Theorem cvgratnnlemrate 11874

Description: Lemma for cvgratnn 11875. (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 9686	. . . . . . 7 |- NN = ( ZZ>= ` 1 )
2		1zzd 9401	. . . . . . 7 |- ( ph -> 1 e. ZZ )
3		cvgratnn.6	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. CC )
4	1, 2, 3	serf 10630	. . . . . 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 9723	. . . . . . 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 5718	. . . . 5 |- ( ph -> ( seq 1 ( + , F ) ` N ) e. CC )
10	4, 5	ffvelcdmd 5718	. . . . 5 |- ( ph -> ( seq 1 ( + , F ) ` M ) e. CC )
11	9, 10	subcld 8385	. . . 4 |- ( ph -> ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) e. CC )
12	11	abscld 11525	. . 3 |- ( ph -> ( abs ` ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) ) e. RR )
13		fveq2 5578	. . . . . . 7 |- ( k = M -> ( F ` k ) = ( F ` M ) )
14	13	eleq1d 2274	. . . . . 6 |- ( k = M -> ( ( F ` k ) e. CC <-> ( F ` M ) e. CC ) )
15	3	ralrimiva 2579	. . . . . 6 |- ( ph -> A. k e. NN ( F ` k ) e. CC )
16	14, 15, 5	rspcdva 2882	. . . . 5 |- ( ph -> ( F ` M ) e. CC )
17	16	abscld 11525	. . . 4 |- ( ph -> ( abs ` ( F ` M ) ) e. RR )
18	5	nnzd 9496	. . . . . . 7 |- ( ph -> M e. ZZ )
19	18	peano2zd 9500	. . . . . 6 |- ( ph -> ( M + 1 ) e. ZZ )
20		eluzelz 9659	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
21	6, 20	syl 14	. . . . . 6 |- ( ph -> N e. ZZ )
22	19, 21	fzfigd 10578	. . . . 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 9051	. . . . . . . . 9 |- ( ph -> M e. RR )
26	25	adantr 276	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> M e. RR )
27		peano2re 8210	. . . . . . . . 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 10149	. . . . . . . . . 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 9497	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> i e. RR )
32	26	lep1d 9006	. . . . . . . 8 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> M <_ ( M + 1 ) )
33		elfzle1 10151	. . . . . . . . 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 8198	. . . . . . 7 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> M <_ i )
36		znn0sub 9440	. . . . . . . 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 10837	. . . . 5 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( A ^ ( i - M ) ) e. RR )
40	22, 39	fsumrecl 11745	. . . 4 |- ( ph -> sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) e. RR )
41	17, 40	remulcld 8105	. . 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 9817	. . . . . . . . . . . 12 |- ( ph -> A e. RR+ )
45	44	reclt1d 9834	. . . . . . . . . . 11 |- ( ph -> ( A < 1 <-> 1 < ( 1 / A ) ) )
46	42, 45	mpbid 147	. . . . . . . . . 10 |- ( ph -> 1 < ( 1 / A ) )
47		1re 8073	. . . . . . . . . . 11 |- 1 e. RR
48	44	rprecred 9832	. . . . . . . . . . 11 |- ( ph -> ( 1 / A ) e. RR )
49		difrp 9816	. . . . . . . . . . 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 9831	. . . . . . . 8 |- ( ph -> ( 1 / ( ( 1 / A ) - 1 ) ) e. RR+ )
53	52, 44	rpdivcld 9838	. . . . . . 7 |- ( ph -> ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) e. RR+ )
54		fveq2 5578	. . . . . . . . . . 11 |- ( k = 1 -> ( F ` k ) = ( F ` 1 ) )
55	54	eleq1d 2274	. . . . . . . . . 10 |- ( k = 1 -> ( ( F ` k ) e. CC <-> ( F ` 1 ) e. CC ) )
56		1nn 9049	. . . . . . . . . . 11 |- 1 e. NN
57	56	a1i 9	. . . . . . . . . 10 |- ( ph -> 1 e. NN )
58	55, 15, 57	rspcdva 2882	. . . . . . . . 9 |- ( ph -> ( F ` 1 ) e. CC )
59	58	abscld 11525	. . . . . . . 8 |- ( ph -> ( abs ` ( F ` 1 ) ) e. RR )
60	58	absge0d 11528	. . . . . . . 8 |- ( ph -> 0 <_ ( abs ` ( F ` 1 ) ) )
61	59, 60	ge0p1rpd 9851	. . . . . . 7 |- ( ph -> ( ( abs ` ( F ` 1 ) ) + 1 ) e. RR+ )
