Theorem cvgratnnlemsumlt 12088

Description: Lemma for cvgratnn 12091. (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 9600	. . . 4 |- ( ph -> M e. ZZ )
3		1zzd 9505	. . . 4 |- ( ph -> 1 e. ZZ )
4		cvgratnn.n	. . . . . 6 |- ( ph -> N e. ( ZZ>= ` M ) )
5		eluzelz 9764	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
6	4, 5	syl 14	. . . . 5 |- ( ph -> N e. ZZ )
7	6, 2	zsubcld 9606	. . . 4 |- ( ph -> ( N - M ) e. ZZ )
8		cvgratnn.3	. . . . . . 7 |- ( ph -> A e. RR )
9	8	recnd 8207	. . . . . 6 |- ( ph -> A e. CC )
10	9	adantr 276	. . . . 5 |- ( ( ph /\ k e. ( 1 ... ( N - M ) ) ) -> A e. CC )
11		elfznn 10288	. . . . . . 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 9454	. . . . 5 |- ( ( ph /\ k e. ( 1 ... ( N - M ) ) ) -> k e. NN0 )
14	10, 13	expcld 10934	. . . 4 |- ( ( ph /\ k e. ( 1 ... ( N - M ) ) ) -> ( A ^ k ) e. CC )
15		oveq2 6025	. . . 4 |- ( k = ( i - M ) -> ( A ^ k ) = ( A ^ ( i - M ) ) )
16	2, 3, 7, 14, 15	fsumshft 12004	. . 3 |- ( ph -> sum_ k e. ( 1 ... ( N - M ) ) ( A ^ k ) = sum_ i e. ( ( 1 + M ) ... ( ( N - M ) + M ) ) ( A ^ ( i - M ) ) )
17		1cnd 8194	. . . . . 6 |- ( ph -> 1 e. CC )
18	1	nncnd 9156	. . . . . 6 |- ( ph -> M e. CC )
19	17, 18	addcomd 8329	. . . . 5 |- ( ph -> ( 1 + M ) = ( M + 1 ) )
20	6	zcnd 9602	. . . . . 6 |- ( ph -> N e. CC )
21	20, 18	npcand 8493	. . . . 5 |- ( ph -> ( ( N - M ) + M ) = N )
22	19, 21	oveq12d 6035	. . . 4 |- ( ph -> ( ( 1 + M ) ... ( ( N - M ) + M ) ) = ( ( M + 1 ) ... N ) )
23	22	sumeq1d 11926	. . 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 2264	. 2 |- ( ph -> sum_ k e. ( 1 ... ( N - M ) ) ( A ^ k ) = sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) )
25		fzval3 10448	. . . . 5 |- ( ( N - M ) e. ZZ -> ( 1 ... ( N - M ) ) = ( 1..^ ( ( N - M ) + 1 ) ) )
26	25	sumeq1d 11926	. . . 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 8193	. . . . . 6 |- ( ph -> 1 e. RR )
29		cvgratnn.4	. . . . . 6 |- ( ph -> A < 1 )
30	8, 28, 29	ltapd 8817	. . . . 5 |- ( ph -> A # 1 )
31		1nn0 9417	. . . . . 6 |- 1 e. NN0
32	31	a1i 9	. . . . 5 |- ( ph -> 1 e. NN0 )
33	7	peano2zd 9604	. . . . . 6 |- ( ph -> ( ( N - M ) + 1 ) e. ZZ )
34		eluzle 9767	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> M <_ N )
35	4, 34	syl 14	. . . . . . . 8 |- ( ph -> M <_ N )
36	6	zred 9601	. . . . . . . . 9 |- ( ph -> N e. RR )
37	1	nnred 9155	. . . . . . . . 9 |- ( ph -> M e. RR )
38	36, 37	subge0d 8714	. . . . . . . 8 |- ( ph -> ( 0 <_ ( N - M ) <-> M <_ N ) )
39	35, 38	mpbird 167	. . . . . . 7 |- ( ph -> 0 <_ ( N - M ) )
40	7	zred 9601	. . . . . . . 8 |- ( ph -> ( N - M ) e. RR )
