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Theorem mertenslemub 10989
Description: Lemma for mertensabs 10992. An upper bound for  T. (Contributed by Jim Kingdon, 3-Dec-2022.)
Hypotheses
Ref Expression
mertenslemub.gb  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( G `  k )  =  B )
mertenslemub.b  |-  ( (
ph  /\  k  e.  NN0 )  ->  B  e.  CC )
mertenslemub.cvg  |-  ( ph  ->  seq 0 (  +  ,  G )  e. 
dom 
~~>  )
mertenslemub.t  |-  T  =  { z  |  E. n  e.  ( 0 ... ( S  - 
1 ) ) z  =  ( abs `  sum_ k  e.  ( ZZ>= `  ( n  +  1
) ) ( G `
 k ) ) }
mertenslemub.elt  |-  ( ph  ->  X  e.  T )
mertenslemub.s  |-  ( ph  ->  S  e.  NN )
Assertion
Ref Expression
mertenslemub  |-  ( ph  ->  X  <_  sum_ n  e.  ( 0 ... ( S  -  1 ) ) ( abs `  sum_ k  e.  ( ZZ>= `  ( n  +  1
) ) ( G `
 k ) ) )
Distinct variable groups:    k, G, n, z    S, k, n, z   
n, X, z    ph, k, n
Allowed substitution hints:    ph( z)    B( z,
k, n)    T( z,
k, n)    X( k)

Proof of Theorem mertenslemub
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 mertenslemub.elt . . . 4  |-  ( ph  ->  X  e.  T )
2 eqeq1 2095 . . . . . . 7  |-  ( z  =  X  ->  (
z  =  ( abs `  sum_ k  e.  (
ZZ>= `  ( n  + 
1 ) ) ( G `  k ) )  <->  X  =  ( abs `  sum_ k  e.  (
ZZ>= `  ( n  + 
1 ) ) ( G `  k ) ) ) )
32rexbidv 2382 . . . . . 6  |-  ( z  =  X  ->  ( E. n  e.  (
0 ... ( S  - 
1 ) ) z  =  ( abs `  sum_ k  e.  ( ZZ>= `  ( n  +  1
) ) ( G `
 k ) )  <->  E. n  e.  (
0 ... ( S  - 
1 ) ) X  =  ( abs `  sum_ k  e.  ( ZZ>= `  ( n  +  1
) ) ( G `
 k ) ) ) )
4 mertenslemub.t . . . . . 6  |-  T  =  { z  |  E. n  e.  ( 0 ... ( S  - 
1 ) ) z  =  ( abs `  sum_ k  e.  ( ZZ>= `  ( n  +  1
) ) ( G `
 k ) ) }
53, 4elab2g 2763 . . . . 5  |-  ( X  e.  T  ->  ( X  e.  T  <->  E. n  e.  ( 0 ... ( S  -  1 ) ) X  =  ( abs `  sum_ k  e.  ( ZZ>= `  ( n  +  1 ) ) ( G `  k
) ) ) )
61, 5syl 14 . . . 4  |-  ( ph  ->  ( X  e.  T  <->  E. n  e.  ( 0 ... ( S  - 
1 ) ) X  =  ( abs `  sum_ k  e.  ( ZZ>= `  ( n  +  1
) ) ( G `
 k ) ) ) )
71, 6mpbid 146 . . 3  |-  ( ph  ->  E. n  e.  ( 0 ... ( S  -  1 ) ) X  =  ( abs `  sum_ k  e.  (
ZZ>= `  ( n  + 
1 ) ) ( G `  k ) ) )
8 fvoveq1 5689 . . . . . . 7  |-  ( n  =  a  ->  ( ZZ>=
`  ( n  + 
