Theorem trirecip 12028

Description: The sum of the reciprocals of the triangle numbers converge to two. This is Metamath 100 proof #42. (Contributed by Scott Fenton, 23-Apr-2014.) (Revised by Mario Carneiro, 22-May-2014.)

Assertion

Ref	Expression
trirecip	|- sum_ k e. NN ( 2 / ( k x. ( k + 1 ) ) ) = 2

Proof of Theorem trirecip

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2cnd 9194	. . . 4 |- ( k e. NN -> 2 e. CC )
2		peano2nn 9133	. . . . . 6 |- ( k e. NN -> ( k + 1 ) e. NN )
3		nnmulcl 9142	. . . . . 6 |- ( ( k e. NN /\ ( k + 1 ) e. NN ) -> ( k x. ( k + 1 ) ) e. NN )
4	2, 3	mpdan 421	. . . . 5 |- ( k e. NN -> ( k x. ( k + 1 ) ) e. NN )
5	4	nncnd 9135	. . . 4 |- ( k e. NN -> ( k x. ( k + 1 ) ) e. CC )
6	4	nnap0d 9167	. . . 4 |- ( k e. NN -> ( k x. ( k + 1 ) ) # 0 )
7	1, 5, 6	divrecapd 8951	. . 3 |- ( k e. NN -> ( 2 / ( k x. ( k + 1 ) ) ) = ( 2 x. ( 1 / ( k x. ( k + 1 ) ) ) ) )
8	7	sumeq2i 11891	. 2 |- sum_ k e. NN ( 2 / ( k x. ( k + 1 ) ) ) = sum_ k e. NN ( 2 x. ( 1 / ( k x. ( k + 1 ) ) ) )
9		nnuz 9770	. . . . 5 |- NN = ( ZZ>= ` 1 )
10		1zzd 9484	. . . . 5 |- ( T. -> 1 e. ZZ )
11		simpr 110	. . . . . 6 |- ( ( T. /\ k e. NN ) -> k e. NN )
12	4	adantl 277	. . . . . . 7 |- ( ( T. /\ k e. NN ) -> ( k x. ( k + 1 ) ) e. NN )
13	12	nnrecred 9168	. . . . . 6 |- ( ( T. /\ k e. NN ) -> ( 1 / ( k x. ( k + 1 ) ) ) e. RR )
14		id 19	. . . . . . . . 9 |- ( n = k -> n = k )
15		oveq1 6014	. . . . . . . . 9 |- ( n = k -> ( n + 1 ) = ( k + 1 ) )
16	14, 15	oveq12d 6025	. . . . . . . 8 |- ( n = k -> ( n x. ( n + 1 ) ) = ( k x. ( k + 1 ) ) )
17	16	oveq2d 6023	. . . . . . 7 |- ( n = k -> ( 1 / ( n x. ( n + 1 ) ) ) = ( 1 / ( k x. ( k + 1 ) ) ) )
18		eqid 2229	. . . . . . 7 |- ( n e. NN |-> ( 1 / ( n x. ( n + 1 ) ) ) ) = ( n e. NN |-> ( 1 / ( n x. ( n + 1 ) ) ) )
19	17, 18	fvmptg 5712	. . . . . 6 |- ( ( k e. NN /\ ( 1 / ( k x. ( k + 1 ) ) ) e. RR ) -> ( ( n e. NN |-> ( 1 / ( n x. ( n + 1 ) ) ) ) ` k ) = ( 1 / ( k x. ( k + 1 ) ) ) )
20	11, 13, 19	syl2anc 411	. . . . 5 |- ( ( T. /\ k e. NN ) -> ( ( n e. NN |-> ( 1 / ( n x. ( n + 1 ) ) ) ) ` k ) = ( 1 / ( k x. ( k + 1 ) ) ) )
21	4	nnrecred 9168	. . . . . . 7 |- ( k e. NN -> ( 1 / ( k x. ( k + 1 ) ) ) e. RR )
22	21	recnd 8186	. . . . . 6 |- ( k e. NN -> ( 1 / ( k x. ( k + 1 ) ) ) e. CC )
