Theorem prmrec 13975

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 2498	. . . . 5 |- ( m = k -> ( m e. Prime <-> k e. Prime ) )
2		oveq2 6094	. . . . 5 |- ( m = k -> ( 1 / m ) = ( 1 / k ) )
3	1, 2	ifbieq1d 3807	. . . 4 |- ( m = k -> if ( m e. Prime , ( 1 / m ) , 0 ) = if ( k e. Prime , ( 1 / k ) , 0 ) )
4	3	cbvmptv 4378	. . 3 |- ( m e. NN |-> if ( m e. Prime , ( 1 / m ) , 0 ) ) = ( k e. NN |-> if ( k e. Prime , ( 1 / k ) , 0 ) )
5	4	prmreclem6 13974	. 2 |- -. seq 1 ( + , ( m e. NN |-> if ( m e. Prime , ( 1 / m ) , 0 ) ) ) e. dom ~~>
6		inss2 3566	. . . . . . . . 9 |- ( Prime i^i ( 1 ... n ) ) C_ ( 1 ... n )
7	6	sseli 3347	. . . . . . . . . . 11 |- ( k e. ( Prime i^i ( 1 ... n ) ) -> k e. ( 1 ... n ) )
8		elfznn 11470	. . . . . . . . . . 11 |- ( k e. ( 1 ... n ) -> k e. NN )
9		nnrecre 10350	. . . . . . . . . . . 12 |- ( k e. NN -> ( 1 / k ) e. RR )
10	9	recnd 9404	. . . . . . . . . . 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 2776	. . . . . . . . 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 11786	. . . . . . . . 9 |- ( 1 ... n ) e. Fin
15	14	olci 391	. . . . . . . 8 |- ( ( 1 ... n ) C_ ( ZZ>= ` 1 ) \/ ( 1 ... n ) e. Fin )
16		sumss2 13195	. . . . . . . 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 3534	. . . . . . . . . 10 |- ( k e. ( Prime i^i ( 1 ... n ) ) <-> ( k e. Prime /\ k e. ( 1 ... n ) ) )
19	18	rbaib 898	. . . . . . . . 9 |- ( k e. ( 1 ... n ) -> ( k e. ( Prime i^i ( 1 ... n ) ) <-> k e. Prime ) )
20	19	ifbid 3806	. . . . . . . 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 13168	. . . . . . 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 2458	. . . . . 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 13758	. . . . . . . . . . . 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 9371	. . . . . . . . . 10 |- ( ( T. /\ -. k e. Prime ) -> 0 e. CC )
28	26, 27	ifclda 3816	. . . . . . . . 9 |- ( T. -> if ( k e. Prime , ( 1 / k ) , 0 ) e. CC )
29	28	trud 1378	. . . . . . . 8 |- if ( k e. Prime , ( 1 / k ) , 0 ) e. CC
30	4	fvmpt2 5776	. . . . . . . 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 10888	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
34	32, 33	syl6eleq 2528	. . . . . . 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 13199	. . . . . 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 2482	. . . . 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 4369	. . . 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 10668	. . . . . . 7 |- 1 e. ZZ
41		seqfn 11810	. . . . . . 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 5501	. . . . . 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 5732	. . . . 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 2468	. . 3 |- F = seq 1 ( + , ( m e. NN |-> if ( m e. Prime , ( 1 / m ) , 0 ) ) )
48	47	eleq1i 2501	. 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 1369   T. wtru 1370   e. wcel 1756   A.wral 2710   i^i cin 3322   C_ wss 3323   ifcif 3786   e. cmpt 4345   dom cdm 4835   Fn wfn 5408   ` cfv 5413  (class class class)co 6086   Fincfn 7302   CCcc 9272   0cc0 9274   1c1 9275   + caddc 9277   / cdiv 9985   NNcn 10314   ZZcz 10638   ZZ>=cuz 10853   ...cfz 11429   seqcseq 11798   ~~> cli 12954   sum_csu 13155   Primecprime 13755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-q 10946  df-rp 10984  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-rlim 12959  df-sum 13156  df-dvds 13528  df-gcd 13683  df-prm 13756  df-pc 13896

This theorem is referenced by: (None)