0  |->  ( B ^
k ) ) )  ~~>  ( 1  /  (
1  -  B ) ) )
15 expcl 11881 . . . . . . 7  |-  ( ( B  e.  CC  /\  a  e.  NN0 )  -> 
( B ^ a
)  e.  CC )
167, 15sylan 471 . . . . . 6  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( B ^ a )  e.  CC )
1713, 16eqeltrd 2515 . . . . 5  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( (
k  e.  NN0  |->  ( B ^ k ) ) `
 a )  e.  CC )
18 geolim3.a . . . . . . . 8  |-  ( ph  ->  A  e.  ZZ )
1918zcnd 10746 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
20 nn0cn 10587 . . . . . . 7  |-  ( a  e.  NN0  ->  a  e.  CC )
21 fvex 5699 . . . . . . . . 9  |-  ( ZZ>= `  A )  e.  _V
2221mptex 5946 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  A
)  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) )  e.  _V
2322shftval4 12564 . . . . . . 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 477 . . . . . 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 10873 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  A  e.  ( ZZ>= `  A )
)
2618, 25syl 16 . . . . . . . 8  |-  ( ph  ->  A  e.  ( ZZ>= `  A ) )
27 uzaddcl 10909 . . . . . . . 8  |-  ( ( A  e.  ( ZZ>= `  A )  /\  a  e.  NN0 )  ->  ( A  +  a )  e.  ( ZZ>= `  A )
)
2826, 27sylan 471 . . . . . . 7  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( A  +  a )  e.  ( ZZ>= `  A )
)
29 oveq1 6096 . . . . . . . . . 10  |-  ( k  =  ( A  +  a )  ->  (
k  -  A )  =  ( ( A  +  a )  -  A ) )
3029oveq2d 6105 . . . . . . . . 9  |-  ( k  =  ( A  +  a )  ->  ( B ^ ( k  -  A ) )  =  ( B ^ (
( A  +  a )  -  A ) ) )
3130oveq2d 6105 . . . . . . . 8  |-  ( k  =  ( A  +  a )  ->  ( C  x.  ( B ^ ( k  -  A ) ) )  =  ( C  x.  ( B ^ ( ( A  +  a )  -  A ) ) ) )
32 eqid 2441 . . . . . . . 8  |-  ( k  e.  ( ZZ>= `  A
)  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) )  =  ( k  e.  ( ZZ>= `  A )  |->  ( C  x.  ( B ^
( k  -  A
) ) ) )
33 ovex 6114 . . . . . . . 8  |-  ( C  x.  ( B ^
( ( A  +  a )  -  A
) ) )  e. 
_V
3431, 32, 33fvmpt 5772 . . . . . . 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 16 . . . . . 6  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( (
k  e.  ( ZZ>= `  A )  |->  ( C  x.  ( B ^
( k  -  A
) ) ) ) `
 ( A  +  a ) )  =  ( C  x.  ( B ^ ( ( A  +  a )  -  A ) ) ) )
36 pncan2 9615 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  a  e.  CC )  ->  ( ( A  +  a )  -  A
)  =  a )
3719, 20, 36syl2an 477 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( ( A  +  a )  -  A )  =  a )
3837oveq2d 6105 . . . . . . . 8  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( B ^ ( ( A  +  a )  -  A ) )  =  ( B ^ a
) )
3938, 13eqtr4d 2476 . . . . . . 7  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( B ^ ( ( A  +  a )  -  A ) )  =  ( ( k  e. 
