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Theorem basellem9 23741
Description: Lemma for basel 23742. Since by basellem8 23740 
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 13610. 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.
StepHypRef Expression
1 nnuz 11161 . . 3  |-  NN  =  ( ZZ>= `  1 )
2 1zzd 10935 . . 3  |-  ( T. 
->  1  e.  ZZ )
3 oveq1 6284 . . . . 5  |-  ( n  =  k  ->  (
n ^ -u 2
)  =  ( k ^ -u 2 ) )
4 eqid 2402 . . . . 5  |-  ( n  e.  NN  |->  ( n ^ -u 2 ) )  =  ( n  e.  NN  |->  ( n ^ -u 2 ) )
5 ovex 6305 . . . . 5  |-  ( k ^ -u 2 )  e.  _V
63, 4, 5fvmpt 5931 . . . 4  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  ( n ^ -u 2
) ) `  k
)  =  ( k ^ -u 2 ) )
76adantl 464 . . 3  |-  ( ( T.  /\  k  e.  NN )  ->  (
( n  e.  NN  |->  ( n ^ -u 2
) ) `  k
)  =  ( k ^ -u 2 ) )
8 nnre 10582 . . . . . . . . 9  |-  ( n  e.  NN  ->  n  e.  RR )
9 nnne0 10608 . . . . . . . . 9  |-  ( n  e.  NN  ->  n  =/=  0 )
10 2z 10936 . . . . . . . . . . 11  |-  2  e.  ZZ
11 znegcl 10939 . . . . . . . . . . 11  |-  ( 2  e.  ZZ  ->  -u 2  e.  ZZ )
1210, 11ax-mp 5 . . . . . . . . . 10  |-  -u 2  e.  ZZ
1312a1i 11 . . . . . . . . 9  |-  ( n  e.  NN  ->  -u 2  e.  ZZ )
148, 9, 13reexpclzd 12377 . . . . . . . 8  |-  ( n  e.  NN  ->  (
n ^ -u 2
)  e.  RR )
1514adantl 464 . . . . . . 7  |-  ( ( T.  /\  n  e.  NN )  ->  (
n ^ -u 2
)  e.  RR )
1615, 4fmptd 6032 . . . . . 6  |-  ( T. 
->  ( n  e.  NN  |->  ( n ^ -u 2
) ) : NN --> RR )
1716ffvelrnda 6008 . . . . 5  |-  ( ( T.  /\  k  e.  NN )  ->  (
( n  e.  NN  |->  ( n ^ -u 2
) ) `  k
)  e.  RR )
187, 17eqeltrrd 2491 . . . 4  |-  ( ( T.  /\  k  e.  NN )  ->  (
k ^ -u 2
)  e.  RR )
1918recnd 9651 . . 3  |-  ( ( T.  /\  k  e.  NN )  ->  (
k ^ -u 2
)  e.  CC )
201, 2, 17serfre 12178 . . . . . . . . . . 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 ) ) )
2221feq1i 5705 . . . . . . . . . . 11  |-  ( F : NN --> RR  <->  seq 1
(  +  ,  ( n  e.  NN  |->  ( n ^ -u 2
) ) ) : NN --> RR )
2320, 22sylibr 212 . . . . . . . . . 10  |-  ( T. 
->  F : NN --> RR )
2423ffvelrnda 6008 . . . . . . . . 9  |-  ( ( T.  /\  n  e.  NN )  ->  ( F `  n )  e.  RR )
2524recnd 9651 . . . . . . . 8  |-  ( ( T.  /\  n  e.  NN )  ->  ( F `  n )  e.  CC )
26 remulcl 9606 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  x.  y
)  e.  RR )
2726adantl 464 . . . . . . . . . . . 12  |-  ( ( T.  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  x.  y
)  e.  RR )
28 ovex 6305 . . . . . . . . . . . . . . . 16  |-  ( ( pi ^ 2 )  /  6 )  e. 
_V
2928fconst 5753 . . . . . . . . . . . . . . 15  |-  ( NN 
X.  { ( ( pi ^ 2 )  /  6 ) } ) : NN --> { ( ( pi ^ 2 )  /  6 ) }
30 pire 23141 . . . . . . . . . . . . . . . . . . 19  |-  pi  e.  RR
3130resqcli 12296 . . . . . . . . . . . . . . . . . 18  |-  ( pi
^ 2 )  e.  RR
32 6re 10656 . . . . . . . . . . . . . . . . . 18  |-  6  e.  RR
33 6nn 10737 . . . . . . . . . . . . . . . . . . 19  |-  6  e.  NN
3433nnne0i 10610 . . . . . . . . . . . . . . . . . 18  |-  6  =/=  0
3531, 32, 34redivcli 10351 . . . . . . . . . . . . . . . . 17  |-  ( ( pi ^ 2 )  /  6 )  e.  RR
3635a1i 11 . . . . . . . . . . . . . . . 16  |-  ( T. 
