Theorem cvgratnnlemsumlt 11527

Description: Lemma for cvgratnn 11530. (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 9368	. . . 4 |- ( ph -> M e. ZZ )
3		1zzd 9274	. . . 4 |- ( ph -> 1 e. ZZ )
4		cvgratnn.n	. . . . . 6 |- ( ph -> N e. ( ZZ>= ` M ) )
5		eluzelz 9531	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
6	4, 5	syl 14	. . . . 5 |- ( ph -> N e. ZZ )
7	6, 2	zsubcld 9374	. . . 4 |- ( ph -> ( N - M ) e. ZZ )
8		cvgratnn.3	. . . . . . 7 |- ( ph -> A e. RR )
9	8	recnd 7980	. . . . . 6 |- ( ph -> A e. CC )
10	9	adantr 276	. . . . 5 |- ( ( ph /\ k e. ( 1 ... ( N - M ) ) ) -> A e. CC )
11		elfznn 10047	. . . . . . 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 9223	. . . . 5 |- ( ( ph /\ k e. ( 1 ... ( N - M ) ) ) -> k e. NN0 )
14	10, 13	expcld 10646	. . . 4 |- ( ( ph /\ k e. ( 1 ... ( N - M ) ) ) -> ( A ^ k ) e. CC )
15		oveq2 5878	. . . 4 |- ( k = ( i - M ) -> ( A ^ k ) = ( A ^ ( i - M ) ) )
16	2, 3, 7, 14, 15	fsumshft 11443	. . 3 |- ( ph -> sum_ k e. ( 1 ... ( N - M ) ) ( A ^ k ) = sum_ i e. ( ( 1 + M ) ... ( ( N - M ) + M ) ) ( A ^ ( i - M ) ) )
17		1cnd 7968	. . . . . 6 |- ( ph -> 1 e. CC )
18	1	nncnd 8927	. . . . . 6 |- ( ph -> M e. CC )
19	17, 18	addcomd 8102	. . . . 5 |- ( ph -> ( 1 + M ) = ( M + 1 ) )
20	6	zcnd 9370	. . . . . 6 |- ( ph -> N e. CC )
21	20, 18	npcand 8266	. . . . 5 |- ( ph -> ( ( N - M ) + M ) = N )
22	19, 21	oveq12d 5888	. . . 4 |- ( ph -> ( ( 1 + M ) ... ( ( N - M ) + M ) ) = ( ( M + 1 ) ... N ) )
23	22	sumeq1d 11365	. . 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 2210	. 2 |- ( ph -> sum_ k e. ( 1 ... ( N - M ) ) ( A ^ k ) = sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) )
25		fzval3 10197	. . . . 5 |- ( ( N - M ) e. ZZ -> ( 1 ... ( N - M ) ) = ( 1..^ ( ( N - M ) + 1 ) ) )
26	25	sumeq1d 11365	. . . 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 7967	. . . . . 6 |- ( ph -> 1 e. RR )
29		cvgratnn.4	. . . . . 6 |- ( ph -> A < 1 )
30	8, 28, 29	ltapd 8589	. . . . 5 |- ( ph -> A # 1 )
31		1nn0 9186	. . . . . 6 |- 1 e. NN0
32	31	a1i 9	. . . . 5 |- ( ph -> 1 e. NN0 )
33	7	peano2zd 9372	. . . . . 6 |- ( ph -> ( ( N - M ) + 1 ) e. ZZ )
34		eluzle 9534	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> M <_ N )
35	4, 34	syl 14	. . . . . . . 8 |- ( ph -> M <_ N )
36	6	zred 9369	. . . . . . . . 9 |- ( ph -> N e. RR )
37	1	nnred 8926	. . . . . . . . 9 |- ( ph -> M e. RR )
38	36, 37	subge0d 8486	. . . . . . . 8 |- ( ph -> ( 0 <_ ( N - M ) <-> M <_ N ) )
39	35, 38	mpbird 167	. . . . . . 7 |- ( ph -> 0 <_ ( N - M ) )
40	7	zred 9369	. . . . . . . 8 |- ( ph -> ( N - M ) e. RR )
41	28, 40	addge02d 8485	. . . . . . 7 |- ( ph -> ( 0 <_ ( N - M ) <-> 1 <_ ( ( N - M ) + 1 ) ) )
42	39, 41	mpbid 147	. . . . . 6 |- ( ph -> 1 <_ ( ( N - M ) + 1 ) )
43		eluz2 9528	. . . . . 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 1181	. . . . 5 |- ( ph -> ( ( N - M ) + 1 ) e. ( ZZ>= ` 1 ) )
45	9, 30, 32, 44	geosergap 11505	. . . 4 |- ( ph -> sum_ k e. ( 1..^ ( ( N - M ) + 1 ) ) ( A ^ k ) = ( ( ( A ^ 1 ) - ( A ^ ( ( N - M ) + 1 ) ) ) / ( 1 - A ) ) )
46	9	exp1d 10641	. . . . . . 7 |- ( ph -> ( A ^ 1 ) = A )
47	46, 8	eqeltrd 2254	. . . . . 6 |- ( ph -> ( A ^ 1 ) e. RR )
48		cvgratnn.gt0	. . . . . . . . 9 |- ( ph -> 0 < A )
49	8, 48	elrpd 9687	. . . . . . . 8 |- ( ph -> A e. RR+ )
50	49, 33	rpexpcld 10670	. . . . . . 7 |- ( ph -> ( A ^ ( ( N - M ) + 1 ) ) e. RR+ )
51	50	rpred 9690	. . . . . 6 |- ( ph -> ( A ^ ( ( N - M ) + 1 ) ) e. RR )
52	47, 51	resubcld 8332	. . . . 5 |- ( ph -> ( ( A ^ 1 ) - ( A ^ ( ( N - M ) + 1 ) ) ) e. RR )
53	28, 8	resubcld 8332	. . . . . 6 |- ( ph -> ( 1 - A ) e. RR )
54	8, 28	posdifd 8483	. . . . . . 7 |- ( ph -> ( A < 1 <-> 0 < ( 1 - A ) ) )
55	29, 54	mpbid 147	. . . . . 6 |- ( ph -> 0 < ( 1 - A ) )
56	53, 55	elrpd 9687	. . . . 5 |- ( ph -> ( 1 - A ) e. RR+ )
57	46	oveq1d 5885	. . . . . 6 |- ( ph -> ( ( A ^ 1 ) - ( A ^ ( ( N - M ) + 1 ) ) ) = ( A - ( A ^ ( ( N - M ) + 1 ) ) ) )
58	8, 50	ltsubrpd 9723	. . . . . 6 |- ( ph -> ( A - ( A ^ ( ( N - M ) + 1 ) ) ) < A )
59	57, 58	eqbrtrd 4023	. . . . 5 |- ( ph -> ( ( A ^ 1 ) - ( A ^ ( ( N - M ) + 1 ) ) ) < A )
60	52, 8, 56, 59	ltdiv1dd 9748	. . . 4 |- ( ph -> ( ( ( A ^ 1 ) - ( A ^ ( ( N - M ) + 1 ) ) ) / ( 1 - A ) ) < ( A / ( 1 - A ) ) )
61	45, 60	eqbrtrd 4023	. . 3 |- ( ph -> sum_ k e. ( 1..^ ( ( N - M ) + 1 ) ) ( A ^ k ) < ( A / ( 1 - A ) ) )
62	27, 61	eqbrtrd 4023	. 2 |- ( ph -> sum_ k e. ( 1 ... ( N - M ) ) ( A ^ k ) < ( A / ( 1 - A ) ) )
63	24, 62	eqbrtrrd 4025	1 |- ( ph -> sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) < ( A / ( 1 - A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   class class class wbr 4001   ` cfv 5213  (class class class)co 5870   CCcc 7804   RRcr 7805   0cc0 7806   1c1 7807   + caddc 7809   x. cmul 7811   < clt 7986   <_ cle 7987   - cmin 8122   / cdiv 8623   NNcn 8913   NN0cn0 9170   ZZcz 9247   ZZ>=cuz 9522   ...cfz 10002  ..^cfzo 10135   ^cexp 10512   abscabs 10997   sum_csu 11352

