Theorem cvgratnnlemrate 11292

Description: Lemma for cvgratnn 11293. (Contributed by Jim Kingdon, 21-Nov-2022.)

Hypotheses

Ref	Expression
cvgratnn.3	|- ( ph -> A e. RR )
cvgratnn.4	|- ( ph -> A < 1 )
cvgratnn.gt0	|- ( ph -> 0 < A )
cvgratnn.6	|- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )
cvgratnn.7	|- ( ( ph /\ k e. NN ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )
cvgratnnlemrate.m	|- ( ph -> M e. NN )
cvgratnnlemrate.n	|- ( ph -> N e. ( ZZ>= ` M ) )

Assertion

Ref	Expression
cvgratnnlemrate	|- ( ph -> ( abs ` ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) ) < ( ( ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) x. ( A / ( 1 - A ) ) ) / M ) )

Distinct variable groups:   A, k   k, F   k, N   ph, k   k, M

Proof of Theorem cvgratnnlemrate

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnuz 9354	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
2		1zzd 9074	. . . . . . 7 |- ( ph -> 1 e. ZZ )
3		cvgratnn.6	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )
4	1, 2, 3	serf 10240	. . . . . 6 |- ( ph -> seq 1 ( + , F ) : NN --> CC )
5		cvgratnnlemrate.m	. . . . . . 7 |- ( ph -> M e. NN )
6		cvgratnnlemrate.n	. . . . . . 7 |- ( ph -> N e. ( ZZ>= ` M ) )
7		eluznn 9387	. . . . . . 7 |- ( ( M e. NN /\ N e. ( ZZ>= ` M ) ) -> N e. NN )
8	5, 6, 7	syl2anc 408	. . . . . 6 |- ( ph -> N e. NN )
9	4, 8	ffvelrnd 5549	. . . . 5 |- ( ph -> ( seq 1 ( + , F ) ` N ) e. CC )
10	4, 5	ffvelrnd 5549	. . . . 5 |- ( ph -> ( seq 1 ( + , F ) ` M ) e. CC )
11	9, 10	subcld 8066	. . . 4 |- ( ph -> ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) e. CC )
12	11	abscld 10946	. . 3 |- ( ph -> ( abs ` ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) ) e. RR )
13		fveq2 5414	. . . . . . 7 |- ( k = M -> ( F ` k ) = ( F ` M ) )
14	13	eleq1d 2206	. . . . . 6 |- ( k = M -> ( ( F ` k ) e. CC <-> ( F ` M ) e. CC ) )
15	3	ralrimiva 2503	. . . . . 6 |- ( ph -> A. k e. NN ( F ` k ) e. CC )
16	14, 15, 5	rspcdva 2789	. . . . 5 |- ( ph -> ( F ` M ) e. CC )
17	16	abscld 10946	. . . 4 |- ( ph -> ( abs ` ( F ` M ) ) e. RR )
18	5	nnzd 9165	. . . . . . 7 |- ( ph -> M e. ZZ )
19	18	peano2zd 9169	. . . . . 6 |- ( ph -> ( M + 1 ) e. ZZ )
20		eluzelz 9328	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
21	6, 20	syl 14	. . . . . 6 |- ( ph -> N e. ZZ )
22	19, 21	fzfigd 10197	. . . . 5 |- ( ph -> ( ( M + 1 ) ... N ) e. Fin )
23		cvgratnn.3	. . . . . . 7 |- ( ph -> A e. RR )
24	23	adantr 274	. . . . . 6 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> A e. RR )
25	5	nnred 8726	. . . . . . . . 9 |- ( ph -> M e. RR )
26	25	adantr 274	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> M e. RR )
27		peano2re 7891	. . . . . . . . 9 |- ( M e. RR -> ( M + 1 ) e. RR )
28	26, 27	syl 14	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( M + 1 ) e. RR )
29		elfzelz 9799	. . . . . . . . . 10 |- ( i e. ( ( M + 1 ) ... N ) -> i e. ZZ )
30	29	adantl 275	. . . . . . . . 9 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> i e. ZZ )
31	30	zred 9166	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> i e. RR )
32	26	lep1d 8682	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> M <_ ( M + 1 ) )
33		elfzle1 9800	. . . . . . . . 9 |- ( i e. ( ( M + 1 ) ... N ) -> ( M + 1 ) <_ i )
34	33	adantl 275	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( M + 1 ) <_ i )
35	26, 28, 31, 32, 34	letrd 7879	. . . . . . 7 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> M <_ i )
36		znn0sub 9112	. . . . . . . 8 |- ( ( M e. ZZ /\ i e. ZZ ) -> ( M <_ i <-> ( i - M ) e. NN0 ) )
