Theorem cvgratnnlemabsle 12272

Description: Lemma for cvgratnn 12276. (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 9746	. . . . . . 7 |- ( ph -> M e. ZZ )
3	2	peano2zd 9750	. . . . . 6 |- ( ph -> ( M + 1 ) e. ZZ )
4		cvgratnn.n	. . . . . . 7 |- ( ph -> N e. ( ZZ>= ` M ) )
5		eluzelz 9910	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
6	4, 5	syl 14	. . . . . 6 |- ( ph -> N e. ZZ )
7	3, 6	fzfigd 10846	. . . . 5 |- ( ph -> ( ( M + 1 ) ... N ) e. Fin )
8		fveq2 5690	. . . . . . 7 |- ( k = i -> ( F ` k ) = ( F ` i ) )
9	8	eleq1d 2307	. . . . . 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 2623	. . . . . . 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 10407	. . . . . . . 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 8317	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> 0 e. RR )
16	1	peano2nnd 9298	. . . . . . . . . 10 |- ( ph -> ( M + 1 ) e. NN )
17	16	adantr 276	. . . . . . . . 9 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( M + 1 ) e. NN )
18	17	nnred 9296	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( M + 1 ) e. RR )
19	14	zred 9747	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> i e. RR )
20	16	nngt0d 9327	. . . . . . . . 9 |- ( ph -> 0 < ( M + 1 ) )
21	20	adantr 276	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> 0 < ( M + 1 ) )
22		elfzle1 10410	. . . . . . . . 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 8741	. . . . . . 7 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> 0 < i )
25		elnnz 9633	. . . . . . 7 |- ( i e. NN <-> ( i e. ZZ /\ 0 < i ) )
26	14, 24, 25	sylanbrc 421	. . . . . 6 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> i e. NN )
27	9, 12, 26	rspcdva 2934	. . . . 5 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( F ` i ) e. CC )
28	7, 27	fsumcl 12145	. . . 4 |- ( ph -> sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) e. CC )
29	28	abscld 11925	. . 3 |- ( ph -> ( abs ` sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) ) e. RR )
30	27	abscld 11925	. . . 4 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( abs ` ( F ` i ) ) e. RR )
31	7, 30	fsumrecl 12146	. . 3 |- ( ph -> sum_ i e. ( ( M + 1 ) ... N ) ( abs ` ( F ` i ) ) e. RR )
32		fveq2 5690	. . . . . . . . 9 |- ( k = M -> ( F ` k ) = ( F ` M ) )
33	32	eleq1d 2307	. . . . . . . 8 |- ( k = M -> ( ( F ` k ) e. CC <-> ( F ` M ) e. CC ) )
34	33, 11, 1	rspcdva 2934	. . . . . . 7 |- ( ph -> ( F ` M ) e. CC )
35	34	adantr 276	. . . . . 6 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( F ` M ) e. CC )
36	35	abscld 11925	. . . . 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 9752	. . . . . . 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 9296	. . . . . . . . 9 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> M e. RR )
43	42	lep1d 9251	. . . . . . . . 9 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> M <_ ( M + 1 ) )
44	42, 18, 19, 43, 23	letrd 8440	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> M <_ i )
45	19, 42	subge0d 8853	. . . . . . . 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 9636	. . . . . . 7 |- ( ( i - M ) e. NN0 <-> ( ( i - M ) e. ZZ /\ 0 <_ ( i - M ) ) )
48	40, 46, 47	sylanbrc 421	. . . . . 6 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( i - M ) e. NN0 )
49	38, 48	reexpcld 11106	. . . . 5 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( A ^ ( i - M ) ) e. RR )
50	36, 49	remulcld 8346	. . . 4 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( ( abs ` ( F ` M ) ) x. ( A ^ ( i - M ) ) ) e. RR )
51	7, 50	fsumrecl 12146	. . 3 |- ( ph -> sum_ i e. ( ( M + 1 ) ... N ) ( ( abs ` ( F ` M ) ) x. ( A ^ ( i - M ) ) ) e. RR )
52	7, 27	fsumabs 12210	. . 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 481	. . . . 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 481	. . . . 5 |- ( ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) /\ k e. NN ) -> ( abs ` ( F ` ( k + 1 ) ) ) <_ ( A x. ( abs ` ( F ` k ) ) ) )
60		eluz2 9906	. . . . . 6 |- ( i e. ( ZZ>= ` M ) <-> ( M e. ZZ /\ i e. ZZ /\ M <_ i ) )
61	39, 14, 44, 60	syl3anbrc 1212	. . . . 5 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> i e. ( ZZ>= ` M ) )
62	38, 54, 56, 57, 59, 41, 61	cvgratnnlemmn 12270	. . . 4 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( abs ` ( F ` i ) ) <_ ( ( abs ` ( F ` M ) ) x. ( A ^ ( i - M ) ) ) )
63	7, 30, 50, 62	fsumle 12208	. . 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 8440	. 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 11925	. . . 4 |- ( ph -> ( abs ` ( F ` M ) ) e. RR )
66	65	recnd 8344	. . 3 |- ( ph -> ( abs ` ( F ` M ) ) e. CC )
67	38	recnd 8344	. . . 4 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> A e. CC )
68	67, 48	expcld 11089	. . 3 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( A ^ ( i - M ) ) e. CC )
69	7, 66, 68	fsummulc2 12193	. 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 4153	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 1402   e. wcel 2209   A.wral 2528   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   CCcc 8167   RRcr 8168   0cc0 8169   1c1 8170   + caddc 8172   x. cmul 8174   < clt 8350   <_ cle 8351   - cmin 8487   NNcn 9283   NN0cn0 9542   ZZcz 9623   ZZ>=cuz 9900   ...cfz 10390   ^cexp 10953   abscabs 11741   sum_csu 12097

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-oadd 6681  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-ico 10275  df-fz 10391  df-fzo 10528  df-seqfrec 10863  df-exp 10954  df-ihash 11193  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-clim 12023  df-sumdc 12098

This theorem is referenced by:  cvgratnnlemrate  12275