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Theorem cvgratnnlemrate 11533
Description: Lemma for cvgratnn 11534. (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 9561 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
2 1zzd 9278 . . . . . . 7  |-  ( ph  ->  1  e.  ZZ )
3 cvgratnn.6 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  e.  CC )
41, 2, 3serf 10471 . . . . . 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 9598 . . . . . . 7  |-  ( ( M  e.  NN  /\  N  e.  ( ZZ>= `  M ) )  ->  N  e.  NN )
85, 6, 7syl2anc 411 . . . . . 6  |-  ( ph  ->  N  e.  NN )
94, 8ffvelcdmd 5652 . . . . 5  |-  ( ph  ->  (  seq 1 (  +  ,  F ) `
 N )  e.  CC )
104, 5ffvelcdmd 5652 . . . . 5  |-  ( ph  ->  (  seq 1 (  +  ,  F ) `
 M )  e.  CC )
119, 10subcld 8266 . . . 4  |-  ( ph  ->  ( (  seq 1
(  +  ,  F
) `  N )  -  (  seq 1
(  +  ,  F
) `  M )
)  e.  CC )
1211abscld 11185 . . 3  |-  ( ph  ->  ( abs `  (
(  seq 1 (  +  ,  F ) `  N )  -  (  seq 1 (  +  ,  F ) `  M
) ) )  e.  RR )
13 fveq2 5515 . . . . . . 7  |-  ( k  =  M  ->  ( F `  k )  =  ( F `  M ) )
1413eleq1d 2246 . . . . . 6  |-  ( k  =  M  ->  (
( F `  k
)  e.  CC  <->  ( F `  M )  e.  CC ) )
153ralrimiva 2550 . . . . . 6  |-  ( ph  ->  A. k  e.  NN  ( F `  k )  e.  CC )
1614, 15, 5rspcdva 2846 . . . . 5  |-  ( ph  ->  ( F `  M
)  e.  CC )
1716abscld 11185 . . . 4  |-  ( ph  ->  ( abs `  ( F `  M )
)  e.  RR )
185nnzd 9372 . . . . . . 7  |-  ( ph  ->  M  e.  ZZ )
1918peano2zd 9376 . . . . . 6  |-  ( ph  ->  ( M  +  1 )  e.  ZZ )
20 eluzelz 9535 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
216, 20syl 14 . . . . . 6  |-  ( ph  ->  N  e.  ZZ )
2219, 21fzfigd 10428 . . . . 5  |-  ( ph  ->  ( ( M  + 
1 ) ... N
)  e.  Fin )
23 cvgratnn.3 . . . . . . 7  |-  ( ph  ->  A  e.  RR )
2423adantr 276 . . . . . 6  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  A  e.  RR )
255nnred 8930 . . . . . . . . 9  |-  ( ph  ->  M  e.  RR )
2625adantr 276 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  M  e.  RR )
27 peano2re 8091 . . . . . . . . 9  |-  ( M  e.  RR  ->  ( M  +  1 )  e.  RR )
2826, 27syl 14 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( M  +  1 )  e.  RR )
29 elfzelz 10022 . . . . . . . . . 10  |-  ( i  e.  ( ( M  +  1 ) ... N )  ->  i  e.  ZZ )
3029adantl 277 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  i  e.  ZZ )
3130zred 9373 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  i  e.  RR )
3226lep1d 8886 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  M  <_  ( M  +  1 ) )
33 elfzle1 10024 . . . . . . . . 9  |-  ( i  e.  ( ( M  +  1 ) ... N )  ->  ( M  +  1 )  <_  i )
3433adantl 277 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( M  +  1 )  <_  i )
3526, 28, 31, 32, 34letrd 8079 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  M  <_  i )
36 znn0sub 9316 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  i  e.  ZZ )  ->  ( M  <_  i  <->  ( i  -  M )  e.  NN0 ) )
