Theorem basellem9 24815

Description: Lemma for basel 24816. Since by basellem8 24814 F is bounded by two expressions that tend to pi ^ 2 / 6, F must also go to pi ^ 2 / 6 by the squeeze theorem climsqz 14371. But the series F is exactly the partial sums of k ^ -u 2, so it follows that this is also the value of the infinite sum sum_ k e. NN ( k ^ -u 2 ). (Contributed by Mario Carneiro, 28-Jul-2014.)

Hypotheses

Ref	Expression
basel.g	|- G = ( n e. NN |-> ( 1 / ( ( 2 x. n ) + 1 ) ) )
basel.f	|- F = seq 1 ( + , ( n e. NN |-> ( n ^ -u 2 ) ) )
basel.h	|- H = ( ( NN X. { ( ( pi ^ 2 ) / 6 ) } ) oF x. ( ( NN X. { 1 } ) oF - G ) )
basel.j	|- J = ( H oF x. ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) )
basel.k	|- K = ( H oF x. ( ( NN X. { 1 } ) oF + G ) )

Assertion

Ref	Expression
basellem9	|- sum_ k e. NN ( k ^ -u 2 ) = ( ( pi ^ 2 ) / 6 )

Distinct variable groups:   k, n, F   k, G   k, H   k, J, n   k, K

Allowed substitution hints:   G( n)   H( n)   K( n)

Proof of Theorem basellem9

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnuz 11723	. . 3 |- NN = ( ZZ>= ` 1 )
2		1zzd 11408	. . 3 |- ( T. -> 1 e. ZZ )
3		oveq1 6657	. . . . 5 |- ( n = k -> ( n ^ -u 2 ) = ( k ^ -u 2 ) )
4		eqid 2622	. . . . 5 |- ( n e. NN |-> ( n ^ -u 2 ) ) = ( n e. NN |-> ( n ^ -u 2 ) )
5		ovex 6678	. . . . 5 |- ( k ^ -u 2 ) e. _V
6	3, 4, 5	fvmpt 6282	. . . 4 |- ( k e. NN -> ( ( n e. NN |-> ( n ^ -u 2 ) ) ` k ) = ( k ^ -u 2 ) )
7	6	adantl 482	. . 3 |- ( ( T. /\ k e. NN ) -> ( ( n e. NN |-> ( n ^ -u 2 ) ) ` k ) = ( k ^ -u 2 ) )
8		nnre 11027	. . . . . . . . 9 |- ( n e. NN -> n e. RR )
9		nnne0 11053	. . . . . . . . 9 |- ( n e. NN -> n =/= 0 )
10		2z 11409	. . . . . . . . . . 11 |- 2 e. ZZ
11		znegcl 11412	. . . . . . . . . . 11 |- ( 2 e. ZZ -> -u 2 e. ZZ )
12	10, 11	ax-mp 5	. . . . . . . . . 10 |- -u 2 e. ZZ
13	12	a1i 11	. . . . . . . . 9 |- ( n e. NN -> -u 2 e. ZZ )
14	8, 9, 13	reexpclzd 13034	. . . . . . . 8 |- ( n e. NN -> ( n ^ -u 2 ) e. RR )
15	14	adantl 482	. . . . . . 7 |- ( ( T. /\ n e. NN ) -> ( n ^ -u 2 ) e. RR )
16	15, 4	fmptd 6385	. . . . . 6 |- ( T. -> ( n e. NN |-> ( n ^ -u 2 ) ) : NN --> RR )
17	16	ffvelrnda 6359	. . . . 5 |- ( ( T. /\ k e. NN ) -> ( ( n e. NN |-> ( n ^ -u 2 ) ) ` k ) e. RR )
18	7, 17	eqeltrrd 2702	. . . 4 |- ( ( T. /\ k e. NN ) -> ( k ^ -u 2 ) e. RR )
19	18	recnd 10068	. . 3 |- ( ( T. /\ k e. NN ) -> ( k ^ -u 2 ) e. CC )
20	1, 2, 17	serfre 12830	. . . . . . . . . . 11 |- ( T. -> seq 1 ( + , ( n e. NN |-> ( n ^ -u 2 ) ) ) : NN --> RR )
21		basel.f	. . . . . . . . . . . 12 |- F = seq 1 ( + , ( n e. NN |-> ( n ^ -u 2 ) ) )
22	21	feq1i 6036	. . . . . . . . . . 11 |- ( F : NN --> RR <-> seq 1 ( + , ( n e. NN |-> ( n ^ -u 2 ) ) ) : NN --> RR )
23	20, 22	sylibr 224	. . . . . . . . . 10 |- ( T. -> F : NN --> RR )
24	23	ffvelrnda 6359	. . . . . . . . 9 |- ( ( T. /\ n e. NN ) -> ( F ` n ) e. RR )
25	24	recnd 10068	. . . . . . . 8 |- ( ( T. /\ n e. NN ) -> ( F ` n ) e. CC )
26		remulcl 10021	. . . . . . . . . . . . 13 |- ( ( x e. RR /\ y e. RR ) -> ( x x. y ) e. RR )
27	26	adantl 482	. . . . . . . . . . . 12 |- ( ( T. /\ ( x e. RR /\ y e. RR ) ) -> ( x x. y ) e. RR )
28		ovex 6678	. . . . . . . . . . . . . . . 16 |- ( ( pi ^ 2 ) / 6 ) e. _V
29	28	fconst 6091	. . . . . . . . . . . . . . 15 |- ( NN X. { ( ( pi ^ 2 ) / 6 ) } ) : NN --> { ( ( pi ^ 2 ) / 6 ) }
30		pire 24210	. . . . . . . . . . . . . . . . . . 19 |- pi e. RR
31	30	resqcli 12949	. . . . . . . . . . . . . . . . . 18 |- ( pi ^ 2 ) e. RR
32		6re 11101	. . . . . . . . . . . . . . . . . 18 |- 6 e. RR
33		6nn 11189	. . . . . . . . . . . . . . . . . . 19 |- 6 e. NN
34	33	nnne0i 11055	. . . . . . . . . . . . . . . . . 18 |- 6 =/= 0
35	31, 32, 34	redivcli 10792	. . . . . . . . . . . . . . . . 17 |- ( ( pi ^ 2 ) / 6 ) e. RR
36	35	a1i 11	. . . . . . . . . . . . . . . 16 |- ( T. -> ( ( pi ^ 2 ) / 6 ) e. RR )
