Theorem prmrec 14854

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 2494	. . . . 5 |- ( m = k -> ( m e. Prime <-> k e. Prime ) )
2		oveq2 6310	. . . . 5 |- ( m = k -> ( 1 / m ) = ( 1 / k ) )
3	1, 2	ifbieq1d 3932	. . . 4 |- ( m = k -> if ( m e. Prime , ( 1 / m ) , 0 ) = if ( k e. Prime , ( 1 / k ) , 0 ) )
4	3	cbvmptv 4513	. . 3 |- ( m e. NN |-> if ( m e. Prime , ( 1 / m ) , 0 ) ) = ( k e. NN |-> if ( k e. Prime , ( 1 / k ) , 0 ) )
5	4	prmreclem6 14853	. 2 |- -. seq 1 ( + , ( m e. NN |-> if ( m e. Prime , ( 1 / m ) , 0 ) ) ) e. dom ~~>
6		inss2 3683	. . . . . . . . 9 |- ( Prime i^i ( 1 ... n ) ) C_ ( 1 ... n )
7	6	sseli 3460	. . . . . . . . . . 11 |- ( k e. ( Prime i^i ( 1 ... n ) ) -> k e. ( 1 ... n ) )
8		elfznn 11829	. . . . . . . . . . 11 |- ( k e. ( 1 ... n ) -> k e. NN )
9		nnrecre 10647	. . . . . . . . . . . 12 |- ( k e. NN -> ( 1 / k ) e. RR )
10	9	recnd 9670	. . . . . . . . . . 11 |- ( k e. NN -> ( 1 / k ) e. CC )
11	7, 8, 10	3syl 18	. . . . . . . . . 10 |- ( k e. ( Prime i^i ( 1 ... n ) ) -> ( 1 / k ) e. CC )
12	11	rgen 2785	. . . . . . . . 9 |- A. k e. ( Prime i^i ( 1 ... n ) ) ( 1 / k ) e. CC
13	6, 12	pm3.2i 456	. . . . . . . 8 |- ( ( Prime i^i ( 1 ... n ) ) C_ ( 1 ... n ) /\ A. k e. ( Prime i^i ( 1 ... n ) ) ( 1 / k ) e. CC )
14		fzfi 12185	. . . . . . . . 9 |- ( 1 ... n ) e. Fin
15	14	olci 392	. . . . . . . 8 |- ( ( 1 ... n ) C_ ( ZZ>= ` 1 ) \/ ( 1 ... n ) e. Fin )
16		sumss2 13780	. . . . . . . 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 676	. . . . . . 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 3649	. . . . . . . . . 10 |- ( k e. ( Prime i^i ( 1 ... n ) ) <-> ( k e. Prime /\ k e. ( 1 ... n ) ) )
19	18	rbaib 914	. . . . . . . . 9 |- ( k e. ( 1 ... n ) -> ( k e. ( Prime i^i ( 1 ... n ) ) <-> k e. Prime ) )
20	19	ifbid 3931	. . . . . . . 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 13753	. . . . . . 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 2451	. . . . . 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 467	. . . . . . . 8 |- ( ( n e. NN /\ k e. ( 1 ... n ) ) -> k e. NN )
24		prmnn 14613	. . . . . . . . . . . 12 |- ( k e. Prime -> k e. NN )
25	24, 10	syl 17	. . . . . . . . . . 11 |- ( k e. Prime -> ( 1 / k ) e. CC )
26	25	adantl 467	. . . . . . . . . 10 |- ( ( T. /\ k e. Prime ) -> ( 1 / k ) e. CC )
27		0cnd 9637	. . . . . . . . . 10 |- ( ( T. /\ -. k e. Prime ) -> 0 e. CC )
28	26, 27	ifclda 3941	. . . . . . . . 9 |- ( T. -> if ( k e. Prime , ( 1 / k ) , 0 ) e. CC )
29	28	trud 1446	. . . . . . . 8 |- if ( k e. Prime , ( 1 / k ) , 0 ) e. CC
30	4	fvmpt2 5970	. . . . . . . 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 666	. . . . . . 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 23	. . . . . . . 8 |- ( n e. NN -> n e. NN )
33		nnuz 11195	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
34	32, 33	syl6eleq 2520	. . . . . . 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 13784	. . . . . 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 2475	. . . . 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 4503	. . . 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 10968	. . . . . . 7 |- 1 e. ZZ
41		seqfn 12225	. . . . . . 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 5686	. . . . . 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 212	. . . . 5 |- seq 1 ( + , ( m e. NN |-> if ( m e. Prime , ( 1 / m ) , 0 ) ) ) Fn NN
45		dffn5 5923	. . . . 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 211	. . . 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 2461	. . 3 |- F = seq 1 ( + , ( m e. NN |-> if ( m e. Prime , ( 1 / m ) , 0 ) ) )
48	47	eleq1i 2499	. 2 |- ( F e. dom ~~> <-> seq 1 ( + , ( m e. NN |-> if ( m e. Prime , ( 1 / m ) , 0 ) ) ) e. dom ~~> )
49	5, 48	mtbir 300	1 |- -. F e. dom ~~>

Syntax hints:   -. wn 3   \/ wo 369   /\ wa 370   = wceq 1437   T. wtru 1438   e. wcel 1868   A.wral 2775   i^i cin 3435   C_ wss 3436   ifcif 3909   |-> cmpt 4479   dom cdm 4850   Fn wfn 5593   ` cfv 5598  (class class class)co 6302   Fincfn 7574   CCcc 9538   0cc0 9540   1c1 9541   + caddc 9543   / cdiv 10270   NNcn 10610   ZZcz 10938   ZZ>=cuz 11160   ...cfz 11785   seqcseq 12213   ~~> cli 13536   sum_csu 13740   Primecprime 14610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-om 6704  df-1st 6804  df-2nd 6805  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-sup 7959  df-inf 7960  df-oi 8028  df-card 8375  df-cda 8599  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-n0 10871  df-z 10939  df-uz 11161  df-q 11266  df-rp 11304  df-fz 11786  df-fzo 11917  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-clim 13540  df-rlim 13541  df-sum 13741  df-dvds 14294  df-gcd 14457  df-prm 14611  df-pc 14775

This theorem is referenced by: (None)