3718, 29, 36syl2an 289 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( M  <_  i  <->  ( i  -  M )  e.  NN0 ) )
3835, 37mpbid 147 . . . . . 6  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  (
i  -  M )  e.  NN0 )
3924, 38reexpcld 10667 . . . . 5  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( A ^ ( i  -  M ) )  e.  RR )
4022, 39fsumrecl 11404 . . . 4  |-  ( ph  -> 
sum_ i  e.  ( ( M  +  1 ) ... N ) ( A ^ (
i  -  M ) )  e.  RR )
4117, 40remulcld 7986 . . 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 9691 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  RR+ )
4544reclt1d 9708 . . . . . . . . . . 11  |-  ( ph  ->  ( A  <  1  <->  1  <  ( 1  /  A ) ) )
4642, 45mpbid 147 . . . . . . . . . 10  |-  ( ph  ->  1  <  ( 1  /  A ) )
47 1re 7955 . . . . . . . . . . 11  |-  1  e.  RR
4844rprecred 9706 . . . . . . . . . . 11  |-  ( ph  ->  ( 1  /  A
)  e.  RR )
49 difrp 9690 . . . . . . . . . . 11  |-  ( ( 1  e.  RR  /\  ( 1  /  A
)  e.  RR )  ->  ( 1  < 
( 1  /  A
)  <->  ( ( 1  /  A )  - 
1 )  e.  RR+ ) )
5047, 48, 49sylancr 414 . . . . . . . . . 10  |-  ( ph  ->  ( 1  <  (
1  /  A )  <-> 
( ( 1  /  A )  -  1 )  e.  RR+ )
)
5146, 50mpbid 147 . . . . . . . . 9  |-  ( ph  ->  ( ( 1  /  A )  -  1 )  e.  RR+ )
5251rpreccld 9705 . . . . . . . 8  |-  ( ph  ->  ( 1  /  (
( 1  /  A
)  -  1 ) )  e.  RR+ )
5352, 44rpdivcld 9712 . . . . . . 7  |-  ( ph  ->  ( ( 1  / 
( ( 1  /  A )  -  1 ) )  /  A
)  e.  RR+ )
54 fveq2 5515 . . . . . . . . . . 11  |-  ( k  =  1  ->  ( F `  k )  =  ( F ` 
1 ) )
5554eleq1d 2246 . . . . . . . . . 10  |-  ( k  =  1  ->  (
( F `  k
)  e.  CC  <->  ( F `  1 )  e.  CC ) )
56 1nn 8928 . . . . . . . . . . 11  |-  1  e.  NN
5756a1i 9 . . . . . . . . . 10  |-  ( ph  ->  1  e.  NN )
5855, 15, 57rspcdva 2846 . . . . . . . . 9  |-  ( ph  ->  ( F `  1
)  e.  CC )
5958abscld 11185 . . . . . . . 8  |-  ( ph  ->  ( abs `  ( F `  1 )
)  e.  RR )
6058absge0d 11188 . . . . . . . 8  |-  ( ph  ->  0  <_  ( abs `  ( F `  1
) ) )
6159, 60ge0p1rpd 9725 . . . . . . 7  |-  ( ph  ->  ( ( abs `  ( F `  1 )
)  +  1 )  e.  RR+ )
6253, 61rpmulcld 9711 . . . . . 6  |-  ( ph  ->  ( ( ( 1  /  ( ( 1  /  A )  - 
1 ) )  /  A )  x.  (
( abs `  ( F `  1 )
)  +  1 ) )  e.  RR+ )
6362rpred 9694 . . . . 5  |-  ( ph  ->  ( ( ( 1  /  ( ( 1  /  A )  - 
1 ) )  /  A )  x.  (
( abs `  ( F `  1 )
)  +  1 ) )  e.  RR )
6463, 5nndivred 8967 . . . 4  |-  ( ph  ->  ( ( ( ( 1  /  ( ( 1  /  A )  -  1 ) )  /  A )  x.  ( ( abs `  ( F `  1 )
)  +  1 ) )  /  M )  e.  RR )
65 1red 7971 . . . . . . . 8  |-  ( ph  ->  1  e.  RR )
6665, 23resubcld 8336 . . . . . . 7  |-  ( ph  ->  ( 1  -  A
)  e.  RR )
6723, 65posdifd 8487 . . . . . . . 8  |-  ( ph  ->  ( A  <  1  <->  0  <  ( 1  -  A ) ) )
6842, 67mpbid 147 . . . . . . 7  |-  ( ph  ->  0  <  ( 1  -  A ) )
6966, 68elrpd 9691 . . . . . 6  |-  ( ph  ->  ( 1  -  A
)  e.  RR+ )
7044, 69rpdivcld 9712 . . . . 5  |-  ( ph  ->  ( A  /  (
1  -  A ) )  e.  RR+ )
7170rpred 9694 . . . 4  |-  ( ph  ->  ( A  /  (
