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Theorem cvgratnnlemsumlt 11571
Description: Lemma for cvgratnn 11574. (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
StepHypRef Expression
1 cvgratnn.m . . . . 5  |-  ( ph  ->  M  e.  NN )
21nnzd 9405 . . . 4  |-  ( ph  ->  M  e.  ZZ )
3 1zzd 9311 . . . 4  |-  ( ph  ->  1  e.  ZZ )
4 cvgratnn.n . . . . . 6  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
5 eluzelz 9568 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
64, 5syl 14 . . . . 5  |-  ( ph  ->  N  e.  ZZ )
76, 2zsubcld 9411 . . . 4  |-  ( ph  ->  ( N  -  M
)  e.  ZZ )
8 cvgratnn.3 . . . . . . 7  |-  ( ph  ->  A  e.  RR )
98recnd 8017 . . . . . 6  |-  ( ph  ->  A  e.  CC )
109adantr 276 . . . . 5  |-  ( (
ph  /\  k  e.  ( 1 ... ( N  -  M )
) )  ->  A  e.  CC )
11 elfznn 10086 . . . . . . 7  |-  ( k  e.  ( 1 ... ( N  -  M
) )  ->  k  e.  NN )
1211adantl 277 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 1 ... ( N  -  M )
) )  ->  k  e.  NN )
1312nnnn0d 9260 . . . . 5  |-  ( (
ph  /\  k  e.  ( 1 ... ( N  -  M )
) )  ->  k  e.  NN0 )
1410, 13expcld 10688 . . . 4  |-  ( (
ph  /\  k  e.  ( 1 ... ( N  -  M )
) )  ->  ( A ^ k )  e.  CC )
15 oveq2 5905 . . . 4  |-  ( k  =  ( i  -  M )  ->  ( A ^ k )  =  ( A ^ (
i  -  M ) ) )
162, 3, 7, 14, 15fsumshft 11487 . . 3  |-  ( ph  -> 
sum_ k  e.  ( 1 ... ( N  -  M ) ) ( A ^ k
)  =  sum_ i  e.  ( ( 1  +  M ) ... (
( N  -  M
)  +  M ) ) ( A ^
( i  -  M
) ) )
17 1cnd 8004 . . . . . 6  |-  ( ph  ->  1  e.  CC )
181nncnd 8964 . . . . . 6  |-  ( ph  ->  M  e.  CC )
1917, 18addcomd 8139 . . . . 5  |-  ( ph  ->  ( 1  +  M
)  =  ( M  +  1 ) )
206zcnd 9407 . . . . . 6  |-  ( ph  ->  N  e.  CC )
2120, 18npcand 8303 . . . . 5  |-  ( ph  ->  ( ( N  -  M )  +  M
)  =  N )
2219, 21oveq12d 5915 . . . 4  |-  ( ph  ->  ( ( 1  +  M ) ... (
( N  -  M
)  +  M ) )  =  ( ( M  +  1 ) ... N ) )
2322sumeq1d 11409 . . 3  |-  ( ph  -> 
sum_ i  e.  ( ( 1  +  M
) ... ( ( N  -  M )  +  M ) ) ( A ^ ( i  -  M ) )  =  sum_ i  e.  ( ( M  +  1 ) ... N ) ( A ^ (
i  -  M ) ) )
2416, 23eqtrd 2222 . 2  |-  ( ph  -> 
sum_ k  e.  ( 1 ... ( N  -  M ) ) ( A ^ k
)  =  sum_ i  e.  ( ( M  + 
1 ) ... N
) ( A ^
( i  -  M
) ) )
25 fzval3 10236 . . . . 5  |-  ( ( N  -  M )  e.  ZZ  ->  (
1 ... ( N  -  M ) )  =  ( 1..^ ( ( N  -  M )  +  1 ) ) )
2625sumeq1d 11409 . . . 4  |-  ( ( N  -  M )  e.  ZZ  ->  sum_ k  e.  ( 1 ... ( N  -  M )
) ( A ^
k )  =  sum_ k  e.  ( 1..^ ( ( N  -  M )  +  1 ) ) ( A ^ k ) )
277, 26syl 14 . . 3  |-  ( ph  -> 
sum_ k  e.  ( 1 ... ( N  -  M ) ) ( A ^ k
)  =  sum_ k  e.  ( 1..^ ( ( N  -  M )  +  1 ) ) ( A ^ k
) )
28 1red 8003 . . . . . 6  |-  ( ph  ->  1  e.  RR )
29 cvgratnn.4 . . . . . 6  |-  ( ph  ->  A  <  1 )
308, 28, 29ltapd 8626 . . . . 5  |-  ( ph  ->  A #  1 )
31 1nn0 9223 . . . . . 6  |-  1  e.  NN0
3231a1i 9 . . . . 5  |-  ( ph  ->  1  e.  NN0 )
337peano2zd 9409 . . . . . 6  |-  ( ph  ->  ( ( N  -  M )  +  1 )  e.  ZZ )
34 eluzle 9571 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  <_  N )
354, 34syl 14 . . . . . . . 8  |-  ( ph  ->  M  <_  N )
366zred 9406 . . . . . . . . 9  |-  ( ph  ->  N  e.  RR )
371nnred 8963 . . . . . . . . 9  |-  ( ph  ->  M  e.  RR )
3836, 37subge0d 8523 . . . . . . . 8  |-  ( ph  ->  ( 0  <_  ( N  -  M )  <->  M  <_  N ) )
3935, 38mpbird 167 . . . . . . 7  |-  ( ph  ->  0  <_  ( N  -  M ) )
407zred 9406 . . . . . . . 8  |-  ( ph  ->  ( N  -  M
)  e.  RR )
