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Theorem cvgratnnlemrate 10920
Description: Lemma for cvgratnn 10921. (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
) ) ) )
cvgratnnlemrate.m  |-  ( ph  ->  M  e.  NN )
cvgratnnlemrate.n  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
Assertion
Ref Expression
cvgratnnlemrate  |-  ( ph  ->  ( abs `  (
(  seq 1 (  +  ,  F ) `  N )  -  (  seq 1 (  +  ,  F ) `  M
) ) )  < 
( ( ( ( ( 1  /  (
( 1  /  A
)  -  1 ) )  /  A )  x.  ( ( abs `  ( F `  1
) )  +  1 ) )  x.  ( A  /  ( 1  -  A ) ) )  /  M ) )
Distinct variable groups:    A, k    k, F    k, N    ph, k    k, M

Proof of Theorem cvgratnnlemrate
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 nnuz 9052 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
2 1zzd 8775 . . . . . . 7  |-  ( ph  ->  1  e.  ZZ )
3 cvgratnn.6 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  e.  CC )
41, 2, 3serf 9896 . . . . . 6  |-  ( ph  ->  seq 1 (  +  ,  F ) : NN --> CC )
5 cvgratnnlemrate.m . . . . . . 7  |-  ( ph  ->  M  e.  NN )
6 cvgratnnlemrate.n . . . . . . 7  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
7 eluznn 9085 . . . . . . 7  |-  ( ( M  e.  NN  /\  N  e.  ( ZZ>= `  M ) )  ->  N  e.  NN )
85, 6, 7syl2anc 403 . . . . . 6  |-  ( ph  ->  N  e.  NN )
94, 8ffvelrnd 5435 . . . . 5  |-  ( ph  ->  (  seq 1 (  +  ,  F ) `
 N )  e.  CC )
104, 5ffvelrnd 5435 . . . . 5  |-  ( ph  ->  (  seq 1 (  +  ,  F ) `
 M )  e.  CC )
119, 10subcld 7791 . . . 4  |-  ( ph  ->  ( (  seq 1
(  +  ,  F
) `  N )  -  (  seq 1
(  +  ,  F
) `  M )
)  e.  CC )
1211abscld 10610 . . 3  |-  ( ph  ->  ( abs `  (
(  seq 1 (  +  ,  F ) `  N )  -  (  seq 1 (  +  ,  F ) `  M
) ) )  e.  RR )
13 fveq2 5305 . . . . . . 7  |-  ( k  =  M  ->  ( F `  k )  =  ( F `  M ) )
1413eleq1d 2156 . . . . . 6  |-  ( k  =  M  ->  (
( F `  k
)  e.  CC  <->  ( F `  M )  e.  CC ) )
153ralrimiva 2446 . . . . . 6  |-  ( ph  ->  A. k  e.  NN  ( F `  k )  e.  CC )
1614, 15, 5rspcdva 2727 . . . . 5  |-  ( ph  ->  ( F `  M
)  e.  CC )
1716abscld 10610 . . . 4  |-  ( ph  ->  ( abs `  ( F `  M )
)  e.  RR )
185nnzd 8865 . . . . . . 7  |-  ( ph  ->  M  e.  ZZ )
1918peano2zd 8869 . . . . . 6  |-  ( ph  ->  ( M  +  1 )  e.  ZZ )
20 eluzelz 9026 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
216, 20syl 14 . . . . . 6  |-  ( ph  ->  N  e.  ZZ )
2219, 21fzfigd 9834 . . . . 5  |-  ( ph  ->  ( ( M  + 
1 ) ... N
)  e.  Fin )
23 cvgratnn.3 . . . . . . 7  |-  ( ph  ->  A  e.  RR )
2423adantr 270 . . . . . 6  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  A  e.  RR )
255nnred 8433 . . . . . . . . 9  |-  ( ph  ->  M  e.  RR )
2625adantr 270 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  M  e.  RR )
27 peano2re 7616 . . . . . . . . 9  |-  ( M  e.  RR  ->  ( M  +  1 )  e.  RR )
2826, 27syl 14 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( M  +  1 )  e.  RR )
29 elfzelz 9438 . . . . . . . . . 10  |-  ( i  e.  ( ( M  +  1 ) ... N )  ->  i  e.  ZZ )
