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Theorem prmrec 13273
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
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 eleq1 2490 . . . . 5  |-  ( m  =  k  ->  (
m  e.  Prime  <->  k  e.  Prime ) )
2 oveq2 6075 . . . . 5  |-  ( m  =  k  ->  (
1  /  m )  =  ( 1  / 
k ) )
3 eqidd 2431 . . . . 5  |-  ( m  =  k  ->  0  =  0 )
41, 2, 3ifbieq12d 3748 . . . 4  |-  ( m  =  k  ->  if ( m  e.  Prime ,  ( 1  /  m
) ,  0 )  =  if ( k  e.  Prime ,  ( 1  /  k ) ,  0 ) )
54cbvmptv 4287 . . 3  |-  ( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m ) ,  0 ) )  =  ( k  e.  NN  |->  if ( k  e.  Prime ,  ( 1  /  k ) ,  0 ) )
65prmreclem6 13272 . 2  |-  -.  seq  1 (  +  , 
( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m
) ,  0 ) ) )  e.  dom  ~~>
7 inss2 3549 . . . . . . . . 9  |-  ( Prime  i^i  ( 1 ... n
) )  C_  (
1 ... n )
87sseli 3331 . . . . . . . . . . 11  |-  ( k  e.  ( Prime  i^i  ( 1 ... n
) )  ->  k  e.  ( 1 ... n
) )
9 elfznn 11064 . . . . . . . . . . 11  |-  ( k  e.  ( 1 ... n )  ->  k  e.  NN )
10 nnrecre 10020 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  (
1  /  k )  e.  RR )
1110recnd 9098 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  (
1  /  k )  e.  CC )
128, 9, 113syl 19 . . . . . . . . . 10  |-  ( k  e.  ( Prime  i^i  ( 1 ... n
) )  ->  (
1  /  k )  e.  CC )
1312rgen 2758 . . . . . . . . 9  |-  A. k  e.  ( Prime  i^i  (
1 ... n ) ) ( 1  /  k
)  e.  CC
147, 13pm3.2i 442 . . . . . . . 8  |-  ( ( Prime  i^i  ( 1 ... n ) ) 
C_  ( 1 ... n )  /\  A. k  e.  ( Prime  i^i  ( 1 ... n
) ) ( 1  /  k )  e.  CC )
15 fzfi 11294 . . . . . . . . 9  |-  ( 1 ... n )  e. 
Fin
1615olci 381 . . . . . . . 8  |-  ( ( 1 ... n ) 
C_  ( ZZ>= `  1
)  \/  ( 1 ... n )  e. 
Fin )
17 sumss2 12503 . . . . . . . 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 654 . . . . . . 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 3517 . . . . . . . . . 10  |-  ( k  e.  ( Prime  i^i  ( 1 ... n
) )  <->  ( k  e.  Prime  /\  k  e.  ( 1 ... n
) ) )
2019rbaib 874 . . . . . . . . 9  |-  ( k  e.  ( 1 ... n )  ->  (
k  e.  ( Prime  i^i  ( 1 ... n
) )  <->  k  e.  Prime ) )
2120ifbid 3744 . . . . . . . 8  |-  ( k  e.  ( 1 ... n )  ->  if ( k  e.  ( Prime  i^i  ( 1 ... n ) ) ,  ( 1  / 
k ) ,  0 )  =  if ( k  e.  Prime ,  ( 1  /  k ) ,  0 ) )
2221sumeq2i 12476 . . . . . . 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 2450 . . . . . 6  |-  sum_ k  e.  ( Prime  i^i  (
1 ... n ) ) ( 1  /  k
)  =  sum_ k  e.  ( 1 ... n
) if ( k  e.  Prime ,  ( 1  /  k ) ,  0 )
249adantl 453 . . . . . . . 8  |-  ( ( n  e.  NN  /\  k  e.  ( 1 ... n ) )  ->  k  e.  NN )
25 prmnn 13065 . . . . . . . . . . . 12  |-  ( k  e.  Prime  ->  k  e.  NN )
2625, 11syl 16 . . . . . . . . . . 11  |-  ( k  e.  Prime  ->  ( 1  /  k )  e.  CC )
2726adantl 453 . . . . . . . . . 10  |-  ( (  T.  /\  k  e. 
Prime )  ->  ( 1  /  k )  e.  CC )
28 0cn 9068 . . . . . . . . . . 11  |-  0  e.  CC
2928a1i 11 . . . . . . . . . 10  |-  ( (  T.  /\  -.  k  e.  Prime )  ->  0  e.  CC )
3027, 29ifclda 3753 . . . . . . . . 9  |-  (  T. 
