Theorem geolimilem 8497

Description: Lemma for geolimi 8498.

Hypotheses

Ref	Expression
geoser.1	|- F = {<.k, y>. | (k e. NN0 /\ y = (A^k))}
geoser.2	|- A e. CC
geolimilem.3	|- (abs` A) < 1
geolimilem.4	|- G Fn NN0
geolimilem.5	|- (n e. NN0 -> (G` n) = (-u(A x. (F` n)) / (1 - A)))

Assertion

Ref	Expression
geolimilem	|- ( + seq0 F) ~~> (1 / (1 - A))

Distinct variable groups:   k,n,y,A   n,F   n,G

Proof of Theorem geolimilem

Step	Hyp	Ref	Expression
1		ax1cn 6422	. . . . . . . . . . 11 |- 1 e. CC
2		geoser.2	. . . . . . . . . . 11 |- A e. CC
3	1, 2	subcli 6523	. . . . . . . . . 10 |- (1 - A) e. CC
4		1re 6598	. . . . . . . . . . 11 |- 1 e. RR
5		geolimilem.3	. . . . . . . . . . 11 |- (abs` A) < 1
6		abssubne0 8134	. . . . . . . . . . 11 |- ((A e. CC /\ 1 e. RR /\ (abs` A) < 1) -> (1 - A) =/= 0)
7	2, 4, 5, 6	mp3an 1191	. . . . . . . . . 10 |- (1 - A) =/= 0
8	3, 7	reccli 6902	. . . . . . . . 9 |- (1 / (1 - A)) e. CC
9	8	negcli 6526	. . . . . . . 8 |- -u(1 / (1 - A)) e. CC
10	9, 2	mulcli 6474	. . . . . . 7 |- (-u(1 / (1 - A)) x. A) e. CC
11		nnnn0 7315	. . . . . . . . . 10 |- (n e. NN -> n e. NN0)
12		opreq2 4890	. . . . . . . . . . 11 |- (k = n -> (A^k) = (A^n))
13		geoser.1	. . . . . . . . . . 11 |- F = {<.k, y>. | (k e. NN0 /\ y = (A^k))}
14		oprex 4907	. . . . . . . . . . 11 |- (A^n) e. _V
15	12, 13, 14	fvopab4 4743	. . . . . . . . . 10 |- (n e. NN0 -> (F` n) = (A^n))
16	11, 15	syl 12	. . . . . . . . 9 |- (n e. NN -> (F` n) = (A^n))
17	16	rgen 2159	. . . . . . . 8 |- A.n e. NN (F` n) = (A^n)
18		nn0ex 7314	. . . . . . . . . 10 |- NN0 e. _V
19	18, 13	fopabex2 4541	. . . . . . . . 9 |- F e. _V
20	19	expcnv 8494	. . . . . . . 8 |- ((A e. CC /\ A.n e. NN (F` n) = (A^n) /\ (abs` A) < 1) -> F ~~> 0)
21	2, 17, 5, 20	mp3an 1191	. . . . . . 7 |- F ~~> 0
22	10, 21	pm3.2i 307	. . . . . 6 |- ((-u(1 / (1 - A)) x. A) e. CC /\ F ~~> 0)
23		0z 7355	. . . . . . 7 |- 0 e. ZZ
24		elnn0uz 7610	. . . . . . . . 9 |- (n e. NN0 <-> n e. (ZZ>=` 0))
25		expcl 7824	. . . . . . . . . . . . 13 |- ((A e. CC /\ k e. NN0) -> (A^k) e. CC)
26	2, 25	mpan 759	. . . . . . . . . . . 12 |- (k e. NN0 -> (A^k) e. CC)
27	13, 26	fopab 4800	. . . . . . . . . . 11 |- F:NN0-->CC
28	27	ffvelrni 4788	. . . . . . . . . 10 |- (n e. NN0 -> (F` n) e. CC)
29		geolimilem.5	. . . . . . . . . . 11 |- (n e. NN0 -> (G` n) = (-u(A x. (F` n)) / (1 - A)))
30	2	negcli 6526	. . . . . . . . . . . . . 14 |- -uA e. CC
31	3, 7	pm3.2i 307	. . . . . . . . . . . . . 14 |- ((1 - A) e. CC /\ (1 - A) =/= 0)
32		div23 6925	. . . . . . . . . . . . . 14 |- ((-uA e. CC /\ (F` n) e. CC /\ ((1 - A) e. CC /\ (1 - A) =/= 0)) -> ((-uA x. (F` n)) / (1 - A)) = ((-uA / (1 - A)) x. (F` n)))
