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Theorem prmrec 12932
Description: The sum of the reciprocals of the primes diverges. This is the "second" proof at http://en.wikipedia.org/wiki/Prime_harmonic_series, attributed to Paul Erdős. (Contributed by Mario Carneiro, 6-Aug-2014.)
Hypothesis
Ref Expression
prmrec.f  |-  F  =  ( n  e.  NN  |->  sum_ k  e.  ( Prime  i^i  ( 1 ... n
) ) ( 1  /  k ) )
Assertion
Ref Expression
prmrec  |-  -.  F  e.  dom  ~~>
Distinct variable group:    k, n
Allowed substitution hints:    F( k, n)

Proof of Theorem prmrec
StepHypRef Expression
1 eleq1 2318 . . . . 5  |-  ( m  =  k  ->  (
m  e.  Prime  <->  k  e.  Prime ) )
2 oveq2 5800 . . . . 5  |-  ( m  =  k  ->  (
1  /  m )  =  ( 1  / 
k ) )
3 eqidd 2259 . . . . 5  |-  ( m  =  k  ->  0  =  0 )
41, 2, 3ifbieq12d 3561 . . . 4  |-  ( m  =  k  ->  if ( m  e.  Prime ,  ( 1  /  m
) ,  0 )  =  if ( k  e.  Prime ,  ( 1  /  k ) ,  0 ) )
54cbvmptv 4085 . . 3  |-  ( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m ) ,  0 ) )  =  ( k  e.  NN  |->  if ( k  e.  Prime ,  ( 1  /  k ) ,  0 ) )
65prmreclem6 12931 . 2  |-  -.  seq  1 (  +  , 
( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m
) ,  0 ) ) )  e.  dom  ~~>
7 inss2 3365 . . . . . . . . 9  |-  ( Prime  i^i  ( 1 ... n
) )  C_  (
1 ... n )
87sseli 3151 . . . . . . . . . . 11  |-  ( k  e.  ( Prime  i^i  ( 1 ... n
) )  ->  k  e.  ( 1 ... n
) )
9 elfznn 10786 . . . . . . . . . . 11  |-  ( k  e.  ( 1 ... n )  ->  k  e.  NN )
10 nnrecre 9750 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  (
1  /  k )  e.  RR )
1110recnd 8829 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  (
1  /  k )  e.  CC )
128, 9, 113syl 20 . . . . . . . . . 10  |-  ( k  e.  ( Prime  i^i  ( 1 ... n
) )  ->  (
1  /  k )  e.  CC )
1312rgen 2583 . . . . . . . . 9  |-  A. k  e.  ( Prime  i^i  (
1 ... n ) ) ( 1  /  k
)  e.  CC
147, 13pm3.2i 443 . . . . . . . 8  |-  ( ( Prime  i^i  ( 1 ... n ) ) 
C_  ( 1 ... n )  /\  A. k  e.  ( Prime  i^i  ( 1 ... n
) ) ( 1  /  k )  e.  CC )
15 fzfi 11001 . . . . . . . . 9  |-  ( 1 ... n )  e. 
Fin
1615olci 382 . . . . . . . 8  |-  ( ( 1 ... n ) 
C_  ( ZZ>= `  1
)  \/  ( 1 ... n )  e. 
Fin )
17 sumss2 12165 . . . . . . . 8  |-  ( ( ( ( Prime  i^i  ( 1 ... n
) )  C_  (
1 ... n )  /\  A. k  e.  ( Prime  i^i  ( 1 ... n
) ) ( 1  /  k )  e.  CC )  /\  (
( 1 ... n
)  C_  ( ZZ>= ` 
1 )  \/  (
1 ... n )  e. 