23	22	adantl 277	. . . . 5 |- ( ( T. /\ k e. NN ) -> ( 1 / ( k x. ( k + 1 ) ) ) e. CC )
24	18	trireciplem 12027	. . . . . . 7 |- seq 1 ( + , ( n e. NN |-> ( 1 / ( n x. ( n + 1 ) ) ) ) ) ~~> 1
25	24	a1i 9	. . . . . 6 |- ( T. -> seq 1 ( + , ( n e. NN |-> ( 1 / ( n x. ( n + 1 ) ) ) ) ) ~~> 1 )
26		climrel 11807	. . . . . . 7 |- Rel ~~>
27	26	releldmi 4963	. . . . . 6 |- ( seq 1 ( + , ( n e. NN |-> ( 1 / ( n x. ( n + 1 ) ) ) ) ) ~~> 1 -> seq 1 ( + , ( n e. NN |-> ( 1 / ( n x. ( n + 1 ) ) ) ) ) e. dom ~~> )
28	25, 27	syl 14	. . . . 5 |- ( T. -> seq 1 ( + , ( n e. NN |-> ( 1 / ( n x. ( n + 1 ) ) ) ) ) e. dom ~~> )
29		2cnd 9194	. . . . 5 |- ( T. -> 2 e. CC )
30	9, 10, 20, 23, 28, 29	isummulc2 11953	. . . 4 |- ( T. -> ( 2 x. sum_ k e. NN ( 1 / ( k x. ( k + 1 ) ) ) ) = sum_ k e. NN ( 2 x. ( 1 / ( k x. ( k + 1 ) ) ) ) )
31	9, 10, 20, 23, 25	isumclim 11948	. . . . 5 |- ( T. -> sum_ k e. NN ( 1 / ( k x. ( k + 1 ) ) ) = 1 )
32	31	oveq2d 6023	. . . 4 |- ( T. -> ( 2 x. sum_ k e. NN ( 1 / ( k x. ( k + 1 ) ) ) ) = ( 2 x. 1 ) )
33	30, 32	eqtr3d 2264	. . 3 |- ( T. -> sum_ k e. NN ( 2 x. ( 1 / ( k x. ( k + 1 ) ) ) ) = ( 2 x. 1 ) )
34	33	mptru 1404	. 2 |- sum_ k e. NN ( 2 x. ( 1 / ( k x. ( k + 1 ) ) ) ) = ( 2 x. 1 )
35		2t1e2 9275	. 2 |- ( 2 x. 1 ) = 2
36	8, 34, 35	3eqtri 2254	1 |- sum_ k e. NN ( 2 / ( k x. ( k + 1 ) ) ) = 2

Syntax hints:   /\ wa 104   = wceq 1395   T. wtru 1396   e. wcel 2200   class class class wbr 4083   |-> cmpt 4145   dom cdm 4719   ` cfv 5318  (class class class)co 6007   CCcc 8008   RRcr 8009   1c1 8011   + caddc 8013   x. cmul 8015   / cdiv 8830   NNcn 9121   2c2 9172   seqcseq 10681   ~~> cli 11805   sum_csu 11880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129  ax-caucvg 8130

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-frec 6543  df-1o 6568  df-oadd 6572  df-er 6688  df-en 6896  df-dom 6897  df-fin 6898  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-n0 9381  df-z 9458  df-uz 9734  df-q 9827  df-rp 9862  df-fz 10217  df-fzo 10351  df-seqfrec 10682  df-exp 10773  df-ihash 11010  df-shft 11342  df-cj 11369  df-re 11370  df-im 11371  df-rsqrt 11525  df-abs 11526  df-clim 11806  df-sumdc 11881

This theorem is referenced by: (None)