Theorem cvgratnnlemrate 10920

Description: Lemma for cvgratnn 10921. (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 9052	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
2		1zzd 8775	. . . . . . 7 |- ( ph -> 1 e. ZZ )
3		cvgratnn.6	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )
4	1, 2, 3	serf 9896	. . . . . 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 9085	. . . . . . 7 |- ( ( M e. NN /\ N e. ( ZZ>= ` M ) ) -> N e. NN )
8	5, 6, 7	syl2anc 403	. . . . . 6 |- ( ph -> N e. NN )
9	4, 8	ffvelrnd 5435	. . . . 5 |- ( ph -> ( seq 1 ( + , F ) ` N ) e. CC )
10	4, 5	ffvelrnd 5435	. . . . 5 |- ( ph -> ( seq 1 ( + , F ) ` M ) e. CC )
11	9, 10	subcld 7791	. . . 4 |- ( ph -> ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) e. CC )
12	11	abscld 10610	. . 3 |- ( ph -> ( abs ` ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) ) e. RR )
13		fveq2 5305	. . . . . . 7 |- ( k = M -> ( F ` k ) = ( F ` M ) )
14	13	eleq1d 2156	. . . . . 6 |- ( k = M -> ( ( F ` k ) e. CC <-> ( F ` M ) e. CC ) )
15	3	ralrimiva 2446	. . . . . 6 |- ( ph -> A. k e. NN ( F ` k ) e. CC )
16	14, 15, 5	rspcdva 2727	. . . . 5 |- ( ph -> ( F ` M ) e. CC )
17	16	abscld 10610	. . . 4 |- ( ph -> ( abs ` ( F ` M ) ) e. RR )
18	5	nnzd 8865	. . . . . . 7 |- ( ph -> M e. ZZ )
19	18	peano2zd 8869	. . . . . 6 |- ( ph -> ( M + 1 ) e. ZZ )
20		eluzelz 9026	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
21	6, 20	syl 14	. . . . . 6 |- ( ph -> N e. ZZ )
22	19, 21	fzfigd 9834	. . . . 5 |- ( ph -> ( ( M + 1 ) ... N ) e. Fin )
23		cvgratnn.3	. . . . . . 7 |- ( ph -> A e. RR )
24	23	adantr 270	. . . . . 6 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> A e. RR )
25	5	nnred 8433	. . . . . . . . 9 |- ( ph -> M e. RR )
26	25	adantr 270	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> M e. RR )
27		peano2re 7616	. . . . . . . . 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 9438	. . . . . . . . . 10 |- ( i e. ( ( M + 1 ) ... N ) -> i e. ZZ )
30	29	adantl 271	. . . . . . . . 9 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> i e. ZZ )
31	30	zred 8866	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> i e. RR )
32	26	lep1d 8390	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> M <_ ( M + 1 ) )
33		elfzle1 9439	. . . . . . . . 9 |- ( i e. ( ( M + 1 ) ... N ) -> ( M + 1 ) <_ i )
34	33	adantl 271	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( M + 1 ) <_ i )
35	26, 28, 31, 32, 34	letrd 7605	. . . . . . 7 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> M <_ i )
36		znn0sub 8813	. . . . . . . 8 |- ( ( M e. ZZ /\ i e. ZZ ) -> ( M <_ i <-> ( i - M ) e. NN0 ) )
37	18, 29, 36	syl2an 283	. . . . . . 7 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( M <_ i <-> ( i - M ) e. NN0 ) )
38	35, 37	mpbid 145	. . . . . 6 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( i - M ) e. NN0 )
39	24, 38	reexpcld 10099	. . . . 5 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( A ^ ( i - M ) ) e. RR )
40	22, 39	fsumrecl 10791	. . . 4 |- ( ph -> sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) e. RR )
41	17, 40	remulcld 7516	. . 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 9169	. . . . . . . . . . . 12 |- ( ph -> A e. RR+ )
45	44	reclt1d 9185	. . . . . . . . . . 11 |- ( ph -> ( A < 1 <-> 1 < ( 1 / A ) ) )
46	42, 45	mpbid 145	. . . . . . . . . 10 |- ( ph -> 1 < ( 1 / A ) )
47		1re 7485	. . . . . . . . . . 11 |- 1 e. RR
48	44	rprecred 9183	. . . . . . . . . . 11 |- ( ph -> ( 1 / A ) e. RR )
49		difrp 9168	. . . . . . . . . . 11 |- ( ( 1 e. RR /\ ( 1 / A ) e. RR ) -> ( 1 < ( 1 / A ) <-> ( ( 1 / A ) - 1 ) e. RR+ ) )
50	47, 48, 49	sylancr 405	. . . . . . . . . 10 |- ( ph -> ( 1 < ( 1 / A ) <-> ( ( 1 / A ) - 1 ) e. RR+ ) )
51	46, 50	mpbid 145	. . . . . . . . 9 |- ( ph -> ( ( 1 / A ) - 1 ) e. RR+ )
52	51	rpreccld 9182	. . . . . . . 8 |- ( ph -> ( 1 / ( ( 1 / A ) - 1 ) ) e. RR+ )
53	52, 44	rpdivcld 9189	. . . . . . 7 |- ( ph -> ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) e. RR+ )
54		fveq2 5305	. . . . . . . . . . 11 |- ( k = 1 -> ( F ` k ) = ( F ` 1 ) )
55	54	eleq1d 2156	. . . . . . . . . 10 |- ( k = 1 -> ( ( F ` k ) e. CC <-> ( F ` 1 ) e. CC ) )
56		1nn 8431	. . . . . . . . . . 11 |- 1 e. NN
57	56	a1i 9	. . . . . . . . . 10 |- ( ph -> 1 e. NN )
58	55, 15, 57	rspcdva 2727	. . . . . . . . 9 |- ( ph -> ( F ` 1 ) e. CC )