3029adantl 271 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  i  e.  ZZ )
3130zred 8866 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  i  e.  RR )
3226lep1d 8390 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  M  <_  ( M  +  1 ) )
33 elfzle1 9439 . . . . . . . . 9  |-  ( i  e.  ( ( M  +  1 ) ... N )  ->  ( M  +  1 )  <_  i )
3433adantl 271 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( M  +  1 )  <_  i )
3526, 28, 31, 32, 34letrd 7605 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  M  <_  i )
36 znn0sub 8813 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  i  e.  ZZ )  ->  ( M  <_  i  <->  ( i  -  M )  e.  NN0 ) )
3718, 29, 36syl2an 283 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( M  <_  i  <->  ( i  -  M )  e.  NN0 ) )
3835, 37mpbid 145 . . . . . 6  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  (
i  -  M )  e.  NN0 )
3924, 38reexpcld 10099 . . . . 5  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( A ^ ( i  -  M ) )  e.  RR )
4022, 39fsumrecl 10791 . . . 4  |-  ( ph  -> 
sum_ i  e.  ( ( M  +  1 ) ... N ) ( A ^ (
i  -  M ) )  e.  RR )
4117, 40remulcld 7516 . . 3  |-  ( ph  ->  ( ( abs `  ( F `  M )
)  x.  sum_ i  e.  ( ( M  + 
1 ) ... N
) ( A ^
( i  -  M
) ) )  e.  RR )
42 cvgratnn.4 . . . . . . . . . . 11  |-  ( ph  ->  A  <  1 )
43 cvgratnn.gt0 . . . . . . . . . . . . 13  |-  ( ph  ->  0  <  A )
4423, 43elrpd 9169 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  RR+ )
4544reclt1d 9185 . . . . . . . . . . 11  |-  ( ph  ->  ( A  <  1  <->  1  <  ( 1  /  A ) ) )
4642, 45mpbid 145 . . . . . . . . . 10  |-  ( ph  ->  1  <  ( 1  /  A ) )
47 1re 7485 . . . . . . . . . . 11  |-  1  e.  RR
4844rprecred 9183 . . . . . . . . . . 11  |-  ( ph  ->  ( 1  /  A
)  e.  RR )
49 difrp 9168 . . . . . . . . . . 11  |-  ( ( 1  e.  RR  /\  ( 1  /  A
)  e.  RR )  ->  ( 1  < 
( 1  /  A
)  <->  ( ( 1  /  A )  - 
1 )  e.  RR+ ) )
5047, 48, 49sylancr 405 . . . . . . . . . 10  |-  ( ph  ->  ( 1  <  (
1  /  A )  <-> 
( ( 1  /  A )  -  1 )  e.  RR+ )
)
5146, 50mpbid 145 . . . . . . . . 9  |-  ( ph  ->  ( ( 1  /  A )  -  1 )  e.  RR+ )
5251rpreccld 9182 . . . . . . . 8  |-  ( ph  ->  ( 1  /  (
( 1  /  A
)  -  1 ) )  e.  RR+ )
5352, 44rpdivcld 9189 . . . . . . 7  |-  ( ph  ->  ( ( 1  / 
( ( 1  /  A )  -  1 ) )  /  A
)  e.  RR+ )
54 fveq2 5305 . . . . . . . . . . 11  |-  ( k  =  1  ->  ( F `  k )  =  ( F ` 
1 ) )
5554eleq1d 2156 . . . . . . . . . 10  |-  ( k  =  1  ->  (
( F `  k
)  e.  CC  <->  ( F `  1 )  e.  CC ) )
56 1nn 8431 . . . . . . . . . . 11  |-  1  e.  NN
5756a1i 9 . . . . . . . . . 10  |-  ( ph  ->  1  e.  NN )
5855, 15, 57rspcdva 2727 . . . . . . . . 9  |-  ( ph  ->  ( F `  1
)  e.  CC )
5958abscld 10610 . . . . . . . 8  |-  ( ph  ->  ( abs `  ( F `  1 )
)  e.  RR )
6058absge0d 10613 . . . . . . . 8  |-  ( ph  ->  0  <_  ( abs `  ( F `  1
) ) )
6159, 60ge0p1rpd 9202 . . . . . . 7  |-  ( ph  ->  ( ( abs `  ( F `  1 )
)  +  1 )  e.  RR+ )
