Theorem cvgratnnlemabsle 12078

Description: Lemma for cvgratnn 12082. (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 ) ) ) )
cvgratnn.m	|- ( ph -> M e. NN )
cvgratnn.n	|- ( ph -> N e. ( ZZ>= ` M ) )

Assertion

Ref	Expression
cvgratnnlemabsle	|- ( ph -> ( abs ` sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) ) <_ ( ( abs ` ( F ` M ) ) x. sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) ) )

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

Allowed substitution hint:   A( i)

Proof of Theorem cvgratnnlemabsle

Step	Hyp	Ref	Expression
1		cvgratnn.m	. . . . . . . 8 |- ( ph -> M e. NN )
2	1	nnzd 9591	. . . . . . 7 |- ( ph -> M e. ZZ )
3	2	peano2zd 9595	. . . . . 6 |- ( ph -> ( M + 1 ) e. ZZ )
4		cvgratnn.n	. . . . . . 7 |- ( ph -> N e. ( ZZ>= ` M ) )
5		eluzelz 9755	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
6	4, 5	syl 14	. . . . . 6 |- ( ph -> N e. ZZ )
7	3, 6	fzfigd 10683	. . . . 5 |- ( ph -> ( ( M + 1 ) ... N ) e. Fin )
8		fveq2 5635	. . . . . . 7 |- ( k = i -> ( F ` k ) = ( F ` i ) )
9	8	eleq1d 2298	. . . . . 6 |- ( k = i -> ( ( F ` k ) e. CC <-> ( F ` i ) e. CC ) )
10		cvgratnn.6	. . . . . . . 8 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )
11	10	ralrimiva 2603	. . . . . . 7 |- ( ph -> A. k e. NN ( F ` k ) e. CC )
12	11	adantr 276	. . . . . 6 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> A. k e. NN ( F ` k ) e. CC )
13		elfzelz 10250	. . . . . . . 8 |- ( i e. ( ( M + 1 ) ... N ) -> i e. ZZ )
14	13	adantl 277	. . . . . . 7 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> i e. ZZ )
15		0red 8170	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> 0 e. RR )
16	1	peano2nnd 9148	. . . . . . . . . 10 |- ( ph -> ( M + 1 ) e. NN )
17	16	adantr 276	. . . . . . . . 9 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( M + 1 ) e. NN )
18	17	nnred 9146	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( M + 1 ) e. RR )
19	14	zred 9592	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> i e. RR )
20	16	nngt0d 9177	. . . . . . . . 9 |- ( ph -> 0 < ( M + 1 ) )
21	20	adantr 276	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> 0 < ( M + 1 ) )
22		elfzle1 10252	. . . . . . . . 9 |- ( i e. ( ( M + 1 ) ... N ) -> ( M + 1 ) <_ i )
23	22	adantl 277	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( M + 1 ) <_ i )
24	15, 18, 19, 21, 23	ltletrd 8593	. . . . . . 7 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> 0 < i )
25		elnnz 9479	. . . . . . 7 |- ( i e. NN <-> ( i e. ZZ /\ 0 < i ) )
26	14, 24, 25	sylanbrc 417	. . . . . 6 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> i e. NN )
27	9, 12, 26	rspcdva 2913	. . . . 5 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( F ` i ) e. CC )
28	7, 27	fsumcl 11951	. . . 4 |- ( ph -> sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) e. CC )
29	28	abscld 11732	. . 3 |- ( ph -> ( abs ` sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) ) e. RR )
30	27	abscld 11732	. . . 4 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( abs ` ( F ` i ) ) e. RR )
31	7, 30	fsumrecl 11952	. . 3 |- ( ph -> sum_ i e. ( ( M + 1 ) ... N ) ( abs ` ( F ` i ) ) e. RR )
32		fveq2 5635	. . . . . . . . 9 |- ( k = M -> ( F ` k ) = ( F ` M ) )
33	32	eleq1d 2298	. . . . . . . 8 |- ( k = M -> ( ( F ` k ) e. CC <-> ( F ` M ) e. CC ) )
34	33, 11, 1	rspcdva 2913	. . . . . . 7 |- ( ph -> ( F ` M ) e. CC )
35	34	adantr 276	. . . . . 6 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( F ` M ) e. CC )
36	35	abscld 11732	. . . . 5 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( abs ` ( F ` M ) ) e. RR )
37		cvgratnn.3	. . . . . . 7 |- ( ph -> A e. RR )
38	37	adantr 276	. . . . . 6 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> A e. RR )
39	2	adantr 276	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> M e. ZZ )
40	14, 39	zsubcld 9597	. . . . . . 7 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( i - M ) e. ZZ )
41	1	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> M e. NN )
42	41	nnred 9146	. . . . . . . . 9 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> M e. RR )