->  ( ( pi ^
2 )  /  6
)  e.  RR )
3736snssd 4116 . . . . . . . . . . . . . . 15  |-  ( T. 
->  { ( ( pi
^ 2 )  / 
6 ) }  C_  RR )
38 fss 5721 . . . . . . . . . . . . . . 15  |-  ( ( ( NN  X.  {
( ( pi ^
2 )  /  6
) } ) : NN --> { ( ( pi ^ 2 )  /  6 ) }  /\  { ( ( pi ^ 2 )  /  6 ) } 
C_  RR )  -> 
( NN  X.  {
( ( pi ^
2 )  /  6
) } ) : NN --> RR )
3929, 37, 38sylancr 661 . . . . . . . . . . . . . 14  |-  ( T. 
->  ( NN  X.  {
( ( pi ^
2 )  /  6
) } ) : NN --> RR )
40 resubcl 9918 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  -  y
)  e.  RR )
4140adantl 464 . . . . . . . . . . . . . . 15  |-  ( ( T.  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  -  y
)  e.  RR )
42 1ex 9620 . . . . . . . . . . . . . . . . 17  |-  1  e.  _V
4342fconst 5753 . . . . . . . . . . . . . . . 16  |-  ( NN 
X.  { 1 } ) : NN --> { 1 }
44 1red 9640 . . . . . . . . . . . . . . . . 17  |-  ( T. 
->  1  e.  RR )
4544snssd 4116 . . . . . . . . . . . . . . . 16  |-  ( T. 
->  { 1 }  C_  RR )
46 fss 5721 . . . . . . . . . . . . . . . 16  |-  ( ( ( NN  X.  {
1 } ) : NN --> { 1 }  /\  { 1 } 
C_  RR )  -> 
( NN  X.  {
1 } ) : NN --> RR )
4743, 45, 46sylancr 661 . . . . . . . . . . . . . . 15  |-  ( T. 
->  ( NN  X.  {
1 } ) : NN --> RR )
48 2nn 10733 . . . . . . . . . . . . . . . . . . . 20  |-  2  e.  NN
4948a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( T. 
->  2  e.  NN )
50 nnmulcl 10598 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 2  e.  NN  /\  n  e.  NN )  ->  ( 2  x.  n
)  e.  NN )
5149, 50sylan 469 . . . . . . . . . . . . . . . . . 18  |-  ( ( T.  /\  n  e.  NN )  ->  (
2  x.  n )  e.  NN )
5251peano2nnd 10592 . . . . . . . . . . . . . . . . 17  |-  ( ( T.  /\  n  e.  NN )  ->  (
( 2  x.  n
)  +  1 )  e.  NN )
5352nnrecred 10621 . . . . . . . . . . . . . . . 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 ) ) )
5553, 54fmptd 6032 . . . . . . . . . . . . . . 15  |-  ( T. 
->  G : NN --> RR )
56 nnex 10581 . . . . . . . . . . . . . . . 16  |-  NN  e.  _V
5756a1i 11 . . . . . . . . . . . . . . 15  |-  ( T. 
->  NN  e.  _V )
58 inidm 3647 . . . . . . . . . . . . . . 15  |-  ( NN 
i^i  NN )  =  NN
5941, 47, 55, 57, 57, 58off 6535 . . . . . . . . . . . . . 14  |-  ( T. 
->  ( ( NN  X.  { 1 } )  oF  -  G
) : NN --> RR )
6027, 39, 59, 57, 57, 58off 6535 . . . . . . . . . . . . 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 ) )
6261feq1i 5705 . . . . . . . . . . . . 13  |-  ( H : NN --> RR  <->  ( ( NN  X.  { ( ( pi ^ 2 )  /  6 ) } )  oF  x.  ( ( NN  X.  { 1 } )  oF  -  G
) ) : NN --> RR )
6360, 62sylibr 212 . . . . . . . . . . . 12  |-  ( T. 
->  H : NN --> RR )
64 readdcl 9604 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
6564adantl 464 . . . . . . . . . . . . 13  |-  ( ( T.  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  +  y )  e.  RR )
66 negex 9853 . . . . . . . . . . . . . . . 16  |-  -u 2  e.  _V
6766fconst 5753 . . . . . . . . . . . . . . 15  |-  ( NN 
X.  { -u 2 } ) : NN --> { -u 2 }
6812zrei 10910 . . . . . . . . . . . . . . . . 17  |-  -u 2  e.  RR
6968a1i 11 . . . . . . . . . . . . . . . 16  |-  ( T. 
->  -u 2  e.  RR )
7069snssd 4116 . . . . . . . . . . . . . . 15  |-  ( T. 
->  { -u 2 } 
C_  RR )
71 fss 5721 . . . . . . . . . . . . . . 15  |-  ( ( ( NN  X.  { -u 2 } ) : NN --> { -u 2 }  /\  { -u 2 }  C_  RR )  -> 
( NN  X.  { -u 2 } ) : NN --> RR )
7267, 70, 71sylancr 661 . . . . . . . . . . . . . 14  |-  ( T. 