37	36	snssd 4340	. . . . . . . . . . . . . . 15 |- ( T. -> { ( ( pi ^ 2 ) / 6 ) } C_ RR )
38		fss 6056	. . . . . . . . . . . . . . 15 |- ( ( ( NN X. { ( ( pi ^ 2 ) / 6 ) } ) : NN --> { ( ( pi ^ 2 ) / 6 ) } /\ { ( ( pi ^ 2 ) / 6 ) } C_ RR ) -> ( NN X. { ( ( pi ^ 2 ) / 6 ) } ) : NN --> RR )
39	29, 37, 38	sylancr 695	. . . . . . . . . . . . . 14 |- ( T. -> ( NN X. { ( ( pi ^ 2 ) / 6 ) } ) : NN --> RR )
40		resubcl 10345	. . . . . . . . . . . . . . . 16 |- ( ( x e. RR /\ y e. RR ) -> ( x - y ) e. RR )
41	40	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( T. /\ ( x e. RR /\ y e. RR ) ) -> ( x - y ) e. RR )
42		1ex 10035	. . . . . . . . . . . . . . . . 17 |- 1 e. _V
43	42	fconst 6091	. . . . . . . . . . . . . . . 16 |- ( NN X. { 1 } ) : NN --> { 1 }
44		1red 10055	. . . . . . . . . . . . . . . . 17 |- ( T. -> 1 e. RR )
45	44	snssd 4340	. . . . . . . . . . . . . . . 16 |- ( T. -> { 1 } C_ RR )
46		fss 6056	. . . . . . . . . . . . . . . 16 |- ( ( ( NN X. { 1 } ) : NN --> { 1 } /\ { 1 } C_ RR ) -> ( NN X. { 1 } ) : NN --> RR )
47	43, 45, 46	sylancr 695	. . . . . . . . . . . . . . 15 |- ( T. -> ( NN X. { 1 } ) : NN --> RR )
48		2nn 11185	. . . . . . . . . . . . . . . . . . . 20 |- 2 e. NN
49	48	a1i 11	. . . . . . . . . . . . . . . . . . 19 |- ( T. -> 2 e. NN )
50		nnmulcl 11043	. . . . . . . . . . . . . . . . . . 19 |- ( ( 2 e. NN /\ n e. NN ) -> ( 2 x. n ) e. NN )
51	49, 50	sylan 488	. . . . . . . . . . . . . . . . . 18 |- ( ( T. /\ n e. NN ) -> ( 2 x. n ) e. NN )
52	51	peano2nnd 11037	. . . . . . . . . . . . . . . . 17 |- ( ( T. /\ n e. NN ) -> ( ( 2 x. n ) + 1 ) e. NN )
53	52	nnrecred 11066	. . . . . . . . . . . . . . . 16 |- ( ( T. /\ n e. NN ) -> ( 1 / ( ( 2 x. n ) + 1 ) ) e. RR )
54		basel.g	. . . . . . . . . . . . . . . 16 |- G = ( n e. NN |-> ( 1 / ( ( 2 x. n ) + 1 ) ) )
55	53, 54	fmptd 6385	. . . . . . . . . . . . . . 15 |- ( T. -> G : NN --> RR )
56		nnex 11026	. . . . . . . . . . . . . . . 16 |- NN e. _V
57	56	a1i 11	. . . . . . . . . . . . . . 15 |- ( T. -> NN e. _V )
58		inidm 3822	. . . . . . . . . . . . . . 15 |- ( NN i^i NN ) = NN
59	41, 47, 55, 57, 57, 58	off 6912	. . . . . . . . . . . . . 14 |- ( T. -> ( ( NN X. { 1 } ) oF - G ) : NN --> RR )
60	27, 39, 59, 57, 57, 58	off 6912	. . . . . . . . . . . . 13 |- ( T. -> ( ( NN X. { ( ( pi ^ 2 ) / 6 ) } ) oF x. ( ( NN X. { 1 } ) oF - G ) ) : NN --> RR )
61		basel.h	. . . . . . . . . . . . . 14 |- H = ( ( NN X. { ( ( pi ^ 2 ) / 6 ) } ) oF x. ( ( NN X. { 1 } ) oF - G ) )
62	61	feq1i 6036	. . . . . . . . . . . . 13 |- ( H : NN --> RR <-> ( ( NN X. { ( ( pi ^ 2 ) / 6 ) } ) oF x. ( ( NN X. { 1 } ) oF - G ) ) : NN --> RR )
63	60, 62	sylibr 224	. . . . . . . . . . . 12 |- ( T. -> H : NN --> RR )
64		readdcl 10019	. . . . . . . . . . . . . 14 |- ( ( x e. RR /\ y e. RR ) -> ( x + y ) e. RR )
65	64	adantl 482	. . . . . . . . . . . . 13 |- ( ( T. /\ ( x e. RR /\ y e. RR ) ) -> ( x + y ) e. RR )
66		negex 10279	. . . . . . . . . . . . . . . 16 |- -u 2 e. _V
67	66	fconst 6091	. . . . . . . . . . . . . . 15 |- ( NN X. { -u 2 } ) : NN --> { -u 2 }
68	12	zrei 11383	. . . . . . . . . . . . . . . . 17 |- -u 2 e. RR
69	68	a1i 11	. . . . . . . . . . . . . . . 16 |- ( T. -> -u 2 e. RR )
70	69	snssd 4340	. . . . . . . . . . . . . . 15 |- ( T. -> { -u 2 } C_ RR )
71		fss 6056	. . . . . . . . . . . . . . 15 |- ( ( ( NN X. { -u 2 } ) : NN --> { -u 2 } /\ { -u 2 } C_ RR ) -> ( NN X. { -u 2 } ) : NN --> RR )
72	67, 70, 71	sylancr 695	. . . . . . . . . . . . . 14 |- ( T. -> ( NN X. { -u 2 } ) : NN --> RR )
73	27, 72, 55, 57, 57, 58	off 6912	. . . . . . . . . . . . 13 |- ( T. -> ( ( NN X. { -u 2 } ) oF x. G ) : NN --> RR )
74	65, 47, 73, 57, 57, 58	off 6912	. . . . . . . . . . . 12 |- ( T. -> ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) : NN --> RR )
