Theorem trirecip 11668

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 9065	. . . 4 |- ( k e. NN -> 2 e. CC )
2		peano2nn 9004	. . . . . 6 |- ( k e. NN -> ( k + 1 ) e. NN )
3		nnmulcl 9013	. . . . . 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 9006	. . . 4 |- ( k e. NN -> ( k x. ( k + 1 ) ) e. CC )
6	4	nnap0d 9038	. . . 4 |- ( k e. NN -> ( k x. ( k + 1 ) ) # 0 )
7	1, 5, 6	divrecapd 8822	. . 3 |- ( k e. NN -> ( 2 / ( k x. ( k + 1 ) ) ) = ( 2 x. ( 1 / ( k x. ( k + 1 ) ) ) ) )
8	7	sumeq2i 11531	. 2 |- sum_ k e. NN ( 2 / ( k x. ( k + 1 ) ) ) = sum_ k e. NN ( 2 x. ( 1 / ( k x. ( k + 1 ) ) ) )
9		nnuz 9639	. . . . 5 |- NN = ( ZZ>= ` 1 )
10		1zzd 9355	. . . . 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 9039	. . . . . 6 |- ( ( T. /\ k e. NN ) -> ( 1 / ( k x. ( k + 1 ) ) ) e. RR )
14		id 19	. . . . . . . . 9 |- ( n = k -> n = k )
15		oveq1 5930	. . . . . . . . 9 |- ( n = k -> ( n + 1 ) = ( k + 1 ) )
16	14, 15	oveq12d 5941	. . . . . . . 8 |- ( n = k -> ( n x. ( n + 1 ) ) = ( k x. ( k + 1 ) ) )
17	16	oveq2d 5939	. . . . . . 7 |- ( n = k -> ( 1 / ( n x. ( n + 1 ) ) ) = ( 1 / ( k x. ( k + 1 ) ) ) )
18		eqid 2196	. . . . . . 7 |- ( n e. NN |-> ( 1 / ( n x. ( n + 1 ) ) ) ) = ( n e. NN |-> ( 1 / ( n x. ( n + 1 ) ) ) )
19	17, 18	fvmptg 5638	. . . . . 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 9039	. . . . . . 7 |- ( k e. NN -> ( 1 / ( k x. ( k + 1 ) ) ) e. RR )
22	21	recnd 8057	. . . . . 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 11667	. . . . . . 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 11447	. . . . . . 7 |- Rel ~~>
27	26	releldmi 4906	. . . . . 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 9065	. . . . 5 |- ( T. -> 2 e. CC )
30	9, 10, 20, 23, 28, 29	isummulc2 11593	. . . 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 11588	. . . . 5 |- ( T. -> sum_ k e. NN ( 1 / ( k x. ( k + 1 ) ) ) = 1 )
32	31	oveq2d 5939	. . . 4 |- ( T. -> ( 2 x. sum_ k e. NN ( 1 / ( k x. ( k + 1 ) ) ) ) = ( 2 x. 1 ) )
33	30, 32	eqtr3d 2231	. . 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 9146	. 2 |- ( 2 x. 1 ) = 2
36	8, 34, 35	3eqtri 2221	1 |- sum_ k e. NN ( 2 / ( k x. ( k + 1 ) ) ) = 2

Syntax hints:   /\ wa 104   = wceq 1364   T. wtru 1365   e. wcel 2167   class class class wbr 4034   |-> cmpt 4095   dom cdm 4664   ` cfv 5259  (class class class)co 5923   CCcc 7879   RRcr 7880   1c1 7882   + caddc 7884   x. cmul 7886   / cdiv 8701   NNcn 8992   2c2 9043   seqcseq 10541   ~~> cli 11445   sum_csu 11520

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7972  ax-resscn 7973  ax-1cn 7974  ax-1re 7975  ax-icn 7976  ax-addcl 7977  ax-addrcl 7978  ax-mulcl 7979  ax-mulrcl 7980  ax-addcom 7981  ax-mulcom 7982  ax-addass 7983  ax-mulass 7984  ax-distr 7985  ax-i2m1 7986  ax-0lt1 7987  ax-1rid 7988  ax-0id 7989  ax-rnegex 7990  ax-precex 7991  ax-cnre 7992  ax-pre-ltirr 7993  ax-pre-ltwlin 7994  ax-pre-lttrn 7995  ax-pre-apti 7996  ax-pre-ltadd 7997  ax-pre-mulgt0 7998  ax-pre-mulext 7999  ax-arch 8000  ax-caucvg 8001

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5878  df-ov 5926  df-oprab 5927  df-mpo 5928  df-1st 6199  df-2nd 6200  df-recs 6364  df-irdg 6429  df-frec 6450  df-1o 6475  df-oadd 6479  df-er 6593  df-en 6801  df-dom 6802  df-fin 6803  df-pnf 8065  df-mnf 8066  df-xr 8067  df-ltxr 8068  df-le 8069  df-sub 8201  df-neg 8202  df-reap 8604  df-ap 8611  df-div 8702  df-inn 8993  df-2 9051  df-3 9052  df-4 9053  df-n0 9252  df-z 9329  df-uz 9604  df-q 9696  df-rp 9731  df-fz 10086  df-fzo 10220  df-seqfrec 10542  df-exp 10633  df-ihash 10870  df-shft 10982  df-cj 11009  df-re 11010  df-im 11011  df-rsqrt 11165  df-abs 11166  df-clim 11446  df-sumdc 11521

This theorem is referenced by: (None)