43	42	lep1d 9101	. . . . . . . . 9 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> M <_ ( M + 1 ) )
44	42, 18, 19, 43, 23	letrd 8293	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> M <_ i )
45	19, 42	subge0d 8705	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( 0 <_ ( i - M ) <-> M <_ i ) )
46	44, 45	mpbird 167	. . . . . . 7 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> 0 <_ ( i - M ) )
47		elnn0z 9482	. . . . . . 7 |- ( ( i - M ) e. NN0 <-> ( ( i - M ) e. ZZ /\ 0 <_ ( i - M ) ) )
48	40, 46, 47	sylanbrc 417	. . . . . 6 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( i - M ) e. NN0 )
49	38, 48	reexpcld 10942	. . . . 5 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( A ^ ( i - M ) ) e. RR )
50	36, 49	remulcld 8200	. . . 4 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( ( abs ` ( F ` M ) ) x. ( A ^ ( i - M ) ) ) e. RR )
51	7, 50	fsumrecl 11952	. . 3 |- ( ph -> sum_ i e. ( ( M + 1 ) ... N ) ( ( abs ` ( F ` M ) ) x. ( A ^ ( i - M ) ) ) e. RR )
52	7, 27	fsumabs 12016	. . 3 |- ( ph -> ( abs ` sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) ) <_ sum_ i e. ( ( M + 1 ) ... N ) ( abs ` ( F ` i ) ) )
53		cvgratnn.4	. . . . . 6 |- ( ph -> A < 1 )
54	53	adantr 276	. . . . 5 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> A < 1 )
55		cvgratnn.gt0	. . . . . 6 |- ( ph -> 0 < A )
56	55	adantr 276	. . . . 5 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> 0 < A )
57	10	adantlr 477	. . . . 5 |- ( ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) /\ k e. NN ) -> ( F ` k ) e. CC )
58		cvgratnn.7	. . . . . 6 |- ( ( ph /\ k e. NN ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )
59	58	adantlr 477	. . . . 5 |- ( ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) /\ k e. NN ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )
60		eluz2 9751	. . . . . 6 |- ( i e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ i e. ZZ /\ M <_ i ) )
61	39, 14, 44, 60	syl3anbrc 1205	. . . . 5 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> i e. ( ZZ>= ` M ) )
62	38, 54, 56, 57, 59, 41, 61	cvgratnnlemmn 12076	. . . 4 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( abs ` ( F ` i ) ) <_ ( ( abs ` ( F ` M ) ) x. ( A ^ ( i - M ) ) ) )
63	7, 30, 50, 62	fsumle 12014	. . 3 |- ( ph -> sum_ i e. ( ( M + 1 ) ... N ) ( abs ` ( F ` i ) ) <_ sum_ i e. ( ( M + 1 ) ... N ) ( ( abs ` ( F ` M ) ) x. ( A ^ ( i - M ) ) ) )
64	29, 31, 51, 52, 63	letrd 8293	. 2 |- ( ph -> ( abs ` sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) ) <_ sum_ i e. ( ( M + 1 ) ... N ) ( ( abs ` ( F ` M ) ) x. ( A ^ ( i - M ) ) ) )
65	34	abscld 11732	. . . 4 |- ( ph -> ( abs ` ( F ` M ) ) e. RR )
66	65	recnd 8198	. . 3 |- ( ph -> ( abs ` ( F ` M ) ) e. CC )
67	38	recnd 8198	. . . 4 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> A e. CC )
68	67, 48	expcld 10925	. . 3 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( A ^ ( i - M ) ) e. CC )
69	7, 66, 68	fsummulc2 11999	. 2 |- ( ph -> ( ( abs ` ( F ` M ) ) x. sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) ) = sum_ i e. ( ( M + 1 ) ... N ) ( ( abs ` ( F ` M ) ) x. ( A ^ ( i - M ) ) ) )
70	64, 69	breqtrrd 4114	1 |- ( ph -> ( abs ` sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) ) <_ ( ( abs ` ( F ` M ) ) x. sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   CCcc 8020   RRcr 8021   0cc0 8022   1c1 8023   + caddc 8025   x. cmul 8027   < clt 8204   <_ cle 8205   - cmin 8340   NNcn 9133   NN0cn0 9392   ZZcz 9469   ZZ>=cuz 9745   ...cfz 10233   ^cexp 10790   abscabs 11548   sum_csu 11904

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141  ax-caucvg 8142

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-frec 6552  df-1o 6577  df-oadd 6581  df-er 6697  df-en 6905  df-dom 6906  df-fin 6907  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-n0 9393  df-z 9470  df-uz 9746  df-q 9844  df-rp 9879  df-ico 10119  df-fz 10234  df-fzo 10368  df-seqfrec 10700  df-exp 10791  df-ihash 11028  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550  df-clim 11830  df-sumdc 11905

This theorem is referenced by:  cvgratnnlemrate  12081