Theorem prmrec 14295

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. This is Metamath 100 proof #81. (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.

Step	Hyp	Ref	Expression
1		eleq1 2539	. . . . 5 |- ( m = k -> ( m e. Prime <-> k e. Prime ) )
2		oveq2 6290	. . . . 5 |- ( m = k -> ( 1 / m ) = ( 1 / k ) )
3	1, 2	ifbieq1d 3962	. . . 4 |- ( m = k -> if ( m e. Prime , ( 1 / m ) , 0 ) = if ( k e. Prime , ( 1 / k ) , 0 ) )
4	3	cbvmptv 4538	. . 3 |- ( m e. NN |-> if ( m e. Prime , ( 1 / m ) , 0 ) ) = ( k e. NN |-> if ( k e. Prime , ( 1 / k ) , 0 ) )
5	4	prmreclem6 14294	. 2 |- -. seq 1 ( + , ( m e. NN |-> if ( m e. Prime , ( 1 / m ) , 0 ) ) ) e. dom ~~>
6		inss2 3719	. . . . . . . . 9 |- ( Prime i^i ( 1 ... n ) ) C_ ( 1 ... n )
7	6	sseli 3500	. . . . . . . . . . 11 |- ( k e. ( Prime i^i ( 1 ... n ) ) -> k e. ( 1 ... n ) )
8		elfznn 11710	. . . . . . . . . . 11 |- ( k e. ( 1 ... n ) -> k e. NN )
9		nnrecre 10568	. . . . . . . . . . . 12 |- ( k e. NN -> ( 1 / k ) e. RR )
10	9	recnd 9618	. . . . . . . . . . 11 |- ( k e. NN -> ( 1 / k ) e. CC )
11	7, 8, 10	3syl 20	. . . . . . . . . 10 |- ( k e. ( Prime i^i ( 1 ... n ) ) -> ( 1 / k ) e. CC )
12	11	rgen 2824	. . . . . . . . 9 |- A. k e. ( Prime i^i ( 1 ... n ) ) ( 1 / k ) e. CC
13	6, 12	pm3.2i 455	. . . . . . . 8 |- ( ( Prime i^i ( 1 ... n ) ) C_ ( 1 ... n ) /\ A. k e. ( Prime i^i ( 1 ... n ) ) ( 1 / k ) e. CC )
14		fzfi 12046	. . . . . . . . 9 |- ( 1 ... n ) e. Fin
15	14	olci 391	. . . . . . . 8 |- ( ( 1 ... n ) C_ ( ZZ>= ` 1 ) \/ ( 1 ... n ) e. Fin )
16		sumss2 13507	. . . . . . . 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 ) )
17	13, 15, 16	mp2an 672	. . . . . . 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 )
18		elin 3687	. . . . . . . . . 10 |- ( k e. ( Prime i^i ( 1 ... n ) ) <-> ( k e. Prime /\ k e. ( 1 ... n ) ) )
19	18	rbaib 904	. . . . . . . . 9 |- ( k e. ( 1 ... n ) -> ( k e. ( Prime i^i ( 1 ... n ) ) <-> k e. Prime ) )
20	19	ifbid 3961	. . . . . . . 8 |- ( k e. ( 1 ... n ) -> if ( k e. ( Prime i^i ( 1 ... n ) ) , ( 1 / k ) , 0 ) = if ( k e. Prime , ( 1 / k ) , 0 ) )
21	20	sumeq2i 13480	. . . . . . 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 )
22	17, 21	eqtri 2496	. . . . . 6 |- sum_ k e. ( Prime i^i ( 1 ... n ) ) ( 1 / k ) = sum_ k e. ( 1 ... n ) if ( k e. Prime , ( 1 / k ) , 0 )
23	8	adantl 466	. . . . . . . 8 |- ( ( n e. NN /\ k e. ( 1 ... n ) ) -> k e. NN )
24		prmnn 14075	. . . . . . . . . . . 12 |- ( k e. Prime -> k e. NN )
25	24, 10	syl 16	. . . . . . . . . . 11 |- ( k e. Prime -> ( 1 / k ) e. CC )
26	25	adantl 466	. . . . . . . . . 10 |- ( ( T. /\ k e. Prime ) -> ( 1 / k ) e. CC )
27		0cnd 9585	. . . . . . . . . 10 |- ( ( T. /\ -. k e. Prime ) -> 0 e. CC )
