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Theorem mertenslemub 11555
Description: Lemma for mertensabs 11558. 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 2194 . . . . . . 7  |-  ( z  =  X  ->  (
z  =  ( abs `  sum_ k  e.  (
ZZ>= `  ( n  + 
1 ) ) ( G `  k ) )  <->  X  =  ( abs `  sum_ k  e.  (
ZZ>= `  ( n  + 
1 ) ) ( G `  k ) ) ) )
32rexbidv 2488 . . . . . 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 2896 . . . . 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 147 . . 3  |-  ( ph  ->  E. n  e.  ( 0 ... ( S  -  1 ) ) X  =  ( abs `  sum_ k  e.  (
ZZ>= `  ( n  + 
1 ) ) ( G `  k ) ) )
8 fvoveq1 5911 . . . . . . 7  |-  ( n  =  a  ->  ( ZZ>=
`  ( n  + 
1 ) )  =  ( ZZ>= `  ( a  +  1 ) ) )
98sumeq1d 11387 . . . . . 6  |-  ( n  =  a  ->  sum_ k  e.  ( ZZ>= `  ( n  +  1 ) ) ( G `  k
)  =  sum_ k  e.  ( ZZ>= `  ( a  +  1 ) ) ( G `  k
) )
109fveq2d 5531 . . . . 5  |-  ( n  =  a  ->  ( abs `  sum_ k  e.  (
ZZ>= `  ( n  + 
1 ) ) ( G `  k ) )  =  ( abs `  sum_ k  e.  (
ZZ>= `  ( a  +  1 ) ) ( G `  k ) ) )
1110eqeq2d 2199 . . . 4  |-  ( n  =  a  ->  ( X  =  ( abs ` 
sum_ k  e.  (
ZZ>= `  ( n  + 
1 ) ) ( G `  k ) )  <->  X  =  ( abs `  sum_ k  e.  (
ZZ>= `  ( a  +  1 ) ) ( G `  k ) ) ) )
1211cbvrexv 2716 . . 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 122 . 2  |-  ( ph  ->  E. a  e.  ( 0 ... ( S  -  1 ) ) X  =  ( abs `  sum_ k  e.  (
ZZ>= `  ( a  +  1 ) ) ( G `  k ) ) )
14 simprr 531 . . 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 9278 . . . . 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 276 . . . . . . 7  |-  ( (
ph  /\  ( a  e.  ( 0 ... ( S  -  1 ) )  /\  X  =  ( abs `  sum_ k  e.  ( ZZ>= `  ( a  +  1 ) ) ( G `
 k ) ) ) )  ->  S  e.  NN )
1817nnzd 9387 . . . . . 6  |-  ( (
ph  /\  ( a  e.  ( 0 ... ( S  -  1 ) )  /\  X  =  ( abs `  sum_ k  e.  ( ZZ>= `  ( a  +  1 ) ) ( G `
 k ) ) ) )  ->  S  e.  ZZ )
19 1zzd 9293 . . . . . 6  |-  ( (
ph  /\  ( a  e.  ( 0 ... ( S  -  1 ) )  /\  X  =  ( abs `  sum_ k  e.  ( ZZ>= `  ( a  +  1 ) ) ( G `
 k ) ) ) )  ->  1  e.  ZZ )
2018, 19zsubcld 9393 . . . . 5  |-  ( (
ph  /\  ( a  e.  ( 0 ... ( S  -  1 ) )  /\  X  =  ( abs `  sum_ k  e.  ( ZZ>= `  ( a  +  1 ) ) ( G `
 k ) ) ) )  ->  ( S  -  1 )  e.  ZZ )
2115, 20fzfigd 10444 . . . 4  |-  ( (
ph  /\  ( a  e.  ( 0 ... ( S  -  1 ) )  /\  X  =  ( abs `  sum_ k  e.  ( ZZ>= `  ( a  +  1 ) ) ( G `
 k ) ) ) )  ->  (
0 ... ( S  - 
1 ) )  e. 