1 ) )  =  ( ZZ>= `  ( a  +  1 ) ) )
98sumeq1d 10816 . . . . . 6  |-  ( n  =  a  ->  sum_ k  e.  ( ZZ>= `  ( n  +  1 ) ) ( G `  k
)  =  sum_ k  e.  ( ZZ>= `  ( a  +  1 ) ) ( G `  k
) )
109fveq2d 5322 . . . . 5  |-  ( n  =  a  ->  ( abs `  sum_ k  e.  (
ZZ>= `  ( n  + 
1 ) ) ( G `  k ) )  =  ( abs `  sum_ k  e.  (
ZZ>= `  ( a  +  1 ) ) ( G `  k ) ) )
1110eqeq2d 2100 . . . 4  |-  ( n  =  a  ->  ( X  =  ( abs ` 
sum_ k  e.  (
ZZ>= `  ( n  + 
1 ) ) ( G `  k ) )  <->  X  =  ( abs `  sum_ k  e.  (
ZZ>= `  ( a  +  1 ) ) ( G `  k ) ) ) )
1211cbvrexv 2592 . . 3  |-  ( E. n  e.  ( 0 ... ( S  - 
1 ) ) X  =  ( abs `  sum_ k  e.  ( ZZ>= `  ( n  +  1
) ) ( G `
 k ) )  <->  E. a  e.  (
0 ... ( S  - 
1 ) ) X  =  ( abs `  sum_ k  e.  ( ZZ>= `  ( a  +  1 ) ) ( G `
 k ) ) )
137, 12sylib 121 . 2  |-  ( ph  ->  E. a  e.  ( 0 ... ( S  -  1 ) ) X  =  ( abs `  sum_ k  e.  (
ZZ>= `  ( a  +  1 ) ) ( G `  k ) ) )
14 simprr 500 . . 3  |-  ( (
ph  /\  ( a  e.  ( 0 ... ( S  -  1 ) )  /\  X  =  ( abs `  sum_ k  e.  ( ZZ>= `  ( a  +  1 ) ) ( G `
 k ) ) ) )  ->  X  =  ( abs `  sum_ k  e.  ( ZZ>= `  ( a  +  1 ) ) ( G `
 k ) ) )
15 0zd 8823 . . . . 5  |-  ( (
ph  /\  ( a  e.  ( 0 ... ( S  -  1 ) )  /\  X  =  ( abs `  sum_ k  e.  ( ZZ>= `  ( a  +  1 ) ) ( G `
 k ) ) ) )  ->  0  e.  ZZ )
16 mertenslemub.s . . . . . . . 8  |-  ( ph  ->  S  e.  NN )
1716adantr 271 . . . . . . 7  |-  ( (
ph  /\  ( a  e.  ( 0 ... ( S  -  1 ) )  /\  X  =  ( abs `  sum_ k  e.  ( ZZ>= `  ( a  +  1 ) ) ( G `
 k ) ) ) )  ->  S  e.  NN )
1817nnzd 8928 . . . . . 6  |-  ( (
ph  /\  ( a  e.  ( 0 ... ( S  -  1 ) )  /\  X  =  ( abs `  sum_ k  e.  ( ZZ>= `  ( a  +  1 ) ) ( G `
 k ) ) ) )  ->  S  e.  ZZ )
19 1zzd 8838 . . . . . 6  |-  ( (
ph  /\  ( a  e.  ( 0 ... ( S  -  1 ) )  /\  X  =  ( abs `  sum_ k  e.  ( ZZ>= `  ( a  +  1 ) ) ( G `
 k ) ) ) )  ->  1  e.  ZZ )
2018, 19zsubcld 8934 . . . . 5  |-  ( (
ph  /\  ( a  e.  ( 0 ... ( S  -  1 ) )  /\  X  =  ( abs `  sum_ k  e.  ( ZZ>= `  ( a  +  1 ) ) ( G `
 k ) ) ) )  ->  ( S  -  1 )  e.  ZZ )
2115, 20fzfigd 9899 . . . 4  |-  ( (
ph  /\  ( a  e.  ( 0 ... ( S  -  1 ) )  /\  X  =  ( abs `  sum_ k  e.  ( ZZ>= `  ( a  +  1 ) ) ( G `
 k ) ) ) )  ->  (
0 ... ( S  - 
1 ) )  e. 
