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Theorem prmrec 12843
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 2313 . . . . 5  |-  ( m  =  k  ->  (
m  e.  Prime  <->  k  e.  Prime ) )
2 oveq2 5718 . . . . 5  |-  ( m  =  k  ->  (
1  /  m )  =  ( 1  / 
k ) )
3 eqidd 2254 . . . . 5  |-  ( m  =  k  ->  0  =  0 )
41, 2, 3ifbieq12d 3492 . . . 4  |-  ( m  =  k  ->  if ( m  e.  Prime ,  ( 1  /  m
) ,  0 )  =  if ( k  e.  Prime ,  ( 1  /  k ) ,  0 ) )
54cbvmptv 4008 . . 3  |-  ( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m ) ,  0 ) )  =  ( k  e.  NN  |->  if ( k  e.  Prime ,  ( 1  /  k ) ,  0 ) )
65prmreclem6 12842 . 2  |-  -.  seq  1 (  +  , 
( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m
) ,  0 ) ) )  e.  dom  ~~>
7 inss2 3297 . . . . . . . . 9  |-  ( Prime  i^i  ( 1 ... n
) )  C_  (
1 ... n )
87sseli 3099 . . . . . . . . . . 11  |-  ( k  e.  ( Prime  i^i  ( 1 ... n
) )  ->  k  e.  ( 1 ... n
) )
9 elfznn 10697 . . . . . . . . . . 11  |-  ( k  e.  ( 1 ... n )  ->  k  e.  NN )
10 nnrecre 9662 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  (
1  /  k )  e.  RR )
1110recnd 8741 . . . . . . . . . . 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 2570 . . . . . . . . 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 10912 . . . . . . . . 9  |-  ( 1 ... n )  e. 
Fin
1615olci 382 . . . . . . . 8  |-  ( ( 1 ... n ) 
C_  ( ZZ>= `  1
)  \/  ( 1 ... n )  e. 
Fin )
17 sumss2 12076 . . . . . . . 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 3266 . . . . . . . . . 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 3488 . . . . . . . 8  |-  ( k  e.  ( 1 ... n )  ->  if ( k  e.  ( Prime  i^i  ( 1 ... n ) ) ,  ( 1  / 
k ) ,  0 )  =  if ( k  e.  Prime ,  ( 1  /  k ) ,  0 ) )
2221sumeq2i 12049 . . . . . . 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 2273 . . . . . 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 12636 . . . . . . . . . . . 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 8711 . . . . . . . . . . 11  |-  0  e.  CC
2928a1i 12 . . . . . . . . . 10  |-  ( (  T.  /\  -.  k  e.  Prime )  ->  0  e.  CC )
3027, 29ifclda 3497 . . . . . . . . 9  |-  (  T. 
->  if ( k  e. 
Prime ,  ( 1  /  k ) ,  0 )  e.  CC )
3130trud 1320 . . . . . . . 8  |-  if ( k  e.  Prime ,  ( 1  /  k ) ,  0 )  e.  CC
325fvmpt2 5460 . . . . . . . 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 10142 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
3634, 35syl6eleq 2343 . . . . . . 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 12080 . . . . . 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 2297 . . . . 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 3999 . . . 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 9932 . . . . . . 7  |-  1  e.  ZZ
43 seqfn 10936 . . . . . . 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 5196 . . . . . 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 5420 . . . . 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 2283 . . 3  |-  F  =  seq  1 (  +  ,  ( m  e.  NN  |->  if ( m  e.  Prime ,  ( 1  /  m ) ,  0 ) ) )
5049eleq1i 2316 . 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 2509    i^i cin 3077    C_ wss 3078   ifcif 3470    e. cmpt 3974   dom cdm 4580    Fn wfn 4587   ` cfv 4592  (class class class)co 5710   Fincfn 6749   CCcc 8615   0cc0 8617   1c1 8618    + caddc 8620    / cdiv 9303   NNcn 9626   ZZcz 9903   ZZ>=cuz 10109   ...cfz 10660    seq cseq 10924    ~~> cli 11835   sum_csu 12035   Primecprime 12632
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-2o 6366  df-oadd 6369  df-er 6546  df-map 6660  df-pm 6661  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-sup 7078  df-oi 7109  df-card 7456  df-cda 7678  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-n0 9845  df-z 9904  df-uz 10110  df-q 10196  df-rp 10234  df-fz 10661  df-fzo 10749  df-fl 10803  df-mod 10852  df-seq 10925  df-exp 10983  df-hash 11216  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-clim 11839  df-rlim 11840  df-sum 12036  df-divides 12406  df-gcd 12560  df-prime 12633  df-pc 12764
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