33	30, 31, 32	mp3an13 1182	. . . . . . . . . . . . 13 |- ((F` n) e. CC -> ((-uA x. (F` n)) / (1 - A)) = ((-uA / (1 - A)) x. (F` n)))
34	28, 33	syl 12	. . . . . . . . . . . 12 |- (n e. NN0 -> ((-uA x. (F` n)) / (1 - A)) = ((-uA / (1 - A)) x. (F` n)))
35		mulneg1 6615	. . . . . . . . . . . . . 14 |- ((A e. CC /\ (F` n) e. CC) -> (-uA x. (F` n)) = -u(A x. (F` n)))
36	35, 2, 28	sylancr 526	. . . . . . . . . . . . 13 |- (n e. NN0 -> (-uA x. (F` n)) = -u(A x. (F` n)))
37	36	opreq1d 4897	. . . . . . . . . . . 12 |- (n e. NN0 -> ((-uA x. (F` n)) / (1 - A)) = (-u(A x. (F` n)) / (1 - A)))
38		mulneg12 6617	. . . . . . . . . . . . . . . . 17 |- (((1 / (1 - A)) e. CC /\ A e. CC) -> (-u(1 / (1 - A)) x. A) = ((1 / (1 - A)) x. -uA))
39	8, 2, 38	mp2an 761	. . . . . . . . . . . . . . . 16 |- (-u(1 / (1 - A)) x. A) = ((1 / (1 - A)) x. -uA)
40		divrec2 6923	. . . . . . . . . . . . . . . . 17 |- ((-uA e. CC /\ (1 - A) e. CC /\ (1 - A) =/= 0) -> (-uA / (1 - A)) = ((1 / (1 - A)) x. -uA))
41	30, 3, 7, 40	mp3an 1191	. . . . . . . . . . . . . . . 16 |- (-uA / (1 - A)) = ((1 / (1 - A)) x. -uA)
42	39, 41	eqtr4i 1911	. . . . . . . . . . . . . . 15 |- (-u(1 / (1 - A)) x. A) = (-uA / (1 - A))
43	42	opreq1i 4892	. . . . . . . . . . . . . 14 |- ((-u(1 / (1 - A)) x. A) x. (F` n)) = ((-uA / (1 - A)) x. (F` n))
44	43	eqcomi 1888	. . . . . . . . . . . . 13 |- ((-uA / (1 - A)) x. (F` n)) = ((-u(1 / (1 - A)) x. A) x. (F` n))
45	44	a1i 8	. . . . . . . . . . . 12 |- (n e. NN0 -> ((-uA / (1 - A)) x. (F` n)) = ((-u(1 / (1 - A)) x. A) x. (F` n)))
46	34, 37, 45	3eqtr3d 1934	. . . . . . . . . . 11 |- (n e. NN0 -> (-u(A x. (F` n)) / (1 - A)) = ((-u(1 / (1 - A)) x. A) x. (F` n)))
47	29, 46	eqtrd 1925	. . . . . . . . . 10 |- (n e. NN0 -> (G` n) = ((-u(1 / (1 - A)) x. A) x. (F` n)))
48	28, 47	jca 310	. . . . . . . . 9 |- (n e. NN0 -> ((F` n) e. CC /\ (G` n) = ((-u(1 / (1 - A)) x. A) x. (F` n))))
49	24, 48	sylbir 218	. . . . . . . 8 |- (n e. (ZZ>=` 0) -> ((F` n) e. CC /\ (G` n) = ((-u(1 / (1 - A)) x. A) x. (F` n))))
50	49	rgen 2159	. . . . . . 7 |- A.n e. (ZZ>=` 0)((F` n) e. CC /\ (G` n) = ((-u(1 / (1 - A)) x. A) x. (F` n)))
51	23, 50	pm3.2i 307	. . . . . 6 |- (0 e. ZZ /\ A.n e. (ZZ>=` 0)((F` n) e. CC /\ (G` n) = ((-u(1 / (1 - A)) x. A) x. (F` n))))
52		geolimilem.4	. . . . . . . 8 |- G Fn NN0
53		fnex 4535	. . . . . . . 8 |- ((G Fn NN0 /\ NN0 e. _V) -> G e. _V)
54	52, 18, 53	mp2an 761	. . . . . . 7 |- G e. _V
55	23	elisseti 2301	. . . . . . 7 |- 0 e. _V
56	19, 54, 55	climmulc2 8389	. . . . . 6 |- ((((-u(1 / (1 - A)) x. A) e. CC /\ F ~~> 0) /\ (0 e. ZZ /\ A.n e. (ZZ>=` 0)((F` n) e. CC /\ (G` n) = ((-u(1 / (1 - A)) x. A) x. (F` n))))) -> G ~~> ((-u(1 / (1 - A)) x. A) x. 0))
