Theorem cvgratnnlemrate 11471

Description: Lemma for cvgratnn 11472. (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 9501	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
2		1zzd 9218	. . . . . . 7 |- ( ph -> 1 e. ZZ )
3		cvgratnn.6	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )
4	1, 2, 3	serf 10409	. . . . . 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 9538	. . . . . . 7 |- ( ( M e. NN /\ N e. ( ZZ>= ` M ) ) -> N e. NN )
8	5, 6, 7	syl2anc 409	. . . . . 6 |- ( ph -> N e. NN )
9	4, 8	ffvelrnd 5621	. . . . 5 |- ( ph -> ( seq 1 ( + , F ) ` N ) e. CC )
10	4, 5	ffvelrnd 5621	. . . . 5 |- ( ph -> ( seq 1 ( + , F ) ` M ) e. CC )
11	9, 10	subcld 8209	. . . 4 |- ( ph -> ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) e. CC )
12	11	abscld 11123	. . 3 |- ( ph -> ( abs ` ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) ) e. RR )
13		fveq2 5486	. . . . . . 7 |- ( k = M -> ( F ` k ) = ( F ` M ) )
14	13	eleq1d 2235	. . . . . 6 |- ( k = M -> ( ( F ` k ) e. CC <-> ( F ` M ) e. CC ) )
15	3	ralrimiva 2539	. . . . . 6 |- ( ph -> A. k e. NN ( F ` k ) e. CC )
16	14, 15, 5	rspcdva 2835	. . . . 5 |- ( ph -> ( F ` M ) e. CC )
17	16	abscld 11123	. . . 4 |- ( ph -> ( abs ` ( F ` M ) ) e. RR )
18	5	nnzd 9312	. . . . . . 7 |- ( ph -> M e. ZZ )
19	18	peano2zd 9316	. . . . . 6 |- ( ph -> ( M + 1 ) e. ZZ )
20		eluzelz 9475	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
21	6, 20	syl 14	. . . . . 6 |- ( ph -> N e. ZZ )
22	19, 21	fzfigd 10366	. . . . 5 |- ( ph -> ( ( M + 1 ) ... N ) e. Fin )
23		cvgratnn.3	. . . . . . 7 |- ( ph -> A e. RR )
24	23	adantr 274	. . . . . 6 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> A e. RR )
25	5	nnred 8870	. . . . . . . . 9 |- ( ph -> M e. RR )
26	25	adantr 274	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> M e. RR )
27		peano2re 8034	. . . . . . . . 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 9960	. . . . . . . . . 10 |- ( i e. ( ( M + 1 ) ... N ) -> i e. ZZ )
30	29	adantl 275	. . . . . . . . 9 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> i e. ZZ )
31	30	zred 9313	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> i e. RR )
32	26	lep1d 8826	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> M <_ ( M + 1 ) )
33		elfzle1 9962	. . . . . . . . 9 |- ( i e. ( ( M + 1 ) ... N ) -> ( M + 1 ) <_ i )
34	33	adantl 275	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( M + 1 ) <_ i )
35	26, 28, 31, 32, 34	letrd 8022	. . . . . . 7 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> M <_ i )
36		znn0sub 9256	. . . . . . . 8 |- ( ( M e. ZZ /\ i e. ZZ ) -> ( M <_ i <-> ( i - M ) e. NN0 ) )
37	18, 29, 36	syl2an 287	. . . . . . 7 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( M <_ i <-> ( i - M ) e. NN0 ) )
38	35, 37	mpbid 146	. . . . . 6 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( i - M ) e. NN0 )
39	24, 38	reexpcld 10605	. . . . 5 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( A ^ ( i - M ) ) e. RR )
40	22, 39	fsumrecl 11342	. . . 4 |- ( ph -> sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) e. RR )
41	17, 40	remulcld 7929	. . 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 9629	. . . . . . . . . . . 12 |- ( ph -> A e. RR+ )
45	44	reclt1d 9646	. . . . . . . . . . 11 |- ( ph -> ( A < 1 <-> 1 < ( 1 / A ) ) )
46	42, 45	mpbid 146	. . . . . . . . . 10 |- ( ph -> 1 < ( 1 / A ) )
47		1re 7898	. . . . . . . . . . 11 |- 1 e. RR
48	44	rprecred 9644	. . . . . . . . . . 11 |- ( ph -> ( 1 / A ) e. RR )
49		difrp 9628	. . . . . . . . . . 11 |- ( ( 1 e. RR /\ ( 1 / A ) e. RR ) -> ( 1 < ( 1 / A ) <-> ( ( 1 / A ) - 1 ) e. RR+ ) )
50	47, 48, 49	sylancr 411	. . . . . . . . . 10 |- ( ph -> ( 1 < ( 1 / A ) <-> ( ( 1 / A ) - 1 ) e. RR+ ) )
51	46, 50	mpbid 146	. . . . . . . . 9 |- ( ph -> ( ( 1 / A ) - 1 ) e. RR+ )
52	51	rpreccld 9643	. . . . . . . 8 |- ( ph -> ( 1 / ( ( 1 / A ) - 1 ) ) e. RR+ )
53	52, 44	rpdivcld 9650	. . . . . . 7 |- ( ph -> ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) e. RR+ )
