Theorem geolim3 24094

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.

Step	Hyp	Ref	Expression
1		geolim3.f	. . 3 |- F = ( k e. ( ZZ>= ` A ) |-> ( C x. ( B ^ ( k - A ) ) ) )
2		seqeq3 12806	. . 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 ) ) ) ) ) )
3	1, 2	ax-mp 5	. 2 |- seq A ( + , F ) = seq A ( + , ( k e. ( ZZ>= ` A ) |-> ( C x. ( B ^ ( k - A ) ) ) ) )
4		nn0uz 11722	. . . . 5 |- NN0 = ( ZZ>= ` 0 )
5		0zd 11389	. . . . 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 6658	. . . . . . . 8 |- ( k = a -> ( B ^ k ) = ( B ^ a ) )
10		eqid 2622	. . . . . . . 8 |- ( k e. NN0 |-> ( B ^ k ) ) = ( k e. NN0 |-> ( B ^ k ) )
11		ovex 6678	. . . . . . . 8 |- ( B ^ a ) e. _V
12	9, 10, 11	fvmpt 6282	. . . . . . 7 |- ( a e. NN0 -> ( ( k e. NN0 |-> ( B ^ k ) ) ` a ) = ( B ^ a ) )
13	12	adantl 482	. . . . . 6 |- ( ( ph /\ a e. NN0 ) -> ( ( k e. NN0 |-> ( B ^ k ) ) ` a ) = ( B ^ a ) )
14	7, 8, 13	geolim 14601	. . . . 5 |- ( ph -> seq 0 ( + , ( k e. NN0 |-> ( B ^ k ) ) ) ~~> ( 1 / ( 1 - B ) ) )
15		expcl 12878	. . . . . . 7 |- ( ( B e. CC /\ a e. NN0 ) -> ( B ^ a ) e. CC )
16	7, 15	sylan 488	. . . . . 6 |- ( ( ph /\ a e. NN0 ) -> ( B ^ a ) e. CC )
17	13, 16	eqeltrd 2701	. . . . 5 |- ( ( ph /\ a e. NN0 ) -> ( ( k e. NN0 |-> ( B ^ k ) ) ` a ) e. CC )
18		geolim3.a	. . . . . . . 8 |- ( ph -> A e. ZZ )
19	18	zcnd 11483	. . . . . . 7 |- ( ph -> A e. CC )
20		nn0cn 11302	. . . . . . 7 |- ( a e. NN0 -> a e. CC )
21		fvex 6201	. . . . . . . . 9 |- ( ZZ>= ` A ) e. _V
22	21	mptex 6486	. . . . . . . 8 |- ( k e. ( ZZ>= ` A ) |-> ( C x. ( B ^ ( k - A ) ) ) ) e. _V
23	22	shftval4 13817	. . . . . . 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 ) ) )
24	19, 20, 23	syl2an 494	. . . . . 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 11702	. . . . . . . . 9 |- ( A e. ZZ -> A e. ( ZZ>= ` A ) )
26	18, 25	syl 17	. . . . . . . 8 |- ( ph -> A e. ( ZZ>= ` A ) )
27		uzaddcl 11744	. . . . . . . 8 |- ( ( A e. ( ZZ>= ` A ) /\ a e. NN0 ) -> ( A + a ) e. ( ZZ>= ` A ) )
28	26, 27	sylan 488	. . . . . . 7 |- ( ( ph /\ a e. NN0 ) -> ( A + a ) e. ( ZZ>= ` A ) )
29		oveq1 6657	. . . . . . . . . 10 |- ( k = ( A + a ) -> ( k - A ) = ( ( A + a ) - A ) )
30	29	oveq2d 6666	. . . . . . . . 9 |- ( k = ( A + a ) -> ( B ^ ( k - A ) ) = ( B ^ ( ( A + a ) - A ) ) )
31	30	oveq2d 6666	. . . . . . . 8 |- ( k = ( A + a ) -> ( C x. ( B ^ ( k - A ) ) ) = ( C x. ( B ^ ( ( A + a ) - A ) ) ) )
32		eqid 2622	. . . . . . . 8 |- ( k e. ( ZZ>= ` A ) |-> ( C x. ( B ^ ( k - A ) ) ) ) = ( k e. ( ZZ>= ` A ) |-> ( C x. ( B ^ ( k - A ) ) ) )
33		ovex 6678	. . . . . . . 8 |- ( C x. ( B ^ ( ( A + a ) - A ) ) ) e. _V
34	31, 32, 33	fvmpt 6282	. . . . . . 7 |- ( ( A + a ) e. ( ZZ>= ` A ) -> ( ( k e. ( ZZ>= ` A ) |-> ( C x. ( B ^ ( k - A ) ) ) ) ` ( A + a ) ) = ( C x. ( B ^ ( ( A + a ) - A ) ) ) )
35	28, 34	syl 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 10288	. . . . . . . . . 10 |- ( ( A e. CC /\ a e. CC ) -> ( ( A + a ) - A ) = a )