4128, 40addge02d 8522 . . . . . . 7  |-  ( ph  ->  ( 0  <_  ( N  -  M )  <->  1  <_  ( ( N  -  M )  +  1 ) ) )
4239, 41mpbid 147 . . . . . 6  |-  ( ph  ->  1  <_  ( ( N  -  M )  +  1 ) )
43 eluz2 9565 . . . . . 6  |-  ( ( ( N  -  M
)  +  1 )  e.  ( ZZ>= `  1
)  <->  ( 1  e.  ZZ  /\  ( ( N  -  M )  +  1 )  e.  ZZ  /\  1  <_ 
( ( N  -  M )  +  1 ) ) )
443, 33, 42, 43syl3anbrc 1183 . . . . 5  |-  ( ph  ->  ( ( N  -  M )  +  1 )  e.  ( ZZ>= ` 
1 ) )
459, 30, 32, 44geosergap 11549 . . . 4  |-  ( ph  -> 
sum_ k  e.  ( 1..^ ( ( N  -  M )  +  1 ) ) ( A ^ k )  =  ( ( ( A ^ 1 )  -  ( A ^
( ( N  -  M )  +  1 ) ) )  / 
( 1  -  A
) ) )
469exp1d 10683 . . . . . . 7  |-  ( ph  ->  ( A ^ 1 )  =  A )
4746, 8eqeltrd 2266 . . . . . 6  |-  ( ph  ->  ( A ^ 1 )  e.  RR )
48 cvgratnn.gt0 . . . . . . . . 9  |-  ( ph  ->  0  <  A )
498, 48elrpd 9725 . . . . . . . 8  |-  ( ph  ->  A  e.  RR+ )
5049, 33rpexpcld 10712 . . . . . . 7  |-  ( ph  ->  ( A ^ (
( N  -  M
)  +  1 ) )  e.  RR+ )
5150rpred 9728 . . . . . 6  |-  ( ph  ->  ( A ^ (
( N  -  M
)  +  1 ) )  e.  RR )
5247, 51resubcld 8369 . . . . 5  |-  ( ph  ->  ( ( A ^
1 )  -  ( A ^ ( ( N  -  M )  +  1 ) ) )  e.  RR )
5328, 8resubcld 8369 . . . . . 6  |-  ( ph  ->  ( 1  -  A
)  e.  RR )
548, 28posdifd 8520 . . . . . . 7  |-  ( ph  ->  ( A  <  1  <->  0  <  ( 1  -  A ) ) )
5529, 54mpbid 147 . . . . . 6  |-  ( ph  ->  0  <  ( 1  -  A ) )
5653, 55elrpd 9725 . . . . 5  |-  ( ph  ->  ( 1  -  A
)  e.  RR+ )
5746oveq1d 5912 . . . . . 6  |-  ( ph  ->  ( ( A ^
1 )  -  ( A ^ ( ( N  -  M )  +  1 ) ) )  =  ( A  -  ( A ^ ( ( N  -  M )  +  1 ) ) ) )
588, 50ltsubrpd 9761 . . . . . 6  |-  ( ph  ->  ( A  -  ( A ^ ( ( N  -  M )  +  1 ) ) )  <  A )
5957, 58eqbrtrd 4040 . . . . 5  |-  ( ph  ->  ( ( A ^
1 )  -  ( A ^ ( ( N  -  M )  +  1 ) ) )  <  A )
6052, 8, 56, 59ltdiv1dd 9786 . . . 4  |-  ( ph  ->  ( ( ( A ^ 1 )  -  ( A ^ ( ( N  -  M )  +  1 ) ) )  /  ( 1  -  A ) )  <  ( A  / 
( 1  -  A
) ) )
6145, 60eqbrtrd 4040 . . 3  |-  ( ph  -> 
sum_ k  e.  ( 1..^ ( ( N  -  M )  +  1 ) ) ( A ^ k )  <  ( A  / 
( 1  -  A
) ) )
6227, 61eqbrtrd 4040 . 2  |-  ( ph  -> 
sum_ k  e.  ( 1 ... ( N  -  M ) ) ( A ^ k
)  <  ( A  /  ( 1  -  A ) ) )
6324, 62eqbrtrrd 4042 1  |-  ( ph  -> 
sum_ i  e.  ( ( M  +  1 ) ... N ) ( A ^ (
i  -  M ) )  <  ( A  /  ( 1  -  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   class class class wbr 4018   ` cfv 5235  (class class class)co 5897   CCcc 7840   RRcr 7841   0cc0 7842   1c1 7843    + caddc 7845    x. cmul 7847    < clt 8023    <_ cle 8024    - cmin 8159    / cdiv 8660   NNcn 8950   NN0cn0 9207   ZZcz 9284   ZZ>=cuz 9559   ...cfz 10040  ..^cfzo 10174   ^cexp 10553   abscabs 11041   sum_csu 11396
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-pre-mulext 7960  ax-arch 7961  ax-caucvg 7962
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 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-isom 5244  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-frec 6417  df-1o 6442  df-oadd 6446  df-er 6560  df-en 6768  df-dom 6769  df-fin 6770  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570  df-div 8661  df-inn 8951  df-2 9009  df-3 9010  df-4 9011  df-n0 9208  df-z 9285  df-uz 9560  df-q 9652  df-rp 9686  df-fz 10041  df-fzo 10175  df-seqfrec 10479  df-exp 10554  df-ihash 10791  df-cj 10886  df-re 10887  df-im 10888  df-rsqrt 11042  df-abs 11043  df-clim 11322  df-sumdc 11397
This theorem is referenced by:  cvgratnnlemrate  11573
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