37	18, 29, 36	syl2an 287	. . . . . . 7 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( M <_ i <-> ( i - M ) e. NN0 ) )
38	35, 37	mpbid 146	. . . . . 6 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( i - M ) e. NN0 )
39	24, 38	reexpcld 10434	. . . . 5 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( A ^ ( i - M ) ) e. RR )
40	22, 39	fsumrecl 11163	. . . 4 |- ( ph -> sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) e. RR )
41	17, 40	remulcld 7789	. . 3 |- ( ph -> ( ( abs ` ( F ` M ) ) x. sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) ) e. RR )
42		cvgratnn.4	. . . . . . . . . . 11 |- ( ph -> A < 1 )
43		cvgratnn.gt0	. . . . . . . . . . . . 13 |- ( ph -> 0 < A )
44	23, 43	elrpd 9474	. . . . . . . . . . . 12 |- ( ph -> A e. RR+ )
45	44	reclt1d 9490	. . . . . . . . . . 11 |- ( ph -> ( A < 1 <-> 1 < ( 1 / A ) ) )
46	42, 45	mpbid 146	. . . . . . . . . 10 |- ( ph -> 1 < ( 1 / A ) )
47		1re 7758	. . . . . . . . . . 11 |- 1 e. RR
48	44	rprecred 9488	. . . . . . . . . . 11 |- ( ph -> ( 1 / A ) e. RR )
49		difrp 9473	. . . . . . . . . . 11 |- ( ( 1 e. RR /\ ( 1 / A ) e. RR ) -> ( 1 < ( 1 / A ) <-> ( ( 1 / A ) - 1 ) e. RR+ ) )
50	47, 48, 49	sylancr 410	. . . . . . . . . 10 |- ( ph -> ( 1 < ( 1 / A ) <-> ( ( 1 / A ) - 1 ) e. RR+ ) )
51	46, 50	mpbid 146	. . . . . . . . 9 |- ( ph -> ( ( 1 / A ) - 1 ) e. RR+ )
52	51	rpreccld 9487	. . . . . . . 8 |- ( ph -> ( 1 / ( ( 1 / A ) - 1 ) ) e. RR+ )
53	52, 44	rpdivcld 9494	. . . . . . 7 |- ( ph -> ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) e. RR+ )
54		fveq2 5414	. . . . . . . . . . 11 |- ( k = 1 -> ( F ` k ) = ( F ` 1 ) )
55	54	eleq1d 2206	. . . . . . . . . 10 |- ( k = 1 -> ( ( F ` k ) e. CC <-> ( F ` 1 ) e. CC ) )
56		1nn 8724	. . . . . . . . . . 11 |- 1 e. NN
57	56	a1i 9	. . . . . . . . . 10 |- ( ph -> 1 e. NN )
58	55, 15, 57	rspcdva 2789	. . . . . . . . 9 |- ( ph -> ( F ` 1 ) e. CC )
59	58	abscld 10946	. . . . . . . 8 |- ( ph -> ( abs ` ( F ` 1 ) ) e. RR )
60	58	absge0d 10949	. . . . . . . 8 |- ( ph -> 0 <_ ( abs ` ( F ` 1 ) ) )
61	59, 60	ge0p1rpd 9507	. . . . . . 7 |- ( ph -> ( ( abs ` ( F ` 1 ) ) + 1 ) e. RR+ )
62	53, 61	rpmulcld 9493	. . . . . 6 |- ( ph -> ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) e. RR+ )
63	62	rpred 9476	. . . . 5 |- ( ph -> ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) e. RR )
64	63, 5	nndivred 8763	. . . 4 |- ( ph -> ( ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) / M ) e. RR )
65		1red 7774	. . . . . . . 8 |- ( ph -> 1 e. RR )
66	65, 23	resubcld 8136	. . . . . . 7 |- ( ph -> ( 1 - A ) e. RR )
67	23, 65	posdifd 8287	. . . . . . . 8 |- ( ph -> ( A < 1 <-> 0 < ( 1 - A ) ) )
68	42, 67	mpbid 146	. . . . . . 7 |- ( ph -> 0 < ( 1 - A ) )
69	66, 68	elrpd 9474	. . . . . 6 |- ( ph -> ( 1 - A ) e. RR+ )
70	44, 69	rpdivcld 9494	. . . . 5 |- ( ph -> ( A / ( 1 - A ) ) e. RR+ )
71	70	rpred 9476	. . . 4 |- ( ph -> ( A / ( 1 - A ) ) e. RR )
72	64, 71	remulcld 7789	. . 3 |- ( ph -> ( ( ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) / M ) x. ( A / ( 1 - A ) ) ) e. RR )
73		cvgratnn.7	. . . . . 6 |- ( ( ph /\ k e. NN ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )
74	23, 42, 43, 3, 73, 5, 6	cvgratnnlemseq 11288	. . . . 5 |- ( ph -> ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) = sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) )
75	74	fveq2d 5418	. . . 4 |- ( ph -> ( abs ` ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) ) = ( abs ` sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) ) )