->  if ( k  e. 
Prime ,  ( 1  /  k ) ,  0 )  e.  CC )
3130trud 1332 . . . . . . . 8  |-  if ( k  e.  Prime ,  ( 1  /  k ) ,  0 )  e.  CC
325fvmpt2 5798 . . . . . . . 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 644 . . . . . . 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 20 . . . . . . . 8  |-  ( n  e.  NN  ->  n  e.  NN )
35 nnuz 10505 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
3634, 35syl6eleq 2520 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  ( ZZ>= `  1 )
)
3731a1i 11 . . . . . . 7  |-  ( ( n  e.  NN  /\  k  e.  ( 1 ... n ) )  ->  if ( k  e.  Prime ,  ( 1  /  k ) ,  0 )  e.  CC )
3833, 36, 37fsumser 12507 . . . . . 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 2474 . . . . 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 4278 . . . 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 10295 . . . . . . 7  |-  1  e.  ZZ
43 seqfn 11318 . . . . . . 7  |-  ( 1  e.  ZZ  ->  seq  1 (  +  , 
( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m
) ,  0 ) ) )  Fn  ( ZZ>=
`  1 ) )
4442, 43ax-mp 8 . . . . . 6  |-  seq  1
(  +  ,  ( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m
) ,  0 ) ) )  Fn  ( ZZ>=
`  1 )
4535fneq2i 5526 . . . . . 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 201 . . . . 5  |-  seq  1
(  +  ,  ( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m
) ,  0 ) ) )  Fn  NN
47 dffn5 5758 . . . . 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 200 . . . 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 2460 . . 3  |-  F  =  seq  1 (  +  ,  ( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m ) ,  0 ) ) )
5049eleq1i 2493 . 2  |-  ( F  e.  dom  ~~>  <->  seq  1
(  +  ,  ( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m
) ,  0 ) ) )  e.  dom  ~~>  )
516, 50mtbir 291 1  |-  -.  F  e.  dom  ~~>
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 358    /\ wa 359    T. wtru 1325    = wceq 1652    e. wcel 1725   A.wral 2692    i^i cin 3306    C_ wss 3307   ifcif 3726    e. cmpt 4253   dom cdm 4864    Fn wfn 5435   ` cfv 5440  (class class class)co 6067   Fincfn 7095   CCcc 8972   0cc0 8974   1c1 8975    + caddc 8977    / cdiv 9661   NNcn 9984   ZZcz 10266   ZZ>=cuz 10472   ...cfz 11027    seq cseq 11306    ~~> cli 12261   sum_csu 12462   Primecprime 13062
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-inf2 7580  ax-cnex 9030  ax-resscn 9031  ax-1cn 9032  ax-icn 9033  ax-addcl 9034  ax-addrcl 9035  ax-mulcl 9036  ax-mulrcl 9037  ax-mulcom 9038  ax-addass 9039  ax-mulass 9040  ax-distr 9041  ax-i2m1 9042  ax-1ne0 9043  ax-1rid 9044  ax-rnegex 9045  ax-rrecex 9046  ax-cnre 9047  ax-pre-lttri 9048  ax-pre-lttrn 9049  ax-pre-ltadd 9050  ax-pre-mulgt0 9051  ax-pre-sup 9052
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-int 4038  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-se 4529  df-we 4530  df-ord 4571  df-on 4572  df-lim 4573  df-suc 4574  df-om 4832  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-isom 5449  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-1st 6335  df-2nd 6336  df-riota 6535  df-recs 6619  df-rdg 6654  df-1o 6710  df-2o 6711  df-oadd 6714  df-er 6891  df-map 7006  df-pm 7007  df-en 7096  df-dom 7097  df-sdom 7098  df-fin 7099  df-sup 7432  df-oi 7463  df-card 7810  df-cda 8032  df-pnf 9106  df-mnf 9107  df-xr 9108  df-ltxr 9109  df-le 9110  df-sub 9277  df-neg 9278  df-div 9662  df-nn 9985  df-2 10042  df-3 10043  df-n0 10206  df-z 10267  df-uz 10473  df-q 10559  df-rp 10597  df-fz 11028  df-fzo 11119  df-fl 11185  df-mod 11234  df-seq 11307  df-exp 11366  df-hash 11602  df-cj 11887  df-re 11888  df-im 11889  df-sqr 12023  df-abs 12024  df-clim 12265  df-rlim 12266  df-sum 12463  df-dvds 12836  df-gcd 12990  df-prm 13063  df-pc 13194
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