ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cvgratnnlemabsle Unicode version

Theorem cvgratnnlemabsle 11236
Description: Lemma for cvgratnn 11240. (Contributed by Jim Kingdon, 21-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
cvgratnnlemabsle  |-  ( ph  ->  ( abs `  sum_ i  e.  ( ( M  +  1 ) ... N ) ( F `  i ) )  <_  ( ( abs `  ( F `  M ) )  x. 
sum_ i  e.  ( ( M  +  1 ) ... N ) ( A ^ (
i  -  M ) ) ) )
Distinct variable groups:    A, k    k, F    k, N    ph, k    i, F, k    i, M, k   
i, N    ph, i
Allowed substitution hint:    A( i)

Proof of Theorem cvgratnnlemabsle
StepHypRef Expression
1 cvgratnn.m . . . . . . . 8  |-  ( ph  ->  M  e.  NN )
21nnzd 9123 . . . . . . 7  |-  ( ph  ->  M  e.  ZZ )
32peano2zd 9127 . . . . . 6  |-  ( ph  ->  ( M  +  1 )  e.  ZZ )
4 cvgratnn.n . . . . . . 7  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
5 eluzelz 9284 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
64, 5syl 14 . . . . . 6  |-  ( ph  ->  N  e.  ZZ )
73, 6fzfigd 10144 . . . . 5  |-  ( ph  ->  ( ( M  + 
1 ) ... N
)  e.  Fin )
8 fveq2 5387 . . . . . . 7  |-  ( k  =  i  ->  ( F `  k )  =  ( F `  i ) )
98eleq1d 2184 . . . . . 6  |-  ( k  =  i  ->  (
( F `  k
)  e.  CC  <->  ( F `  i )  e.  CC ) )
10 cvgratnn.6 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  e.  CC )
1110ralrimiva 2480 . . . . . . 7  |-  ( ph  ->  A. k  e.  NN  ( F `  k )  e.  CC )
1211adantr 272 . . . . . 6  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  A. k  e.  NN  ( F `  k )  e.  CC )
13 elfzelz 9746 . . . . . . . 8  |-  ( i  e.  ( ( M  +  1 ) ... N )  ->  i  e.  ZZ )
1413adantl 273 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  i  e.  ZZ )
15 0red 7731 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  0  e.  RR )
161peano2nnd 8692 . . . . . . . . . 10  |-  ( ph  ->  ( M  +  1 )  e.  NN )
1716adantr 272 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( M  +  1 )  e.  NN )
1817nnred 8690 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( M  +  1 )  e.  RR )
1914zred 9124 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  i  e.  RR )
2016nngt0d 8721 . . . . . . . . 9  |-  ( ph  ->  0  <  ( M  +  1 ) )
2120adantr 272 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  0  <  ( M  +  1 ) )
22 elfzle1 9747 . . . . . . . . 9  |-  ( i  e.  ( ( M  +  1 ) ... N )  ->  ( M  +  1 )  <_  i )
2322adantl 273 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( M  +  1 )  <_  i )
2415, 18, 19, 21, 23ltletrd 8149 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  0  <  i )
25 elnnz 9015 . . . . . . 7  |-  ( i  e.  NN  <->  ( i  e.  ZZ  /\  0  < 
i ) )
2614, 24, 25sylanbrc 411 . . . . . 6  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  i  e.  NN )
279, 12, 26rspcdva 2766 . . . . 5  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( F `  i )  e.  CC )
287, 27fsumcl 11109 . . . 4  |-  ( ph  -> 
sum_ i  e.  ( ( M  +  1 ) ... N ) ( F `  i
)  e.  CC )
2928abscld 10893 . . 3  |-  ( ph  ->  ( abs `  sum_ i  e.  ( ( M  +  1 ) ... N ) ( F `  i ) )  e.  RR )
3027abscld 10893 . . . 4  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( abs `  ( F `  i ) )  e.  RR )
317, 30fsumrecl 11110 . . 3  |-  ( ph  -> 
sum_ i  e.  ( ( M  +  1 ) ... N ) ( abs `  ( F `  i )
)  e.  RR )
32 fveq2 5387 . . . . . . . . 9  |-  ( k  =  M  ->  ( F `  k )  =  ( F `  M ) )
3332eleq1d 2184 . . . . . . . 8  |-  ( k  =  M  ->  (
( F `  k
)  e.  CC  <->  ( F `  M )  e.  CC ) )
3433, 11, 1rspcdva 2766 . . . . . . 7  |-  ( ph  ->  ( F `  M
)  e.  CC )
3534adantr 272 . . . . . 6  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( F `  M )  e.  CC )
3635abscld 10893 . . . . 5  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( abs `  ( F `  M ) )  e.  RR )
37 cvgratnn.3 . . . . . . 7  |-  ( ph  ->  A  e.  RR )
3837adantr 272 . . . . . 6  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  A  e.  RR )
392adantr 272 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  M  e.  ZZ )
4014, 39zsubcld 9129 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  (