6253, 61rpmulcld 9188 . . . . . 6  |-  ( ph  ->  ( ( ( 1  /  ( ( 1  /  A )  - 
1 ) )  /  A )  x.  (
( abs `  ( F `  1 )
)  +  1 ) )  e.  RR+ )
6362rpred 9171 . . . . 5  |-  ( ph  ->  ( ( ( 1  /  ( ( 1  /  A )  - 
1 ) )  /  A )  x.  (
( abs `  ( F `  1 )
)  +  1 ) )  e.  RR )
6463, 5nndivred 8470 . . . 4  |-  ( ph  ->  ( ( ( ( 1  /  ( ( 1  /  A )  -  1 ) )  /  A )  x.  ( ( abs `  ( F `  1 )
)  +  1 ) )  /  M )  e.  RR )
65 1red 7501 . . . . . . . 8  |-  ( ph  ->  1  e.  RR )
6665, 23resubcld 7857 . . . . . . 7  |-  ( ph  ->  ( 1  -  A
)  e.  RR )
6723, 65posdifd 8007 . . . . . . . 8  |-  ( ph  ->  ( A  <  1  <->  0  <  ( 1  -  A ) ) )
6842, 67mpbid 145 . . . . . . 7  |-  ( ph  ->  0  <  ( 1  -  A ) )
6966, 68elrpd 9169 . . . . . 6  |-  ( ph  ->  ( 1  -  A
)  e.  RR+ )
7044, 69rpdivcld 9189 . . . . 5  |-  ( ph  ->  ( A  /  (
1  -  A ) )  e.  RR+ )
7170rpred 9171 . . . 4  |-  ( ph  ->  ( A  /  (
1  -  A ) )  e.  RR )
7264, 71remulcld 7516 . . 3  |-  ( ph  ->  ( ( ( ( ( 1  /  (
( 1  /  A
)  -  1 ) )  /  A )  x.  ( ( abs `  ( F `  1
) )  +  1 ) )  /  M
)  x.  ( A  /  ( 1  -  A ) ) )  e.  RR )
73 cvgratnn.7 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( abs `  ( F `  (
k  +  1 ) ) )  <_  ( A  x.  ( abs `  ( F `  k
) ) ) )
7423, 42, 43, 3, 73, 5, 6cvgratnnlemseq 10916 . . . . 5  |-  ( ph  ->  ( (  seq 1
(  +  ,  F
) `  N )  -  (  seq 1
(  +  ,  F
) `  M )
)  =  sum_ i  e.  ( ( M  + 
1 ) ... N
) ( F `  i ) )
7574fveq2d 5309 . . . 4  |-  ( ph  ->  ( abs `  (
(  seq 1 (  +  ,  F ) `  N )  -  (  seq 1 (  +  ,  F ) `  M
) ) )  =  ( abs `  sum_ i  e.  ( ( M  +  1 ) ... N ) ( F `  i ) ) )
7623, 42, 43, 3, 73, 5, 6cvgratnnlemabsle 10917 . . . 4  |-  ( ph  ->  ( abs `  sum_ i  e.  ( ( M  +  1 ) ... N ) ( F `  i ) )  <_  ( ( abs `  ( F `  M ) )  x. 
sum_ i  e.  ( ( M  +  1 ) ... N ) ( A ^ (
i  -  M ) ) ) )
7775, 76eqbrtrd 3865 . . 3  |-  ( ph  ->  ( abs `  (
(  seq 1 (  +  ,  F ) `  N )  -  (  seq 1 (  +  ,  F ) `  M
) ) )  <_ 
( ( abs `  ( F `  M )
)  x.  sum_ i  e.  ( ( M  + 
1 ) ... N
) ( A ^
( i  -  M
) ) ) )
7816absge0d 10613 . . . 4  |-  ( ph  ->  0  <_  ( abs `  ( F `  M
) ) )
7923, 42, 43, 3, 73, 5cvgratnnlemfm 10919 . . . 4  |-  ( ph  ->  ( abs `  ( F `  M )
)  <  ( (
( ( 1  / 
( ( 1  /  A )  -  1 ) )  /  A
)  x.  ( ( abs `  ( F `
 1 ) )  +  1 ) )  /  M ) )
8044adantr 270 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  A  e.  RR+ )
8138nn0zd 8864 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  (
i  -  M )  e.  ZZ )
8280, 81rpexpcld 10106 . . . . . 6  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( A ^ ( i  -  M ) )  e.  RR+ )
8382rpge0d 9175 . . . . 5  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  0  <_  ( A ^ (
i  -  M ) ) )
8422, 39, 83fsumge0 10849 . . . 4  |-  ( ph  ->  0  <_  sum_ i  e.  ( ( M  + 
1 ) ... N
) ( A ^
( i  -  M
) ) )
8523, 42, 43, 3, 73, 5, 6cvgratnnlemsumlt 10918 . . . 4  |-  ( ph  -> 
sum_ i  e.  ( ( M  +  1 ) ... N ) ( A ^ (
i  -  M ) )  <  ( A  /  ( 1  -  A ) ) )