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4116  ax-sep 4119  ax-nul 4127  ax-pow 4172  ax-pr 4207  ax-un 4431  ax-setind 4534  ax-iinf 4585  ax-cnex 7897  ax-resscn 7898  ax-1cn 7899  ax-1re 7900  ax-icn 7901  ax-addcl 7902  ax-addrcl 7903  ax-mulcl 7904  ax-mulrcl 7905  ax-addcom 7906  ax-mulcom 7907  ax-addass 7908  ax-mulass 7909  ax-distr 7910  ax-i2m1 7911  ax-0lt1 7912  ax-1rid 7913  ax-0id 7914  ax-rnegex 7915  ax-precex 7916  ax-cnre 7917  ax-pre-ltirr 7918  ax-pre-ltwlin 7919  ax-pre-lttrn 7920  ax-pre-apti 7921  ax-pre-ltadd 7922  ax-pre-mulgt0 7923  ax-pre-mulext 7924  ax-arch 7925  ax-caucvg 7926

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3809  df-int 3844  df-iun 3887  df-br 4002  df-opab 4063  df-mpt 4064  df-tr 4100  df-id 4291  df-po 4294  df-iso 4295  df-iord 4364  df-on 4366  df-ilim 4367  df-suc 4369  df-iom 4588  df-xp 4630  df-rel 4631  df-cnv 4632  df-co 4633  df-dm 4634  df-rn 4635  df-res 4636  df-ima 4637  df-iota 5175  df-fun 5215  df-fn 5216  df-f 5217  df-f1 5218  df-fo 5219  df-f1o 5220  df-fv 5221  df-isom 5222  df-riota 5826  df-ov 5873  df-oprab 5874  df-mpo 5875  df-1st 6136  df-2nd 6137  df-recs 6301  df-irdg 6366  df-frec 6387  df-1o 6412  df-oadd 6416  df-er 6530  df-en 6736  df-dom 6737  df-fin 6738  df-pnf 7988  df-mnf 7989  df-xr 7990  df-ltxr 7991  df-le 7992  df-sub 8124  df-neg 8125  df-reap 8526  df-ap 8533  df-div 8624  df-inn 8914  df-2 8972  df-3 8973  df-4 8974  df-n0 9171  df-z 9248  df-uz 9523  df-q 9614  df-rp 9648  df-fz 10003  df-fzo 10136  df-seqfrec 10439  df-exp 10513  df-ihash 10747  df-cj 10842  df-re 10843  df-im 10844  df-rsqrt 10998  df-abs 10999  df-clim 11278  df-sumdc 11353

This theorem is referenced by:  cvgratnnlemrate  11529