Fin )
22 eqid 2089 . . . . . . 7  |-  ( ZZ>= `  ( n  +  1
) )  =  (
ZZ>= `  ( n  + 
1 ) )
23 elfzelz 9501 . . . . . . . . 9  |-  ( n  e.  ( 0 ... ( S  -  1 ) )  ->  n  e.  ZZ )
2423adantl 272 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 0 ... ( S  -  1 ) ) )  ->  n  e.  ZZ )
2524peano2zd 8932 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 0 ... ( S  -  1 ) ) )  ->  (
n  +  1 )  e.  ZZ )
26 eqidd 2090 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  ( 0 ... ( S  -  1 ) ) )  /\  k  e.  ( ZZ>= `  ( n  +  1 ) ) )  ->  ( G `  k )  =  ( G `  k ) )
27 simpll 497 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  ( 0 ... ( S  -  1 ) ) )  /\  k  e.  ( ZZ>= `  ( n  +  1 ) ) )  ->  ph )
28 elfznn0 9589 . . . . . . . . . . 11  |-  ( n  e.  ( 0 ... ( S  -  1 ) )  ->  n  e.  NN0 )
2928ad2antlr 474 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  ( 0 ... ( S  -  1 ) ) )  /\  k  e.  ( ZZ>= `  ( n  +  1 ) ) )  ->  n  e.  NN0 )
30 peano2nn0 8774 . . . . . . . . . 10  |-  ( n  e.  NN0  ->  ( n  +  1 )  e. 
NN0 )
3129, 30syl 14 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  ( 0 ... ( S  -  1 ) ) )  /\  k  e.  ( ZZ>= `  ( n  +  1 ) ) )  ->  ( n  +  1 )  e. 
NN0 )
32 eluznn0 9147 . . . . . . . . 9  |-  ( ( ( n  +  1 )  e.  NN0  /\  k  e.  ( ZZ>= `  ( n  +  1
) ) )  -> 
k  e.  NN0 )
3331, 32sylancom 412 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  ( 0 ... ( S  -  1 ) ) )  /\  k  e.  ( ZZ>= `  ( n  +  1 ) ) )  ->  k  e.  NN0 )
34 mertenslemub.gb . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( G `  k )  =  B )
35 mertenslemub.b . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN0 )  ->  B  e.  CC )
3634, 35eqeltrd 2165 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( G `  k )  e.  CC )
3727, 33, 36syl2anc 404 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  ( 0 ... ( S  -  1 ) ) )  /\  k  e.  ( ZZ>= `  ( n  +  1 ) ) )  ->  ( G `  k )  e.  CC )
38 mertenslemub.cvg . . . . . . . . 9  |-  ( ph  ->  seq 0 (  +  ,  G )  e. 
dom 
~~>  )
3938adantr 271 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 0 ... ( S  -  1 ) ) )  ->  seq 0 (  +  ,  G )  e.  dom  ~~>  )
40 nn0uz 9114 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
4128adantl 272 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( 0 ... ( S  -  1 ) ) )  ->  n  e.  NN0 )
4241, 30syl 14 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 0 ... ( S  -  1 ) ) )  ->  (
n  +  1 )  e.  NN0 )
4336adantlr 462 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  ( 0 ... ( S  -  1 ) ) )  /\  k  e.  NN0 )  ->  ( G `  k )  e.  CC )
4440, 42, 43iserex 10788 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 0 ... ( S  -  1 ) ) )  ->  (  seq 0 (  +  ,  G )  e.  dom  ~~>  <->  seq ( n  +  1
) (  +  ,  G )  e.  dom  ~~>  ) )
4539, 44mpbid 146 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 0 ... ( S  -  1 ) ) )  ->  seq ( n  +  1
) (  +  ,  G )  e.  dom  ~~>  )
4622, 25, 26, 37, 45isumcl 10880 . . . . . 6  |-  ( (
ph  /\  n  e.  ( 0 ... ( S  -  1 ) ) )  ->  sum_ k  e.  ( ZZ>= `  ( n  +  1 ) ) ( G `  k
)  e.  CC )
4746adantlr 462 . . . . 5  |-  ( ( ( ph  /\  (
a  e.  ( 0 ... ( S  - 