->  ( NN  X.  { -u 2 } ) : NN --> RR )
7327, 72, 55, 57, 57, 58off 6535 . . . . . . . . . . . . 13  |-  ( T. 
->  ( ( NN  X.  { -u 2 } )  oF  x.  G
) : NN --> RR )
7465, 47, 73, 57, 57, 58off 6535 . . . . . . . . . . . 12  |-  ( T. 
->  ( ( NN  X.  { 1 } )  oF  +  ( ( NN  X.  { -u 2 } )  oF  x.  G ) ) : NN --> RR )
7527, 63, 74, 57, 57, 58off 6535 . . . . . . . . . . 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
) ) )
7776feq1i 5705 . . . . . . . . . . 11  |-  ( J : NN --> RR  <->  ( H  oF  x.  (
( NN  X.  {
1 } )  oF  +  ( ( NN  X.  { -u
2 } )  oF  x.  G ) ) ) : NN --> RR )
7875, 77sylibr 212 . . . . . . . . . 10  |-  ( T. 
->  J : NN --> RR )
7978ffvelrnda 6008 . . . . . . . . 9  |-  ( ( T.  /\  n  e.  NN )  ->  ( J `  n )  e.  RR )
8079recnd 9651 . . . . . . . 8  |-  ( ( T.  /\  n  e.  NN )  ->  ( J `  n )  e.  CC )
8125, 80npcand 9970 . . . . . . 7  |-  ( ( T.  /\  n  e.  NN )  ->  (
( ( F `  n )  -  ( J `  n )
)  +  ( J `
 n ) )  =  ( F `  n ) )
8281mpteq2dva 4480 . . . . . 6  |-  ( T. 
->  ( n  e.  NN  |->  ( ( ( F `
 n )  -  ( J `  n ) )  +  ( J `
 n ) ) )  =  ( n  e.  NN  |->  ( F `
 n ) ) )
83 ovex 6305 . . . . . . . 8  |-  ( ( F `  n )  -  ( J `  n ) )  e. 
_V
8483a1i 11 . . . . . . 7  |-  ( ( T.  /\  n  e.  NN )  ->  (
( F `  n
)  -  ( J `
 n ) )  e.  _V )
8523feqmptd 5901 . . . . . . . 8  |-  ( T. 
->  F  =  (
n  e.  NN  |->  ( F `  n ) ) )
8678feqmptd 5901 . . . . . . . 8  |-  ( T. 
->  J  =  (
n  e.  NN  |->  ( J `  n ) ) )
8757, 24, 79, 85, 86offval2 6537 . . . . . . 7  |-  ( T. 
->  ( F  oF  -  J )  =  ( n  e.  NN  |->  ( ( F `  n )  -  ( J `  n )
) ) )
8857, 84, 79, 87, 86offval2 6537 . . . . . 6  |-  ( T. 
->  ( ( F  oF  -  J )  oF  +  J
)  =  ( n  e.  NN  |->  ( ( ( F `  n
)  -  ( J `
 n ) )  +  ( J `  n ) ) ) )
8982, 88, 853eqtr4d 2453 . . . . 5  |-  ( T. 
->  ( ( F  oF  -  J )  oF  +  J
)  =  F )
9065, 47, 55, 57, 57, 58off 6535 . . . . . . . . . 10  |-  ( T. 
->  ( ( NN  X.  { 1 } )  oF  +  G
) : NN --> RR )
91 recn 9611 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  x  e.  CC )
92 recn 9611 . . . . . . . . . . . 12  |-  ( y  e.  RR  ->  y  e.  CC )
93 recn 9611 . . . . . . . . . . . 12  |-  ( z  e.  RR  ->  z  e.  CC )
94 subdi 10030 . . . . . . . . . . . 12  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
x  x.  ( y  -  z ) )  =  ( ( x  x.  y )  -  ( x  x.  z
) ) )
9591, 92, 93, 94syl3an 1272 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  z  e.  RR )  ->  (
x  x.  ( y  -  z ) )  =  ( ( x  x.  y )  -  ( x  x.  z
) ) )
9695adantl 464 . . . . . . . . . 10  |-  ( ( T.  /\  ( x  e.  RR  /\  y  e.  RR  /\  z  e.  RR ) )  -> 
( x  x.  (
y  -  z ) )  =  ( ( x  x.  y )  -  ( x  x.  z ) ) )
9757, 63, 90, 74, 96caofdi 6557 . . . . . . . . 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 ) ) ) ) )
98 basel.k . . . . . . . . . 10  |-  K  =  ( H  oF  x.  ( ( NN 
X.  { 1 } )  oF  +  G ) )
9998, 76oveq12i 6289 . . . . . . . . 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
) ) ) )
10097, 99syl6eqr 2461 . . . . . . . 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 ) )
10135recni 9637 . . . . . . . . . . . . . 14  |-  ( ( pi ^ 2 )  /  6 )  e.  CC
1021eqimss2i 3496 . . . . . . . . . . . . . . 15  |-  ( ZZ>= ` 
1 )  C_  NN
103102, 56climconst2 13518 . . . . . . . . . . . . . 14  |-  ( ( ( ( pi ^
2 )  /  6
)  e.  CC  /\  1  e.  ZZ )  ->  ( NN  X.  {
( ( pi ^
2 )  /  6
) } )  ~~>  ( ( pi ^ 2 )  /  6 ) )
104101, 2, 103sylancr 661 . . . . . . . . . . . . 13  |-  ( T. 
