Theorem prmrec 14094

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 2523	. . . . 5 |- ( m = k -> ( m e. Prime <-> k e. Prime ) )
2		oveq2 6201	. . . . 5 |- ( m = k -> ( 1 / m ) = ( 1 / k ) )
3	1, 2	ifbieq1d 3913	. . . 4 |- ( m = k -> if ( m e. Prime , ( 1 / m ) , 0 ) = if ( k e. Prime , ( 1 / k ) , 0 ) )
4	3	cbvmptv 4484	. . 3 |- ( m e. NN |-> if ( m e. Prime , ( 1 / m ) , 0 ) ) = ( k e. NN |-> if ( k e. Prime , ( 1 / k ) , 0 ) )
5	4	prmreclem6 14093	. 2 |- -. seq 1 ( + , ( m e. NN |-> if ( m e. Prime , ( 1 / m ) , 0 ) ) ) e. dom ~~>
6		inss2 3672	. . . . . . . . 9 |- ( Prime i^i ( 1 ... n ) ) C_ ( 1 ... n )
7	6	sseli 3453	. . . . . . . . . . 11 |- ( k e. ( Prime i^i ( 1 ... n ) ) -> k e. ( 1 ... n ) )
8		elfznn 11588	. . . . . . . . . . 11 |- ( k e. ( 1 ... n ) -> k e. NN )
9		nnrecre 10462	. . . . . . . . . . . 12 |- ( k e. NN -> ( 1 / k ) e. RR )
10	9	recnd 9516	. . . . . . . . . . 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 2892	. . . . . . . . 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 11904	. . . . . . . . 9 |- ( 1 ... n ) e. Fin
15	14	olci 391	. . . . . . . 8 |- ( ( 1 ... n ) C_ ( ZZ>= ` 1 ) \/ ( 1 ... n ) e. Fin )
16		sumss2 13314	. . . . . . . 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 3640	. . . . . . . . . 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 3912	. . . . . . . 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 13287	. . . . . . 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 2480	. . . . . 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 13877	. . . . . . . . . . . 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 9483	. . . . . . . . . 10 |- ( ( T. /\ -. k e. Prime ) -> 0 e. CC )
28	26, 27	ifclda 3922	. . . . . . . . 9 |- ( T. -> if ( k e. Prime , ( 1 / k ) , 0 ) e. CC )
29	28	trud 1379	. . . . . . . 8 |- if ( k e. Prime , ( 1 / k ) , 0 ) e. CC
30	4	fvmpt2 5883	. . . . . . . 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 11000	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
34	32, 33	syl6eleq 2549	. . . . . . 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 13318	. . . . . 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 2504	. . . . 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 4475	. . . 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 10780	. . . . . . 7 |- 1 e. ZZ
41		seqfn 11928	. . . . . . 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 5607	. . . . . 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 5839	. . . . 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 2490	. . 3 |- F = seq 1 ( + , ( m e. NN |-> if ( m e. Prime , ( 1 / m ) , 0 ) ) )
48	47	eleq1i 2528	. 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 1370   T. wtru 1371   e. wcel 1758   A.wral 2795   i^i cin 3428   C_ wss 3429   ifcif 3892   |-> cmpt 4451   dom cdm 4941   Fn wfn 5514   ` cfv 5519  (class class class)co 6193   Fincfn 7413   CCcc 9384   0cc0 9386   1c1 9387   + caddc 9389   / cdiv 10097   NNcn 10426   ZZcz 10750   ZZ>=cuz 10965   ...cfz 11547   seqcseq 11916   ~~> cli 13073   sum_csu 13274   Primecprime 13874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-er 7204  df-map 7319  df-pm 7320  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-sup 7795  df-oi 7828  df-card 8213  df-cda 8441  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-n0 10684  df-z 10751  df-uz 10966  df-q 11058  df-rp 11096  df-fz 11548  df-fzo 11659  df-fl 11752  df-mod 11819  df-seq 11917  df-exp 11976  df-hash 12214  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-clim 13077  df-rlim 13078  df-sum 13275  df-dvds 13647  df-gcd 13802  df-prm 13875  df-pc 14015

This theorem is referenced by: (None)