Theorem cvgratnnlemsumlt 11297

Description: Lemma for cvgratnn 11300. (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 9172	. . . 4 |- ( ph -> M e. ZZ )
3		1zzd 9081	. . . 4 |- ( ph -> 1 e. ZZ )
4		cvgratnn.n	. . . . . 6 |- ( ph -> N e. ( ZZ>= ` M ) )
5		eluzelz 9335	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
6	4, 5	syl 14	. . . . 5 |- ( ph -> N e. ZZ )
7	6, 2	zsubcld 9178	. . . 4 |- ( ph -> ( N - M ) e. ZZ )
8		cvgratnn.3	. . . . . . 7 |- ( ph -> A e. RR )
9	8	recnd 7794	. . . . . 6 |- ( ph -> A e. CC )
10	9	adantr 274	. . . . 5 |- ( ( ph /\ k e. ( 1 ... ( N - M ) ) ) -> A e. CC )
11		elfznn 9834	. . . . . . 7 |- ( k e. ( 1 ... ( N - M ) ) -> k e. NN )
12	11	adantl 275	. . . . . 6 |- ( ( ph /\ k e. ( 1 ... ( N - M ) ) ) -> k e. NN )
13	12	nnnn0d 9030	. . . . 5 |- ( ( ph /\ k e. ( 1 ... ( N - M ) ) ) -> k e. NN0 )
14	10, 13	expcld 10424	. . . 4 |- ( ( ph /\ k e. ( 1 ... ( N - M ) ) ) -> ( A ^ k ) e. CC )
15		oveq2 5782	. . . 4 |- ( k = ( i - M ) -> ( A ^ k ) = ( A ^ ( i - M ) ) )
16	2, 3, 7, 14, 15	fsumshft 11213	. . 3 |- ( ph -> sum_ k e. ( 1 ... ( N - M ) ) ( A ^ k ) = sum_ i e. ( ( 1 + M ) ... ( ( N - M ) + M ) ) ( A ^ ( i - M ) ) )
17		1cnd 7782	. . . . . 6 |- ( ph -> 1 e. CC )
18	1	nncnd 8734	. . . . . 6 |- ( ph -> M e. CC )
19	17, 18	addcomd 7913	. . . . 5 |- ( ph -> ( 1 + M ) = ( M + 1 ) )
20	6	zcnd 9174	. . . . . 6 |- ( ph -> N e. CC )
21	20, 18	npcand 8077	. . . . 5 |- ( ph -> ( ( N - M ) + M ) = N )
22	19, 21	oveq12d 5792	. . . 4 |- ( ph -> ( ( 1 + M ) ... ( ( N - M ) + M ) ) = ( ( M + 1 ) ... N ) )
23	22	sumeq1d 11135	. . 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 2172	. 2 |- ( ph -> sum_ k e. ( 1 ... ( N - M ) ) ( A ^ k ) = sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) )
25		fzval3 9981	. . . . 5 |- ( ( N - M ) e. ZZ -> ( 1 ... ( N - M ) ) = ( 1..^ ( ( N - M ) + 1 ) ) )
26	25	sumeq1d 11135	. . . 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 7781	. . . . . 6 |- ( ph -> 1 e. RR )
29		cvgratnn.4	. . . . . 6 |- ( ph -> A < 1 )
30	8, 28, 29	ltapd 8400	. . . . 5 |- ( ph -> A # 1 )
31		1nn0 8993	. . . . . 6 |- 1 e. NN0
32	31	a1i 9	. . . . 5 |- ( ph -> 1 e. NN0 )
33	7	peano2zd 9176	. . . . . 6 |- ( ph -> ( ( N - M ) + 1 ) e. ZZ )
34		eluzle 9338	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> M <_ N )
35	4, 34	syl 14	. . . . . . . 8 |- ( ph -> M <_ N )
36	6	zred 9173	. . . . . . . . 9 |- ( ph -> N e. RR )
37	1	nnred 8733	. . . . . . . . 9 |- ( ph -> M e. RR )
38	36, 37	subge0d 8297	. . . . . . . 8 |- ( ph -> ( 0 <_ ( N - M ) <-> M <_ N ) )
39	35, 38	mpbird 166	. . . . . . 7 |- ( ph -> 0 <_ ( N - M ) )
