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Theorem cvgratnnlemabsle 10908
Description: Lemma for cvgratnn 10912. (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 8857 . . . . . . 7  |-  ( ph  ->  M  e.  ZZ )
32peano2zd 8861 . . . . . 6  |-  ( ph  ->  ( M  +  1 )  e.  ZZ )
4 cvgratnn.n . . . . . . 7  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
5 eluzelz 9018 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
64, 5syl 14 . . . . . 6  |-  ( ph  ->  N  e.  ZZ )
73, 6fzfigd 9826 . . . . 5  |-  ( ph  ->  ( ( M  + 
1 ) ... N
)  e.  Fin )
8 fveq2 5299 . . . . . . 7  |-  ( k  =  i  ->  ( F `  k )  =  ( F `  i ) )
98eleq1d 2156 . . . . . 6  |-  ( k  =  i  ->  (
( F `  k
)  e.  CC  <->  ( F `  i )  e.  CC ) )
10 cvgratnn.6 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  e.  CC )
1110ralrimiva 2446 . . . . . . 7  |-  ( ph  ->  A. k  e.  NN  ( F `  k )  e.  CC )
1211adantr 270 . . . . . 6  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  A. k  e.  NN  ( F `  k )  e.  CC )
13 elfzelz 9430 . . . . . . . 8  |-  ( i  e.  ( ( M  +  1 ) ... N )  ->  i  e.  ZZ )
1413adantl 271 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  i  e.  ZZ )
15 0red 7479 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  0  e.  RR )
161peano2nnd 8427 . . . . . . . . . 10  |-  ( ph  ->  ( M  +  1 )  e.  NN )
1716adantr 270 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( M  +  1 )  e.  NN )
1817nnred 8425 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( M  +  1 )  e.  RR )
1914zred 8858 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  i  e.  RR )
2016nngt0d 8456 . . . . . . . . 9  |-  ( ph  ->  0  <  ( M  +  1 ) )
2120adantr 270 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  0  <  ( M  +  1 ) )
22 elfzle1 9431 . . . . . . . . 9  |-  ( i  e.  ( ( M  +  1 ) ... N )  ->  ( M  +  1 )  <_  i )
2322adantl 271 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( M  +  1 )  <_  i )
2415, 18, 19, 21, 23ltletrd 7891 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  0  <  i )
25 elnnz 8750 . . . . . . 7  |-  ( i  e.  NN  <->  ( i  e.  ZZ  /\  0  < 
i ) )
2614, 24, 25sylanbrc 408 . . . . . 6  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  i  e.  NN )
279, 12, 26rspcdva 2727 . . . . 5  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( F `  i )  e.  CC )
287, 27fsumcl 10781 . . . 4  |-  ( ph  -> 
sum_ i  e.  ( ( M  +  1 ) ... N ) ( F `  i
)  e.  CC )
2928abscld 10602 . . 3  |-  ( ph  ->  ( abs `  sum_ i  e.  ( ( M  +  1 ) ... N ) ( F `  i ) )  e.  RR )
3027abscld 10602 . . . 4  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( abs `  ( F `  i ) )  e.  RR )
317, 30fsumrecl 10782 . . 3  |-  ( ph  -> 
sum_ i  e.  ( ( M  +  1 ) ... N ) ( abs `  ( F `  i )
)  e.  RR )
32 fveq2 5299 . . . . . . . . 9  |-  ( k  =  M  ->  ( F `  k )  =  ( F `  M ) )
3332eleq1d 2156 . . . . . . . 8  |-  ( k  =  M  ->  (
( F `  k
)  e.  CC  <->  ( F `  M )  e.  CC ) )
3433, 11, 1rspcdva 2727 . . . . . . 7  |-  ( ph  ->  ( F `  M
)  e.  CC )
3534adantr 270 . . . . . 6  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( F `  M )  e.  CC )
3635abscld 10602 . . . . 5  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( abs `  ( F `  M ) )  e.  RR )
37 cvgratnn.3 . . . . . . 7  |-  ( ph  ->  A  e.  RR )
3837adantr 270 . . . . . 6  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  A  e.  RR )
392adantr 270 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  M  e.  ZZ )
4014, 39zsubcld 8863 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  (
i  -  M )  e.  ZZ )
411adantr 270 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  M  e.  NN )
4241nnred 8425 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  M  e.  RR )
4342lep1d 8382 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  M  <_  ( M  +  1 ) )
4442, 18, 19, 43, 23letrd 7597 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  M  <_  i )
4519, 42subge0d 8002 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  (
0  <_  ( i  -  M )  <->  M  <_  i ) )
4644, 45mpbird 165 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  0  <_  ( i  -  M
) )
47 elnn0z 8753 . . . . . . 7  |-  ( ( i  -  M )  e.  NN0  <->  ( ( i  -  M )  e.  ZZ  /\  0  <_ 
