Theorem cvgratnnlemrate 12090

Description: Lemma for cvgratnn 12091. (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 9791	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
2		1zzd 9505	. . . . . . 7 |- ( ph -> 1 e. ZZ )
3		cvgratnn.6	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )
4	1, 2, 3	serf 10744	. . . . . 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 9833	. . . . . . 7 |- ( ( M e. NN /\ N e. ( ZZ>= ` M ) ) -> N e. NN )
8	5, 6, 7	syl2anc 411	. . . . . 6 |- ( ph -> N e. NN )
9	4, 8	ffvelcdmd 5783	. . . . 5 |- ( ph -> ( seq 1 ( + , F ) ` N ) e. CC )
10	4, 5	ffvelcdmd 5783	. . . . 5 |- ( ph -> ( seq 1 ( + , F ) ` M ) e. CC )
11	9, 10	subcld 8489	. . . 4 |- ( ph -> ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) e. CC )
12	11	abscld 11741	. . 3 |- ( ph -> ( abs ` ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) ) e. RR )
13		fveq2 5639	. . . . . . 7 |- ( k = M -> ( F ` k ) = ( F ` M ) )
14	13	eleq1d 2300	. . . . . 6 |- ( k = M -> ( ( F ` k ) e. CC <-> ( F ` M ) e. CC ) )
15	3	ralrimiva 2605	. . . . . 6 |- ( ph -> A. k e. NN ( F ` k ) e. CC )
16	14, 15, 5	rspcdva 2915	. . . . 5 |- ( ph -> ( F ` M ) e. CC )
17	16	abscld 11741	. . . 4 |- ( ph -> ( abs ` ( F ` M ) ) e. RR )
18	5	nnzd 9600	. . . . . . 7 |- ( ph -> M e. ZZ )
19	18	peano2zd 9604	. . . . . 6 |- ( ph -> ( M + 1 ) e. ZZ )
20		eluzelz 9764	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
21	6, 20	syl 14	. . . . . 6 |- ( ph -> N e. ZZ )
22	19, 21	fzfigd 10692	. . . . 5 |- ( ph -> ( ( M + 1 ) ... N ) e. Fin )
23		cvgratnn.3	. . . . . . 7 |- ( ph -> A e. RR )
24	23	adantr 276	. . . . . 6 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> A e. RR )
25	5	nnred 9155	. . . . . . . . 9 |- ( ph -> M e. RR )
26	25	adantr 276	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> M e. RR )
27		peano2re 8314	. . . . . . . . 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 10259	. . . . . . . . . 10 |- ( i e. ( ( M + 1 ) ... N ) -> i e. ZZ )
30	29	adantl 277	. . . . . . . . 9 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> i e. ZZ )
31	30	zred 9601	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> i e. RR )
32	26	lep1d 9110	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> M <_ ( M + 1 ) )
33		elfzle1 10261	. . . . . . . . 9 |- ( i e. ( ( M + 1 ) ... N ) -> ( M + 1 ) <_ i )
34	33	adantl 277	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( M + 1 ) <_ i )
35	26, 28, 31, 32, 34	letrd 8302	. . . . . . 7 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> M <_ i )
36		znn0sub 9544	. . . . . . . 8 |- ( ( M e. ZZ /\ i e. ZZ ) -> ( M <_ i <-> ( i - M ) e. NN0 ) )
37	18, 29, 36	syl2an 289	. . . . . . 7 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( M <_ i <-> ( i - M ) e. NN0 ) )
38	35, 37	mpbid 147	. . . . . 6 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( i - M ) e. NN0 )
39	24, 38	reexpcld 10951	. . . . 5 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( A ^ ( i - M ) ) e. RR )
40	22, 39	fsumrecl 11961	. . . 4 |- ( ph -> sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) e. RR )
41	17, 40	remulcld 8209	. . 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 9927	. . . . . . . . . . . 12 |- ( ph -> A e. RR+ )
45	44	reclt1d 9944	. . . . . . . . . . 11 |- ( ph -> ( A < 1 <-> 1 < ( 1 / A ) ) )
46	42, 45	mpbid 147	. . . . . . . . . 10 |- ( ph -> 1 < ( 1 / A ) )
47		1re 8177	. . . . . . . . . . 11 |- 1 e. RR
48	44	rprecred 9942	. . . . . . . . . . 11 |- ( ph -> ( 1 / A ) e. RR )
49		difrp 9926	. . . . . . . . . . 11 |- ( ( 1 e. RR /\ ( 1 / A ) e. RR ) -> ( 1 < ( 1 / A ) <-> ( ( 1 / A ) - 1 ) e. RR+ ) )
50	47, 48, 49	sylancr 414	. . . . . . . . . 10 |- ( ph -> ( 1 < ( 1 / A ) <-> ( ( 1 / A ) - 1 ) e. RR+ ) )
51	46, 50	mpbid 147	. . . . . . . . 9 |- ( ph -> ( ( 1 / A ) - 1 ) e. RR+ )
52	51	rpreccld 9941	. . . . . . . 8 |- ( ph -> ( 1 / ( ( 1 / A ) - 1 ) ) e. RR+ )
53	52, 44	rpdivcld 9948	. . . . . . 7 |- ( ph -> ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) e. RR+ )
