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Theorem cvgratnnlemrate 11540
Description: Lemma for cvgratnn 11541. (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 9565 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
2 1zzd 9282 . . . . . . 7  |-  ( ph  ->  1  e.  ZZ )
3 cvgratnn.6 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( F `
 k )  e.  CC )
41, 2, 3serf 10476 . . . . . 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 9602 . . . . . . 7  |-  ( ( M  e.  NN  /\  N  e.  ( ZZ>= `  M ) )  ->  N  e.  NN )
85, 6, 7syl2anc 411 . . . . . 6  |-  ( ph  ->  N  e.  NN )
94, 8ffvelcdmd 5654 . . . . 5  |-  ( ph  ->  (  seq 1 (  +  ,  F ) `
 N )  e.  CC )
104, 5ffvelcdmd 5654 . . . . 5  |-  ( ph  ->  (  seq 1 (  +  ,  F ) `
 M )  e.  CC )
119, 10subcld 8270 . . . 4  |-  ( ph  ->  ( (  seq 1
(  +  ,  F
) `  N )  -  (  seq 1
(  +  ,  F
) `  M )
)  e.  CC )
1211abscld 11192 . . 3  |-  ( ph  ->  ( abs `  (
(  seq 1 (  +  ,  F ) `  N )  -  (  seq 1 (  +  ,  F ) `  M
) ) )  e.  RR )
13 fveq2 5517 . . . . . . 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 2848 . . . . 5  |-  ( ph  ->  ( F `  M
)  e.  CC )
1716abscld 11192 . . . 4  |-  ( ph  ->  ( abs `  ( F `  M )
)  e.  RR )
185nnzd 9376 . . . . . . 7  |-  ( ph  ->  M  e.  ZZ )
1918peano2zd 9380 . . . . . 6  |-  ( ph  ->  ( M  +  1 )  e.  ZZ )
20 eluzelz 9539 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
216, 20syl 14 . . . . . 6  |-  ( ph  ->  N  e.  ZZ )
2219, 21fzfigd 10433 . . . . 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 8934 . . . . . . . . 9  |-  ( ph  ->  M  e.  RR )
2625adantr 276 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  M  e.  RR )
27 peano2re 8095 . . . . . . . . 9  |-  ( M  e.  RR  ->  ( M  +  1 )  e.  RR )
2826, 27syl 14 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( M  +  1 )  e.  RR )
29 elfzelz 10027 . . . . . . . . . 10  |-  ( i  e.  ( ( M  +  1 ) ... N )  ->  i  e.  ZZ )
3029adantl 277 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  i  e.  ZZ )
3130zred 9377 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  i  e.  RR )
3226lep1d 8890 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  M  <_  ( M  +  1 ) )
33 elfzle1 10029 . . . . . . . . 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 8083 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  M  <_  i )
36 znn0sub 9320 . . . . . . . 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 10673 . . . . 5  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( A ^ ( i  -  M ) )  e.  RR )
4022, 39fsumrecl 11411 . . . 4  |-  ( ph  -> 
sum_ i  e.  ( ( M  +  1 ) ... N ) ( A ^ (
i  -  M ) )  e.  RR )
4117, 40remulcld 7990 . . 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 9695 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  RR+ )
4544reclt1d 9712 . . . . . . . . . . 11  |-  ( ph  ->  ( A  <  1  <->  1  <  ( 1  /  A ) ) )
4642, 45mpbid 147 . . . . . . . . . 10  |-  ( ph  ->  1  <  ( 1  /  A ) )
47 1re 7958 . . . . . . . . . . 11  |-  1  e.  RR
4844rprecred 9710 . . . . . . . . . . 11  |-  ( ph  ->  ( 1  /  A
)  e.  RR )
49 difrp 9694 . . . . . . . . . . 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 9709 . . . . . . . 8  |-  ( ph  ->  ( 1  /  (
( 1  /  A
)  -  1 ) )  e.  RR+ )
5352, 44rpdivcld 9716 . . . . . . 7  |-  ( ph  ->  ( ( 1  / 
( ( 1  /  A )  -  1 ) )  /  A
)  e.  RR+ )
54 fveq2 5517 . . . . . . . . . . 11  |-  ( k  =  1  ->  ( F `  k )  =  ( F ` 
1 ) )
5554eleq1d 2246 . . . . . . . . . 10  |-  ( k  =  1  ->  (
( F `  k
)  e.  CC  <->  ( F `  1 )  e.  CC ) )
56 1nn 8932 . . . . . . . . . . 11  |-  1  e.  NN
5756a1i 9 . . . . . . . . . 10  |-  ( ph  ->  1  e.  NN )
5855, 15, 57rspcdva 2848 . . . . . . . . 9  |-  ( ph  ->  ( F `  1
)  e.  CC )
5958abscld 11192 . . . . . . . 8  |-  ( ph  ->  ( abs `  ( F `  1 )
)  e.  RR )