1  -  A ) )  e.  RR )
7264, 71remulcld 7986 . . 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 11529 . . . . 5  |-  ( ph  ->  ( (  seq 1
(  +  ,  F
) `  N )  -  (  seq 1
(  +  ,  F
) `  M )
)  =  sum_ i  e.  ( ( M  + 
1 ) ... N
) ( F `  i ) )
7574fveq2d 5519 . . . 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 11530 . . . 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 4025 . . 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 11188 . . . 4  |-  ( ph  ->  0  <_  ( abs `  ( F `  M
) ) )
7923, 42, 43, 3, 73, 5cvgratnnlemfm 11532 . . . 4  |-  ( ph  ->  ( abs `  ( F `  M )
)  <  ( (
( ( 1  / 
( ( 1  /  A )  -  1 ) )  /  A
)  x.  ( ( abs `  ( F `
 1 ) )  +  1 ) )  /  M ) )
8044adantr 276 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  A  e.  RR+ )
8138nn0zd 9371 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  (
i  -  M )  e.  ZZ )
8280, 81rpexpcld 10674 . . . . . 6  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( A ^ ( i  -  M ) )  e.  RR+ )
8382rpge0d 9698 . . . . 5  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  0  <_  ( A ^ (
i  -  M ) ) )
8422, 39, 83fsumge0 11462 . . . 4  |-  ( ph  ->  0  <_  sum_ i  e.  ( ( M  + 
1 ) ... N
) ( A ^
( i  -  M
) ) )
8523, 42, 43, 3, 73, 5, 6cvgratnnlemsumlt 11531 . . . 4  |-  ( ph  -> 
sum_ i  e.  ( ( M  +  1 ) ... N ) ( A ^ (
i  -  M ) )  <  ( A  /  ( 1  -  A ) ) )
8617, 64, 40, 71, 78, 79, 84, 85ltmul12ad 8896 . . 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 8080 . 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 7984 . . 3  |-  ( ph  ->  ( ( ( 1  /  ( ( 1  /  A )  - 
1 ) )  /  A )  x.  (
( abs `  ( F `  1 )
)  +  1 ) )  e.  CC )
8971recnd 7984 . . 3  |-  ( ph  ->  ( A  /  (
1  -  A ) )  e.  CC )
905nncnd 8931 . . 3  |-  ( ph  ->  M  e.  CC )
915nnap0d 8963 . . 3  |-  ( ph  ->  M #  0 )
9288, 89, 90, 91div23apd 8783 . 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 4031 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 104    <-> wb 105    = wceq 1353    e. wcel 2148   class class class wbr 4003   ` cfv 5216  (class class class)co 5874   CCcc 7808   RRcr 7809   0cc0 7810   1c1 7811    + caddc 7813    x. cmul 7815    < clt 7990    <_ cle 7991    - cmin 8126    / cdiv 8627   NNcn 8917   NN0cn0 9174   ZZcz 9251   ZZ>=cuz 9526   RR+crp 9651   ...cfz 10006    seqcseq 10442   ^cexp 10516   abscabs 11001   sum_csu 11356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-frec 6391  df-1o 6416  df-oadd 6420  df-er 6534  df-en 6740  df-dom 6741  df-fin 6742  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537  df-div 8628  df-inn 8918  df-2 8976  df-3 8977  df-4 8978  df-n0 9175  df-z 9252  df-uz 9527  df-q 9618  df-rp 9652  df-ico 9892  df-fz 10007  df-fzo 10140  df-seqfrec 10443  df-exp 10517  df-ihash 10751  df-cj 10846  df-re 10847  df-im 10848  df-rsqrt 11002  df-abs 11003  df-clim 11282  df-sumdc 11357
This theorem is referenced by:  cvgratnn  11534
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