28	26, 27	ifclda 3971	. . . . . . . . 9 |- ( T. -> if ( k e. Prime , ( 1 / k ) , 0 ) e. CC )
29	28	trud 1388	. . . . . . . 8 |- if ( k e. Prime , ( 1 / k ) , 0 ) e. CC
30	4	fvmpt2 5955	. . . . . . . 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 ) )
31	23, 29, 30	sylancl 662	. . . . . . 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 ) )
32		id 22	. . . . . . . 8 |- ( n e. NN -> n e. NN )
33		nnuz 11113	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
34	32, 33	syl6eleq 2565	. . . . . . 7 |- ( n e. NN -> n e. ( ZZ>= ` 1 ) )
35	29	a1i 11	. . . . . . 7 |- ( ( n e. NN /\ k e. ( 1 ... n ) ) -> if ( k e. Prime , ( 1 / k ) , 0 ) e. CC )
36	31, 34, 35	fsumser 13511	. . . . . 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 ) )
37	22, 36	syl5eq 2520	. . . . 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 ) )
38	37	mpteq2ia 4529	. . . 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 ) )
39		prmrec.f	. . . 4 |- F = ( n e. NN |-> sum_ k e. ( Prime i^i ( 1 ... n ) ) ( 1 / k ) )
40		1z 10890	. . . . . . 7 |- 1 e. ZZ
41		seqfn 12083	. . . . . . 7 |- ( 1 e. ZZ -> seq 1 ( + , ( m e. NN |-> if ( m e. Prime , ( 1 / m ) , 0 ) ) ) Fn ( ZZ>= ` 1 ) )
42	40, 41	ax-mp 5	. . . . . 6 |- seq 1 ( + , ( m e. NN |-> if ( m e. Prime , ( 1 / m ) , 0 ) ) ) Fn ( ZZ>= ` 1 )
43	33	fneq2i 5674	. . . . . 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 ) )
44	42, 43	mpbir 209	. . . . 5 |- seq 1 ( + , ( m e. NN |-> if ( m e. Prime , ( 1 / m ) , 0 ) ) ) Fn NN
45		dffn5 5911	. . . . 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 ) ) )
46	44, 45	mpbi 208	. . . 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 ) )
47	38, 39, 46	3eqtr4i 2506	. . 3 |- F = seq 1 ( + , ( m e. NN |-> if ( m e. Prime , ( 1 / m ) , 0 ) ) )
48	47	eleq1i 2544	. 2 |- ( F e. dom ~~> <-> seq 1 ( + , ( m e. NN |-> if ( m e. Prime , ( 1 / m ) , 0 ) ) ) e. dom ~~> )
49	5, 48	mtbir 299	1 |- -. F e. dom ~~>

Syntax hints:   -. wn 3   \/ wo 368   /\ wa 369   = wceq 1379   T. wtru 1380   e. wcel 1767   A.wral 2814   i^i cin 3475   C_ wss 3476   ifcif 3939   |-> cmpt 4505   dom cdm 4999   Fn wfn 5581   ` cfv 5586  (class class class)co 6282   Fincfn 7513   CCcc 9486   0cc0 9488   1c1 9489   + caddc 9491   / cdiv 10202   NNcn 10532   ZZcz 10860   ZZ>=cuz 11078   ...cfz 11668   seqcseq 12071   ~~> cli 13266   sum_csu 13467   Primecprime 14072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-q 11179  df-rp 11217  df-fz 11669  df-fzo 11789  df-fl 11893  df-mod 11961  df-seq 12072  df-exp 12131  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-rlim 13271  df-sum 13468  df-dvds 13844  df-gcd 14000  df-prm 14073  df-pc 14216

This theorem is referenced by: (None)