75	27, 63, 74, 57, 57, 58	off 6912	. . . . . . . . . . 11 |- ( T. -> ( H oF x. ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) ) : NN --> RR )
76		basel.j	. . . . . . . . . . . 12 |- J = ( H oF x. ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) )
77	76	feq1i 6036	. . . . . . . . . . 11 |- ( J : NN --> RR <-> ( H oF x. ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) ) : NN --> RR )
78	75, 77	sylibr 224	. . . . . . . . . 10 |- ( T. -> J : NN --> RR )
79	78	ffvelrnda 6359	. . . . . . . . 9 |- ( ( T. /\ n e. NN ) -> ( J ` n ) e. RR )
80	79	recnd 10068	. . . . . . . 8 |- ( ( T. /\ n e. NN ) -> ( J ` n ) e. CC )
81	25, 80	npcand 10396	. . . . . . 7 |- ( ( T. /\ n e. NN ) -> ( ( ( F ` n ) - ( J ` n ) ) + ( J ` n ) ) = ( F ` n ) )
82	81	mpteq2dva 4744	. . . . . 6 |- ( T. -> ( n e. NN |-> ( ( ( F ` n ) - ( J ` n ) ) + ( J ` n ) ) ) = ( n e. NN |-> ( F ` n ) ) )
83		ovexd 6680	. . . . . . 7 |- ( ( T. /\ n e. NN ) -> ( ( F ` n ) - ( J ` n ) ) e. _V )
84	23	feqmptd 6249	. . . . . . . 8 |- ( T. -> F = ( n e. NN |-> ( F ` n ) ) )
85	78	feqmptd 6249	. . . . . . . 8 |- ( T. -> J = ( n e. NN |-> ( J ` n ) ) )
86	57, 24, 79, 84, 85	offval2 6914	. . . . . . 7 |- ( T. -> ( F oF - J ) = ( n e. NN |-> ( ( F ` n ) - ( J ` n ) ) ) )
87	57, 83, 79, 86, 85	offval2 6914	. . . . . 6 |- ( T. -> ( ( F oF - J ) oF + J ) = ( n e. NN |-> ( ( ( F ` n ) - ( J ` n ) ) + ( J ` n ) ) ) )
88	82, 87, 84	3eqtr4d 2666	. . . . 5 |- ( T. -> ( ( F oF - J ) oF + J ) = F )
89	65, 47, 55, 57, 57, 58	off 6912	. . . . . . . . . 10 |- ( T. -> ( ( NN X. { 1 } ) oF + G ) : NN --> RR )
90		recn 10026	. . . . . . . . . . . 12 |- ( x e. RR -> x e. CC )
91		recn 10026	. . . . . . . . . . . 12 |- ( y e. RR -> y e. CC )
92		recn 10026	. . . . . . . . . . . 12 |- ( z e. RR -> z e. CC )
93		subdi 10463	. . . . . . . . . . . 12 |- ( ( x e. CC /\ y e. CC /\ z e. CC ) -> ( x x. ( y - z ) ) = ( ( x x. y ) - ( x x. z ) ) )
94	90, 91, 92, 93	syl3an 1368	. . . . . . . . . . 11 |- ( ( x e. RR /\ y e. RR /\ z e. RR ) -> ( x x. ( y - z ) ) = ( ( x x. y ) - ( x x. z ) ) )
95	94	adantl 482	. . . . . . . . . 10 |- ( ( T. /\ ( x e. RR /\ y e. RR /\ z e. RR ) ) -> ( x x. ( y - z ) ) = ( ( x x. y ) - ( x x. z ) ) )
96	57, 63, 89, 74, 95	caofdi 6933	. . . . . . . . 9 |- ( T. -> ( H oF x. ( ( ( NN X. { 1 } ) oF + G ) oF - ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) ) ) = ( ( H oF x. ( ( NN X. { 1 } ) oF + G ) ) oF - ( H oF x. ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) ) ) )
97		basel.k	. . . . . . . . . 10 |- K = ( H oF x. ( ( NN X. { 1 } ) oF + G ) )
98	97, 76	oveq12i 6662	. . . . . . . . 9 |- ( K oF - J ) = ( ( H oF x. ( ( NN X. { 1 } ) oF + G ) ) oF - ( H oF x. ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) ) )
99	96, 98	syl6eqr 2674	. . . . . . . 8 |- ( T. -> ( H oF x. ( ( ( NN X. { 1 } ) oF + G ) oF - ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) ) ) = ( K oF - J ) )
100	35	recni 10052	. . . . . . . . . . . . . 14 |- ( ( pi ^ 2 ) / 6 ) e. CC
101	1	eqimss2i 3660	. . . . . . . . . . . . . . 15 |- ( ZZ>= ` 1 ) C_ NN
102	101, 56	climconst2 14279	. . . . . . . . . . . . . 14 |- ( ( ( ( pi ^ 2 ) / 6 ) e. CC /\ 1 e. ZZ ) -> ( NN X. { ( ( pi ^ 2 ) / 6 ) } ) ~~> ( ( pi ^ 2 ) / 6 ) )
103	100, 2, 102	sylancr 695	. . . . . . . . . . . . 13 |- ( T. -> ( NN X. { ( ( pi ^ 2 ) / 6 ) } ) ~~> ( ( pi ^ 2 ) / 6 ) )
104		ovexd 6680	. . . . . . . . . . . . 13 |- ( T. -> ( ( NN X. { ( ( pi ^ 2 ) / 6 ) } ) oF x. ( ( NN X. { 1 } ) oF - G ) ) e. _V )
105		ax-resscn 9993	. . . . . . . . . . . . . . . 16 |- RR C_ CC
106		fss 6056	. . . . . . . . . . . . . . . 16 |- ( ( ( NN X. { 1 } ) : NN --> RR /\ RR C_ CC ) -> ( NN X. { 1 } ) : NN --> CC )
107	47, 105, 106	sylancl 694	. . . . . . . . . . . . . . 15 |- ( T. -> ( NN X. { 1 } ) : NN --> CC )