62	53, 61	rpmulcld 9837	. . . . . 6 |- ( ph -> ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) e. RR+ )
63	62	rpred 9820	. . . . 5 |- ( ph -> ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) e. RR )
64	63, 5	nndivred 9088	. . . 4 |- ( ph -> ( ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) / M ) e. RR )
65		1red 8089	. . . . . . . 8 |- ( ph -> 1 e. RR )
66	65, 23	resubcld 8455	. . . . . . 7 |- ( ph -> ( 1 - A ) e. RR )
67	23, 65	posdifd 8607	. . . . . . . 8 |- ( ph -> ( A < 1 <-> 0 < ( 1 - A ) ) )
68	42, 67	mpbid 147	. . . . . . 7 |- ( ph -> 0 < ( 1 - A ) )
69	66, 68	elrpd 9817	. . . . . 6 |- ( ph -> ( 1 - A ) e. RR+ )
70	44, 69	rpdivcld 9838	. . . . 5 |- ( ph -> ( A / ( 1 - A ) ) e. RR+ )
71	70	rpred 9820	. . . 4 |- ( ph -> ( A / ( 1 - A ) ) e. RR )
72	64, 71	remulcld 8105	. . 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 11870	. . . . 5 |- ( ph -> ( ( seq 1 ( + , F ) ` N ) - ( seq 1 ( + , F ) ` M ) ) = sum_ i e. ( ( M + 1 ) ... N ) ( F ` i ) )
75	74	fveq2d 5582	. . . 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 11871	. . . 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 4067	. . 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 11528	. . . 4 |- ( ph -> 0 <_ ( abs ` ( F ` M ) ) )
79	23, 42, 43, 3, 73, 5	cvgratnnlemfm 11873	. . . 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 9495	. . . . . . 7 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( i - M ) e. ZZ )
82	80, 81	rpexpcld 10844	. . . . . 6 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> ( A ^ ( i - M ) ) e. RR+ )
83	82	rpge0d 9824	. . . . 5 |- ( ( ph /\ i e. ( ( M + 1 ) ... N ) ) -> 0 <_ ( A ^ ( i - M ) ) )
84	22, 39, 83	fsumge0 11803	. . . 4 |- ( ph -> 0 <_ sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) )
85	23, 42, 43, 3, 73, 5, 6	cvgratnnlemsumlt 11872	. . . 4 |- ( ph -> sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) < ( A / ( 1 - A ) ) )
86	17, 64, 40, 71, 78, 79, 84, 85	ltmul12ad 9016	. . 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 8199	. 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 8103	. . 3 |- ( ph -> ( ( ( 1 / ( ( 1 / A ) - 1 ) ) / A ) x. ( ( abs ` ( F ` 1 ) ) + 1 ) ) e. CC )
89	71	recnd 8103	. . 3 |- ( ph -> ( A / ( 1 - A ) ) e. CC )
90	5	nncnd 9052	. . 3 |- ( ph -> M e. CC )
91	5	nnap0d 9084	. . 3 |- ( ph -> M # 0 )
92	88, 89, 90, 91	div23apd 8903	. 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 4073	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 1373   e. wcel 2176   class class class wbr 4045   ` cfv 5272  (class class class)co 5946   CCcc 7925   RRcr 7926   0cc0 7927   1c1 7928   + caddc 7930   x. cmul 7932   < clt 8109   <_ cle 8110   - cmin 8245   / cdiv 8747   NNcn 9038   NN0cn0 9297   ZZcz 9374   ZZ>=cuz 9650   RR+crp 9777   ...cfz 10132   seqcseq 10594   ^cexp 10685   abscabs 11341   sum_csu 11697

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044  ax-pre-mulext 8045  ax-arch 8046  ax-caucvg 8047

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-po 4344  df-iso 4345  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-isom 5281  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-irdg 6458  df-frec 6479  df-1o 6504  df-oadd 6508  df-er 6622  df-en 6830  df-dom 6831  df-fin 6832  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-div 8748  df-inn 9039  df-2 9097  df-3 9098  df-4 9099  df-n0 9298  df-z 9375  df-uz 9651  df-q 9743  df-rp 9778  df-ico 10018  df-fz 10133  df-fzo 10267  df-seqfrec 10595  df-exp 10686  df-ihash 10923  df-cj 11186  df-re 11187  df-im 11188  df-rsqrt 11342  df-abs 11343  df-clim 11623  df-sumdc 11698

This theorem is referenced by:  cvgratnn  11875