1 ) )  /\  X  =  ( abs ` 
sum_ k  e.  (
ZZ>= `  ( a  +  1 ) ) ( G `  k ) ) ) )  /\  n  e.  ( 0 ... ( S  - 
1 ) ) )  ->  sum_ k  e.  (
ZZ>= `  ( n  + 
1 ) ) ( G `  k )  e.  CC )
4847abscld 10675 . . . 4  |-  ( ( ( ph  /\  (
a  e.  ( 0 ... ( S  - 
1 ) )  /\  X  =  ( abs ` 
sum_ k  e.  (
ZZ>= `  ( a  +  1 ) ) ( G `  k ) ) ) )  /\  n  e.  ( 0 ... ( S  - 
1 ) ) )  ->  ( abs `  sum_ k  e.  ( ZZ>= `  ( n  +  1
) ) ( G `
 k ) )  e.  RR )
4947absge0d 10678 . . . 4  |-  ( ( ( ph  /\  (
a  e.  ( 0 ... ( S  - 
1 ) )  /\  X  =  ( abs ` 
sum_ k  e.  (
ZZ>= `  ( a  +  1 ) ) ( G `  k ) ) ) )  /\  n  e.  ( 0 ... ( S  - 
1 ) ) )  ->  0  <_  ( abs `  sum_ k  e.  (
ZZ>= `  ( n  + 
1 ) ) ( G `  k ) ) )
50 simprl 499 . . . 4  |-  ( (
ph  /\  ( a  e.  ( 0 ... ( S  -  1 ) )  /\  X  =  ( abs `  sum_ k  e.  ( ZZ>= `  ( a  +  1 ) ) ( G `
 k ) ) ) )  ->  a  e.  ( 0 ... ( S  -  1 ) ) )
5121, 48, 49, 10, 50fsumge1 10916 . . 3  |-  ( (
ph  /\  ( a  e.  ( 0 ... ( S  -  1 ) )  /\  X  =  ( abs `  sum_ k  e.  ( ZZ>= `  ( a  +  1 ) ) ( G `
 k ) ) ) )  ->  ( abs `  sum_ k  e.  (
ZZ>= `  ( a  +  1 ) ) ( G `  k ) )  <_  sum_ n  e.  ( 0 ... ( S  -  1 ) ) ( abs `  sum_ k  e.  ( ZZ>= `  ( n  +  1
) ) ( G `
 k ) ) )
5214, 51eqbrtrd 3871 . 2  |-  ( (
ph  /\  ( a  e.  ( 0 ... ( S  -  1 ) )  /\  X  =  ( abs `  sum_ k  e.  ( ZZ>= `  ( a  +  1 ) ) ( G `
 k ) ) ) )  ->  X  <_ 
sum_ n  e.  (
0 ... ( S  - 
1 ) ) ( abs `  sum_ k  e.  ( ZZ>= `  ( n  +  1 ) ) ( G `  k
) ) )
5313, 52rexlimddv 2494 1  |-  ( ph  ->  X  <_  sum_ n  e.  ( 0 ... ( S  -  1 ) ) ( abs `  sum_ k  e.  ( ZZ>= `  ( n  +  1
) ) ( G `
 k ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1290    e. wcel 1439   {cab 2075   E.wrex 2361   class class class wbr 3851   dom cdm 4452   ` cfv 5028  (class class class)co 5666   CCcc 7409   0cc0 7411   1c1 7412    + caddc 7414    <_ cle 7584    - cmin 7714   NNcn 8483   NN0cn0 8734   ZZcz 8811   ZZ>=cuz 9080   ...cfz 9485    seqcseq 9913   abscabs 10491    ~~> cli 10727   sum_csu 10803
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-mulrcl 7505  ax-addcom 7506  ax-mulcom 7507  ax-addass 7508  ax-mulass 7509  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-1rid 7513  ax-0id 7514  ax-rnegex 7515  ax-precex 7516  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-apti 7521  ax-pre-ltadd 7522  ax-pre-mulgt0 7523  ax-pre-mulext 7524  ax-arch 7525  ax-caucvg 7526
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-if 3398  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-isom 5037  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-frec 6170  df-1o 6195  df-oadd 6199  df-er 6306  df-en 6512  df-dom 6513  df-fin 6514  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-reap 8113  df-ap 8120  df-div 8201  df-inn 8484  df-2 8542  df-3 8543  df-4 8544  df-n0 8735  df-z 8812  df-uz 9081  df-q 9166  df-rp 9196  df-ico 9373  df-fz 9486  df-fzo 9615  df-iseq 9914  df-seq3 9915  df-exp 10016  df-ihash 10245  df-cj 10337  df-re 10338  df-im 10339  df-rsqrt 10492  df-abs 10493  df-clim 10728  df-isum 10804
This theorem is referenced by:  mertenslem2  10991
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