108		fss 6056	. . . . . . . . . . . . . . . 16 |- ( ( G : NN --> RR /\ RR C_ CC ) -> G : NN --> CC )
109	55, 105, 108	sylancl 694	. . . . . . . . . . . . . . 15 |- ( T. -> G : NN --> CC )
110		ofnegsub 11018	. . . . . . . . . . . . . . 15 |- ( ( NN e. _V /\ ( NN X. { 1 } ) : NN --> CC /\ G : NN --> CC ) -> ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 1 } ) oF x. G ) ) = ( ( NN X. { 1 } ) oF - G ) )
111	57, 107, 109, 110	syl3anc 1326	. . . . . . . . . . . . . 14 |- ( T. -> ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 1 } ) oF x. G ) ) = ( ( NN X. { 1 } ) oF - G ) )
112		neg1cn 11124	. . . . . . . . . . . . . . 15 |- -u 1 e. CC
113	54, 112	basellem7 24813	. . . . . . . . . . . . . 14 |- ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 1 } ) oF x. G ) ) ~~> 1
114	111, 113	syl6eqbrr 4693	. . . . . . . . . . . . 13 |- ( T. -> ( ( NN X. { 1 } ) oF - G ) ~~> 1 )
115	39	ffvelrnda 6359	. . . . . . . . . . . . . 14 |- ( ( T. /\ k e. NN ) -> ( ( NN X. { ( ( pi ^ 2 ) / 6 ) } ) ` k ) e. RR )
116	115	recnd 10068	. . . . . . . . . . . . 13 |- ( ( T. /\ k e. NN ) -> ( ( NN X. { ( ( pi ^ 2 ) / 6 ) } ) ` k ) e. CC )
117	59	ffvelrnda 6359	. . . . . . . . . . . . . 14 |- ( ( T. /\ k e. NN ) -> ( ( ( NN X. { 1 } ) oF - G ) ` k ) e. RR )
118	117	recnd 10068	. . . . . . . . . . . . 13 |- ( ( T. /\ k e. NN ) -> ( ( ( NN X. { 1 } ) oF - G ) ` k ) e. CC )
119		ffn 6045	. . . . . . . . . . . . . . 15 |- ( ( NN X. { ( ( pi ^ 2 ) / 6 ) } ) : NN --> RR -> ( NN X. { ( ( pi ^ 2 ) / 6 ) } ) Fn NN )
120	39, 119	syl 17	. . . . . . . . . . . . . 14 |- ( T. -> ( NN X. { ( ( pi ^ 2 ) / 6 ) } ) Fn NN )
121		fnconstg 6093	. . . . . . . . . . . . . . . 16 |- ( 1 e. ZZ -> ( NN X. { 1 } ) Fn NN )
122	2, 121	syl 17	. . . . . . . . . . . . . . 15 |- ( T. -> ( NN X. { 1 } ) Fn NN )
123		ffn 6045	. . . . . . . . . . . . . . . 16 |- ( G : NN --> RR -> G Fn NN )
124	55, 123	syl 17	. . . . . . . . . . . . . . 15 |- ( T. -> G Fn NN )
125	122, 124, 57, 57, 58	offn 6908	. . . . . . . . . . . . . 14 |- ( T. -> ( ( NN X. { 1 } ) oF - G ) Fn NN )
126		eqidd 2623	. . . . . . . . . . . . . 14 |- ( ( T. /\ k e. NN ) -> ( ( NN X. { ( ( pi ^ 2 ) / 6 ) } ) ` k ) = ( ( NN X. { ( ( pi ^ 2 ) / 6 ) } ) ` k ) )
127		eqidd 2623	. . . . . . . . . . . . . 14 |- ( ( T. /\ k e. NN ) -> ( ( ( NN X. { 1 } ) oF - G ) ` k ) = ( ( ( NN X. { 1 } ) oF - G ) ` k ) )
128	120, 125, 57, 57, 58, 126, 127	ofval 6906	. . . . . . . . . . . . 13 |- ( ( T. /\ k e. NN ) -> ( ( ( NN X. { ( ( pi ^ 2 ) / 6 ) } ) oF x. ( ( NN X. { 1 } ) oF - G ) ) ` k ) = ( ( ( NN X. { ( ( pi ^ 2 ) / 6 ) } ) ` k ) x. ( ( ( NN X. { 1 } ) oF - G ) ` k ) ) )
129	1, 2, 103, 104, 114, 116, 118, 128	climmul 14363	. . . . . . . . . . . 12 |- ( T. -> ( ( NN X. { ( ( pi ^ 2 ) / 6 ) } ) oF x. ( ( NN X. { 1 } ) oF - G ) ) ~~> ( ( ( pi ^ 2 ) / 6 ) x. 1 ) )
130	100	mulid1i 10042	. . . . . . . . . . . 12 |- ( ( ( pi ^ 2 ) / 6 ) x. 1 ) = ( ( pi ^ 2 ) / 6 )
131	129, 130	syl6breq 4694	. . . . . . . . . . 11 |- ( T. -> ( ( NN X. { ( ( pi ^ 2 ) / 6 ) } ) oF x. ( ( NN X. { 1 } ) oF - G ) ) ~~> ( ( pi ^ 2 ) / 6 ) )
132	61, 131	syl5eqbr 4688	. . . . . . . . . 10 |- ( T. -> H ~~> ( ( pi ^ 2 ) / 6 ) )
133		ovexd 6680	. . . . . . . . . 10 |- ( T. -> ( H oF x. ( ( ( NN X. { 1 } ) oF + G ) oF - ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) ) ) e. _V )
134		3cn 11095	. . . . . . . . . . . . 13 |- 3 e. CC
135	101, 56	climconst2 14279	. . . . . . . . . . . . 13 |- ( ( 3 e. CC /\ 1 e. ZZ ) -> ( NN X. { 3 } ) ~~> 3 )
136	134, 2, 135	sylancr 695	. . . . . . . . . . . 12 |- ( T. -> ( NN X. { 3 } ) ~~> 3 )
137		ovexd 6680	. . . . . . . . . . . 12 |- ( T. -> ( ( NN X. { 3 } ) oF x. G ) e. _V )
138	54	basellem6 24812	. . . . . . . . . . . . 13 |- G ~~> 0
139	138	a1i 11	. . . . . . . . . . . 12 |- ( T. -> G ~~> 0 )
140		3ex 11096	. . . . . . . . . . . . . . . 16 |- 3 e. _V
