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Theorem cvgratnnlemabsle 11435
Description: Lemma for cvgratnn 11439. (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 9290 . . . . . . 7  |-  ( ph  ->  M  e.  ZZ )
32peano2zd 9294 . . . . . 6  |-  ( ph  ->  ( M  +  1 )  e.  ZZ )
4 cvgratnn.n . . . . . . 7  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
5 eluzelz 9453 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
64, 5syl 14 . . . . . 6  |-  ( ph  ->  N  e.  ZZ )
73, 6fzfigd 10339 . . . . 5  |-  ( ph  ->  ( ( M  + 
1 ) ... N
)  e.  Fin )
8 fveq2 5470 . . . . . . 7  |-  ( k  =  i  ->  ( F `  k )  =  ( F `  i ) )
98eleq1d 2226 . . . . . 6  |-  ( k  =  i  ->  (
( F `  k
)  e.  CC  <->  ( F `  i )  e.  CC ) )
10 cvgratnn.6 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  e.  CC )
1110ralrimiva 2530 . . . . . . 7  |-  ( ph  ->  A. k  e.  NN  ( F `  k )  e.  CC )
1211adantr 274 . . . . . 6  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  A. k  e.  NN  ( F `  k )  e.  CC )
13 elfzelz 9934 . . . . . . . 8  |-  ( i  e.  ( ( M  +  1 ) ... N )  ->  i  e.  ZZ )
1413adantl 275 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  i  e.  ZZ )
15 0red 7881 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  0  e.  RR )
161peano2nnd 8853 . . . . . . . . . 10  |-  ( ph  ->  ( M  +  1 )  e.  NN )
1716adantr 274 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( M  +  1 )  e.  NN )
1817nnred 8851 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( M  +  1 )  e.  RR )
1914zred 9291 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  i  e.  RR )
2016nngt0d 8882 . . . . . . . . 9  |-  ( ph  ->  0  <  ( M  +  1 ) )
2120adantr 274 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  0  <  ( M  +  1 ) )
22 elfzle1 9935 . . . . . . . . 9  |-  ( i  e.  ( ( M  +  1 ) ... N )  ->  ( M  +  1 )  <_  i )
2322adantl 275 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( M  +  1 )  <_  i )
2415, 18, 19, 21, 23ltletrd 8302 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  0  <  i )
25 elnnz 9182 . . . . . . 7  |-  ( i  e.  NN  <->  ( i  e.  ZZ  /\  0  < 
i ) )
2614, 24, 25sylanbrc 414 . . . . . 6  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  i  e.  NN )
279, 12, 26rspcdva 2821 . . . . 5  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( F `  i )  e.  CC )
287, 27fsumcl 11308 . . . 4  |-  ( ph  -> 
sum_ i  e.  ( ( M  +  1 ) ... N ) ( F `  i
)  e.  CC )
2928abscld 11092 . . 3  |-  ( ph  ->  ( abs `  sum_ i  e.  ( ( M  +  1 ) ... N ) ( F `  i ) )  e.  RR )
3027abscld 11092 . . . 4  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( abs `  ( F `  i ) )  e.  RR )
317, 30fsumrecl 11309 . . 3  |-  ( ph  -> 
sum_ i  e.  ( ( M  +  1 ) ... N ) ( abs `  ( F `  i )
)  e.  RR )
32 fveq2 5470 . . . . . . . . 9  |-  ( k  =  M  ->  ( F `  k )  =  ( F `  M ) )
3332eleq1d 2226 . . . . . . . 8  |-  ( k  =  M  ->  (
( F `  k
)  e.  CC  <->  ( F `  M )  e.  CC ) )
3433, 11, 1rspcdva 2821 . . . . . . 7  |-  ( ph  ->  ( F `  M
)  e.  CC )
3534adantr 274 . . . . . 6  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( F `  M )  e.  CC )
3635abscld 11092 . . . . 5  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( abs `  ( F `  M ) )  e.  RR )
37 cvgratnn.3 . . . . . . 7  |-  ( ph  ->  A  e.  RR )
3837adantr 274 . . . . . 6  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  A  e.  RR )
392adantr 274 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  M  e.  ZZ )
4014, 39zsubcld 9296 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  (
i  -  M )  e.  ZZ )
411adantr 274 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  M  e.  NN )
4241nnred 8851 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  M  e.  RR )
4342lep1d 8807 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  M  <_  ( M  +  1 ) )
4442, 18, 19, 43, 23letrd 8003 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  M  <_  i )
4519, 42subge0d 8414 . . . . . . . 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 9185 . . . . . . 7  |-  ( ( i  -  M )  e.  NN0  <->  ( ( i  -  M )  e.  ZZ  /\  0  <_ 