59	58	abscld 10610	. . . . . . . 8 |- ( ph -> ( abs ` ( F ` 1 ) ) e. RR )
60	58	absge0d 10613	. . . . . . . 8 |- ( ph -> 0 <_ ( abs ` ( F ` 1 ) ) )
61	59, 60	ge0p1rpd 9202	. . . . . . 7 |- ( ph -> ( ( abs ` ( F ` 1 ) ) + 1 ) e. RR+ )
62	53, 61	rpmulcld 9188	. . . . . 6 |- ( ph -> ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) e. RR+ )
63	62	rpred 9171	. . . . 5 |- ( ph -> ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) e. RR )
64	63, 5	nndivred 8470	. . . 4 |- ( ph -> ( ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) / M ) e. RR )
65		1red 7501	. . . . . . . 8 |- ( ph -> 1 e. RR )
66	65, 23	resubcld 7857	. . . . . . 7 |- ( ph -> ( 1 - A ) e. RR )
67	23, 65	posdifd 8007	. . . . . . . 8 |- ( ph -> ( A < 1 <-> 0 < ( 1 - A ) ) )
68	42, 67	mpbid 145	. . . . . . 7 |- ( ph -> 0 < ( 1 - A ) )
69	66, 68	elrpd 9169	. . . . . 6 |- ( ph -> ( 1 - A ) e. RR+ )
70	44, 69	rpdivcld 9189	. . . . 5 |- ( ph -> ( A / ( 1 - A ) ) e. RR+ )
71	70	rpred 9171	. . . 4 |- ( ph -> ( A / ( 1 - A ) ) e. RR )
72	64, 71	remulcld 7516	. . 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 10916	. . . . 5 |- ( ph -> ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) = sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) )
75	74	fveq2d 5309	. . . 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 10917	. . . 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 3865	. . 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 10613	. . . 4 |- ( ph -> 0 <_ ( abs ` ( F ` M ) ) )
79	23, 42, 43, 3, 73, 5	cvgratnnlemfm 10919	. . . 4 |- ( ph -> ( abs ` ( F ` M ) ) < ( ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) / M ) )
80	44	adantr 270	. . . . . . 7 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> A e. RR+ )
81	38	nn0zd 8864	. . . . . . 7 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( i - M ) e. ZZ )
82	80, 81	rpexpcld 10106	. . . . . 6 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( A ^ ( i - M ) ) e. RR+ )
83	82	rpge0d 9175	. . . . 5 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> 0 <_ ( A ^ ( i - M ) ) )
84	22, 39, 83	fsumge0 10849	. . . 4 |- ( ph -> 0 <_ sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) )
85	23, 42, 43, 3, 73, 5, 6	cvgratnnlemsumlt 10918	. . . 4 |- ( ph -> sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) < ( A / ( 1 - A ) ) )
86	17, 64, 40, 71, 78, 79, 84, 85	ltmul12ad 8400	. . 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 7606	. 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 7514	. . 3 |- ( ph -> ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) e. CC )
89	71	recnd 7514	. . 3 |- ( ph -> ( A / ( 1 - A ) ) e. CC )
90	5	nncnd 8434	. . 3 |- ( ph -> M e. CC )
91	5	nnap0d 8466	. . 3 |- ( ph -> M # 0 )
92	88, 89, 90, 91	div23apd 8293	. 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 3871	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 102   <-> wb 103   = wceq 1289   e. wcel 1438   class class class wbr 3845   ` cfv 5015  (class class class)co 5652   CCcc 7346   RRcr 7347   0cc0 7348   1c1 7349   + caddc 7351   x. cmul 7353   < clt 7520   <_ cle 7521   - cmin 7651   / cdiv 8137   NNcn 8420   NN0cn0 8671   ZZcz 8748   ZZ>=cuz 9017   RR+crp 9132   ...cfz 9422   seqcseq 9848   ^cexp 9950   abscabs 10426   sum_csu 10738

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-mulrcl 7442  ax-addcom 7443  ax-mulcom 7444  ax-addass 7445  ax-mulass 7446  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-1rid 7450  ax-0id 7451  ax-rnegex 7452  ax-precex 7453  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-apti 7458  ax-pre-ltadd 7459  ax-pre-mulgt0 7460  ax-pre-mulext 7461  ax-arch 7462  ax-caucvg 7463

This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-isom 5024  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-frec 6156  df-1o 6181  df-oadd 6185  df-er 6290  df-en 6456  df-dom 6457  df-fin 6458  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-reap 8050  df-ap 8057  df-div 8138  df-inn 8421  df-2 8479  df-3 8480  df-4 8481  df-n0 8672  df-z 8749  df-uz 9018  df-q 9103  df-rp 9133  df-ico 9310  df-fz 9423  df-fzo 9550  df-iseq 9849  df-seq3 9850  df-exp 9951  df-ihash 10180  df-cj 10272  df-re 10273  df-im 10274  df-rsqrt 10427  df-abs 10428  df-clim 10663  df-isum 10739

This theorem is referenced by:  cvgratnn  10921