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Theorem prmrec 13249
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 2468 . . . . 5  |-  ( m  =  k  ->  (
m  e.  Prime  <->  k  e.  Prime ) )
2 oveq2 6052 . . . . 5  |-  ( m  =  k  ->  (
1  /  m )  =  ( 1  / 
k ) )
3 eqidd 2409 . . . . 5  |-  ( m  =  k  ->  0  =  0 )
41, 2, 3ifbieq12d 3725 . . . 4  |-  ( m  =  k  ->  if ( m  e.  Prime ,  ( 1  /  m
) ,  0 )  =  if ( k  e.  Prime ,  ( 1  /  k ) ,  0 ) )
54cbvmptv 4264 . . 3  |-  ( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m ) ,  0 ) )  =  ( k  e.  NN  |->  if ( k  e.  Prime ,  ( 1  /  k ) ,  0 ) )
65prmreclem6 13248 . 2  |-  -.  seq  1 (  +  , 
( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m
) ,  0 ) ) )  e.  dom  ~~>
7 inss2 3526 . . . . . . . . 9  |-  ( Prime  i^i  ( 1 ... n
) )  C_  (
1 ... n )
87sseli 3308 . . . . . . . . . . 11  |-  ( k  e.  ( Prime  i^i  ( 1 ... n
) )  ->  k  e.  ( 1 ... n
) )
9 elfznn 11040 . . . . . . . . . . 11  |-  ( k  e.  ( 1 ... n )  ->  k  e.  NN )
10 nnrecre 9996 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  (
1  /  k )  e.  RR )
1110recnd 9074 . . . . . . . . . . 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 2735 . . . . . . . . 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 11270 . . . . . . . . 9  |-  ( 1 ... n )  e. 
Fin
1615olci 381 . . . . . . . 8  |-  ( ( 1 ... n ) 
C_  ( ZZ>= `  1
)  \/  ( 1 ... n )  e. 
Fin )
17 sumss2 12479 . . . . . . . 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 3494 . . . . . . . . . 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 3721 . . . . . . . 8  |-  ( k  e.  ( 1 ... n )  ->  if ( k  e.  ( Prime  i^i  ( 1 ... n ) ) ,  ( 1  / 
k ) ,  0 )  =  if ( k  e.  Prime ,  ( 1  /  k ) ,  0 ) )
2221sumeq2i 12452 . . . . . . 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 2428 . . . . . 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 13041 . . . . . . . . . . . 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 9044 . . . . . . . . . . 11  |-  0  e.  CC
2928a1i 11 . . . . . . . . . 10  |-  ( (  T.  /\  -.  k  e.  Prime )  ->  0  e.  CC )
3027, 29ifclda 3730 . . . . . . . . 9  |-  (  T. 
->  if ( k  e. 
Prime ,  ( 1  /  k ) ,  0 )  e.  CC )
3130trud 1329 . . . . . . . 8  |-  if ( k  e.  Prime ,  ( 1  /  k ) ,  0 )  e.  CC
325fvmpt2 5775 . . . . . . . 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 10481 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
3634, 35syl6eleq 2498 . . . . . . 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 12483 . . . . . 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 2452 . . . . 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 4255 . . . 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 10271 . . . . . . 7  |-  1  e.  ZZ
43 seqfn 11294 . . . . . . 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 5503 . . . . . 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 5735 . . . . 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 2438 . . 3  |-  F  =  seq  1 (  +  ,  ( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m ) ,  0 ) ) )
5049eleq1i 2471 . 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 1322    = wceq 1649    e. wcel 1721   A.wral 2670    i^i cin 3283    C_ wss 3284   ifcif 3703    e. cmpt 4230   dom cdm 4841    Fn wfn 5412   ` cfv 5417  (class class class)co 6044   Fincfn 7072   CCcc 8948   0cc0 8950   1c1 8951    + caddc 8953    / cdiv 9637   NNcn 9960   ZZcz 10242   ZZ>=cuz 10448   ...cfz 11003    seq cseq 11282    ~~> cli 12237   sum_csu 12438   Primecprime 13038
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6868  df-map 6983  df-pm 6984  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-sup 7408  df-oi 7439  df-card 7786  df-cda 8008  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-n0 10182  df-z 10243  df-uz 10449  df-q 10535  df-rp 10573  df-fz 11004  df-fzo 11095  df-fl 11161  df-mod 11210  df-seq 11283  df-exp 11342  df-hash 11578  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-clim 12241  df-rlim 12242  df-sum 12439  df-dvds 12812  df-gcd 12966  df-prm 13039  df-pc 13170
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