( i  -  M
) ) )
4840, 46, 47sylanbrc 414 . . . . . 6  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  (
i  -  M )  e.  NN0 )
4938, 48reexpcld 10577 . . . . 5  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( A ^ ( i  -  M ) )  e.  RR )
5036, 49remulcld 7910 . . . 4  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  (
( abs `  ( F `  M )
)  x.  ( A ^ ( i  -  M ) ) )  e.  RR )
517, 50fsumrecl 11309 . . 3  |-  ( ph  -> 
sum_ i  e.  ( ( M  +  1 ) ... N ) ( ( abs `  ( F `  M )
)  x.  ( A ^ ( i  -  M ) ) )  e.  RR )
527, 27fsumabs 11373 . . 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 274 . . . . 5  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  A  <  1 )
55 cvgratnn.gt0 . . . . . 6  |-  ( ph  ->  0  <  A )
5655adantr 274 . . . . 5  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  0  <  A )
5710adantlr 469 . . . . 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 469 . . . . 5  |-  ( ( ( ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  /\  k  e.  NN )  ->  ( abs `  ( F `  ( k  +  1 ) ) )  <_ 
( A  x.  ( abs `  ( F `  k ) ) ) )
60 eluz2 9450 . . . . . 6  |-  ( i  e.  ( ZZ>= `  M
)  <->  ( M  e.  ZZ  /\  i  e.  ZZ  /\  M  <_ 
i ) )
6139, 14, 44, 60syl3anbrc 1166 . . . . 5  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  i  e.  ( ZZ>= `  M )
)
6238, 54, 56, 57, 59, 41, 61cvgratnnlemmn 11433 . . . 4  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( abs `  ( F `  i ) )  <_ 
( ( abs `  ( F `  M )
)  x.  ( A ^ ( i  -  M ) ) ) )
637, 30, 50, 62fsumle 11371 . . 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 8003 . 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 11092 . . . 4  |-  ( ph  ->  ( abs `  ( F `  M )
)  e.  RR )
6665recnd 7908 . . 3  |-  ( ph  ->  ( abs `  ( F `  M )
)  e.  CC )
6738recnd 7908 . . . 4  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  A  e.  CC )
6867, 48expcld 10560 . . 3  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( A ^ ( i  -  M ) )  e.  CC )
697, 66, 68fsummulc2 11356 . 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 3994 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 1335    e. wcel 2128   A.wral 2435   class class class wbr 3967   ` cfv 5172  (class class class)co 5826   CCcc 7732   RRcr 7733   0cc0 7734   1c1 7735    + caddc 7737    x. cmul 7739    < clt 7914    <_ cle 7915    - cmin 8050   NNcn 8838   NN0cn0 9095   ZZcz 9172   ZZ>=cuz 9444   ...cfz 9918   ^cexp 10427   abscabs 10908   sum_csu 11261
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-iinf 4549  ax-cnex 7825  ax-resscn 7826  ax-1cn 7827  ax-1re 7828  ax-icn 7829  ax-addcl 7830  ax-addrcl 7831  ax-mulcl 7832  ax-mulrcl 7833  ax-addcom 7834  ax-mulcom 7835  ax-addass 7836  ax-mulass 7837  ax-distr 7838  ax-i2m1 7839  ax-0lt1 7840  ax-1rid 7841  ax-0id 7842  ax-rnegex 7843  ax-precex 7844  ax-cnre 7845  ax-pre-ltirr 7846  ax-pre-ltwlin 7847  ax-pre-lttrn 7848  ax-pre-apti 7849  ax-pre-ltadd 7850  ax-pre-mulgt0 7851  ax-pre-mulext 7852  ax-arch 7853  ax-caucvg 7854
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-if 3507  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-id 4255  df-po 4258  df-iso 4259  df-iord 4328  df-on 4330  df-ilim 4331  df-suc 4333  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-isom 5181  df-riota 5782  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-recs 6254  df-irdg 6319  df-frec 6340  df-1o 6365  df-oadd 6369  df-er 6482  df-en 6688  df-dom 6689  df-fin 6690  df-pnf 7916  df-mnf 7917  df-xr 7918  df-ltxr 7919  df-le 7920  df-sub 8052  df-neg 8053  df-reap 8454  df-ap 8461  df-div 8550  df-inn 8839  df-2 8897  df-3 8898  df-4 8899  df-n0 9096  df-z 9173  df-uz 9445  df-q 9535  df-rp 9567  df-ico 9804  df-fz 9919  df-fzo 10051  df-seqfrec 10354  df-exp 10428  df-ihash 10661  df-cj 10753  df-re 10754  df-im 10755  df-rsqrt 10909  df-abs 10910  df-clim 11187  df-sumdc 11262
This theorem is referenced by:  cvgratnnlemrate  11438
  Copyright terms: Public domain W3C validator