8617, 64, 40, 71, 78, 79, 84, 85ltmul12ad 8400 . . 3  |-  ( ph  ->  ( ( abs `  ( F `  M )
)  x.  sum_ i  e.  ( ( M  + 
1 ) ... N
) ( A ^
( i  -  M
) ) )  < 
( ( ( ( ( 1  /  (
( 1  /  A
)  -  1 ) )  /  A )  x.  ( ( abs `  ( F `  1
) )  +  1 ) )  /  M
)  x.  ( A  /  ( 1  -  A ) ) ) )
8712, 41, 72, 77, 86lelttrd 7606 . 2  |-  ( ph  ->  ( abs `  (
(  seq 1 (  +  ,  F ) `  N )  -  (  seq 1 (  +  ,  F ) `  M
) ) )  < 
( ( ( ( ( 1  /  (
( 1  /  A
)  -  1 ) )  /  A )  x.  ( ( abs `  ( F `  1
) )  +  1 ) )  /  M
)  x.  ( A  /  ( 1  -  A ) ) ) )
8863recnd 7514 . . 3  |-  ( ph  ->  ( ( ( 1  /  ( ( 1  /  A )  - 
1 ) )  /  A )  x.  (
( abs `  ( F `  1 )
)  +  1 ) )  e.  CC )
8971recnd 7514 . . 3  |-  ( ph  ->  ( A  /  (
1  -  A ) )  e.  CC )
905nncnd 8434 . . 3  |-  ( ph  ->  M  e.  CC )
915nnap0d 8466 . . 3  |-  ( ph  ->  M #  0 )
9288, 89, 90, 91div23apd 8293 . 2  |-  ( ph  ->  ( ( ( ( ( 1  /  (
( 1  /  A
)  -  1 ) )  /  A )  x.  ( ( abs `  ( F `  1
) )  +  1 ) )  x.  ( A  /  ( 1  -  A ) ) )  /  M )  =  ( ( ( ( ( 1  /  (
( 1  /  A
)  -  1 ) )  /  A )  x.  ( ( abs `  ( F `  1
) )  +  1 ) )  /  M
)  x.  ( A  /  ( 1  -  A ) ) ) )
9387, 92breqtrrd 3871 1  |-  ( ph  ->  ( abs `  (
(  seq 1 (  +  ,  F ) `  N )  -  (  seq 1 (  +  ,  F ) `  M
) ) )  < 
( ( ( ( ( 1  /  (
( 1  /  A
)  -  1 ) )  /  A )  x.  ( ( abs `  ( F `  1
) )  +  1 ) )  x.  ( A  /  ( 1  -  A ) ) )  /  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   class class class wbr 3845   ` cfv 5015  (class class class)co 5652   CCcc 7346   RRcr 7347   0cc0 7348   1c1 7349    + caddc 7351    x. cmul 7353    < clt 7520    <_ cle 7521    - cmin 7651    / cdiv 8137   NNcn 8420   NN0cn0 8671   ZZcz 8748   ZZ>=cuz 9017   RR+crp 9132   ...cfz 9422    seqcseq 9848   ^cexp 9950   abscabs 10426   sum_csu 10738
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 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-mulrcl 7442  ax-addcom 7443  ax-mulcom 7444  ax-addass 7445  ax-mulass 7446  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-1rid 7450  ax-0id 7451  ax-rnegex 7452  ax-precex 7453  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-apti 7458  ax-pre-ltadd 7459  ax-pre-mulgt0 7460  ax-pre-mulext 7461  ax-arch 7462  ax-caucvg 7463
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 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-isom 5024  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-frec 6156  df-1o 6181  df-oadd 6185  df-er 6290  df-en 6456  df-dom 6457  df-fin 6458  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-reap 8050  df-ap 8057  df-div 8138  df-inn 8421  df-2 8479  df-3 8480  df-4 8481  df-n0 8672  df-z 8749  df-uz 9018  df-q 9103  df-rp 9133  df-ico 9310  df-fz 9423  df-fzo 9550  df-iseq 9849  df-seq3 9850  df-exp 9951  df-ihash 10180  df-cj 10272  df-re 10273  df-im 10274  df-rsqrt 10427  df-abs 10428  df-clim 10663  df-isum 10739
This theorem is referenced by:  cvgratnn  10921
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