141	140	fconst 6091	. . . . . . . . . . . . . . 15 |- ( NN X. { 3 } ) : NN --> { 3 }
142		3re 11094	. . . . . . . . . . . . . . . . 17 |- 3 e. RR
143	142	a1i 11	. . . . . . . . . . . . . . . 16 |- ( T. -> 3 e. RR )
144	143	snssd 4340	. . . . . . . . . . . . . . 15 |- ( T. -> { 3 } C_ RR )
145		fss 6056	. . . . . . . . . . . . . . 15 |- ( ( ( NN X. { 3 } ) : NN --> { 3 } /\ { 3 } C_ RR ) -> ( NN X. { 3 } ) : NN --> RR )
146	141, 144, 145	sylancr 695	. . . . . . . . . . . . . 14 |- ( T. -> ( NN X. { 3 } ) : NN --> RR )
147	146	ffvelrnda 6359	. . . . . . . . . . . . 13 |- ( ( T. /\ k e. NN ) -> ( ( NN X. { 3 } ) ` k ) e. RR )
148	147	recnd 10068	. . . . . . . . . . . 12 |- ( ( T. /\ k e. NN ) -> ( ( NN X. { 3 } ) ` k ) e. CC )
149	55	ffvelrnda 6359	. . . . . . . . . . . . 13 |- ( ( T. /\ k e. NN ) -> ( G ` k ) e. RR )
150	149	recnd 10068	. . . . . . . . . . . 12 |- ( ( T. /\ k e. NN ) -> ( G ` k ) e. CC )
151		ffn 6045	. . . . . . . . . . . . . 14 |- ( ( NN X. { 3 } ) : NN --> RR -> ( NN X. { 3 } ) Fn NN )
152	146, 151	syl 17	. . . . . . . . . . . . 13 |- ( T. -> ( NN X. { 3 } ) Fn NN )
153		eqidd 2623	. . . . . . . . . . . . 13 |- ( ( T. /\ k e. NN ) -> ( ( NN X. { 3 } ) ` k ) = ( ( NN X. { 3 } ) ` k ) )
154		eqidd 2623	. . . . . . . . . . . . 13 |- ( ( T. /\ k e. NN ) -> ( G ` k ) = ( G ` k ) )
155	152, 124, 57, 57, 58, 153, 154	ofval 6906	. . . . . . . . . . . 12 |- ( ( T. /\ k e. NN ) -> ( ( ( NN X. { 3 } ) oF x. G ) ` k ) = ( ( ( NN X. { 3 } ) ` k ) x. ( G ` k ) ) )
156	1, 2, 136, 137, 139, 148, 150, 155	climmul 14363	. . . . . . . . . . 11 |- ( T. -> ( ( NN X. { 3 } ) oF x. G ) ~~> ( 3 x. 0 ) )
157	134	mul01i 10226	. . . . . . . . . . 11 |- ( 3 x. 0 ) = 0
158	156, 157	syl6breq 4694	. . . . . . . . . 10 |- ( T. -> ( ( NN X. { 3 } ) oF x. G ) ~~> 0 )
159	63	ffvelrnda 6359	. . . . . . . . . . 11 |- ( ( T. /\ k e. NN ) -> ( H ` k ) e. RR )
160	159	recnd 10068	. . . . . . . . . 10 |- ( ( T. /\ k e. NN ) -> ( H ` k ) e. CC )
161	27, 146, 55, 57, 57, 58	off 6912	. . . . . . . . . . . 12 |- ( T. -> ( ( NN X. { 3 } ) oF x. G ) : NN --> RR )
162	161	ffvelrnda 6359	. . . . . . . . . . 11 |- ( ( T. /\ k e. NN ) -> ( ( ( NN X. { 3 } ) oF x. G ) ` k ) e. RR )
163	162	recnd 10068	. . . . . . . . . 10 |- ( ( T. /\ k e. NN ) -> ( ( ( NN X. { 3 } ) oF x. G ) ` k ) e. CC )
164		ffn 6045	. . . . . . . . . . . 12 |- ( H : NN --> RR -> H Fn NN )
165	63, 164	syl 17	. . . . . . . . . . 11 |- ( T. -> H Fn NN )
166	41, 89, 74, 57, 57, 58	off 6912	. . . . . . . . . . . 12 |- ( T. -> ( ( ( NN X. { 1 } ) oF + G ) oF - ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) ) : NN --> RR )
167		ffn 6045	. . . . . . . . . . . 12 |- ( ( ( ( NN X. { 1 } ) oF + G ) oF - ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) ) : NN --> RR -> ( ( ( NN X. { 1 } ) oF + G ) oF - ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) ) Fn NN )
168	166, 167	syl 17	. . . . . . . . . . 11 |- ( T. -> ( ( ( NN X. { 1 } ) oF + G ) oF - ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) ) Fn NN )
169		eqidd 2623	. . . . . . . . . . 11 |- ( ( T. /\ k e. NN ) -> ( H ` k ) = ( H ` k ) )
170	150	mulid2d 10058	. . . . . . . . . . . . . . 15 |- ( ( T. /\ k e. NN ) -> ( 1 x. ( G ` k ) ) = ( G ` k ) )
171		2cn 11091	. . . . . . . . . . . . . . . . . 18 |- 2 e. CC
172		mulneg1 10466	. . . . . . . . . . . . . . . . . 18 |- ( ( 2 e. CC /\ ( G ` k ) e. CC ) -> ( -u 2 x. ( G ` k ) ) = -u ( 2 x. ( G ` k ) ) )
173	171, 150, 172	sylancr 695	. . . . . . . . . . . . . . . . 17 |- ( ( T. /\ k e. NN ) -> ( -u 2 x. ( G ` k ) ) = -u ( 2 x. ( G ` k ) ) )
174	173	negeqd 10275	. . . . . . . . . . . . . . . 16 |- ( ( T. /\ k e. NN ) -> -u ( -u 2 x. ( G ` k ) ) = -u -u ( 2 x. ( G ` k ) ) )
175		mulcl 10020	. . . . . . . . . . . . . . . . . 18 |- ( ( 2 e. CC /\ ( G ` k ) e. CC ) -> ( 2 x. ( G ` k ) ) e. CC )