( i  -  M
) ) )
4840, 46, 47sylanbrc 408 . . . . . 6  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  (
i  -  M )  e.  NN0 )
4938, 48reexpcld 10091 . . . . 5  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( A ^ ( i  -  M ) )  e.  RR )
5036, 49remulcld 7508 . . . 4  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  (
( abs `  ( F `  M )
)  x.  ( A ^ ( i  -  M ) ) )  e.  RR )
517, 50fsumrecl 10782 . . 3  |-  ( ph  -> 
sum_ i  e.  ( ( M  +  1 ) ... N ) ( ( abs `  ( F `  M )
)  x.  ( A ^ ( i  -  M ) ) )  e.  RR )
527, 27fsumabs 10846 . . 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 270 . . . . 5  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  A  <  1 )
55 cvgratnn.gt0 . . . . . 6  |-  ( ph  ->  0  <  A )
5655adantr 270 . . . . 5  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  0  <  A )
5710adantlr 461 . . . . 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 461 . . . . 5  |-  ( ( ( ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  /\  k  e.  NN )  ->  ( abs `  ( F `  ( k  +  1 ) ) )  <_ 
( A  x.  ( abs `  ( F `  k ) ) ) )
60 eluz2 9015 . . . . . 6  |-  ( i  e.  ( ZZ>= `  M
)  <->  ( M  e.  ZZ  /\  i  e.  ZZ  /\  M  <_ 
i ) )
6139, 14, 44, 60syl3anbrc 1127 . . . . 5  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  i  e.  ( ZZ>= `  M )
)
6238, 54, 56, 57, 59, 41, 61cvgratnnlemmn 10906 . . . 4  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( abs `  ( F `  i ) )  <_ 
( ( abs `  ( F `  M )
)  x.  ( A ^ ( i  -  M ) ) ) )
637, 30, 50, 62fsumle 10844 . . 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 7597 . 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 10602 . . . 4  |-  ( ph  ->  ( abs `  ( F `  M )
)  e.  RR )
6665recnd 7506 . . 3  |-  ( ph  ->  ( abs `  ( F `  M )
)  e.  CC )
6738recnd 7506 . . . 4  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  A  e.  CC )
6867, 48expcld 10074 . . 3  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( A ^ ( i  -  M ) )  e.  CC )
697, 66, 68fsummulc2 10829 . 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 3869 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 102    = wceq 1289    e. wcel 1438   A.wral 2359   class class class wbr 3843   ` cfv 5010  (class class class)co 5644   CCcc 7338   RRcr 7339   0cc0 7340   1c1 7341    + caddc 7343    x. cmul 7345    < clt 7512    <_ cle 7513    - cmin 7643   NNcn 8412   NN0cn0 8663   ZZcz 8740   ZZ>=cuz 9009   ...cfz 9414   ^cexp 9942   abscabs 10418   sum_csu 10729
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3952  ax-sep 3955  ax-nul 3963  ax-pow 4007  ax-pr 4034  ax-un 4258  ax-setind 4351  ax-iinf 4401  ax-cnex 7426  ax-resscn 7427  ax-1cn 7428  ax-1re 7429  ax-icn 7430  ax-addcl 7431  ax-addrcl 7432  ax-mulcl 7433  ax-mulrcl 7434  ax-addcom 7435  ax-mulcom 7436  ax-addass 7437  ax-mulass 7438  ax-distr 7439  ax-i2m1 7440  ax-0lt1 7441  ax-1rid 7442  ax-0id 7443  ax-rnegex 7444  ax-precex 7445  ax-cnre 7446  ax-pre-ltirr 7447  ax-pre-ltwlin 7448  ax-pre-lttrn 7449  ax-pre-apti 7450  ax-pre-ltadd 7451  ax-pre-mulgt0 7452  ax-pre-mulext 7453  ax-arch 7454  ax-caucvg 7455
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3392  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-int 3687  df-iun 3730  df-br 3844  df-opab 3898  df-mpt 3899  df-tr 3935  df-id 4118  df-po 4121  df-iso 4122  df-iord 4191  df-on 4193  df-ilim 4194  df-suc 4196  df-iom 4404  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017  df-fv 5018  df-isom 5019  df-riota 5600  df-ov 5647  df-oprab 5648  df-mpt2 5649  df-1st 5903  df-2nd 5904  df-recs 6062  df-irdg 6127  df-frec 6148  df-1o 6173  df-oadd 6177  df-er 6282  df-en 6448  df-dom 6449  df-fin 6450  df-pnf 7514  df-mnf 7515  df-xr 7516  df-ltxr 7517  df-le 7518  df-sub 7645  df-neg 7646  df-reap 8042  df-ap 8049  df-div 8130  df-inn 8413  df-2 8471  df-3 8472  df-4 8473  df-n0 8664  df-z 8741  df-uz 9010  df-q 9095  df-rp 9125  df-ico 9302  df-fz 9415  df-fzo 9542  df-iseq 9841  df-seq3 9842  df-exp 9943  df-ihash 10172  df-cj 10264  df-re 10265  df-im 10266  df-rsqrt 10419  df-abs 10420  df-clim 10654  df-isum 10730
This theorem is referenced by:  cvgratnnlemrate  10911
  Copyright terms: Public domain W3C validator