Theorem prmrec 14647

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 2474	. . . . 5 |- ( m = k -> ( m e. Prime <-> k e. Prime ) )
2		oveq2 6285	. . . . 5 |- ( m = k -> ( 1 / m ) = ( 1 / k ) )
3	1, 2	ifbieq1d 3907	. . . 4 |- ( m = k -> if ( m e. Prime , ( 1 / m ) , 0 ) = if ( k e. Prime , ( 1 / k ) , 0 ) )
4	3	cbvmptv 4486	. . 3 |- ( m e. NN |-> if ( m e. Prime , ( 1 / m ) , 0 ) ) = ( k e. NN |-> if ( k e. Prime , ( 1 / k ) , 0 ) )
5	4	prmreclem6 14646	. 2 |- -. seq 1 ( + , ( m e. NN |-> if ( m e. Prime , ( 1 / m ) , 0 ) ) ) e. dom ~~>
6		inss2 3659	. . . . . . . . 9 |- ( Prime i^i ( 1 ... n ) ) C_ ( 1 ... n )
7	6	sseli 3437	. . . . . . . . . . 11 |- ( k e. ( Prime i^i ( 1 ... n ) ) -> k e. ( 1 ... n ) )
8		elfznn 11766	. . . . . . . . . . 11 |- ( k e. ( 1 ... n ) -> k e. NN )
9		nnrecre 10612	. . . . . . . . . . . 12 |- ( k e. NN -> ( 1 / k ) e. RR )
10	9	recnd 9651	. . . . . . . . . . 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 2763	. . . . . . . . 9 |- A. k e. ( Prime i^i ( 1 ... n ) ) ( 1 / k ) e. CC
13	6, 12	pm3.2i 453	. . . . . . . 8 |- ( ( Prime i^i ( 1 ... n ) ) C_ ( 1 ... n ) /\ A. k e. ( Prime i^i ( 1 ... n ) ) ( 1 / k ) e. CC )
14		fzfi 12121	. . . . . . . . 9 |- ( 1 ... n ) e. Fin
15	14	olci 389	. . . . . . . 8 |- ( ( 1 ... n ) C_ ( ZZ>= ` 1 ) \/ ( 1 ... n ) e. Fin )
16		sumss2 13695	. . . . . . . 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 670	. . . . . . 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 3625	. . . . . . . . . 10 |- ( k e. ( Prime i^i ( 1 ... n ) ) <-> ( k e. Prime /\ k e. ( 1 ... n ) ) )
19	18	rbaib 907	. . . . . . . . 9 |- ( k e. ( 1 ... n ) -> ( k e. ( Prime i^i ( 1 ... n ) ) <-> k e. Prime ) )
20	19	ifbid 3906	. . . . . . . 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 13668	. . . . . . 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 2431	. . . . . 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 464	. . . . . . . 8 |- ( ( n e. NN /\ k e. ( 1 ... n ) ) -> k e. NN )
24		prmnn 14427	. . . . . . . . . . . 12 |- ( k e. Prime -> k e. NN )
25	24, 10	syl 17	. . . . . . . . . . 11 |- ( k e. Prime -> ( 1 / k ) e. CC )
26	25	adantl 464	. . . . . . . . . 10 |- ( ( T. /\ k e. Prime ) -> ( 1 / k ) e. CC )
27		0cnd 9618	. . . . . . . . . 10 |- ( ( T. /\ -. k e. Prime ) -> 0 e. CC )
28	26, 27	ifclda 3916	. . . . . . . . 9 |- ( T. -> if ( k e. Prime , ( 1 / k ) , 0 ) e. CC )
29	28	trud 1414	. . . . . . . 8 |- if ( k e. Prime , ( 1 / k ) , 0 ) e. CC
30	4	fvmpt2 5940	. . . . . . . 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 660	. . . . . . 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 11161	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
34	32, 33	syl6eleq 2500	. . . . . . 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 13699	. . . . . 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 2455	. . . . 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 4476	. . . 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 10934	. . . . . . 7 |- 1 e. ZZ
41		seqfn 12161	. . . . . . 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 5656	. . . . . 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 5893	. . . . 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 2441	. . 3 |- F = seq 1 ( + , ( m e. NN |-> if ( m e. Prime , ( 1 / m ) , 0 ) ) )
48	47	eleq1i 2479	. 2 |- ( F e. dom ~~> <-> seq 1 ( + , ( m e. NN |-> if ( m e. Prime , ( 1 / m ) , 0 ) ) ) e. dom ~~> )
49	5, 48	mtbir 297	1 |- -. F e. dom ~~>

Syntax hints:   -. wn 3   \/ wo 366   /\ wa 367   = wceq 1405   T. wtru 1406   e. wcel 1842   A.wral 2753   i^i cin 3412   C_ wss 3413   ifcif 3884   |-> cmpt 4452   dom cdm 4822   Fn wfn 5563   ` cfv 5568  (class class class)co 6277   Fincfn 7553   CCcc 9519   0cc0 9521   1c1 9522   + caddc 9524   / cdiv 10246   NNcn 10575   ZZcz 10904   ZZ>=cuz 11126   ...cfz 11724   seqcseq 12149   ~~> cli 13454   sum_csu 13655   Primecprime 14424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-inf2 8090  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-2o 7167  df-oadd 7170  df-er 7347  df-map 7458  df-pm 7459  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-sup 7934  df-oi 7968  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-n0 10836  df-z 10905  df-uz 11127  df-q 11227  df-rp 11265  df-fz 11725  df-fzo 11853  df-fl 11964  df-mod 12033  df-seq 12150  df-exp 12209  df-hash 12451  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-clim 13458  df-rlim 13459  df-sum 13656  df-dvds 14194  df-gcd 14352  df-prm 14425  df-pc 14568

This theorem is referenced by: (None)