Theorem cvgratnnlemsumlt 12107

Description: Lemma for cvgratnn 12110. (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 9601	. . . 4 |- ( ph -> M e. ZZ )
3		1zzd 9506	. . . 4 |- ( ph -> 1 e. ZZ )
4		cvgratnn.n	. . . . . 6 |- ( ph -> N e. ( ZZ>= ` M ) )
5		eluzelz 9765	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
6	4, 5	syl 14	. . . . 5 |- ( ph -> N e. ZZ )
7	6, 2	zsubcld 9607	. . . 4 |- ( ph -> ( N - M ) e. ZZ )
8		cvgratnn.3	. . . . . . 7 |- ( ph -> A e. RR )
9	8	recnd 8208	. . . . . 6 |- ( ph -> A e. CC )
10	9	adantr 276	. . . . 5 |- ( ( ph /\ k e. ( 1 ... ( N - M ) ) ) -> A e. CC )
11		elfznn 10289	. . . . . . 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 9455	. . . . 5 |- ( ( ph /\ k e. ( 1 ... ( N - M ) ) ) -> k e. NN0 )
14	10, 13	expcld 10936	. . . 4 |- ( ( ph /\ k e. ( 1 ... ( N - M ) ) ) -> ( A ^ k ) e. CC )
15		oveq2 6026	. . . 4 |- ( k = ( i - M ) -> ( A ^ k ) = ( A ^ ( i - M ) ) )
16	2, 3, 7, 14, 15	fsumshft 12023	. . 3 |- ( ph -> sum_ k e. ( 1 ... ( N - M ) ) ( A ^ k ) = sum_ i e. ( ( 1 + M ) ... ( ( N - M ) + M ) ) ( A ^ ( i - M ) ) )
17		1cnd 8195	. . . . . 6 |- ( ph -> 1 e. CC )
18	1	nncnd 9157	. . . . . 6 |- ( ph -> M e. CC )
19	17, 18	addcomd 8330	. . . . 5 |- ( ph -> ( 1 + M ) = ( M + 1 ) )
20	6	zcnd 9603	. . . . . 6 |- ( ph -> N e. CC )
21	20, 18	npcand 8494	. . . . 5 |- ( ph -> ( ( N - M ) + M ) = N )
22	19, 21	oveq12d 6036	. . . 4 |- ( ph -> ( ( 1 + M ) ... ( ( N - M ) + M ) ) = ( ( M + 1 ) ... N ) )
23	22	sumeq1d 11944	. . 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 10450	. . . . 5 |- ( ( N - M ) e. ZZ -> ( 1 ... ( N - M ) ) = ( 1..^ ( ( N - M ) + 1 ) ) )
26	25	sumeq1d 11944	. . . 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 8194	. . . . . 6 |- ( ph -> 1 e. RR )
29		cvgratnn.4	. . . . . 6 |- ( ph -> A < 1 )
30	8, 28, 29	ltapd 8818	. . . . 5 |- ( ph -> A # 1 )
31		1nn0 9418	. . . . . 6 |- 1 e. NN0
32	31	a1i 9	. . . . 5 |- ( ph -> 1 e. NN0 )
33	7	peano2zd 9605	. . . . . 6 |- ( ph -> ( ( N - M ) + 1 ) e. ZZ )
34		eluzle 9768	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> M <_ N )
35	4, 34	syl 14	. . . . . . . 8 |- ( ph -> M <_ N )
36	6	zred 9602	. . . . . . . . 9 |- ( ph -> N e. RR )
37	1	nnred 9156	. . . . . . . . 9 |- ( ph -> M e. RR )
38	36, 37	subge0d 8715	. . . . . . . 8 |- ( ph -> ( 0 <_ ( N - M ) <-> M <_ N ) )
39	35, 38	mpbird 167	. . . . . . 7 |- ( ph -> 0 <_ ( N - M ) )
40	7	zred 9602	. . . . . . . 8 |- ( ph -> ( N - M ) e. RR )
41	28, 40	addge02d 8714	. . . . . . 7 |- ( ph -> ( 0 <_ ( N - M ) <-> 1 <_ ( ( N - M ) + 1 ) ) )
42	39, 41	mpbid 147	. . . . . 6 |- ( ph -> 1 <_ ( ( N - M ) + 1 ) )
43		eluz2 9761	. . . . . 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 12085	. . . 4 |- ( ph -> sum_ k e. ( 1..^ ( ( N - M ) + 1 ) ) ( A ^ k ) = ( ( ( A ^ 1 ) - ( A ^ ( ( N - M ) + 1 ) ) ) / ( 1 - A ) ) )
46	9	exp1d 10931	. . . . . . 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 9928	. . . . . . . 8 |- ( ph -> A e. RR+ )
50	49, 33	rpexpcld 10960	. . . . . . 7 |- ( ph -> ( A ^ ( ( N - M ) + 1 ) ) e. RR+ )
51	50	rpred 9931	. . . . . 6 |- ( ph -> ( A ^ ( ( N - M ) + 1 ) ) e. RR )
52	47, 51	resubcld 8560	. . . . 5 |- ( ph -> ( ( A ^ 1 ) - ( A ^ ( ( N - M ) + 1 ) ) ) e. RR )
53	28, 8	resubcld 8560	. . . . . 6 |- ( ph -> ( 1 - A ) e. RR )
54	8, 28	posdifd 8712	. . . . . . 7 |- ( ph -> ( A < 1 <-> 0 < ( 1 - A ) ) )
55	29, 54	mpbid 147	. . . . . 6 |- ( ph -> 0 < ( 1 - A ) )
56	53, 55	elrpd 9928	. . . . 5 |- ( ph -> ( 1 - A ) e. RR+ )
57	46	oveq1d 6033	. . . . . 6 |- ( ph -> ( ( A ^ 1 ) - ( A ^ ( ( N - M ) + 1 ) ) ) = ( A - ( A ^ ( ( N - M ) + 1 ) ) ) )
58	8, 50	ltsubrpd 9964	. . . . . 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 9989	. . . 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 6018   CCcc 8030   RRcr 8031   0cc0 8032   1c1 8033   + caddc 8035   x. cmul 8037   < clt 8214   <_ cle 8215   - cmin 8350   / cdiv 8852   NNcn 9143   NN0cn0 9402   ZZcz 9479   ZZ>=cuz 9755   ...cfz 10243  ..^cfzo 10377   ^cexp 10801   abscabs 11575   sum_csu 11931

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 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

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 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-frec 6557  df-1o 6582  df-oadd 6586  df-er 6702  df-en 6910  df-dom 6911  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-fz 10244  df-fzo 10378  df-seqfrec 10711  df-exp 10802  df-ihash 11039  df-cj 11420  df-re 11421  df-im 11422  df-rsqrt 11576  df-abs 11577  df-clim 11857  df-sumdc 11932

This theorem is referenced by:  cvgratnnlemrate  12109