176	171, 150, 175	sylancr 695	. . . . . . . . . . . . . . . . 17 |- ( ( T. /\ k e. NN ) -> ( 2 x. ( G ` k ) ) e. CC )
177	176	negnegd 10383	. . . . . . . . . . . . . . . 16 |- ( ( T. /\ k e. NN ) -> -u -u ( 2 x. ( G ` k ) ) = ( 2 x. ( G ` k ) ) )
178	174, 177	eqtr2d 2657	. . . . . . . . . . . . . . 15 |- ( ( T. /\ k e. NN ) -> ( 2 x. ( G ` k ) ) = -u ( -u 2 x. ( G ` k ) ) )
179	170, 178	oveq12d 6668	. . . . . . . . . . . . . 14 |- ( ( T. /\ k e. NN ) -> ( ( 1 x. ( G ` k ) ) + ( 2 x. ( G ` k ) ) ) = ( ( G ` k ) + -u ( -u 2 x. ( G ` k ) ) ) )
180		remulcl 10021	. . . . . . . . . . . . . . . . 17 |- ( ( -u 2 e. RR /\ ( G ` k ) e. RR ) -> ( -u 2 x. ( G ` k ) ) e. RR )
181	68, 149, 180	sylancr 695	. . . . . . . . . . . . . . . 16 |- ( ( T. /\ k e. NN ) -> ( -u 2 x. ( G ` k ) ) e. RR )
182	181	recnd 10068	. . . . . . . . . . . . . . 15 |- ( ( T. /\ k e. NN ) -> ( -u 2 x. ( G ` k ) ) e. CC )
183	150, 182	negsubd 10398	. . . . . . . . . . . . . 14 |- ( ( T. /\ k e. NN ) -> ( ( G ` k ) + -u ( -u 2 x. ( G ` k ) ) ) = ( ( G ` k ) - ( -u 2 x. ( G ` k ) ) ) )
184	179, 183	eqtrd 2656	. . . . . . . . . . . . 13 |- ( ( T. /\ k e. NN ) -> ( ( 1 x. ( G ` k ) ) + ( 2 x. ( G ` k ) ) ) = ( ( G ` k ) - ( -u 2 x. ( G ` k ) ) ) )
185		df-3 11080	. . . . . . . . . . . . . . . 16 |- 3 = ( 2 + 1 )
186		ax-1cn 9994	. . . . . . . . . . . . . . . . 17 |- 1 e. CC
187	171, 186	addcomi 10227	. . . . . . . . . . . . . . . 16 |- ( 2 + 1 ) = ( 1 + 2 )
188	185, 187	eqtri 2644	. . . . . . . . . . . . . . 15 |- 3 = ( 1 + 2 )
189	188	oveq1i 6660	. . . . . . . . . . . . . 14 |- ( 3 x. ( G ` k ) ) = ( ( 1 + 2 ) x. ( G ` k ) )
190		1cnd 10056	. . . . . . . . . . . . . . 15 |- ( ( T. /\ k e. NN ) -> 1 e. CC )
191	171	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( T. /\ k e. NN ) -> 2 e. CC )
192	190, 191, 150	adddird 10065	. . . . . . . . . . . . . 14 |- ( ( T. /\ k e. NN ) -> ( ( 1 + 2 ) x. ( G ` k ) ) = ( ( 1 x. ( G ` k ) ) + ( 2 x. ( G ` k ) ) ) )
193	189, 192	syl5eq 2668	. . . . . . . . . . . . 13 |- ( ( T. /\ k e. NN ) -> ( 3 x. ( G ` k ) ) = ( ( 1 x. ( G ` k ) ) + ( 2 x. ( G ` k ) ) ) )
194	190, 150, 182	pnpcand 10429	. . . . . . . . . . . . 13 |- ( ( T. /\ k e. NN ) -> ( ( 1 + ( G ` k ) ) - ( 1 + ( -u 2 x. ( G ` k ) ) ) ) = ( ( G ` k ) - ( -u 2 x. ( G ` k ) ) ) )
195	184, 193, 194	3eqtr4rd 2667	. . . . . . . . . . . 12 |- ( ( T. /\ k e. NN ) -> ( ( 1 + ( G ` k ) ) - ( 1 + ( -u 2 x. ( G ` k ) ) ) ) = ( 3 x. ( G ` k ) ) )
196	122, 124, 57, 57, 58	offn 6908	. . . . . . . . . . . . 13 |- ( T. -> ( ( NN X. { 1 } ) oF + G ) Fn NN )
197	12	a1i 11	. . . . . . . . . . . . . . . 16 |- ( T. -> -u 2 e. ZZ )
198		fnconstg 6093	. . . . . . . . . . . . . . . 16 |- ( -u 2 e. ZZ -> ( NN X. { -u 2 } ) Fn NN )
199	197, 198	syl 17	. . . . . . . . . . . . . . 15 |- ( T. -> ( NN X. { -u 2 } ) Fn NN )
200	199, 124, 57, 57, 58	offn 6908	. . . . . . . . . . . . . 14 |- ( T. -> ( ( NN X. { -u 2 } ) oF x. G ) Fn NN )
201	122, 200, 57, 57, 58	offn 6908	. . . . . . . . . . . . 13 |- ( T. -> ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) Fn NN )
202	57, 44, 124, 154	ofc1 6920	. . . . . . . . . . . . 13 |- ( ( T. /\ k e. NN ) -> ( ( ( NN X. { 1 } ) oF + G ) ` k ) = ( 1 + ( G ` k ) ) )
203	57, 69, 124, 154	ofc1 6920	. . . . . . . . . . . . . 14 |- ( ( T. /\ k e. NN ) -> ( ( ( NN X. { -u 2 } ) oF x. G ) ` k ) = ( -u 2 x. ( G ` k ) ) )
204	57, 44, 200, 203	ofc1 6920	. . . . . . . . . . . . 13 |- ( ( T. /\ k e. NN ) -> ( ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) ` k ) = ( 1 + ( -u 2 x. ( G ` k ) ) ) )
205	196, 201, 57, 57, 58, 202, 204	ofval 6906	. . . . . . . . . . . 12 |- ( ( T. /\ k e. NN ) -> ( ( ( ( NN X. { 1 } ) oF + G ) oF - ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) ) ` k ) = ( ( 1 + ( G ` k ) ) - ( 1 + ( -u 2 x. ( G ` k ) ) ) ) )
