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Theorem cvgratnnlemsumlt 10918
Description: Lemma for cvgratnn 10921. (Contributed by Jim Kingdon, 23-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
cvgratnnlemsumlt  |-  ( ph  -> 
sum_ i  e.  ( ( M  +  1 ) ... N ) ( A ^ (
i  -  M ) )  <  ( A  /  ( 1  -  A ) ) )
Distinct variable groups:    A, k    k, F    k, N    ph, k    A, i, k    i, M, k   
i, N    ph, i
Allowed substitution hint:    F( i)

Proof of Theorem cvgratnnlemsumlt
StepHypRef Expression
1 cvgratnn.m . . . . 5  |-  ( ph  ->  M  e.  NN )
21nnzd 8865 . . . 4  |-  ( ph  ->  M  e.  ZZ )
3 1zzd 8775 . . . 4  |-  ( ph  ->  1  e.  ZZ )
4 cvgratnn.n . . . . . 6  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
5 eluzelz 9026 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ZZ )
64, 5syl 14 . . . . 5  |-  ( ph  ->  N  e.  ZZ )
76, 2zsubcld 8871 . . . 4  |-  ( ph  ->  ( N  -  M
)  e.  ZZ )
8 cvgratnn.3 . . . . . . 7  |-  ( ph  ->  A  e.  RR )
98recnd 7514 . . . . . 6  |-  ( ph  ->  A  e.  CC )
109adantr 270 . . . . 5  |-  ( (
ph  /\  k  e.  ( 1 ... ( N  -  M )
) )  ->  A  e.  CC )
11 elfznn 9466 . . . . . . 7  |-  ( k  e.  ( 1 ... ( N  -  M
) )  ->  k  e.  NN )
1211adantl 271 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 1 ... ( N  -  M )
) )  ->  k  e.  NN )
1312nnnn0d 8724 . . . . 5  |-  ( (
ph  /\  k  e.  ( 1 ... ( N  -  M )
) )  ->  k  e.  NN0 )
1410, 13expcld 10082 . . . 4  |-  ( (
ph  /\  k  e.  ( 1 ... ( N  -  M )
) )  ->  ( A ^ k )  e.  CC )
15 oveq2 5660 . . . 4  |-  ( k  =  ( i  -  M )  ->  ( A ^ k )  =  ( A ^ (
i  -  M ) ) )
162, 3, 7, 14, 15fsumshft 10834 . . 3  |-  ( ph  -> 
sum_ k  e.  ( 1 ... ( N  -  M ) ) ( A ^ k
)  =  sum_ i  e.  ( ( 1  +  M ) ... (
( N  -  M
)  +  M ) ) ( A ^
( i  -  M
) ) )
17 1cnd 7502 . . . . . 6  |-  ( ph  ->  1  e.  CC )
181nncnd 8434 . . . . . 6  |-  ( ph  ->  M  e.  CC )
1917, 18addcomd 7631 . . . . 5  |-  ( ph  ->  ( 1  +  M
)  =  ( M  +  1 ) )
206zcnd 8867 . . . . . 6  |-  ( ph  ->  N  e.  CC )
2120, 18npcand 7795 . . . . 5  |-  ( ph  ->  ( ( N  -  M )  +  M
)  =  N )
2219, 21oveq12d 5670 . . . 4  |-  ( ph  ->  ( ( 1  +  M ) ... (
( N  -  M
)  +  M ) )  =  ( ( M  +  1 ) ... N ) )
2322sumeq1d 10751 . . 3  |-  ( ph  -> 
sum_ i  e.  ( ( 1  +  M
) ... ( ( N  -  M )  +  M ) ) ( A ^ ( i  -  M ) )  =  sum_ i  e.  ( ( M  +  1 ) ... N ) ( A ^ (
i  -  M ) ) )
2416, 23eqtrd 2120 . 2  |-  ( ph  -> 
sum_ k  e.  ( 1 ... ( N  -  M ) ) ( A ^ k
)  =  sum_ i  e.  ( ( M  + 
1 ) ... N
) ( A ^
( i  -  M
) ) )
25 fzval3 9611 . . . . 5  |-  ( ( N  -  M )  e.  ZZ  ->  (
1 ... ( N  -  M ) )  =  ( 1..^ ( ( N  -  M )  +  1 ) ) )
2625sumeq1d 10751 . . . 4  |-  ( ( N  -  M )  e.  ZZ  ->  sum_ k  e.  ( 1 ... ( N  -  M )
) ( A ^
k )  =  sum_ k  e.  ( 1..^ ( ( N  -  M )  +  1 ) ) ( A ^ k ) )
277, 26syl 14 . . 3  |-  ( ph  -> 
sum_ k  e.  ( 1 ... ( N  -  M ) ) ( A ^ k
)  =  sum_ k  e.  ( 1..^ ( ( N  -  M )  +  1 ) ) ( A ^ k
) )
28 1red 7501 . . . . . 6  |-  ( ph  ->  1  e.  RR )
29 cvgratnn.4 . . . . . 6  |-  ( ph  ->  A  <  1 )
308, 28, 29ltapd 8111 . . . . 5  |-  ( ph  ->  A #  1 )
31 1nn0 8687 . . . . . 6  |-  1  e.  NN0
3231a1i 9 . . . . 5  |-  ( ph  ->  1  e.  NN0 )
