Theorem cvgratnnlemabsle 12038

Description: Lemma for cvgratnn 12042. (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 9568	. . . . . . 7 |- ( ph -> M e. ZZ )
3	2	peano2zd 9572	. . . . . 6 |- ( ph -> ( M + 1 ) e. ZZ )
4		cvgratnn.n	. . . . . . 7 |- ( ph -> N e. ( ZZ>= ` M ) )
5		eluzelz 9731	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
6	4, 5	syl 14	. . . . . 6 |- ( ph -> N e. ZZ )
7	3, 6	fzfigd 10653	. . . . 5 |- ( ph -> ( ( M + 1 ) ... N ) e. Fin )
8		fveq2 5627	. . . . . . 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 10221	. . . . . . . 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 8147	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> 0 e. RR )
16	1	peano2nnd 9125	. . . . . . . . . 10 |- ( ph -> ( M + 1 ) e. NN )
17	16	adantr 276	. . . . . . . . 9 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( M + 1 ) e. NN )
18	17	nnred 9123	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( M + 1 ) e. RR )
19	14	zred 9569	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> i e. RR )
20	16	nngt0d 9154	. . . . . . . . 9 |- ( ph -> 0 < ( M + 1 ) )
21	20	adantr 276	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> 0 < ( M + 1 ) )
22		elfzle1 10223	. . . . . . . . 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 8570	. . . . . . 7 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> 0 < i )
25		elnnz 9456	. . . . . . 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 2912	. . . . 5 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( F ` i ) e. CC )
28	7, 27	fsumcl 11911	. . . 4 |- ( ph -> sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) e. CC )
29	28	abscld 11692	. . 3 |- ( ph -> ( abs ` sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) ) e. RR )
30	27	abscld 11692	. . . 4 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( abs ` ( F ` i ) ) e. RR )
31	7, 30	fsumrecl 11912	. . 3 |- ( ph -> sum_ i e. ( ( M + 1 ) ... N ) ( abs ` ( F ` i ) ) e. RR )
32		fveq2 5627	. . . . . . . . 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 2912	. . . . . . 7 |- ( ph -> ( F ` M ) e. CC )
35	34	adantr 276	. . . . . 6 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( F ` M ) e. CC )
36	35	abscld 11692	. . . . 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 9574	. . . . . . 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 9123	. . . . . . . . 9 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> M e. RR )
43	42	lep1d 9078	. . . . . . . . 9 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> M <_ ( M + 1 ) )
44	42, 18, 19, 43, 23	letrd 8270	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> M <_ i )
45	19, 42	subge0d 8682	. . . . . . . 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 9459	. . . . . . 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 10912	. . . . 5 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( A ^ ( i - M ) ) e. RR )
50	36, 49	remulcld 8177	. . . 4 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( ( abs ` ( F ` M ) ) x. ( A ^ ( i - M ) ) ) e. RR )
51	7, 50	fsumrecl 11912	. . 3 |- ( ph -> sum_ i e. ( ( M + 1 ) ... N ) ( ( abs ` ( F ` M ) ) x. ( A ^ ( i - M ) ) ) e. RR )
52	7, 27	fsumabs 11976	. . 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 9728	. . . . . 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 12036	. . . 4 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( abs ` ( F ` i ) ) <_ ( ( abs ` ( F ` M ) ) x. ( A ^ ( i - M ) ) ) )
63	7, 30, 50, 62	fsumle 11974	. . 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 8270	. 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 11692	. . . 4 |- ( ph -> ( abs ` ( F ` M ) ) e. RR )
66	65	recnd 8175	. . 3 |- ( ph -> ( abs ` ( F ` M ) ) e. CC )
67	38	recnd 8175	. . . 4 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> A e. CC )
68	67, 48	expcld 10895	. . 3 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( A ^ ( i - M ) ) e. CC )
69	7, 66, 68	fsummulc2 11959	. 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 4111	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 4083   ` cfv 5318  (class class class)co 6001   CCcc 7997   RRcr 7998   0cc0 7999   1c1 8000   + caddc 8002   x. cmul 8004   < clt 8181   <_ cle 8182   - cmin 8317   NNcn 9110   NN0cn0 9369   ZZcz 9446   ZZ>=cuz 9722   ...cfz 10204   ^cexp 10760   abscabs 11508   sum_csu 11864

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 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-frec 6537  df-1o 6562  df-oadd 6566  df-er 6680  df-en 6888  df-dom 6889  df-fin 6890  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-ico 10090  df-fz 10205  df-fzo 10339  df-seqfrec 10670  df-exp 10761  df-ihash 10998  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-clim 11790  df-sumdc 11865

This theorem is referenced by:  cvgratnnlemrate  12041