206	57, 143, 124, 154	ofc1 6920	. . . . . . . . . . . 12 |- ( ( T. /\ k e. NN ) -> ( ( ( NN X. { 3 } ) oF x. G ) ` k ) = ( 3 x. ( G ` k ) ) )
207	195, 205, 206	3eqtr4d 2666	. . . . . . . . . . 11 |- ( ( T. /\ k e. NN ) -> ( ( ( ( NN X. { 1 } ) oF + G ) oF - ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) ) ` k ) = ( ( ( NN X. { 3 } ) oF x. G ) ` k ) )
208	165, 168, 57, 57, 58, 169, 207	ofval 6906	. . . . . . . . . 10 |- ( ( T. /\ k e. NN ) -> ( ( H oF x. ( ( ( NN X. { 1 } ) oF + G ) oF - ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) ) ) ` k ) = ( ( H ` k ) x. ( ( ( NN X. { 3 } ) oF x. G ) ` k ) ) )
209	1, 2, 132, 133, 158, 160, 163, 208	climmul 14363	. . . . . . . . 9 |- ( T. -> ( H oF x. ( ( ( NN X. { 1 } ) oF + G ) oF - ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) ) ) ~~> ( ( ( pi ^ 2 ) / 6 ) x. 0 ) )
210	100	mul01i 10226	. . . . . . . . 9 |- ( ( ( pi ^ 2 ) / 6 ) x. 0 ) = 0
211	209, 210	syl6breq 4694	. . . . . . . 8 |- ( T. -> ( H oF x. ( ( ( NN X. { 1 } ) oF + G ) oF - ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) ) ) ~~> 0 )
212	99, 211	eqbrtrrd 4677	. . . . . . 7 |- ( T. -> ( K oF - J ) ~~> 0 )
213		ovexd 6680	. . . . . . 7 |- ( T. -> ( F oF - J ) e. _V )
214	27, 63, 89, 57, 57, 58	off 6912	. . . . . . . . . 10 |- ( T. -> ( H oF x. ( ( NN X. { 1 } ) oF + G ) ) : NN --> RR )
215	97	feq1i 6036	. . . . . . . . . 10 |- ( K : NN --> RR <-> ( H oF x. ( ( NN X. { 1 } ) oF + G ) ) : NN --> RR )
216	214, 215	sylibr 224	. . . . . . . . 9 |- ( T. -> K : NN --> RR )
217	41, 216, 78, 57, 57, 58	off 6912	. . . . . . . 8 |- ( T. -> ( K oF - J ) : NN --> RR )
218	217	ffvelrnda 6359	. . . . . . 7 |- ( ( T. /\ k e. NN ) -> ( ( K oF - J ) ` k ) e. RR )
219	41, 23, 78, 57, 57, 58	off 6912	. . . . . . . 8 |- ( T. -> ( F oF - J ) : NN --> RR )
220	219	ffvelrnda 6359	. . . . . . 7 |- ( ( T. /\ k e. NN ) -> ( ( F oF - J ) ` k ) e. RR )
221	23	ffvelrnda 6359	. . . . . . . . 9 |- ( ( T. /\ k e. NN ) -> ( F ` k ) e. RR )
222	216	ffvelrnda 6359	. . . . . . . . 9 |- ( ( T. /\ k e. NN ) -> ( K ` k ) e. RR )
223	78	ffvelrnda 6359	. . . . . . . . 9 |- ( ( T. /\ k e. NN ) -> ( J ` k ) e. RR )
224		eqid 2622	. . . . . . . . . . . 12 |- ( ( 2 x. k ) + 1 ) = ( ( 2 x. k ) + 1 )
225	54, 21, 61, 76, 97, 224	basellem8 24814	. . . . . . . . . . 11 |- ( k e. NN -> ( ( J ` k ) <_ ( F ` k ) /\ ( F ` k ) <_ ( K ` k ) ) )
226	225	adantl 482	. . . . . . . . . 10 |- ( ( T. /\ k e. NN ) -> ( ( J ` k ) <_ ( F ` k ) /\ ( F ` k ) <_ ( K ` k ) ) )
227	226	simprd 479	. . . . . . . . 9 |- ( ( T. /\ k e. NN ) -> ( F ` k ) <_ ( K ` k ) )
228	221, 222, 223, 227	lesub1dd 10643	. . . . . . . 8 |- ( ( T. /\ k e. NN ) -> ( ( F ` k ) - ( J ` k ) ) <_ ( ( K ` k ) - ( J ` k ) ) )
229		ffn 6045	. . . . . . . . . 10 |- ( F : NN --> RR -> F Fn NN )
230	23, 229	syl 17	. . . . . . . . 9 |- ( T. -> F Fn NN )
231		ffn 6045	. . . . . . . . . 10 |- ( J : NN --> RR -> J Fn NN )
232	78, 231	syl 17	. . . . . . . . 9 |- ( T. -> J Fn NN )
233		eqidd 2623	. . . . . . . . 9 |- ( ( T. /\ k e. NN ) -> ( F ` k ) = ( F ` k ) )
234		eqidd 2623	. . . . . . . . 9 |- ( ( T. /\ k e. NN ) -> ( J ` k ) = ( J ` k ) )
235	230, 232, 57, 57, 58, 233, 234	ofval 6906	. . . . . . . 8 |- ( ( T. /\ k e. NN ) -> ( ( F oF - J ) ` k ) = ( ( F ` k ) - ( J ` k ) ) )
236		ffn 6045	. . . . . . . . . 10 |- ( K : NN --> RR -> K Fn NN )
237	216, 236	syl 17	. . . . . . . . 9 |- ( T. -> K Fn NN )
238		eqidd 2623	. . . . . . . . 9 |- ( ( T. /\ k e. NN ) -> ( K ` k ) = ( K ` k ) )
239	237, 232, 57, 57, 58, 238, 234	ofval 6906	. . . . . . . 8 |- ( ( T. /\ k e. NN ) -> ( ( K oF - J ) ` k ) = ( ( K ` k ) - ( J ` k ) ) )
240	228, 235, 239	3brtr4d 4685	. . . . . . 7 |- ( ( T. /\ k e. NN ) -> ( ( F oF - J ) ` k ) <_ ( ( K oF - J ) ` k ) )
