Theorem cvgratnnlemsumlt 11693

Description: Lemma for cvgratnn 11696. (Contributed by Jim Kingdon, 23-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
cvgratnnlemsumlt	|- ( ph -> sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) < ( A / ( 1 - A ) ) )

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

Allowed substitution hint:   F( i)

Proof of Theorem cvgratnnlemsumlt

Step	Hyp	Ref	Expression
1		cvgratnn.m	. . . . 5 |- ( ph -> M e. NN )
2	1	nnzd 9447	. . . 4 |- ( ph -> M e. ZZ )
3		1zzd 9353	. . . 4 |- ( ph -> 1 e. ZZ )
4		cvgratnn.n	. . . . . 6 |- ( ph -> N e. ( ZZ>= ` M ) )
5		eluzelz 9610	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
6	4, 5	syl 14	. . . . 5 |- ( ph -> N e. ZZ )
7	6, 2	zsubcld 9453	. . . 4 |- ( ph -> ( N - M ) e. ZZ )
8		cvgratnn.3	. . . . . . 7 |- ( ph -> A e. RR )
9	8	recnd 8055	. . . . . 6 |- ( ph -> A e. CC )
10	9	adantr 276	. . . . 5 |- ( ( ph /\ k e. ( 1 ... ( N - M ) ) ) -> A e. CC )
11		elfznn 10129	. . . . . . 7 |- ( k e. ( 1 ... ( N - M ) ) -> k e. NN )
12	11	adantl 277	. . . . . 6 |- ( ( ph /\ k e. ( 1 ... ( N - M ) ) ) -> k e. NN )
13	12	nnnn0d 9302	. . . . 5 |- ( ( ph /\ k e. ( 1 ... ( N - M ) ) ) -> k e. NN0 )
14	10, 13	expcld 10765	. . . 4 |- ( ( ph /\ k e. ( 1 ... ( N - M ) ) ) -> ( A ^ k ) e. CC )
15		oveq2 5930	. . . 4 |- ( k = ( i - M ) -> ( A ^ k ) = ( A ^ ( i - M ) ) )
16	2, 3, 7, 14, 15	fsumshft 11609	. . 3 |- ( ph -> sum_ k e. ( 1 ... ( N - M ) ) ( A ^ k ) = sum_ i e. ( ( 1 + M ) ... ( ( N - M ) + M ) ) ( A ^ ( i - M ) ) )
17		1cnd 8042	. . . . . 6 |- ( ph -> 1 e. CC )
18	1	nncnd 9004	. . . . . 6 |- ( ph -> M e. CC )
19	17, 18	addcomd 8177	. . . . 5 |- ( ph -> ( 1 + M ) = ( M + 1 ) )
20	6	zcnd 9449	. . . . . 6 |- ( ph -> N e. CC )
21	20, 18	npcand 8341	. . . . 5 |- ( ph -> ( ( N - M ) + M ) = N )
22	19, 21	oveq12d 5940	. . . 4 |- ( ph -> ( ( 1 + M ) ... ( ( N - M ) + M ) ) = ( ( M + 1 ) ... N ) )
23	22	sumeq1d 11531	. . 3 |- ( ph -> sum_ i e. ( ( 1 + M ) ... ( ( N - M ) + M ) ) ( A ^ ( i - M ) ) = sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) )
24	16, 23	eqtrd 2229	. 2 |- ( ph -> sum_ k e. ( 1 ... ( N - M ) ) ( A ^ k ) = sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) )
25		fzval3 10280	. . . . 5 |- ( ( N - M ) e. ZZ -> ( 1 ... ( N - M ) ) = ( 1..^ ( ( N - M ) + 1 ) ) )
26	25	sumeq1d 11531	. . . 4 |- ( ( N - M ) e. ZZ -> sum_ k e. ( 1 ... ( N - M ) ) ( A ^ k ) = sum_ k e. ( 1..^ ( ( N - M ) + 1 ) ) ( A ^ k ) )
27	7, 26	syl 14	. . 3 |- ( ph -> sum_ k e. ( 1 ... ( N - M ) ) ( A ^ k ) = sum_ k e. ( 1..^ ( ( N - M ) + 1 ) ) ( A ^ k ) )
28		1red 8041	. . . . . 6 |- ( ph -> 1 e. RR )
29		cvgratnn.4	. . . . . 6 |- ( ph -> A < 1 )
30	8, 28, 29	ltapd 8665	. . . . 5 |- ( ph -> A # 1 )
31		1nn0 9265	. . . . . 6 |- 1 e. NN0
32	31	a1i 9	. . . . 5 |- ( ph -> 1 e. NN0 )
33	7	peano2zd 9451	. . . . . 6 |- ( ph -> ( ( N - M ) + 1 ) e. ZZ )
34		eluzle 9613	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> M <_ N )
35	4, 34	syl 14	. . . . . . . 8 |- ( ph -> M <_ N )
36	6	zred 9448	. . . . . . . . 9 |- ( ph -> N e. RR )