54		fveq2 5486	. . . . . . . . . . 11 |- ( k = 1 -> ( F ` k ) = ( F ` 1 ) )
55	54	eleq1d 2235	. . . . . . . . . 10 |- ( k = 1 -> ( ( F ` k ) e. CC <-> ( F ` 1 ) e. CC ) )
56		1nn 8868	. . . . . . . . . . 11 |- 1 e. NN
57	56	a1i 9	. . . . . . . . . 10 |- ( ph -> 1 e. NN )
58	55, 15, 57	rspcdva 2835	. . . . . . . . 9 |- ( ph -> ( F ` 1 ) e. CC )
59	58	abscld 11123	. . . . . . . 8 |- ( ph -> ( abs ` ( F ` 1 ) ) e. RR )
60	58	absge0d 11126	. . . . . . . 8 |- ( ph -> 0 <_ ( abs ` ( F ` 1 ) ) )
61	59, 60	ge0p1rpd 9663	. . . . . . 7 |- ( ph -> ( ( abs ` ( F ` 1 ) ) + 1 ) e. RR+ )
62	53, 61	rpmulcld 9649	. . . . . 6 |- ( ph -> ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) e. RR+ )
63	62	rpred 9632	. . . . 5 |- ( ph -> ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) e. RR )
64	63, 5	nndivred 8907	. . . 4 |- ( ph -> ( ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) / M ) e. RR )
65		1red 7914	. . . . . . . 8 |- ( ph -> 1 e. RR )
66	65, 23	resubcld 8279	. . . . . . 7 |- ( ph -> ( 1 - A ) e. RR )
67	23, 65	posdifd 8430	. . . . . . . 8 |- ( ph -> ( A < 1 <-> 0 < ( 1 - A ) ) )
68	42, 67	mpbid 146	. . . . . . 7 |- ( ph -> 0 < ( 1 - A ) )
69	66, 68	elrpd 9629	. . . . . 6 |- ( ph -> ( 1 - A ) e. RR+ )
70	44, 69	rpdivcld 9650	. . . . 5 |- ( ph -> ( A / ( 1 - A ) ) e. RR+ )
71	70	rpred 9632	. . . 4 |- ( ph -> ( A / ( 1 - A ) ) e. RR )
72	64, 71	remulcld 7929	. . 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 11467	. . . . 5 |- ( ph -> ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) = sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) )
75	74	fveq2d 5490	. . . 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 11468	. . . 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 4004	. . 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 11126	. . . 4 |- ( ph -> 0 <_ ( abs ` ( F ` M ) ) )
79	23, 42, 43, 3, 73, 5	cvgratnnlemfm 11470	. . . 4 |- ( ph -> ( abs ` ( F ` M ) ) < ( ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) / M ) )
80	44	adantr 274	. . . . . . 7 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> A e. RR+ )
81	38	nn0zd 9311	. . . . . . 7 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( i - M ) e. ZZ )
82	80, 81	rpexpcld 10612	. . . . . 6 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( A ^ ( i - M ) ) e. RR+ )
83	82	rpge0d 9636	. . . . 5 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> 0 <_ ( A ^ ( i - M ) ) )
84	22, 39, 83	fsumge0 11400	. . . 4 |- ( ph -> 0 <_ sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) )
85	23, 42, 43, 3, 73, 5, 6	cvgratnnlemsumlt 11469	. . . 4 |- ( ph -> sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) < ( A / ( 1 - A ) ) )
86	17, 64, 40, 71, 78, 79, 84, 85	ltmul12ad 8836	. . 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 8023	. 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 7927	. . 3 |- ( ph -> ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) e. CC )
89	71	recnd 7927	. . 3 |- ( ph -> ( A / ( 1 - A ) ) e. CC )
90	5	nncnd 8871	. . 3 |- ( ph -> M e. CC )
91	5	nnap0d 8903	. . 3 |- ( ph -> M # 0 )
92	88, 89, 90, 91	div23apd 8724	. 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 4010	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 103   <-> wb 104   = wceq 1343   e. wcel 2136   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   CCcc 7751   RRcr 7752   0cc0 7753   1c1 7754   + caddc 7756   x. cmul 7758   < clt 7933   <_ cle 7934   - cmin 8069   / cdiv 8568   NNcn 8857   NN0cn0 9114   ZZcz 9191   ZZ>=cuz 9466   RR+crp 9589   ...cfz 9944   seqcseq 10380   ^cexp 10454   abscabs 10939   sum_csu 11294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-frec 6359  df-1o 6384  df-oadd 6388  df-er 6501  df-en 6707  df-dom 6708  df-fin 6709  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-ico 9830  df-fz 9945  df-fzo 10078  df-seqfrec 10381  df-exp 10455  df-ihash 10689  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220  df-sumdc 11295

This theorem is referenced by:  cvgratnn  11472