57	22, 51, 56	mp2an 761	. . . . 5 |- G ~~> ((-u(1 / (1 - A)) x. A) x. 0)
58	10	mul01i 6594	. . . . 5 |- ((-u(1 / (1 - A)) x. A) x. 0) = 0
59	57, 58	breqtri 3360	. . . 4 |- G ~~> 0
60	59, 8	pm3.2i 307	. . 3 |- (G ~~> 0 /\ (1 / (1 - A)) e. CC)
61		mulcl 6456	. . . . . . . . . . 11 |- ((A e. CC /\ (F` n) e. CC) -> (A x. (F` n)) e. CC)
62	61, 2, 28	sylancr 526	. . . . . . . . . 10 |- (n e. NN0 -> (A x. (F` n)) e. CC)
63		negcl 6525	. . . . . . . . . 10 |- ((A x. (F` n)) e. CC -> -u(A x. (F` n)) e. CC)
64	62, 63	syl 12	. . . . . . . . 9 |- (n e. NN0 -> -u(A x. (F` n)) e. CC)
65		divcl 6901	. . . . . . . . . 10 |- ((-u(A x. (F` n)) e. CC /\ (1 - A) e. CC /\ (1 - A) =/= 0) -> (-u(A x. (F` n)) / (1 - A)) e. CC)
66	3, 7, 65	mp3an23 1183	. . . . . . . . 9 |- (-u(A x. (F` n)) e. CC -> (-u(A x. (F` n)) / (1 - A)) e. CC)
67	64, 66	syl 12	. . . . . . . 8 |- (n e. NN0 -> (-u(A x. (F` n)) / (1 - A)) e. CC)
68	29, 67	eqeltrd 1971	. . . . . . 7 |- (n e. NN0 -> (G` n) e. CC)
69		mulcom 6459	. . . . . . . . . . . . . 14 |- ((A e. CC /\ (A^n) e. CC) -> (A x. (A^n)) = ((A^n) x. A))
70		expcl 7824	. . . . . . . . . . . . . . 15 |- ((A e. CC /\ n e. NN0) -> (A^n) e. CC)
71	2, 70	mpan 759	. . . . . . . . . . . . . 14 |- (n e. NN0 -> (A^n) e. CC)
72	69, 2, 71	sylancr 526	. . . . . . . . . . . . 13 |- (n e. NN0 -> (A x. (A^n)) = ((A^n) x. A))
73	15	opreq2d 4898	. . . . . . . . . . . . 13 |- (n e. NN0 -> (A x. (F` n)) = (A x. (A^n)))
74		expp1 7817	. . . . . . . . . . . . . 14 |- ((A e. CC /\ n e. NN0) -> (A^(n + 1)) = ((A^n) x. A))
75	2, 74	mpan 759	. . . . . . . . . . . . 13 |- (n e. NN0 -> (A^(n + 1)) = ((A^n) x. A))
76	72, 73, 75	3eqtr4rd 1939	. . . . . . . . . . . 12 |- (n e. NN0 -> (A^(n + 1)) = (A x. (F` n)))
77	76	opreq2d 4898	. . . . . . . . . . 11 |- (n e. NN0 -> (1 - (A^(n + 1))) = (1 - (A x. (F` n))))
78		negsub 6540	. . . . . . . . . . . 12 |- ((1 e. CC /\ (A x. (F` n)) e. CC) -> (1 + -u(A x. (F` n))) = (1 - (A x. (F` n))))
79	78, 1, 62	sylancr 526	. . . . . . . . . . 11 |- (n e. NN0 -> (1 + -u(A x. (F` n))) = (1 - (A x. (F` n))))
80	77, 79	eqtr4d 1928	. . . . . . . . . 10 |- (n e. NN0 -> (1 - (A^(n + 1))) = (1 + -u(A x. (F` n))))
81	80	opreq1d 4897	. . . . . . . . 9 |- (n e. NN0 -> ((1 - (A^(n + 1))) / (1 - A)) = ((1 + -u(A x. (F` n))) / (1 - A)))
82		divdir 6933	. . . . . . . . . . 11 |- ((1 e. CC /\ -u(A x. (F` n)) e. CC /\ ((1 - A) e. CC /\ (1 - A) =/= 0)) -> ((1 + -u(A x. (F` n))) / (1 - A)) = ((1 / (1 - A)) + (-u(A x. (F` n)) / (1 - A))))
83	1, 31, 82	mp3an13 1182	. . . . . . . . . 10 |- (-u(A x. (F` n)) e. CC -> ((1 + -u(A x. (F` n))) / (1 - A)) = ((1 / (1 - A)) + (-u(A x. (F` n)) / (1 - A))))