76	23, 42, 43, 3, 73, 5, 6	cvgratnnlemabsle 11289	. . . 4 |- ( ph -> ( abs ` sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) ) <_ ( ( abs ` ( F ` M ) ) x. sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) ) )
77	75, 76	eqbrtrd 3945	. . 3 |- ( ph -> ( abs ` ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) ) <_ ( ( abs ` ( F ` M ) ) x. sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) ) )
78	16	absge0d 10949	. . . 4 |- ( ph -> 0 <_ ( abs ` ( F ` M ) ) )
79	23, 42, 43, 3, 73, 5	cvgratnnlemfm 11291	. . . 4 |- ( ph -> ( abs ` ( F ` M ) ) < ( ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) / M ) )
80	44	adantr 274	. . . . . . 7 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> A e. RR+ )
81	38	nn0zd 9164	. . . . . . 7 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( i - M ) e. ZZ )
82	80, 81	rpexpcld 10441	. . . . . 6 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( A ^ ( i - M ) ) e. RR+ )
83	82	rpge0d 9480	. . . . 5 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> 0 <_ ( A ^ ( i - M ) ) )
84	22, 39, 83	fsumge0 11221	. . . 4 |- ( ph -> 0 <_ sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) )
85	23, 42, 43, 3, 73, 5, 6	cvgratnnlemsumlt 11290	. . . 4 |- ( ph -> sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) < ( A / ( 1 - A ) ) )
86	17, 64, 40, 71, 78, 79, 84, 85	ltmul12ad 8692	. . 3 |- ( ph -> ( ( abs ` ( F ` M ) ) x. sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) ) < ( ( ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) / M ) x. ( A / ( 1 - A ) ) ) )
87	12, 41, 72, 77, 86	lelttrd 7880	. 2 |- ( ph -> ( abs ` ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) ) < ( ( ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) / M ) x. ( A / ( 1 - A ) ) ) )
88	63	recnd 7787	. . 3 |- ( ph -> ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) e. CC )
89	71	recnd 7787	. . 3 |- ( ph -> ( A / ( 1 - A ) ) e. CC )
90	5	nncnd 8727	. . 3 |- ( ph -> M e. CC )
91	5	nnap0d 8759	. . 3 |- ( ph -> M # 0 )
92	88, 89, 90, 91	div23apd 8581	. 2 |- ( ph -> ( ( ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) x. ( A / ( 1 - A ) ) ) / M ) = ( ( ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) / M ) x. ( A / ( 1 - A ) ) ) )
93	87, 92	breqtrrd 3951	1 |- ( ph -> ( abs ` ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) ) < ( ( ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) x. ( A / ( 1 - A ) ) ) / M ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   class class class wbr 3924   ` cfv 5118  (class class class)co 5767   CCcc 7611   RRcr 7612   0cc0 7613   1c1 7614   + caddc 7616   x. cmul 7618   < clt 7793   <_ cle 7794   - cmin 7926   / cdiv 8425   NNcn 8713   NN0cn0 8970   ZZcz 9047   ZZ>=cuz 9319   RR+crp 9434   ...cfz 9783   seqcseq 10211   ^cexp 10285   abscabs 10762   sum_csu 11115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732  ax-caucvg 7733

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-isom 5127  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-frec 6281  df-1o 6306  df-oadd 6310  df-er 6422  df-en 6628  df-dom 6629  df-fin 6630  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-3 8773  df-4 8774  df-n0 8971  df-z 9048  df-uz 9320  df-q 9405  df-rp 9435  df-ico 9670  df-fz 9784  df-fzo 9913  df-seqfrec 10212  df-exp 10286  df-ihash 10515  df-cj 10607  df-re 10608  df-im 10609  df-rsqrt 10763  df-abs 10764  df-clim 11041  df-sumdc 11116

This theorem is referenced by:  cvgratnn  11293