Theorem trirecip 11644

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 9055	. . . 4 |- ( k e. NN -> 2 e. CC )
2		peano2nn 8994	. . . . . 6 |- ( k e. NN -> ( k + 1 ) e. NN )
3		nnmulcl 9003	. . . . . 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 8996	. . . 4 |- ( k e. NN -> ( k x. ( k + 1 ) ) e. CC )
6	4	nnap0d 9028	. . . 4 |- ( k e. NN -> ( k x. ( k + 1 ) ) # 0 )
7	1, 5, 6	divrecapd 8812	. . 3 |- ( k e. NN -> ( 2 / ( k x. ( k + 1 ) ) ) = ( 2 x. ( 1 / ( k x. ( k + 1 ) ) ) ) )
8	7	sumeq2i 11507	. 2 |- sum_ k e. NN ( 2 / ( k x. ( k + 1 ) ) ) = sum_ k e. NN ( 2 x. ( 1 / ( k x. ( k + 1 ) ) ) )
9		nnuz 9628	. . . . 5 |- NN = ( ZZ>= ` 1 )
10		1zzd 9344	. . . . 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 9029	. . . . . 6 |- ( ( T. /\ k e. NN ) -> ( 1 / ( k x. ( k + 1 ) ) ) e. RR )
14		id 19	. . . . . . . . 9 |- ( n = k -> n = k )
15		oveq1 5925	. . . . . . . . 9 |- ( n = k -> ( n + 1 ) = ( k + 1 ) )
16	14, 15	oveq12d 5936	. . . . . . . 8 |- ( n = k -> ( n x. ( n + 1 ) ) = ( k x. ( k + 1 ) ) )
17	16	oveq2d 5934	. . . . . . 7 |- ( n = k -> ( 1 / ( n x. ( n + 1 ) ) ) = ( 1 / ( k x. ( k + 1 ) ) ) )
18		eqid 2193	. . . . . . 7 |- ( n e. NN |-> ( 1 / ( n x. ( n + 1 ) ) ) ) = ( n e. NN |-> ( 1 / ( n x. ( n + 1 ) ) ) )
19	17, 18	fvmptg 5633	. . . . . 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 9029	. . . . . . 7 |- ( k e. NN -> ( 1 / ( k x. ( k + 1 ) ) ) e. RR )
22	21	recnd 8048	. . . . . 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 11643	. . . . . . 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 11423	. . . . . . 7 |- Rel ~~>
27	26	releldmi 4901	. . . . . 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 9055	. . . . 5 |- ( T. -> 2 e. CC )
30	9, 10, 20, 23, 28, 29	isummulc2 11569	. . . 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 11564	. . . . 5 |- ( T. -> sum_ k e. NN ( 1 / ( k x. ( k + 1 ) ) ) = 1 )
32	31	oveq2d 5934	. . . 4 |- ( T. -> ( 2 x. sum_ k e. NN ( 1 / ( k x. ( k + 1 ) ) ) ) = ( 2 x. 1 ) )
33	30, 32	eqtr3d 2228	. . 3 |- ( T. -> sum_ k e. NN ( 2 x. ( 1 / ( k x. ( k + 1 ) ) ) ) = ( 2 x. 1 ) )
34	33	mptru 1373	. 2 |- sum_ k e. NN ( 2 x. ( 1 / ( k x. ( k + 1 ) ) ) ) = ( 2 x. 1 )
35		2t1e2 9135	. 2 |- ( 2 x. 1 ) = 2
36	8, 34, 35	3eqtri 2218	1 |- sum_ k e. NN ( 2 / ( k x. ( k + 1 ) ) ) = 2

Syntax hints:   /\ wa 104   = wceq 1364   T. wtru 1365   e. wcel 2164   class class class wbr 4029   |-> cmpt 4090   dom cdm 4659   ` cfv 5254  (class class class)co 5918   CCcc 7870   RRcr 7871   1c1 7873   + caddc 7875   x. cmul 7877   / cdiv 8691   NNcn 8982   2c2 9033   seqcseq 10518   ~~> cli 11421   sum_csu 11496

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-frec 6444  df-1o 6469  df-oadd 6473  df-er 6587  df-en 6795  df-dom 6796  df-fin 6797  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-seqfrec 10519  df-exp 10610  df-ihash 10847  df-shft 10959  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-clim 11422  df-sumdc 11497

This theorem is referenced by: (None)