241	226	simpld 475	. . . . . . . . 9 |- ( ( T. /\ k e. NN ) -> ( J ` k ) <_ ( F ` k ) )
242	221, 223	subge0d 10617	. . . . . . . . 9 |- ( ( T. /\ k e. NN ) -> ( 0 <_ ( ( F ` k ) - ( J ` k ) ) <-> ( J ` k ) <_ ( F ` k ) ) )
243	241, 242	mpbird 247	. . . . . . . 8 |- ( ( T. /\ k e. NN ) -> 0 <_ ( ( F ` k ) - ( J ` k ) ) )
244	243, 235	breqtrrd 4681	. . . . . . 7 |- ( ( T. /\ k e. NN ) -> 0 <_ ( ( F oF - J ) ` k ) )
245	1, 2, 212, 213, 218, 220, 240, 244	climsqz2 14372	. . . . . 6 |- ( T. -> ( F oF - J ) ~~> 0 )
246		ovexd 6680	. . . . . 6 |- ( T. -> ( ( F oF - J ) oF + J ) e. _V )
247		ovexd 6680	. . . . . . . . 9 |- ( T. -> ( H oF x. ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) ) e. _V )
248	68	recni 10052	. . . . . . . . . . 11 |- -u 2 e. CC
249	54, 248	basellem7 24813	. . . . . . . . . 10 |- ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) ~~> 1
250	249	a1i 11	. . . . . . . . 9 |- ( T. -> ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) ~~> 1 )
251	74	ffvelrnda 6359	. . . . . . . . . 10 |- ( ( T. /\ k e. NN ) -> ( ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) ` k ) e. RR )
252	251	recnd 10068	. . . . . . . . 9 |- ( ( T. /\ k e. NN ) -> ( ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) ` k ) e. CC )
253		eqidd 2623	. . . . . . . . . 10 |- ( ( T. /\ k e. NN ) -> ( ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) ` k ) = ( ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) ` k ) )
254	165, 201, 57, 57, 58, 169, 253	ofval 6906	. . . . . . . . 9 |- ( ( T. /\ k e. NN ) -> ( ( H oF x. ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) ) ` k ) = ( ( H ` k ) x. ( ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) ` k ) ) )
255	1, 2, 132, 247, 250, 160, 252, 254	climmul 14363	. . . . . . . 8 |- ( T. -> ( H oF x. ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) ) ~~> ( ( ( pi ^ 2 ) / 6 ) x. 1 ) )
256	255, 130	syl6breq 4694	. . . . . . 7 |- ( T. -> ( H oF x. ( ( NN X. { 1 } ) oF + ( ( NN X. { -u 2 } ) oF x. G ) ) ) ~~> ( ( pi ^ 2 ) / 6 ) )
257	76, 256	syl5eqbr 4688	. . . . . 6 |- ( T. -> J ~~> ( ( pi ^ 2 ) / 6 ) )
258	220	recnd 10068	. . . . . 6 |- ( ( T. /\ k e. NN ) -> ( ( F oF - J ) ` k ) e. CC )
259	223	recnd 10068	. . . . . 6 |- ( ( T. /\ k e. NN ) -> ( J ` k ) e. CC )
260		ffn 6045	. . . . . . . 8 |- ( ( F oF - J ) : NN --> RR -> ( F oF - J ) Fn NN )
261	219, 260	syl 17	. . . . . . 7 |- ( T. -> ( F oF - J ) Fn NN )
262		eqidd 2623	. . . . . . 7 |- ( ( T. /\ k e. NN ) -> ( ( F oF - J ) ` k ) = ( ( F oF - J ) ` k ) )
263	261, 232, 57, 57, 58, 262, 234	ofval 6906	. . . . . 6 |- ( ( T. /\ k e. NN ) -> ( ( ( F oF - J ) oF + J ) ` k ) = ( ( ( F oF - J ) ` k ) + ( J ` k ) ) )
264	1, 2, 245, 246, 257, 258, 259, 263	climadd 14362	. . . . 5 |- ( T. -> ( ( F oF - J ) oF + J ) ~~> ( 0 + ( ( pi ^ 2 ) / 6 ) ) )
265	88, 264	eqbrtrrd 4677	. . . 4 |- ( T. -> F ~~> ( 0 + ( ( pi ^ 2 ) / 6 ) ) )
266	100	addid2i 10224	. . . 4 |- ( 0 + ( ( pi ^ 2 ) / 6 ) ) = ( ( pi ^ 2 ) / 6 )
267	265, 21, 266	3brtr3g 4686	. . 3 |- ( T. -> seq 1 ( + , ( n e. NN |-> ( n ^ -u 2 ) ) ) ~~> ( ( pi ^ 2 ) / 6 ) )
268	1, 2, 7, 19, 267	isumclim 14488	. 2 |- ( T. -> sum_ k e. NN ( k ^ -u 2 ) = ( ( pi ^ 2 ) / 6 ) )
269	268	trud 1493	1 |- sum_ k e. NN ( k ^ -u 2 ) = ( ( pi ^ 2 ) / 6 )

Syntax hints:   /\ wa 384   /\ w3a 1037   = wceq 1483   T. wtru 1484   e. wcel 1990   _Vcvv 3200   C_ wss 3574   {csn 4177   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   3c3 11071   6c6 11074   ZZcz 11377   ZZ>=cuz 11687   seqcseq 12801   ^cexp 12860   ~~> cli 14215   sum_csu 14416   picpi 14797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-tan 14802  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-0p 23437  df-limc 23630  df-dv 23631  df-ply 23944  df-idp 23945  df-coe 23946  df-dgr 23947  df-quot 24046

This theorem is referenced by:  basel  24816