Fin ) )  ->  sum_ k  e.  ( Prime  i^i  ( 1 ... n
) ) ( 1  /  k )  = 
sum_ k  e.  ( 1 ... n ) if ( k  e.  ( Prime  i^i  (
1 ... n ) ) ,  ( 1  / 
k ) ,  0 ) )
1814, 16, 17mp2an 656 . . . . . . 7  |-  sum_ k  e.  ( Prime  i^i  (
1 ... n ) ) ( 1  /  k
)  =  sum_ k  e.  ( 1 ... n
) if ( k  e.  ( Prime  i^i  ( 1 ... n
) ) ,  ( 1  /  k ) ,  0 )
19 elin 3333 . . . . . . . . . 10  |-  ( k  e.  ( Prime  i^i  ( 1 ... n
) )  <->  ( k  e.  Prime  /\  k  e.  ( 1 ... n
) ) )
2019rbaib 878 . . . . . . . . 9  |-  ( k  e.  ( 1 ... n )  ->  (
k  e.  ( Prime  i^i  ( 1 ... n
) )  <->  k  e.  Prime ) )
2120ifbid 3557 . . . . . . . 8  |-  ( k  e.  ( 1 ... n )  ->  if ( k  e.  ( Prime  i^i  ( 1 ... n ) ) ,  ( 1  / 
k ) ,  0 )  =  if ( k  e.  Prime ,  ( 1  /  k ) ,  0 ) )
2221sumeq2i 12138 . . . . . . 7  |-  sum_ k  e.  ( 1 ... n
) if ( k  e.  ( Prime  i^i  ( 1 ... n
) ) ,  ( 1  /  k ) ,  0 )  = 
sum_ k  e.  ( 1 ... n ) if ( k  e. 
Prime ,  ( 1  /  k ) ,  0 )
2318, 22eqtri 2278 . . . . . 6  |-  sum_ k  e.  ( Prime  i^i  (
1 ... n ) ) ( 1  /  k
)  =  sum_ k  e.  ( 1 ... n
) if ( k  e.  Prime ,  ( 1  /  k ) ,  0 )
249adantl 454 . . . . . . . 8  |-  ( ( n  e.  NN  /\  k  e.  ( 1 ... n ) )  ->  k  e.  NN )
25 prmnn 12725 . . . . . . . . . . . 12  |-  ( k  e.  Prime  ->  k  e.  NN )
2625, 11syl 17 . . . . . . . . . . 11  |-  ( k  e.  Prime  ->  ( 1  /  k )  e.  CC )
2726adantl 454 . . . . . . . . . 10  |-  ( (  T.  /\  k  e. 
Prime )  ->  ( 1  /  k )  e.  CC )
28 0cn 8799 . . . . . . . . . . 11  |-  0  e.  CC
2928a1i 12 . . . . . . . . . 10  |-  ( (  T.  /\  -.  k  e.  Prime )  ->  0  e.  CC )
3027, 29ifclda 3566 . . . . . . . . 9  |-  (  T. 
->  if ( k  e. 
Prime ,  ( 1  /  k ) ,  0 )  e.  CC )
3130trud 1320 . . . . . . . 8  |-  if ( k  e.  Prime ,  ( 1  /  k ) ,  0 )  e.  CC
325fvmpt2 5542 . . . . . . . 8  |-  ( ( k  e.  NN  /\  if ( k  e.  Prime ,  ( 1  /  k
) ,  0 )  e.  CC )  -> 
( ( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m ) ,  0 ) ) `  k )  =  if ( k  e.  Prime ,  ( 1  /  k
) ,  0 ) )
3324, 31, 32sylancl 646 . . . . . . 7  |-  ( ( n  e.  NN  /\  k  e.  ( 1 ... n ) )  ->  ( ( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m ) ,  0 ) ) `
 k )  =  if ( k  e. 