41	28, 40	addge02d 8713	. . . . . . 7 |- ( ph -> ( 0 <_ ( N - M ) <-> 1 <_ ( ( N - M ) + 1 ) ) )
42	39, 41	mpbid 147	. . . . . 6 |- ( ph -> 1 <_ ( ( N - M ) + 1 ) )
43		eluz2 9760	. . . . . 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 1207	. . . . 5 |- ( ph -> ( ( N - M ) + 1 ) e. ( ZZ>= ` 1 ) )
45	9, 30, 32, 44	geosergap 12066	. . . 4 |- ( ph -> sum_ k e. ( 1..^ ( ( N - M ) + 1 ) ) ( A ^ k ) = ( ( ( A ^ 1 ) - ( A ^ ( ( N - M ) + 1 ) ) ) / ( 1 - A ) ) )
46	9	exp1d 10929	. . . . . . 7 |- ( ph -> ( A ^ 1 ) = A )
47	46, 8	eqeltrd 2308	. . . . . 6 |- ( ph -> ( A ^ 1 ) e. RR )
48		cvgratnn.gt0	. . . . . . . . 9 |- ( ph -> 0 < A )
49	8, 48	elrpd 9927	. . . . . . . 8 |- ( ph -> A e. RR+ )
50	49, 33	rpexpcld 10958	. . . . . . 7 |- ( ph -> ( A ^ ( ( N - M ) + 1 ) ) e. RR+ )
51	50	rpred 9930	. . . . . 6 |- ( ph -> ( A ^ ( ( N - M ) + 1 ) ) e. RR )
52	47, 51	resubcld 8559	. . . . 5 |- ( ph -> ( ( A ^ 1 ) - ( A ^ ( ( N - M ) + 1 ) ) ) e. RR )
53	28, 8	resubcld 8559	. . . . . 6 |- ( ph -> ( 1 - A ) e. RR )
54	8, 28	posdifd 8711	. . . . . . 7 |- ( ph -> ( A < 1 <-> 0 < ( 1 - A ) ) )
55	29, 54	mpbid 147	. . . . . 6 |- ( ph -> 0 < ( 1 - A ) )
56	53, 55	elrpd 9927	. . . . 5 |- ( ph -> ( 1 - A ) e. RR+ )
57	46	oveq1d 6032	. . . . . 6 |- ( ph -> ( ( A ^ 1 ) - ( A ^ ( ( N - M ) + 1 ) ) ) = ( A - ( A ^ ( ( N - M ) + 1 ) ) ) )
58	8, 50	ltsubrpd 9963	. . . . . 6 |- ( ph -> ( A - ( A ^ ( ( N - M ) + 1 ) ) ) < A )
59	57, 58	eqbrtrd 4110	. . . . 5 |- ( ph -> ( ( A ^ 1 ) - ( A ^ ( ( N - M ) + 1 ) ) ) < A )
60	52, 8, 56, 59	ltdiv1dd 9988	. . . 4 |- ( ph -> ( ( ( A ^ 1 ) - ( A ^ ( ( N - M ) + 1 ) ) ) / ( 1 - A ) ) < ( A / ( 1 - A ) ) )
61	45, 60	eqbrtrd 4110	. . 3 |- ( ph -> sum_ k e. ( 1..^ ( ( N - M ) + 1 ) ) ( A ^ k ) < ( A / ( 1 - A ) ) )
62	27, 61	eqbrtrd 4110	. 2 |- ( ph -> sum_ k e. ( 1 ... ( N - M ) ) ( A ^ k ) < ( A / ( 1 - A ) ) )
63	24, 62	eqbrtrrd 4112	1 |- ( ph -> sum_ i e. ( ( M + 1 ) ... N ) ( A ^ ( i - M ) ) < ( A / ( 1 - A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   CCcc 8029   RRcr 8030   0cc0 8031   1c1 8032   + caddc 8034   x. cmul 8036   < clt 8213   <_ cle 8214   - cmin 8349   / cdiv 8851   NNcn 9142   NN0cn0 9401   ZZcz 9478   ZZ>=cuz 9754   ...cfz 10242  ..^cfzo 10376   ^cexp 10799   abscabs 11557   sum_csu 11913

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-frec 6556  df-1o 6581  df-oadd 6585  df-er 6701  df-en 6909  df-dom 6910  df-fin 6911  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-fz 10243  df-fzo 10377  df-seqfrec 10709  df-exp 10800  df-ihash 11037  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-clim 11839  df-sumdc 11914

This theorem is referenced by:  cvgratnnlemrate  12090