NN0  |->  ( B ^
k ) ) `  a ) )
4039oveq2d 6105 . . . . . 6  |-  ( (
ph  /\  a  e.  NN0 )  ->  ( C  x.  ( B ^ (
( A  +  a )  -  A ) ) )  =  ( C  x.  ( ( k  e.  NN0  |->  ( B ^ k ) ) `
 a ) ) )
4124, 35, 403eqtrd 2477 . . . . 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 13133 . . . 4  |-  ( ph  ->  seq 0 (  +  ,  ( ( k  e.  ( ZZ>= `  A
)  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) )  shift  -u A
) )  ~~>  ( C  x.  ( 1  / 
( 1  -  B
) ) ) )
4319negidd 9707 . . . . 5  |-  ( ph  ->  ( A  +  -u A )  =  0 )
4443seqeq1d 11810 . . . 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 9338 . . . . . 6  |-  1  e.  CC
46 subcl 9607 . . . . . 6  |-  ( ( 1  e.  CC  /\  B  e.  CC )  ->  ( 1  -  B
)  e.  CC )
4745, 7, 46sylancr 663 . . . . 5  |-  ( ph  ->  ( 1  -  B
)  e.  CC )
48 abs1 12784 . . . . . . . . 9  |-  ( abs `  1 )  =  1
4948a1i 11 . . . . . . . 8  |-  ( ph  ->  ( abs `  1
)  =  1 )
507abscld 12920 . . . . . . . . 9  |-  ( ph  ->  ( abs `  B
)  e.  RR )
5150, 8gtned 9507 . . . . . . . 8  |-  ( ph  ->  1  =/=  ( abs `  B ) )
5249, 51eqnetrd 2624 . . . . . . 7  |-  ( ph  ->  ( abs `  1
)  =/=  ( abs `  B ) )
53 fveq2 5689 . . . . . . . 8  |-  ( 1  =  B  ->  ( abs `  1 )  =  ( abs `  B
) )
5453necon3i 2648 . . . . . . 7  |-  ( ( abs `  1 )  =/=  ( abs `  B
)  ->  1  =/=  B )
5552, 54syl 16 . . . . . 6  |-  ( ph  ->  1  =/=  B )
56 subeq0 9633 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  B  e.  CC )  ->  ( ( 1  -  B )  =  0  <->  1  =  B ) )
5745, 7, 56sylancr 663 . . . . . . 7  |-  ( ph  ->  ( ( 1  -  B )  =  0  <->  1  =  B ) )
5857necon3bid 2641 . . . . . 6  |-  ( ph  ->  ( ( 1  -  B )  =/=  0  <->  1  =/=  B ) )
5955, 58mpbird 232 . . . . 5  |-  ( ph  ->  ( 1  -  B
)  =/=  0 )
606, 47, 59divrecd 10108 . . . 4  |-  ( ph  ->  ( C  /  (
1  -  B ) )  =  ( C  x.  ( 1  / 
( 1  -  B
) ) ) )
6142, 44, 603brtr4d 4320 . . 3  |-  ( ph  ->  seq ( A  +  -u A ) (  +  ,  ( ( k  e.  ( ZZ>= `  A
)  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) )  shift  -u A
) )  ~~>  ( C  /  ( 1  -  B ) ) )
6218znegcld 10747 . . . 4  |-  ( ph  -> 
-u A  e.  ZZ )
6322isershft 13139 . . . 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 661 . . 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 232 . 2  |-  ( ph  ->  seq A (  +  ,  ( k  e.  ( ZZ>= `  A )  |->  ( C  x.  ( B ^ ( k  -  A ) ) ) ) )  ~~>  ( C  /  ( 1  -  B ) ) )
663, 65syl5eqbr 4323 1  |-  ( ph  ->  seq A (  +  ,  F )  ~~>  ( C  /  ( 1  -  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2604   class class class wbr 4290    e. cmpt 4348   ` cfv 5416  (class class class)co 6089   CCcc 9278   0cc0 9280   1c1 9281    + caddc 9283    x. cmul 9285    < clt 9416    - cmin 9593   -ucneg 9594    / cdiv 9991   NN0cn0 10577   ZZcz 10644   ZZ>=cuz 10859    seqcseq 11804   ^cexp 11863    shift cshi 12553   abscabs 12721    ~~> cli 12960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-inf2 7845  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-se 4678  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-isom 5425  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-pm 7215  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-sup 7689  df-oi 7722  df-card 8107  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-3 10379  df-n0 10578  df-z 10645  df-uz 10860  df-rp 10990  df-fz 11436  df-fzo 11547  df-fl 11640  df-seq 11805  df-exp 11864  df-hash 12102  df-shft 12554  df-cj 12586  df-re 12587  df-im 12588  df-sqr 12722  df-abs 12723  df-clim 12964  df-rlim 12965  df-sum 13162
This theorem is referenced by:  aaliou3lem3  21808
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