337peano2zd 8869 . . . . . 6  |-  ( ph  ->  ( ( N  -  M )  +  1 )  e.  ZZ )
34 eluzle 9029 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  <_  N )
354, 34syl 14 . . . . . . . 8  |-  ( ph  ->  M  <_  N )
366zred 8866 . . . . . . . . 9  |-  ( ph  ->  N  e.  RR )
371nnred 8433 . . . . . . . . 9  |-  ( ph  ->  M  e.  RR )
3836, 37subge0d 8010 . . . . . . . 8  |-  ( ph  ->  ( 0  <_  ( N  -  M )  <->  M  <_  N ) )
3935, 38mpbird 165 . . . . . . 7  |-  ( ph  ->  0  <_  ( N  -  M ) )
407zred 8866 . . . . . . . 8  |-  ( ph  ->  ( N  -  M
)  e.  RR )
4128, 40addge02d 8009 . . . . . . 7  |-  ( ph  ->  ( 0  <_  ( N  -  M )  <->  1  <_  ( ( N  -  M )  +  1 ) ) )
4239, 41mpbid 145 . . . . . 6  |-  ( ph  ->  1  <_  ( ( N  -  M )  +  1 ) )
43 eluz2 9023 . . . . . 6  |-  ( ( ( N  -  M
)  +  1 )  e.  ( ZZ>= `  1
)  <->  ( 1  e.  ZZ  /\  ( ( N  -  M )  +  1 )  e.  ZZ  /\  1  <_ 
( ( N  -  M )  +  1 ) ) )
443, 33, 42, 43syl3anbrc 1127 . . . . 5  |-  ( ph  ->  ( ( N  -  M )  +  1 )  e.  ( ZZ>= ` 
1 ) )
459, 30, 32, 44geosergap 10896 . . . 4  |-  ( ph  -> 
sum_ k  e.  ( 1..^ ( ( N  -  M )  +  1 ) ) ( A ^ k )  =  ( ( ( A ^ 1 )  -  ( A ^
( ( N  -  M )  +  1 ) ) )  / 
( 1  -  A
) ) )
469exp1d 10077 . . . . . . 7  |-  ( ph  ->  ( A ^ 1 )  =  A )
4746, 8eqeltrd 2164 . . . . . 6  |-  ( ph  ->  ( A ^ 1 )  e.  RR )
48 cvgratnn.gt0 . . . . . . . . 9  |-  ( ph  ->  0  <  A )
498, 48elrpd 9169 . . . . . . . 8  |-  ( ph  ->  A  e.  RR+ )
5049, 33rpexpcld 10106 . . . . . . 7  |-  ( ph  ->  ( A ^ (
( N  -  M
)  +  1 ) )  e.  RR+ )
5150rpred 9171 . . . . . 6  |-  ( ph  ->  ( A ^ (
( N  -  M
)  +  1 ) )  e.  RR )
5247, 51resubcld 7857 . . . . 5  |-  ( ph  ->  ( ( A ^
1 )  -  ( A ^ ( ( N  -  M )  +  1 ) ) )  e.  RR )
5328, 8resubcld 7857 . . . . . 6  |-  ( ph  ->  ( 1  -  A
)  e.  RR )
548, 28posdifd 8007 . . . . . . 7  |-  ( ph  ->  ( A  <  1  <->  0  <  ( 1  -  A ) ) )
5529, 54mpbid 145 . . . . . 6  |-  ( ph  ->  0  <  ( 1  -  A ) )
5653, 55elrpd 9169 . . . . 5  |-  ( ph  ->  ( 1  -  A
)  e.  RR+ )
5746oveq1d 5667 . . . . . 6  |-  ( ph  ->  ( ( A ^
1 )  -  ( A ^ ( ( N  -  M )  +  1 ) ) )  =  ( A  -  ( A ^ ( ( N  -  M )  +  1 ) ) ) )
588, 50ltsubrpd 9204 . . . . . 6  |-  ( ph  ->  ( A  -  ( A ^ ( ( N  -  M )  +  1 ) ) )  <  A )
5957, 58eqbrtrd 3865 . . . . 5  |-  ( ph  ->  ( ( A ^
1 )  -  ( A ^ ( ( N  -  M )  +  1 ) ) )  <  A )
6052, 8, 56, 59ltdiv1dd 9229 . . . 4  |-  ( ph  ->  ( ( ( A ^ 1 )  -  ( A ^ ( ( N  -  M )  +  1 ) ) )  /  ( 1  -  A ) )  <  ( A  / 
( 1  -  A
) ) )
6145, 60eqbrtrd 3865 . . 3  |-  ( ph  -> 
sum_ k  e.  ( 1..^ ( ( N  -  M )  +  1 ) ) ( A ^ k )  <  ( A  / 
( 1  -  A
) ) )
6227, 61eqbrtrd 3865 . 2  |-  ( ph  -> 
sum_ k  e.  ( 1 ... ( N  -  M ) ) ( A ^ k
)  <  ( A  /  ( 1  -  A ) ) )
6324, 62eqbrtrrd 3867 1  |-  ( ph  -> 
sum_ i  e.  ( ( M  +  1 ) ... N ) ( A ^ (
i  -  M ) )  <  ( A  /  ( 1  -  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = 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   ...cfz 9422  ..^cfzo 9549   ^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-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:  cvgratnnlemrate  10920
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