40	7	zred 9173	. . . . . . . 8 |- ( ph -> ( N - M ) e. RR )
41	28, 40	addge02d 8296	. . . . . . 7 |- ( ph -> ( 0 <_ ( N - M ) <-> 1 <_ ( ( N - M ) + 1 ) ) )
42	39, 41	mpbid 146	. . . . . 6 |- ( ph -> 1 <_ ( ( N - M ) + 1 ) )
43		eluz2 9332	. . . . . 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 1165	. . . . 5 |- ( ph -> ( ( N - M ) + 1 ) e. ( ZZ>= ` 1 ) )
45	9, 30, 32, 44	geosergap 11275	. . . 4 |- ( ph -> sum_ k e. ( 1..^ ( ( N - M ) + 1 ) ) ( A ^ k ) = ( ( ( A ^ 1 ) - ( A ^ ( ( N - M ) + 1 ) ) ) / ( 1 - A ) ) )
46	9	exp1d 10419	. . . . . . 7 |- ( ph -> ( A ^ 1 ) = A )
47	46, 8	eqeltrd 2216	. . . . . 6 |- ( ph -> ( A ^ 1 ) e. RR )
48		cvgratnn.gt0	. . . . . . . . 9 |- ( ph -> 0 < A )
49	8, 48	elrpd 9481	. . . . . . . 8 |- ( ph -> A e. RR+ )
50	49, 33	rpexpcld 10448	. . . . . . 7 |- ( ph -> ( A ^ ( ( N - M ) + 1 ) ) e. RR+ )
51	50	rpred 9483	. . . . . 6 |- ( ph -> ( A ^ ( ( N - M ) + 1 ) ) e. RR )
52	47, 51	resubcld 8143	. . . . 5 |- ( ph -> ( ( A ^ 1 ) - ( A ^ ( ( N - M ) + 1 ) ) ) e. RR )
53	28, 8	resubcld 8143	. . . . . 6 |- ( ph -> ( 1 - A ) e. RR )
54	8, 28	posdifd 8294	. . . . . . 7 |- ( ph -> ( A < 1 <-> 0 < ( 1 - A ) ) )
55	29, 54	mpbid 146	. . . . . 6 |- ( ph -> 0 < ( 1 - A ) )
56	53, 55	elrpd 9481	. . . . 5 |- ( ph -> ( 1 - A ) e. RR+ )
57	46	oveq1d 5789	. . . . . 6 |- ( ph -> ( ( A ^ 1 ) - ( A ^ ( ( N - M ) + 1 ) ) ) = ( A - ( A ^ ( ( N - M ) + 1 ) ) ) )
58	8, 50	ltsubrpd 9516	. . . . . 6 |- ( ph -> ( A - ( A ^ ( ( N - M ) + 1 ) ) ) < A )
59	57, 58	eqbrtrd 3950	. . . . 5 |- ( ph -> ( ( A ^ 1 ) - ( A ^ ( ( N - M ) + 1 ) ) ) < A )
60	52, 8, 56, 59	ltdiv1dd 9541	. . . 4 |- ( ph -> ( ( ( A ^ 1 ) - ( A ^ ( ( N - M ) + 1 ) ) ) / ( 1 - A ) ) < ( A / ( 1 - A ) ) )
61	45, 60	eqbrtrd 3950	. . 3 |- ( ph -> sum_ k e. ( 1..^ ( ( N - M ) + 1 ) ) ( A ^ k ) < ( A / ( 1 - A ) ) )
62	27, 61	eqbrtrd 3950	. 2 |- ( ph -> sum_ k e. ( 1 ... ( N - M ) ) ( A ^ k ) < ( A / ( 1 - A ) ) )
63	24, 62	eqbrtrrd 3952	1 |- ( ph -> sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) < ( A / ( 1 - A ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7618   RRcr 7619   0cc0 7620   1c1 7621   + caddc 7623   x. cmul 7625   < clt 7800   <_ cle 7801   - cmin 7933   / cdiv 8432   NNcn 8720   NN0cn0 8977   ZZcz 9054   ZZ>=cuz 9326   ...cfz 9790  ..^cfzo 9919   ^cexp 10292   abscabs 10769   sum_csu 11122

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-fz 9791  df-fzo 9920  df-seqfrec 10219  df-exp 10293  df-ihash 10522  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-clim 11048  df-sumdc 11123

This theorem is referenced by:  cvgratnnlemrate  11299