->  ( NN  X.  {
( ( pi ^
2 )  /  6
) } )  ~~>  ( ( pi ^ 2 )  /  6 ) )
105 ovex 6305 . . . . . . . . . . . . . 14  |-  ( ( NN  X.  { ( ( pi ^ 2 )  /  6 ) } )  oF  x.  ( ( NN 
X.  { 1 } )  oF  -  G ) )  e. 
_V
106105a1i 11 . . . . . . . . . . . . 13  |-  ( T. 
->  ( ( NN  X.  { ( ( pi
^ 2 )  / 
6 ) } )  oF  x.  (
( NN  X.  {
1 } )  oF  -  G ) )  e.  _V )
107 ax-resscn 9578 . . . . . . . . . . . . . . . 16  |-  RR  C_  CC
108 fss 5721 . . . . . . . . . . . . . . . 16  |-  ( ( ( NN  X.  {
1 } ) : NN --> RR  /\  RR  C_  CC )  ->  ( NN  X.  { 1 } ) : NN --> CC )
10947, 107, 108sylancl 660 . . . . . . . . . . . . . . 15  |-  ( T. 
->  ( NN  X.  {
1 } ) : NN --> CC )
110 fss 5721 . . . . . . . . . . . . . . . 16  |-  ( ( G : NN --> RR  /\  RR  C_  CC )  ->  G : NN --> CC )
11155, 107, 110sylancl 660 . . . . . . . . . . . . . . 15  |-  ( T. 
->  G : NN --> CC )
112 ofnegsub 10573 . . . . . . . . . . . . . . 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 ) )
11357, 109, 111, 112syl3anc 1230 . . . . . . . . . . . . . 14  |-  ( T. 
->  ( ( NN  X.  { 1 } )  oF  +  ( ( NN  X.  { -u 1 } )  oF  x.  G ) )  =  ( ( NN  X.  { 1 } )  oF  -  G ) )
114 neg1cn 10679 . . . . . . . . . . . . . . 15  |-  -u 1  e.  CC
11554, 114basellem7 23739 . . . . . . . . . . . . . 14  |-  ( ( NN  X.  { 1 } )  oF  +  ( ( NN 
X.  { -u 1 } )  oF  x.  G ) )  ~~>  1
116113, 115syl6eqbrr 4432 . . . . . . . . . . . . 13  |-  ( T. 
->  ( ( NN  X.  { 1 } )  oF  -  G
)  ~~>  1 )
11739ffvelrnda 6008 . . . . . . . . . . . . . 14  |-  ( ( T.  /\  k  e.  NN )  ->  (
( NN  X.  {
( ( pi ^
2 )  /  6
) } ) `  k )  e.  RR )
118117recnd 9651 . . . . . . . . . . . . 13  |-  ( ( T.  /\  k  e.  NN )  ->  (
( NN  X.  {
( ( pi ^
2 )  /  6
) } ) `  k )  e.  CC )
11959ffvelrnda 6008 . . . . . . . . . . . . . 14  |-  ( ( T.  /\  k  e.  NN )  ->  (
( ( NN  X.  { 1 } )  oF  -  G
) `  k )  e.  RR )
120119recnd 9651 . . . . . . . . . . . . 13  |-  ( ( T.  /\  k  e.  NN )  ->  (
( ( NN  X.  { 1 } )  oF  -  G
) `  k )  e.  CC )
121 ffn 5713 . . . . . . . . . . . . . . 15  |-  ( ( NN  X.  { ( ( pi ^ 2 )  /  6 ) } ) : NN --> RR  ->  ( NN  X.  { ( ( pi
^ 2 )  / 
6 ) } )  Fn  NN )
12239, 121syl 17 . . . . . . . . . . . . . 14  |-  ( T. 
->  ( NN  X.  {
( ( pi ^
2 )  /  6
) } )  Fn  NN )
123 fnconstg 5755 . . . . . . . . . . . . . . . 16  |-  ( 1  e.  ZZ  ->  ( NN  X.  { 1 } )  Fn  NN )
1242, 123syl 17 . . . . . . . . . . . . . . 15  |-  ( T. 
->  ( NN  X.  {
1 } )  Fn  NN )
125 ffn 5713 . . . . . . . . . . . . . . . 16  |-  ( G : NN --> RR  ->  G  Fn  NN )
12655, 125syl 17 . . . . . . . . . . . . . . 15  |-  ( T. 