84	64, 83	syl 12	. . . . . . . . 9 |- (n e. NN0 -> ((1 + -u(A x. (F` n))) / (1 - A)) = ((1 / (1 - A)) + (-u(A x. (F` n)) / (1 - A))))
85	81, 84	eqtrd 1925	. . . . . . . 8 |- (n e. NN0 -> ((1 - (A^(n + 1))) / (1 - A)) = ((1 / (1 - A)) + (-u(A x. (F` n)) / (1 - A))))
86	1, 2	subeq0i 6565	. . . . . . . . . . . 12 |- ((1 - A) = 0 <-> 1 = A)
87	86	necon3bii 2032	. . . . . . . . . . 11 |- ((1 - A) =/= 0 <-> 1 =/= A)
88	7, 87	mpbi 206	. . . . . . . . . 10 |- 1 =/= A
89		necom 2094	. . . . . . . . . 10 |- (1 =/= A <-> A =/= 1)
90	88, 89	mpbi 206	. . . . . . . . 9 |- A =/= 1
91	13, 2	geoseri 8496	. . . . . . . . 9 |- ((n e. NN0 /\ A =/= 1) -> (( + seq0 F)` n) = ((1 - (A^(n + 1))) / (1 - A)))
92	90, 91	mpan2 760	. . . . . . . 8 |- (n e. NN0 -> (( + seq0 F)` n) = ((1 - (A^(n + 1))) / (1 - A)))
93	29	opreq2d 4898	. . . . . . . 8 |- (n e. NN0 -> ((1 / (1 - A)) + (G` n)) = ((1 / (1 - A)) + (-u(A x. (F` n)) / (1 - A))))
94	85, 92, 93	3eqtr4d 1937	. . . . . . 7 |- (n e. NN0 -> (( + seq0 F)` n) = ((1 / (1 - A)) + (G` n)))
95	68, 94	jca 310	. . . . . 6 |- (n e. NN0 -> ((G` n) e. CC /\ (( + seq0 F)` n) = ((1 / (1 - A)) + (G` n))))
96	24, 95	sylbir 218	. . . . 5 |- (n e. (ZZ>=` 0) -> ((G` n) e. CC /\ (( + seq0 F)` n) = ((1 / (1 - A)) + (G` n))))
97	96	rgen 2159	. . . 4 |- A.n e. (ZZ>=` 0)((G` n) e. CC /\ (( + seq0 F)` n) = ((1 / (1 - A)) + (G` n)))
98	23, 97	pm3.2i 307	. . 3 |- (0 e. ZZ /\ A.n e. (ZZ>=` 0)((G` n) e. CC /\ (( + seq0 F)` n) = ((1 / (1 - A)) + (G` n))))
99		oprex 4907	. . . 4 |- ( + seq0 F) e. _V
100		oprex 4907	. . . 4 |- (1 / (1 - A)) e. _V
101	54, 99, 55, 100	climaddc2 8379	. . 3 |- (((G ~~> 0 /\ (1 / (1 - A)) e. CC) /\ (0 e. ZZ /\ A.n e. (ZZ>=` 0)((G` n) e. CC /\ (( + seq0 F)` n) = ((1 / (1 - A)) + (G` n))))) -> ( + seq0 F) ~~> ((1 / (1 - A)) + 0))
102	60, 98, 101	mp2an 761	. 2 |- ( + seq0 F) ~~> ((1 / (1 - A)) + 0)
103	8	addid1i 6483	. 2 |- ((1 / (1 - A)) + 0) = (1 / (1 - A))
104	102, 103	breqtri 3360	1 |- ( + seq0 F) ~~> (1 / (1 - A))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  _Vcvv 2292   class class class wbr 3338  {copab 3395   Fn wfn 3993  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   x. cmul 6391   - cmin 6445  -ucneg 6446   / cdiv 6447  NNcn 6449  NN0cn0 6450  ZZcz 6451   < clt 6653  ZZ>=cuz 7586   seq0 cseq0 7775  ^cexp 7811  abscabs 8000   ~~> cli 8234

This theorem is referenced by:  geolimi 8498

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-n0 7309  df-z 7345  df-fl 7463  df-uz 7587  df-seq1 7721  df-shft 7754  df-seq0 7777  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-clim 8235

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