37	1	nnred 9003	. . . . . . . . 9 |- ( ph -> M e. RR )
38	36, 37	subge0d 8562	. . . . . . . 8 |- ( ph -> ( 0 <_ ( N - M ) <-> M <_ N ) )
39	35, 38	mpbird 167	. . . . . . 7 |- ( ph -> 0 <_ ( N - M ) )
40	7	zred 9448	. . . . . . . 8 |- ( ph -> ( N - M ) e. RR )
41	28, 40	addge02d 8561	. . . . . . 7 |- ( ph -> ( 0 <_ ( N - M ) <-> 1 <_ ( ( N - M ) + 1 ) ) )
42	39, 41	mpbid 147	. . . . . 6 |- ( ph -> 1 <_ ( ( N - M ) + 1 ) )
43		eluz2 9607	. . . . . 6 |- ( ( ( N - M ) + 1 ) e. ( ZZ>= ` 1 ) <-> ( 1 e. ZZ /\ ( ( N - M ) + 1 ) e. ZZ /\ 1 <_ ( ( N - M ) + 1 ) ) )
44	3, 33, 42, 43	syl3anbrc 1183	. . . . 5 |- ( ph -> ( ( N - M ) + 1 ) e. ( ZZ>= ` 1 ) )
45	9, 30, 32, 44	geosergap 11671	. . . 4 |- ( ph -> sum_ k e. ( 1..^ ( ( N - M ) + 1 ) ) ( A ^ k ) = ( ( ( A ^ 1 ) - ( A ^ ( ( N - M ) + 1 ) ) ) / ( 1 - A ) ) )
46	9	exp1d 10760	. . . . . . 7 |- ( ph -> ( A ^ 1 ) = A )
47	46, 8	eqeltrd 2273	. . . . . 6 |- ( ph -> ( A ^ 1 ) e. RR )
48		cvgratnn.gt0	. . . . . . . . 9 |- ( ph -> 0 < A )
49	8, 48	elrpd 9768	. . . . . . . 8 |- ( ph -> A e. RR+ )
50	49, 33	rpexpcld 10789	. . . . . . 7 |- ( ph -> ( A ^ ( ( N - M ) + 1 ) ) e. RR+ )
51	50	rpred 9771	. . . . . 6 |- ( ph -> ( A ^ ( ( N - M ) + 1 ) ) e. RR )
52	47, 51	resubcld 8407	. . . . 5 |- ( ph -> ( ( A ^ 1 ) - ( A ^ ( ( N - M ) + 1 ) ) ) e. RR )
53	28, 8	resubcld 8407	. . . . . 6 |- ( ph -> ( 1 - A ) e. RR )
54	8, 28	posdifd 8559	. . . . . . 7 |- ( ph -> ( A < 1 <-> 0 < ( 1 - A ) ) )
55	29, 54	mpbid 147	. . . . . 6 |- ( ph -> 0 < ( 1 - A ) )
56	53, 55	elrpd 9768	. . . . 5 |- ( ph -> ( 1 - A ) e. RR+ )
57	46	oveq1d 5937	. . . . . 6 |- ( ph -> ( ( A ^ 1 ) - ( A ^ ( ( N - M ) + 1 ) ) ) = ( A - ( A ^ ( ( N - M ) + 1 ) ) ) )
58	8, 50	ltsubrpd 9804	. . . . . 6 |- ( ph -> ( A - ( A ^ ( ( N - M ) + 1 ) ) ) < A )
59	57, 58	eqbrtrd 4055	. . . . 5 |- ( ph -> ( ( A ^ 1 ) - ( A ^ ( ( N - M ) + 1 ) ) ) < A )
60	52, 8, 56, 59	ltdiv1dd 9829	. . . 4 |- ( ph -> ( ( ( A ^ 1 ) - ( A ^ ( ( N - M ) + 1 ) ) ) / ( 1 - A ) ) < ( A / ( 1 - A ) ) )
61	45, 60	eqbrtrd 4055	. . 3 |- ( ph -> sum_ k e. ( 1..^ ( ( N - M ) + 1 ) ) ( A ^ k ) < ( A / ( 1 - A ) ) )
62	27, 61	eqbrtrd 4055	. 2 |- ( ph -> sum_ k e. ( 1 ... ( N - M ) ) ( A ^ k ) < ( A / ( 1 - A ) ) )
63	24, 62	eqbrtrrd 4057	1 |- ( ph -> sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) < ( A / ( 1 - A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   CCcc 7877   RRcr 7878   0cc0 7879   1c1 7880   + caddc 7882   x. cmul 7884   < clt 8061   <_ cle 8062   - cmin 8197   / cdiv 8699   NNcn 8990   NN0cn0 9249   ZZcz 9326   ZZ>=cuz 9601   ...cfz 10083  ..^cfzo 10217   ^cexp 10630   abscabs 11162   sum_csu 11518

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-frec 6449  df-1o 6474  df-oadd 6478  df-er 6592  df-en 6800  df-dom 6801  df-fin 6802  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-fz 10084  df-fzo 10218  df-seqfrec 10540  df-exp 10631  df-ihash 10868  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-clim 11444  df-sumdc 11519

This theorem is referenced by:  cvgratnnlemrate  11695