->  G  Fn  NN )
127124, 126, 57, 57, 58offn 6531 . . . . . . . . . . . . . 14  |-  ( T. 
->  ( ( NN  X.  { 1 } )  oF  -  G
)  Fn  NN )
128 eqidd 2403 . . . . . . . . . . . . . 14  |-  ( ( T.  /\  k  e.  NN )  ->  (
( NN  X.  {
( ( pi ^
2 )  /  6
) } ) `  k )  =  ( ( NN  X.  {
( ( pi ^
2 )  /  6
) } ) `  k ) )
129 eqidd 2403 . . . . . . . . . . . . . 14  |-  ( ( T.  /\  k  e.  NN )  ->  (
( ( NN  X.  { 1 } )  oF  -  G
) `  k )  =  ( ( ( NN  X.  { 1 } )  oF  -  G ) `  k ) )
130122, 127, 57, 57, 58, 128, 129ofval 6529 . . . . . . . . . . . . 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 ) ) )
1311, 2, 104, 106, 116, 118, 120, 130climmul 13602 . . . . . . . . . . . 12  |-  ( T. 
->  ( ( NN  X.  { ( ( pi
^ 2 )  / 
6 ) } )  oF  x.  (
( NN  X.  {
1 } )  oF  -  G ) )  ~~>  ( ( ( pi ^ 2 )  /  6 )  x.  1 ) )
132101mulid1i 9627 . . . . . . . . . . . 12  |-  ( ( ( pi ^ 2 )  /  6 )  x.  1 )  =  ( ( pi ^
2 )  /  6
)
133131, 132syl6breq 4433 . . . . . . . . . . 11  |-  ( T. 
->  ( ( NN  X.  { ( ( pi
^ 2 )  / 
6 ) } )  oF  x.  (
( NN  X.  {
1 } )  oF  -  G ) )  ~~>  ( ( pi
^ 2 )  / 
6 ) )
13461, 133syl5eqbr 4427 . . . . . . . . . 10  |-  ( T. 
->  H  ~~>  ( (
pi ^ 2 )  /  6 ) )
135 ovex 6305 . . . . . . . . . . 11  |-  ( H  oF  x.  (
( ( NN  X.  { 1 } )  oF  +  G
)  oF  -  ( ( NN  X.  { 1 } )  oF  +  ( ( NN  X.  { -u 2 } )  oF  x.  G ) ) ) )  e. 
_V
136135a1i 11 . . . . . . . . . 10  |-  ( T. 
->  ( H  oF  x.  ( ( ( NN  X.  { 1 } )  oF  +  G )  oF  -  ( ( NN  X.  { 1 } )  oF  +  ( ( NN 
X.  { -u 2 } )  oF  x.  G ) ) ) )  e.  _V )
137 3cn 10650 . . . . . . . . . . . . 13  |-  3  e.  CC
138102, 56climconst2 13518 . . . . . . . . . . . . 13  |-  ( ( 3  e.  CC  /\  1  e.  ZZ )  ->  ( NN  X.  {
3 } )  ~~>  3 )
139137, 2, 138sylancr 661 . . . . . . . . . . . 12  |-  ( T. 
->  ( NN  X.  {
3 } )  ~~>  3 )
140 ovex 6305 . . . . . . . . . . . . 13  |-  ( ( NN  X.  { 3 } )  oF  x.  G )  e. 
_V
141140a1i 11 . . . . . . . . . . . 12  |-  ( T. 
->  ( ( NN  X.  { 3 } )  oF  x.  G
)  e.  _V )
14254basellem6 23738 . . . . . . . . . . . . 13  |-  G  ~~>  0
143142a1i 11 . . . . . . . . . . . 12  |-  ( T. 
->  G  ~~>  0 )
144 3ex 10651 . . . . . . . . . . . . . . . 16  |-  3  e.  _V
145144fconst 5753 . . . . . . . . . . . . . . 15  |-  ( NN 
X.  { 3 } ) : NN --> { 3 }
146 3re 10649 . . . . . . . . . . . . . . . . 17  |-  3  e.  RR
147146a1i 11 . . . . . . . . . . . . . . . 16  |-  ( T. 
->  3  e.  RR )
148147snssd 4116 . . . . . . . . . . . . . . 15  |-  ( T. 
->  { 3 }  C_  RR )
149 fss 5721 . . . . . . . . . . . . . . 15  |-  ( ( ( NN  X.  {
3 } ) : NN --> { 3 }  /\  { 3 } 
C_  RR )  -> 
( NN  X.  {
3 } ) : NN --> RR )
150145, 148, 149sylancr 661 . . . . . . . . . . . . . 14  |-  ( T. 
