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Theorem cvgratnnlemsumlt 11237
Description: Lemma for cvgratnn 11240. (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 9123 . . . 4  |-  ( ph  ->  M  e.  ZZ )
3 1zzd 9032 . . . 4  |-  ( ph  ->  1  e.  ZZ )
4 cvgratnn.n . . . . . 6  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
5 eluzelz 9284 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
64, 5syl 14 . . . . 5  |-  ( ph  ->  N  e.  ZZ )
76, 2zsubcld 9129 . . . 4  |-  ( ph  ->  ( N  -  M
)  e.  ZZ )
8 cvgratnn.3 . . . . . . 7  |-  ( ph  ->  A  e.  RR )
98recnd 7758 . . . . . 6  |-  ( ph  ->  A  e.  CC )
109adantr 272 . . . . 5  |-  ( (
ph  /\  k  e.  ( 1 ... ( N  -  M )
) )  ->  A  e.  CC )
11 elfznn 9774 . . . . . . 7  |-  ( k  e.  ( 1 ... ( N  -  M
) )  ->  k  e.  NN )
1211adantl 273 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 1 ... ( N  -  M )
) )  ->  k  e.  NN )
1312nnnn0d 8981 . . . . 5  |-  ( (
ph  /\  k  e.  ( 1 ... ( N  -  M )
) )  ->  k  e.  NN0 )
1410, 13expcld 10364 . . . 4  |-  ( (
ph  /\  k  e.  ( 1 ... ( N  -  M )
) )  ->  ( A ^ k )  e.  CC )
15 oveq2 5748 . . . 4  |-  ( k  =  ( i  -  M )  ->  ( A ^ k )  =  ( A ^ (
i  -  M ) ) )
162, 3, 7, 14, 15fsumshft 11153 . . 3  |-  ( ph  -> 
sum_ k  e.  ( 1 ... ( N  -  M ) ) ( A ^ k
)  =  sum_ i  e.  ( ( 1  +  M ) ... (
( N  -  M
)  +  M ) ) ( A ^
( i  -  M
) ) )
17 1cnd 7746 . . . . . 6  |-  ( ph  ->  1  e.  CC )
181nncnd 8691 . . . . . 6  |-  ( ph  ->  M  e.  CC )
1917, 18addcomd 7877 . . . . 5  |-  ( ph  ->  ( 1  +  M
)  =  ( M  +  1 ) )
206zcnd 9125 . . . . . 6  |-  ( ph  ->  N  e.  CC )
2120, 18npcand 8041 . . . . 5  |-  ( ph  ->  ( ( N  -  M )  +  M
)  =  N )
2219, 21oveq12d 5758 . . . 4  |-  ( ph  ->  ( ( 1  +  M ) ... (
( N  -  M
)  +  M ) )  =  ( ( M  +  1 ) ... N ) )
2322sumeq1d 11075 . . 3  |-  ( ph  -> 
sum_ i  e.  ( ( 1  +  M
) ... ( ( N  -  M )  +  M ) ) ( A ^ ( i  -  M ) )  =  sum_ i  e.  ( ( M  +  1 ) ... N ) ( A ^ (
i  -  M ) ) )
2416, 23eqtrd 2148 . 2  |-  ( ph  -> 
sum_ k  e.  ( 1 ... ( N  -  M ) ) ( A ^ k
)  =  sum_ i  e.  ( ( M  + 
1 ) ... N
) ( A ^
( i  -  M
) ) )
25 fzval3 9921 . . . . 5  |-  ( ( N  -  M )  e.  ZZ  ->  (
1 ... ( N  -  M ) )  =  ( 1..^ ( ( N  -  M )  +  1 ) ) )
2625sumeq1d 11075 . . . 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 7745 . . . . . 6  |-  ( ph  ->  1  e.  RR )
29 cvgratnn.4 . . . . . 6  |-  ( ph  ->  A  <  1 )
308, 28, 29ltapd 8362 . . . . 5  |-  ( ph  ->  A #  1 )
31 1nn0 8944 . . . . . 6  |-  1  e.  NN0
3231a1i 9 . . . . 5  |-  ( ph  ->  1  e.  NN0 )
337peano2zd 9127 . . . . . 6  |-  ( ph  ->  ( ( N  -  M )  +  1 )  e.  ZZ )
34 eluzle 9287 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  <_  N )
354, 34syl 14 . . . . . . . 8  |-  ( ph  ->  M  <_  N )
366zred 9124 . . . . . . . . 9  |-  ( ph  ->  N  e.  RR )
371nnred 8690 . . . . . . . . 9  |-  ( ph  ->  M  e.  RR )
3836, 37subge0d 8260 . . . . . . . 8  |-  ( ph  ->  ( 0  <_  ( N  -  M )  <->  M  <_  N ) )
3935, 38mpbird 166 . . . . . . 7  |-  ( ph  ->  0  <_  ( N  -  M ) )
407zred 9124 . . . . . . . 8  |-  ( ph  ->  ( N  -  M
)  e.  RR )
4128, 40addge02d 8259 . . . . . . 7  |-  ( ph  ->  ( 0  <_  ( N  -  M )  <->  1  <_  ( ( N  -  M )  +  1 ) ) )