Fin )
22 eqid 2187 . . . . . . 7  |-  ( ZZ>= `  ( n  +  1
) )  =  (
ZZ>= `  ( n  + 
1 ) )
23 elfzelz 10038 . . . . . . . . 9  |-  ( n  e.  ( 0 ... ( S  -  1 ) )  ->  n  e.  ZZ )
2423adantl 277 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 0 ... ( S  -  1 ) ) )  ->  n  e.  ZZ )
2524peano2zd 9391 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 0 ... ( S  -  1 ) ) )  ->  (
n  +  1 )  e.  ZZ )
26 eqidd 2188 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  ( 0 ... ( S  -  1 ) ) )  /\  k  e.  ( ZZ>= `  ( n  +  1 ) ) )  ->  ( G `  k )  =  ( G `  k ) )
27 simpll 527 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  ( 0 ... ( S  -  1 ) ) )  /\  k  e.  ( ZZ>= `  ( n  +  1 ) ) )  ->  ph )
28 elfznn0 10127 . . . . . . . . . . 11  |-  ( n  e.  ( 0 ... ( S  -  1 ) )  ->  n  e.  NN0 )
2928ad2antlr 489 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  ( 0 ... ( S  -  1 ) ) )  /\  k  e.  ( ZZ>= `  ( n  +  1 ) ) )  ->  n  e.  NN0 )
30 peano2nn0 9229 . . . . . . . . . 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 9612 . . . . . . . . 9  |-  ( ( ( n  +  1 )  e.  NN0  /\  k  e.  ( ZZ>= `  ( n  +  1
) ) )  -> 
k  e.  NN0 )
3331, 32sylancom 420 . . . . . . . 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 2264 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( G `  k )  e.  CC )
3727, 33, 36syl2anc 411 . . . . . . 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 276 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 0 ... ( S  -  1 ) ) )  ->  seq 0 (  +  ,  G )  e.  dom  ~~>  )
40 nn0uz 9575 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
4128adantl 277 . . . . . . . . . 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 477 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  ( 0 ... ( S  -  1 ) ) )  /\  k  e.  NN0 )  ->  ( G `  k )  e.  CC )
4440, 42, 43iserex 11360 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 0 ... ( S  -  1 ) ) )  ->  (  seq 0 (  +  ,  G )  e.  dom  ~~>  <->  seq ( n  +  1
) (  +  ,  G )  e.  dom  ~~>  ) )
4539, 44mpbid 147 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 0 ... ( S  -  1 ) ) )  ->  seq ( n  +  1
) (  +  ,  G )  e.  dom  ~~>  )
4622, 25, 26, 37, 45isumcl 11446 . . . . . 6  |-  ( (
ph  /\  n  e.  ( 0 ... ( S  -  1 ) ) )  ->  sum_ k  e.  ( ZZ>= `  ( n  +  1 ) ) ( G `  k
)  e.  CC )
4746adantlr 477 . . . . 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 11203 . . . 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 11206 . . . 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 529 . . . 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 11482 . . 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 4037 . 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 2609 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 104    <-> wb 105    = wceq 1363    e. wcel 2158   {cab 2173   E.wrex 2466   class class class wbr 4015   dom cdm 4638   ` cfv 5228  (class class class)co 5888   CCcc 7822   0cc0 7824   1c1 7825    + caddc 7827    <_ cle 8006    - cmin 8141   NNcn 8932   NN0cn0 9189   ZZcz 9266   ZZ>=cuz 9541   ...cfz 10021    seqcseq 10458   abscabs 11019    ~~> cli 11299   sum_csu 11374
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942  ax-arch 7943  ax-caucvg 7944
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-isom 5237  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-irdg 6384  df-frec 6405  df-1o 6430  df-oadd 6434  df-er 6548  df-en 6754  df-dom 6755  df-fin 6756  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-inn 8933  df-2 8991  df-3 8992  df-4 8993  df-n0 9190  df-z 9267  df-uz 9542  df-q 9633  df-rp 9667  df-ico 9907  df-fz 10022  df-fzo 10156  df-seqfrec 10459  df-exp 10533  df-ihash 10769  df-cj 10864  df-re 10865  df-im 10866  df-rsqrt 11020  df-abs 11021  df-clim 11300  df-sumdc 11375
This theorem is referenced by:  mertenslem2  11557
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