i  -  M )  e.  ZZ )
411adantr 272 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  M  e.  NN )
4241nnred 8690 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  M  e.  RR )
4342lep1d 8646 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  M  <_  ( M  +  1 ) )
4442, 18, 19, 43, 23letrd 7850 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  M  <_  i )
4519, 42subge0d 8260 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  (
0  <_  ( i  -  M )  <->  M  <_  i ) )
4644, 45mpbird 166 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  0  <_  ( i  -  M
) )
47 elnn0z 9018 . . . . . . 7  |-  ( ( i  -  M )  e.  NN0  <->  ( ( i  -  M )  e.  ZZ  /\  0  <_ 
( i  -  M
) ) )
4840, 46, 47sylanbrc 411 . . . . . 6  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  (
i  -  M )  e.  NN0 )
4938, 48reexpcld 10381 . . . . 5  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( A ^ ( i  -  M ) )  e.  RR )
5036, 49remulcld 7760 . . . 4  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  (
( abs `  ( F `  M )
)  x.  ( A ^ ( i  -  M ) ) )  e.  RR )
517, 50fsumrecl 11110 . . 3  |-  ( ph  -> 
sum_ i  e.  ( ( M  +  1 ) ... N ) ( ( abs `  ( F `  M )
)  x.  ( A ^ ( i  -  M ) ) )  e.  RR )
527, 27fsumabs 11174 . . 3  |-  ( ph  ->  ( abs `  sum_ i  e.  ( ( M  +  1 ) ... N ) ( F `  i ) )  <_  sum_ i  e.  ( ( M  + 
1 ) ... N
) ( abs `  ( F `  i )
) )
53 cvgratnn.4 . . . . . 6  |-  ( ph  ->  A  <  1 )
5453adantr 272 . . . . 5  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  A  <  1 )
55 cvgratnn.gt0 . . . . . 6  |-  ( ph  ->  0  <  A )
5655adantr 272 . . . . 5  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  0  <  A )
5710adantlr 466 . . . . 5  |-  ( ( ( ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  /\  k  e.  NN )  ->  ( F `  k )  e.  CC )
58 cvgratnn.7 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( abs `  ( F `  (
k  +  1 ) ) )  <_  ( A  x.  ( abs `  ( F `  k
) ) ) )
5958adantlr 466 . . . . 5  |-  ( ( ( ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  /\  k  e.  NN )  ->  ( abs `  ( F `  ( k  +  1 ) ) )  <_ 
( A  x.  ( abs `  ( F `  k ) ) ) )
60 eluz2 9281 . . . . . 6  |-  ( i  e.  ( ZZ>= `  M
)  <->  ( M  e.  ZZ  /\  i  e.  ZZ  /\  M  <_ 
i ) )
6139, 14, 44, 60syl3anbrc 1148 . . . . 5  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  i  e.  ( ZZ>= `  M )
)
6238, 54, 56, 57, 59, 41, 61cvgratnnlemmn 11234 . . . 4  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( abs `  ( F `  i ) )  <_ 
( ( abs `  ( F `  M )
)  x.  ( A ^ ( i  -  M ) ) ) )
637, 30, 50, 62fsumle 11172 . . 3  |-  ( ph  -> 
sum_ i  e.  ( ( M  +  1 ) ... N ) ( abs `  ( F `  i )
)  <_  sum_ i  e.  ( ( M  + 
1 ) ... N
) ( ( abs `  ( F `  M
) )  x.  ( A ^ ( i  -  M ) ) ) )
6429, 31, 51, 52, 63letrd 7850 . 2  |-  ( ph  ->  ( abs `  sum_ i  e.  ( ( M  +  1 ) ... N ) ( F `  i ) )  <_  sum_ i  e.  ( ( M  + 
1 ) ... N
) ( ( abs `  ( F `  M
) )  x.  ( A ^ ( i  -  M ) ) ) )
6534abscld 10893 . . . 4  |-  ( ph  ->  ( abs `  ( F `  M )
)  e.  RR )
6665recnd 7758 . . 3  |-  ( ph  ->  ( abs `  ( F `  M )
)  e.  CC )
6738recnd 7758 . . . 4  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  A  e.  CC )
6867, 48expcld 10364 . . 3  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( A ^ ( i  -  M ) )  e.  CC )
697, 66, 68fsummulc2 11157 . 2  |-  ( ph  ->  ( ( abs `  ( F `  M )
)  x.  sum_ i  e.  ( ( M  + 
1 ) ... N
) ( A ^
( i  -  M
) ) )  = 
sum_ i  e.  ( ( M  +  1 ) ... N ) ( ( abs `  ( F `  M )
)  x.  ( A ^ ( i  -  M ) ) ) )
7064, 69breqtrrd 3924 1  |-  ( ph  ->  ( abs `  sum_ i  e.  ( ( M  +  1 ) ... N ) ( F `  i ) )  <_  ( ( abs `  ( F `  M ) )  x. 
sum_ i  e.  ( ( M  +  1 ) ... N ) ( A ^ (
i  -  M ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463   A.wral 2391   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   NNcn 8677   NN0cn0 8928   ZZcz 9005   ZZ>=cuz 9275   ...cfz 9730   ^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-ico 9617  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
  Copyright terms: Public domain W3C validator