4239, 41mpbid 146 . . . . . 6  |-  ( ph  ->  1  <_  ( ( N  -  M )  +  1 ) )
43 eluz2 9281 . . . . . 6  |-  ( ( ( N  -  M
)  +  1 )  e.  ( ZZ>= `  1
)  <->  ( 1  e.  ZZ  /\  ( ( N  -  M )  +  1 )  e.  ZZ  /\  1  <_ 
( ( N  -  M )  +  1 ) ) )
443, 33, 42, 43syl3anbrc 1148 . . . . 5  |-  ( ph  ->  ( ( N  -  M )  +  1 )  e.  ( ZZ>= ` 
1 ) )
459, 30, 32, 44geosergap 11215 . . . 4  |-  ( ph  -> 
sum_ k  e.  ( 1..^ ( ( N  -  M )  +  1 ) ) ( A ^ k )  =  ( ( ( A ^ 1 )  -  ( A ^
( ( N  -  M )  +  1 ) ) )  / 
( 1  -  A
) ) )
469exp1d 10359 . . . . . . 7  |-  ( ph  ->  ( A ^ 1 )  =  A )
4746, 8eqeltrd 2192 . . . . . 6  |-  ( ph  ->  ( A ^ 1 )  e.  RR )
48 cvgratnn.gt0 . . . . . . . . 9  |-  ( ph  ->  0  <  A )
498, 48elrpd 9427 . . . . . . . 8  |-  ( ph  ->  A  e.  RR+ )
5049, 33rpexpcld 10388 . . . . . . 7  |-  ( ph  ->  ( A ^ (
( N  -  M
)  +  1 ) )  e.  RR+ )
5150rpred 9429 . . . . . 6  |-  ( ph  ->  ( A ^ (
( N  -  M
)  +  1 ) )  e.  RR )
5247, 51resubcld 8107 . . . . 5  |-  ( ph  ->  ( ( A ^
1 )  -  ( A ^ ( ( N  -  M )  +  1 ) ) )  e.  RR )
5328, 8resubcld 8107 . . . . . 6  |-  ( ph  ->  ( 1  -  A
)  e.  RR )
548, 28posdifd 8257 . . . . . . 7  |-  ( ph  ->  ( A  <  1  <->  0  <  ( 1  -  A ) ) )
5529, 54mpbid 146 . . . . . 6  |-  ( ph  ->  0  <  ( 1  -  A ) )
5653, 55elrpd 9427 . . . . 5  |-  ( ph  ->  ( 1  -  A
)  e.  RR+ )
5746oveq1d 5755 . . . . . 6  |-  ( ph  ->  ( ( A ^
1 )  -  ( A ^ ( ( N  -  M )  +  1 ) ) )  =  ( A  -  ( A ^ ( ( N  -  M )  +  1 ) ) ) )
588, 50ltsubrpd 9462 . . . . . 6  |-  ( ph  ->  ( A  -  ( A ^ ( ( N  -  M )  +  1 ) ) )  <  A )
5957, 58eqbrtrd 3918 . . . . 5  |-  ( ph  ->  ( ( A ^
1 )  -  ( A ^ ( ( N  -  M )  +  1 ) ) )  <  A )
6052, 8, 56, 59ltdiv1dd 9487 . . . 4  |-  ( ph  ->  ( ( ( A ^ 1 )  -  ( A ^ ( ( N  -  M )  +  1 ) ) )  /  ( 1  -  A ) )  <  ( A  / 
( 1  -  A
) ) )
6145, 60eqbrtrd 3918 . . 3  |-  ( ph  -> 
sum_ k  e.  ( 1..^ ( ( N  -  M )  +  1 ) ) ( A ^ k )  <  ( A  / 
( 1  -  A
) ) )
6227, 61eqbrtrd 3918 . 2  |-  ( ph  -> 
sum_ k  e.  ( 1 ... ( N  -  M ) ) ( A ^ k
)  <  ( A  /  ( 1  -  A ) ) )
6324, 62eqbrtrrd 3920 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 103    = wceq 1314    e. wcel 1463   class class class wbr 3897   ` cfv 5091  (class class class)co 5740   CCcc 7582   RRcr 7583   0cc0 7584   1c1 7585    + caddc 7587    x. cmul 7589    < clt 7764    <_ cle 7765    - cmin 7897    / cdiv 8392   NNcn 8677   NN0cn0 8928   ZZcz 9005   ZZ>=cuz 9275   ...cfz 9730  ..^cfzo 9859   ^cexp 10232   abscabs 10709   sum_csu 11062
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703  ax-caucvg 7704
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-isom 5100  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-frec 6254  df-1o 6279  df-oadd 6283  df-er 6395  df-en 6601  df-dom 6602  df-fin 6603  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8393  df-inn 8678  df-2 8736  df-3 8737  df-4 8738  df-n0 8929  df-z 9006  df-uz 9276  df-q 9361  df-rp 9391  df-fz 9731  df-fzo 9860  df-seqfrec 10159  df-exp 10233  df-ihash 10462  df-cj 10554  df-re 10555  df-im 10556  df-rsqrt 10710  df-abs 10711  df-clim 10988  df-sumdc 11063
This theorem is referenced by:  cvgratnnlemrate  11239
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