Prime ,  ( 1  /  k ) ,  0 ) )
34 id 21 . . . . . . . 8  |-  ( n  e.  NN  ->  n  e.  NN )
35 nnuz 10231 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
3634, 35syl6eleq 2348 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  ( ZZ>= `  1 )
)
3731a1i 12 . . . . . . 7  |-  ( ( n  e.  NN  /\  k  e.  ( 1 ... n ) )  ->  if ( k  e.  Prime ,  ( 1  /  k ) ,  0 )  e.  CC )
3833, 36, 37fsumser 12169 . . . . . 6  |-  ( n  e.  NN  ->  sum_ k  e.  ( 1 ... n
) if ( k  e.  Prime ,  ( 1  /  k ) ,  0 )  =  (  seq  1 (  +  ,  ( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m ) ,  0 ) ) ) `
 n ) )
3923, 38syl5eq 2302 . . . . 5  |-  ( n  e.  NN  ->  sum_ k  e.  ( Prime  i^i  (
1 ... n ) ) ( 1  /  k
)  =  (  seq  1 (  +  , 
( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m
) ,  0 ) ) ) `  n
) )
4039mpteq2ia 4076 . . . 4  |-  ( n  e.  NN  |->  sum_ k  e.  ( Prime  i^i  (
1 ... n ) ) ( 1  /  k
) )  =  ( n  e.  NN  |->  (  seq  1 (  +  ,  ( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m ) ,  0 ) ) ) `
 n ) )
41 prmrec.f . . . 4  |-  F  =  ( n  e.  NN  |->  sum_ k  e.  ( Prime  i^i  ( 1 ... n
) ) ( 1  /  k ) )
42 1z 10021 . . . . . . 7  |-  1  e.  ZZ
43 seqfn 11025 . . . . . . 7  |-  ( 1  e.  ZZ  ->  seq  1 (  +  , 
( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m
) ,  0 ) ) )  Fn  ( ZZ>=
`  1 ) )
4442, 43ax-mp 10 . . . . . 6  |-  seq  1
(  +  ,  ( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m
) ,  0 ) ) )  Fn  ( ZZ>=
`  1 )
4535fneq2i 5277 . . . . . 6  |-  (  seq  1 (  +  , 
( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m
) ,  0 ) ) )  Fn  NN  <->  seq  1 (  +  , 
( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m
) ,  0 ) ) )  Fn  ( ZZ>=
`  1 ) )
4644, 45mpbir 202 . . . . 5  |-  seq  1
(  +  ,  ( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m
) ,  0 ) ) )  Fn  NN
47 dffn5 5502 . . . . 5  |-  (  seq  1 (  +  , 
( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m
) ,  0 ) ) )  Fn  NN  <->  seq  1 (  +  , 
( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m
) ,  0 ) ) )  =  ( n  e.  NN  |->  (  seq  1 (  +  ,  ( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m ) ,  0 ) ) ) `
 n ) ) )
4846, 47mpbi 201 . . . 4  |-  seq  1
(  +  ,  ( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m
) ,  0 ) ) )  =  ( n  e.  NN  |->  (  seq  1 (  +  ,  ( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m ) ,  0 ) ) ) `
 n ) )
4940, 41, 483eqtr4i 2288 . . 3  |-  F  =  seq  1 (  +  ,  ( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m ) ,  0 ) ) )
5049eleq1i 2321 . 2  |-  ( F  e.  dom  ~~>  <->  seq  1
(  +  ,  ( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m
) ,  0 ) ) )  e.  dom  ~~>  )
516, 50mtbir 292 1  |-  -.  F  e.  dom  ~~>
Colors of variables: wff set class
Syntax hints:   -. wn 5    \/ wo 359    /\ wa 360    T. wtru 1312    = wceq 1619    e. wcel 1621   A.wral 2518    i^i cin 3126    C_ wss 3127   ifcif 3539    e. cmpt 4051   dom cdm 4661    Fn wfn 4668   ` cfv 4673  (class class class)co 5792   Fincfn 6831   CCcc 8703   0cc0 8705   1c1 8706    + caddc 8708    / cdiv 9391   NNcn 9714   ZZcz 9992   ZZ>=cuz 10198   ...cfz 10749    seq cseq 11013    ~~> cli 11924   sum_csu 12124   Primecprime 12721
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-2o 6448  df-oadd 6451  df-er 6628  df-map 6742  df-pm 6743  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-sup 7162  df-oi 7193  df-card 7540  df-cda 7762  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-n0 9934  df-z 9993  df-uz 10199  df-q 10285  df-rp 10323  df-fz 10750  df-fzo 10838  df-fl 10892  df-mod 10941  df-seq 11014  df-exp 11072  df-hash 11305  df-cj 11550  df-re 11551  df-im 11552  df-sqr 11686  df-abs 11687  df-clim 11928  df-rlim 11929  df-sum 12125  df-divides 12495  df-gcd 12649  df-prime 12722  df-pc 12853
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