37	19, 20, 36	syl2an 494	. . . . . . . . 9 |- ( ( ph /\ a e. NN0 ) -> ( ( A + a ) - A ) = a )
38	37	oveq2d 6666	. . . . . . . 8 |- ( ( ph /\ a e. NN0 ) -> ( B ^ ( ( A + a ) - A ) ) = ( B ^ a ) )
39	38, 13	eqtr4d 2659	. . . . . . 7 |- ( ( ph /\ a e. NN0 ) -> ( B ^ ( ( A + a ) - A ) ) = ( ( k e. NN0 |-> ( B ^ k ) ) ` a ) )
40	39	oveq2d 6666	. . . . . 6 |- ( ( ph /\ a e. NN0 ) -> ( C x. ( B ^ ( ( A + a ) - A ) ) ) = ( C x. ( ( k e. NN0 |-> ( B ^ k ) ) ` a ) ) )
41	24, 35, 40	3eqtrd 2660	. . . . 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 ) ) )
42	4, 5, 6, 14, 17, 41	isermulc2 14388	. . . 4 |- ( ph -> seq 0 ( + , ( ( k e. ( ZZ>= ` A ) |-> ( C x. ( B ^ ( k - A ) ) ) ) shift -u A ) ) ~~> ( C x. ( 1 / ( 1 - B ) ) ) )
43	19	negidd 10382	. . . . 5 |- ( ph -> ( A + -u A ) = 0 )
44	43	seqeq1d 12807	. . . 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 9994	. . . . . 6 |- 1 e. CC
46		subcl 10280	. . . . . 6 |- ( ( 1 e. CC /\ B e. CC ) -> ( 1 - B ) e. CC )
47	45, 7, 46	sylancr 695	. . . . 5 |- ( ph -> ( 1 - B ) e. CC )
48		abs1 14037	. . . . . . . . 9 |- ( abs ` 1 ) = 1
49	48	a1i 11	. . . . . . . 8 |- ( ph -> ( abs ` 1 ) = 1 )
50	7	abscld 14175	. . . . . . . . 9 |- ( ph -> ( abs ` B ) e. RR )
51	50, 8	gtned 10172	. . . . . . . 8 |- ( ph -> 1 =/= ( abs ` B ) )
52	49, 51	eqnetrd 2861	. . . . . . 7 |- ( ph -> ( abs ` 1 ) =/= ( abs ` B ) )
53		fveq2 6191	. . . . . . . 8 |- ( 1 = B -> ( abs ` 1 ) = ( abs ` B ) )
54	53	necon3i 2826	. . . . . . 7 |- ( ( abs ` 1 ) =/= ( abs ` B ) -> 1 =/= B )
55	52, 54	syl 17	. . . . . 6 |- ( ph -> 1 =/= B )
56		subeq0 10307	. . . . . . . 8 |- ( ( 1 e. CC /\ B e. CC ) -> ( ( 1 - B ) = 0 <-> 1 = B ) )
57	45, 7, 56	sylancr 695	. . . . . . 7 |- ( ph -> ( ( 1 - B ) = 0 <-> 1 = B ) )
58	57	necon3bid 2838	. . . . . 6 |- ( ph -> ( ( 1 - B ) =/= 0 <-> 1 =/= B ) )
59	55, 58	mpbird 247	. . . . 5 |- ( ph -> ( 1 - B ) =/= 0 )
60	6, 47, 59	divrecd 10804	. . . 4 |- ( ph -> ( C / ( 1 - B ) ) = ( C x. ( 1 / ( 1 - B ) ) ) )
61	42, 44, 60	3brtr4d 4685	. . 3 |- ( ph -> seq ( A + -u A ) ( + , ( ( k e. ( ZZ>= ` A ) |-> ( C x. ( B ^ ( k - A ) ) ) ) shift -u A ) ) ~~> ( C / ( 1 - B ) ) )
62	18	znegcld 11484	. . . 4 |- ( ph -> -u A e. ZZ )
63	22	isershft 14394	. . . 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 ) ) ) )
64	18, 62, 63	syl2anc 693	. . 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 ) ) ) )
65	61, 64	mpbird 247	. 2 |- ( ph -> seq A ( + , ( k e. ( ZZ>= ` A ) |-> ( C x. ( B ^ ( k - A ) ) ) ) ) ~~> ( C / ( 1 - B ) ) )
66	3, 65	syl5eqbr 4688	1 |- ( ph -> seq A ( + , F ) ~~> ( C / ( 1 - B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   -ucneg 10267   / cdiv 10684   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   seqcseq 12801   ^cexp 12860   shift cshi 13806   abscabs 13974   ~~> cli 14215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417

This theorem is referenced by:  aaliou3lem3  24099