->  ( NN  X.  {
3 } ) : NN --> RR )
151150ffvelrnda 6008 . . . . . . . . . . . . 13  |-  ( ( T.  /\  k  e.  NN )  ->  (
( NN  X.  {
3 } ) `  k )  e.  RR )
152151recnd 9651 . . . . . . . . . . . 12  |-  ( ( T.  /\  k  e.  NN )  ->  (
( NN  X.  {
3 } ) `  k )  e.  CC )
15355ffvelrnda 6008 . . . . . . . . . . . . 13  |-  ( ( T.  /\  k  e.  NN )  ->  ( G `  k )  e.  RR )
154153recnd 9651 . . . . . . . . . . . 12  |-  ( ( T.  /\  k  e.  NN )  ->  ( G `  k )  e.  CC )
155 ffn 5713 . . . . . . . . . . . . . 14  |-  ( ( NN  X.  { 3 } ) : NN --> RR  ->  ( NN  X.  { 3 } )  Fn  NN )
156150, 155syl 17 . . . . . . . . . . . . 13  |-  ( T. 
->  ( NN  X.  {
3 } )  Fn  NN )
157 eqidd 2403 . . . . . . . . . . . . 13  |-  ( ( T.  /\  k  e.  NN )  ->  (
( NN  X.  {
3 } ) `  k )  =  ( ( NN  X.  {
3 } ) `  k ) )
158 eqidd 2403 . . . . . . . . . . . . 13  |-  ( ( T.  /\  k  e.  NN )  ->  ( G `  k )  =  ( G `  k ) )
159156, 126, 57, 57, 58, 157, 158ofval 6529 . . . . . . . . . . . 12  |-  ( ( T.  /\  k  e.  NN )  ->  (
( ( NN  X.  { 3 } )  oF  x.  G
) `  k )  =  ( ( ( NN  X.  { 3 } ) `  k
)  x.  ( G `
 k ) ) )
1601, 2, 139, 141, 143, 152, 154, 159climmul 13602 . . . . . . . . . . 11  |-  ( T. 
->  ( ( NN  X.  { 3 } )  oF  x.  G
)  ~~>  ( 3  x.  0 ) )
161137mul01i 9803 . . . . . . . . . . 11  |-  ( 3  x.  0 )  =  0
162160, 161syl6breq 4433 . . . . . . . . . 10  |-  ( T. 
->  ( ( NN  X.  { 3 } )  oF  x.  G
)  ~~>  0 )
16363ffvelrnda 6008 . . . . . . . . . . 11  |-  ( ( T.  /\  k  e.  NN )  ->  ( H `  k )  e.  RR )
164163recnd 9651 . . . . . . . . . 10  |-  ( ( T.  /\  k  e.  NN )  ->  ( H `  k )  e.  CC )
16527, 150, 55, 57, 57, 58off 6535 . . . . . . . . . . . 12  |-  ( T. 
->  ( ( NN  X.  { 3 } )  oF  x.  G
) : NN --> RR )
166165ffvelrnda 6008 . . . . . . . . . . 11  |-  ( ( T.  /\  k  e.  NN )  ->  (
( ( NN  X.  { 3 } )  oF  x.  G
) `  k )  e.  RR )
167166recnd 9651 . . . . . . . . . 10  |-  ( ( T.  /\  k  e.  NN )  ->  (
( ( NN  X.  { 3 } )  oF  x.  G
) `  k )  e.  CC )
168 ffn 5713 . . . . . . . . . . . 12  |-  ( H : NN --> RR  ->  H  Fn  NN )
16963, 168syl 17 . . . . . . . . . . 11  |-  ( T. 
->  H  Fn  NN )
17041, 90, 74, 57, 57, 58off 6535 . . . . . . . . . . . 12  |-  ( T. 
->  ( ( ( NN 
X.  { 1 } )  oF  +  G )  oF  -  ( ( NN 
X.  { 1 } )  oF  +  ( ( NN  X.  { -u 2 } )  oF  x.  G
) ) ) : NN --> RR )
171 ffn 5713 . . . . . . . . . . . 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 )
172170, 171syl 17 . . . . . . . . . . 11  |-  ( T. 