54		fveq2 5639	. . . . . . . . . . 11 |- ( k = 1 -> ( F ` k ) = ( F ` 1 ) )
55	54	eleq1d 2300	. . . . . . . . . 10 |- ( k = 1 -> ( ( F ` k ) e. CC <-> ( F ` 1 ) e. CC ) )
56		1nn 9153	. . . . . . . . . . 11 |- 1 e. NN
57	56	a1i 9	. . . . . . . . . 10 |- ( ph -> 1 e. NN )
58	55, 15, 57	rspcdva 2915	. . . . . . . . 9 |- ( ph -> ( F ` 1 ) e. CC )
59	58	abscld 11741	. . . . . . . 8 |- ( ph -> ( abs ` ( F ` 1 ) ) e. RR )
60	58	absge0d 11744	. . . . . . . 8 |- ( ph -> 0 <_ ( abs ` ( F ` 1 ) ) )
61	59, 60	ge0p1rpd 9961	. . . . . . 7 |- ( ph -> ( ( abs ` ( F ` 1 ) ) + 1 ) e. RR+ )
62	53, 61	rpmulcld 9947	. . . . . 6 |- ( ph -> ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) e. RR+ )
63	62	rpred 9930	. . . . 5 |- ( ph -> ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) e. RR )
64	63, 5	nndivred 9192	. . . 4 |- ( ph -> ( ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) / M ) e. RR )
65		1red 8193	. . . . . . . 8 |- ( ph -> 1 e. RR )
66	65, 23	resubcld 8559	. . . . . . 7 |- ( ph -> ( 1 - A ) e. RR )
67	23, 65	posdifd 8711	. . . . . . . 8 |- ( ph -> ( A < 1 <-> 0 < ( 1 - A ) ) )
68	42, 67	mpbid 147	. . . . . . 7 |- ( ph -> 0 < ( 1 - A ) )
69	66, 68	elrpd 9927	. . . . . 6 |- ( ph -> ( 1 - A ) e. RR+ )
70	44, 69	rpdivcld 9948	. . . . 5 |- ( ph -> ( A / ( 1 - A ) ) e. RR+ )
71	70	rpred 9930	. . . 4 |- ( ph -> ( A / ( 1 - A ) ) e. RR )
72	64, 71	remulcld 8209	. . 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 12086	. . . . 5 |- ( ph -> ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) = sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) )
75	74	fveq2d 5643	. . . 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 12087	. . . 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 4110	. . 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 11744	. . . 4 |- ( ph -> 0 <_ ( abs ` ( F ` M ) ) )
79	23, 42, 43, 3, 73, 5	cvgratnnlemfm 12089	. . . 4 |- ( ph -> ( abs ` ( F ` M ) ) < ( ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) / M ) )
80	44	adantr 276	. . . . . . 7 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> A e. RR+ )
81	38	nn0zd 9599	. . . . . . 7 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( i - M ) e. ZZ )
82	80, 81	rpexpcld 10958	. . . . . 6 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( A ^ ( i - M ) ) e. RR+ )
83	82	rpge0d 9934	. . . . 5 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> 0 <_ ( A ^ ( i - M ) ) )
84	22, 39, 83	fsumge0 12019	. . . 4 |- ( ph -> 0 <_ sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) )
85	23, 42, 43, 3, 73, 5, 6	cvgratnnlemsumlt 12088	. . . 4 |- ( ph -> sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) < ( A / ( 1 - A ) ) )
86	17, 64, 40, 71, 78, 79, 84, 85	ltmul12ad 9120	. . 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 8303	. 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 8207	. . 3 |- ( ph -> ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) e. CC )
89	71	recnd 8207	. . 3 |- ( ph -> ( A / ( 1 - A ) ) e. CC )
90	5	nncnd 9156	. . 3 |- ( ph -> M e. CC )
91	5	nnap0d 9188	. . 3 |- ( ph -> M # 0 )
92	88, 89, 90, 91	div23apd 9007	. 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 4116	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 104   <-> wb 105   = wceq 1397   e. wcel 2202   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   CCcc 8029   RRcr 8030   0cc0 8031   1c1 8032   + caddc 8034   x. cmul 8036   < clt 8213   <_ cle 8214   - cmin 8349   / cdiv 8851   NNcn 9142   NN0cn0 9401   ZZcz 9478   ZZ>=cuz 9754   RR+crp 9887   ...cfz 10242   seqcseq 10708   ^cexp 10799   abscabs 11557   sum_csu 11913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-frec 6556  df-1o 6581  df-oadd 6585  df-er 6701  df-en 6909  df-dom 6910  df-fin 6911  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-ico 10128  df-fz 10243  df-fzo 10377  df-seqfrec 10709  df-exp 10800  df-ihash 11037  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-clim 11839  df-sumdc 11914

This theorem is referenced by:  cvgratnn  12091