6058absge0d 11195 . . . . . . . 8  |-  ( ph  ->  0  <_  ( abs `  ( F `  1
) ) )
6159, 60ge0p1rpd 9729 . . . . . . 7  |-  ( ph  ->  ( ( abs `  ( F `  1 )
)  +  1 )  e.  RR+ )
6253, 61rpmulcld 9715 . . . . . 6  |-  ( ph  ->  ( ( ( 1  /  ( ( 1  /  A )  - 
1 ) )  /  A )  x.  (
( abs `  ( F `  1 )
)  +  1 ) )  e.  RR+ )
6362rpred 9698 . . . . 5  |-  ( ph  ->  ( ( ( 1  /  ( ( 1  /  A )  - 
1 ) )  /  A )  x.  (
( abs `  ( F `  1 )
)  +  1 ) )  e.  RR )
6463, 5nndivred 8971 . . . 4  |-  ( ph  ->  ( ( ( ( 1  /  ( ( 1  /  A )  -  1 ) )  /  A )  x.  ( ( abs `  ( F `  1 )
)  +  1 ) )  /  M )  e.  RR )
65 1red 7974 . . . . . . . 8  |-  ( ph  ->  1  e.  RR )
6665, 23resubcld 8340 . . . . . . 7  |-  ( ph  ->  ( 1  -  A
)  e.  RR )
6723, 65posdifd 8491 . . . . . . . 8  |-  ( ph  ->  ( A  <  1  <->  0  <  ( 1  -  A ) ) )
6842, 67mpbid 147 . . . . . . 7  |-  ( ph  ->  0  <  ( 1  -  A ) )
6966, 68elrpd 9695 . . . . . 6  |-  ( ph  ->  ( 1  -  A
)  e.  RR+ )
7044, 69rpdivcld 9716 . . . . 5  |-  ( ph  ->  ( A  /  (
1  -  A ) )  e.  RR+ )
7170rpred 9698 . . . 4  |-  ( ph  ->  ( A  /  (
1  -  A ) )  e.  RR )
7264, 71remulcld 7990 . . 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 11536 . . . . 5  |-  ( ph  ->  ( (  seq 1
(  +  ,  F
) `  N )  -  (  seq 1
(  +  ,  F
) `  M )
)  =  sum_ i  e.  ( ( M  + 
1 ) ... N
) ( F `  i ) )
7574fveq2d 5521 . . . 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 11537 . . . 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 4027 . . 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 11195 . . . 4  |-  ( ph  ->  0  <_  ( abs `  ( F `  M
) ) )
7923, 42, 43, 3, 73, 5cvgratnnlemfm 11539 . . . 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 9375 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  (
i  -  M )  e.  ZZ )
8280, 81rpexpcld 10680 . . . . . 6  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  ( A ^ ( i  -  M ) )  e.  RR+ )
8382rpge0d 9702 . . . . 5  |-  ( (
ph  /\  i  e.  ( ( M  + 
1 ) ... N
) )  ->  0  <_  ( A ^ (
i  -  M ) ) )
8422, 39, 83fsumge0 11469 . . . 4  |-  ( ph  ->  0  <_  sum_ i  e.  ( ( M  + 
1 ) ... N
) ( A ^
( i  -  M
) ) )
8523, 42, 43, 3, 73, 5, 6cvgratnnlemsumlt 11538 . . . 4  |-  ( ph  -> 
sum_ i  e.  ( ( M  +  1 ) ... N ) ( A ^ (
i  -  M ) )  <  ( A  /  ( 1  -  A ) ) )
8617, 64, 40, 71, 78, 79, 84, 85ltmul12ad 8900 . . 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 8084 . 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 7988 . . 3  |-  ( ph  ->  ( ( ( 1  /  ( ( 1  /  A )  - 
1 ) )  /  A )  x.  (
( abs `  ( F `  1 )
)  +  1 ) )  e.  CC )
8971recnd 7988 . . 3  |-  ( ph  ->  ( A  /  (
1  -  A ) )  e.  CC )
905nncnd 8935 . . 3  |-  ( ph  ->  M  e.  CC )
915nnap0d 8967 . . 3  |-  ( ph  ->  M #  0 )
9288, 89, 90, 91div23apd 8787 . 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 4033 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 4005   ` cfv 5218  (class class class)co 5877   CCcc 7811   RRcr 7812   0cc0 7813   1c1 7814    + caddc 7816    x. cmul 7818    < clt 7994    <_ cle 7995    - cmin 8130    / cdiv 8631   NNcn 8921   NN0cn0 9178   ZZcz 9255   ZZ>=cuz 9530   RR+crp 9655   ...cfz 10010    seqcseq 10447   ^cexp 10521   abscabs 11008   sum_csu 11363
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 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933
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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-frec 6394  df-1o 6419  df-oadd 6423  df-er 6537  df-en 6743  df-dom 6744  df-fin 6745  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-ico 9896  df-fz 10011  df-fzo 10145  df-seqfrec 10448  df-exp 10522  df-ihash 10758  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-clim 11289  df-sumdc 11364
This theorem is referenced by:  cvgratnn  11541
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