->  ( ( ( NN 
X.  { 1 } )  oF  +  G )  oF  -  ( ( NN 
X.  { 1 } )  oF  +  ( ( NN  X.  { -u 2 } )  oF  x.  G
) ) )  Fn  NN )
173 eqidd 2403 . . . . . . . . . . 11  |-  ( ( T.  /\  k  e.  NN )  ->  ( H `  k )  =  ( H `  k ) )
174154mulid2d 9643 . . . . . . . . . . . . . . 15  |-  ( ( T.  /\  k  e.  NN )  ->  (
1  x.  ( G `
 k ) )  =  ( G `  k ) )
175 2cn 10646 . . . . . . . . . . . . . . . . . 18  |-  2  e.  CC
176 mulneg1 10033 . . . . . . . . . . . . . . . . . 18  |-  ( ( 2  e.  CC  /\  ( G `  k )  e.  CC )  -> 
( -u 2  x.  ( G `  k )
)  =  -u (
2  x.  ( G `
 k ) ) )
177175, 154, 176sylancr 661 . . . . . . . . . . . . . . . . 17  |-  ( ( T.  /\  k  e.  NN )  ->  ( -u 2  x.  ( G `
 k ) )  =  -u ( 2  x.  ( G `  k
) ) )
178177negeqd 9849 . . . . . . . . . . . . . . . 16  |-  ( ( T.  /\  k  e.  NN )  ->  -u ( -u 2  x.  ( G `
 k ) )  =  -u -u ( 2  x.  ( G `  k
) ) )
179 mulcl 9605 . . . . . . . . . . . . . . . . . 18  |-  ( ( 2  e.  CC  /\  ( G `  k )  e.  CC )  -> 
( 2  x.  ( G `  k )
)  e.  CC )
180175, 154, 179sylancr 661 . . . . . . . . . . . . . . . . 17  |-  ( ( T.  /\  k  e.  NN )  ->  (
2  x.  ( G `
 k ) )  e.  CC )
181180negnegd 9957 . . . . . . . . . . . . . . . 16  |-  ( ( T.  /\  k  e.  NN )  ->  -u -u (
2  x.  ( G `
 k ) )  =  ( 2  x.  ( G `  k
) ) )
182178, 181eqtr2d 2444 . . . . . . . . . . . . . . 15  |-  ( ( T.  /\  k  e.  NN )  ->  (
2  x.  ( G `
 k ) )  =  -u ( -u 2  x.  ( G `  k
) ) )
183174, 182oveq12d 6295 . . . . . . . . . . . . . 14  |-  ( ( T.  /\  k  e.  NN )  ->  (
( 1  x.  ( G `  k )
)  +  ( 2  x.  ( G `  k ) ) )  =  ( ( G `
 k )  + 
-u ( -u 2  x.  ( G `  k
) ) ) )
184 remulcl 9606 . . . . . . . . . . . . . . . . 17  |-  ( (
-u 2  e.  RR  /\  ( G `  k
)  e.  RR )  ->  ( -u 2  x.  ( G `  k
) )  e.  RR )
18568, 153, 184sylancr 661 . . . . . . . . . . . . . . . 16  |-  ( ( T.  /\  k  e.  NN )  ->  ( -u 2  x.  ( G `
 k ) )  e.  RR )
186185recnd 9651 . . . . . . . . . . . . . . 15  |-  ( ( T.  /\  k  e.  NN )  ->  ( -u 2  x.  ( G `
 k ) )  e.  CC )
187154, 186negsubd 9972 . . . . . . . . . . . . . 14  |-  ( ( T.  /\  k  e.  NN )  ->  (
( G `  k
)  +  -u ( -u 2  x.  ( G `
 k ) ) )  =  ( ( G `  k )  -  ( -u 2  x.  ( G `  k
) ) ) )
188183, 187eqtrd 2443 . . . . . . . . . . . . 13  |-  ( ( T.  /\  k  e.  NN )  ->  (
( 1  x.  ( G `  k )
)  +  ( 2  x.  ( G `  k ) ) )  =  ( ( G `
 k )  -  ( -u 2  x.  ( G `  k )
) ) )
189 df-3 10635 . . . . . . . . . . . . . . . 16  |-  3  =  ( 2  +  1 )
190 ax-1cn 9579 . . . . . . . . . . . . . . . . 17  |-  1  e.  CC
191175, 190addcomi 9804 . . . . . . . . . . . . . . . 16  |-  ( 2  +  1 )  =  ( 1  +  2 )
192189, 191eqtri 2431 . . . . . . . . . . . . . . 15  |-  3  =  ( 1  +  2 )
193192oveq1i 6287 . . . . . . . . . . . . . 14  |-  ( 3  x.  ( G `  k ) )  =  ( ( 1  +  2 )  x.  ( G `  k )
)
194 1cnd 9641 . . . . . . . . . . . . . . 15  |-  ( ( T.  /\  k  e.  NN )  ->  1  e.  CC )
195175a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( T.  /\  k  e.  NN )  ->  2  e.  CC )
196194, 195, 154adddird 9650 . . . . . . . . . . . . . 14  |-  ( ( T.  /\  k  e.  NN )  ->  (
( 1  +  2 )  x.  ( G `
 k ) )  =  ( ( 1  x.  ( G `  k ) )  +  ( 2  x.  ( G `  k )
) ) )
197193, 196syl5eq 2455 . . . . . . . . . . . . 13  |-  ( ( T.  /\  k  e.  NN )  ->  (
3  x.  ( G `
 k ) )  =  ( ( 1  x.  ( G `  k ) )  +  ( 2  x.  ( G `  k )
) ) )
198194, 154, 186pnpcand 10003 . . . . . . . . . . . . 13  |-  ( ( T.  /\  k  e.  NN )  ->  (
( 1  +  ( G `  k ) )  -  ( 1  +  ( -u 2  x.  ( G `  k
) ) ) )  =  ( ( G `
 k )  -  ( -u 2  x.  ( G `  k )
) ) )
199188, 197, 1983eqtr4rd 2454 . . . . . . . . . . . 12  |-  ( ( T.  /\  k  e.  NN )  ->  (
( 1  +  ( G `  k ) )  -  ( 1  +  ( -u 2  x.  ( G `  k
) ) ) )  =  ( 3  x.  ( G `  k
) ) )
200124, 126, 57, 57, 58offn 6531 . . . . . . . . . . . . 13  |-  ( T. 
->  ( ( NN  X.  { 1 } )  oF  +  G
)  Fn  NN )
20112a1i 11 . . . . . . . . . . . . . . . 16  |-  ( T. 
->  -u 2  e.  ZZ )
202 fnconstg 5755 . . . . . . . . . . . . . . . 16  |-  ( -u
2  e.  ZZ  ->  ( NN  X.  { -u
2 } )  Fn  NN )
203201, 202syl 17 . . . . . . . . . . . . . . 15  |-  ( T. 
->  ( NN  X.  { -u 2 } )  Fn  NN )
204203, 126, 57, 57, 58offn 6531 . . . . . . . . . . . . . 14  |-  ( T. 
->  ( ( NN  X.  { -u 2 } )  oF  x.  G
)  Fn  NN )
205124, 204, 57, 57, 58offn 6531 . . . . . . . . . . . . 13  |-  ( T. 
->  ( ( NN  X.  { 1 } )  oF  +  ( ( NN  X.  { -u 2 } )  oF  x.  G ) )  Fn  NN )
20657, 44, 126, 158ofc1 6544 . . . . . . . . . . . . 13  |-  ( ( T.  /\  k  e.  NN )  ->  (
( ( NN  X.  { 1 } )  oF  +  G
) `  k )  =  ( 1  +  ( G `  k
) ) )
20757, 69, 126, 158ofc1 6544 . . . . . . . . . . . . . 14  |-  ( ( T.  /\  k  e.  NN )  ->  (
( ( NN  X.  { -u 2 } )  oF  x.  G
) `  k )  =  ( -u 2  x.  ( G `  k
) ) )
20857, 44, 204, 207ofc1 6544 . . . . . . . . . . . . 13  |-  ( ( T.  /\  k  e.  NN )  ->  (
( ( NN  X.  { 1 } )  oF  +  ( ( NN  X.  { -u 2 } )  oF  x.  G ) ) `  k )  =  ( 1  +  ( -u 2  x.  ( G `  k
) ) ) )
209200, 205, 57, 57, 58, 206, 208ofval 6529 . . . . . . . . . . . 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
) ) ) ) )
21057, 147, 126, 158ofc1 6544 . . . . . . . . . . . 12  |-  ( ( T.  /\  k  e.  NN )  ->  (
( ( NN  X.  { 3 } )  oF  x.  G
) `  k )  =  ( 3  x.  ( G `  k
) ) )
211199, 209, 2103eqtr4d 2453 . . . . . . . . . . 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 )
)
212169, 172, 57, 57, 58, 173, 211ofval 6529 . . . . . . . . . 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 ) ) )
2131, 2, 134, 136, 162, 164, 167, 212climmul 13602 . . . . . . . . 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 ) )
214101mul01i 9803 . . . . . . . . 9  |-  ( ( ( pi ^ 2 )  /  6 )  x.  0 )  =  0
215213, 214syl6breq 4433 . . . . . . . 8  |-  ( T. 
->  ( H  oF  x.  ( ( ( NN  X.  { 1 } )  oF  +  G )  oF  -  ( ( NN  X.  { 1 } )  oF  +  ( ( NN 
X.  { -u 2 } )  oF  x.  G ) ) ) )  ~~>  0 )
216100, 215eqbrtrrd 4416 . . . . . . 7  |-  ( T. 
->  ( K  oF  -  J )  ~~>  0 )
217 ovex 6305 . . . . . . . 8  |-  ( F  oF  -  J
)  e.  _V
218217a1i 11 . . . . . . 7  |-  ( T. 
->  ( F  oF  -  J )  e. 
_V )
21927, 63, 90, 57, 57, 58off 6535 . . . . . . . . . 10  |-  ( T. 
->  ( H  oF  x.  ( ( NN 
X.  { 1 } )  oF  +  G ) ) : NN --> RR )
22098feq1i 5705 . . . . . . . . . 10  |-  ( K : NN --> RR  <->  ( H  oF  x.  (
( NN  X.  {
1 } )  oF  +  G ) ) : NN --> RR )
221219, 220sylibr 212 . . . . . . . . 9  |-  ( T. 
->  K : NN --> RR )
22241, 221, 78, 57, 57, 58off 6535 . . . . . . . 8  |-  ( T. 
->  ( K  oF  -  J ) : NN --> RR )
223222ffvelrnda 6008 . . . . . . 7  |-  ( ( T.  /\  k  e.  NN )  ->  (
( K  oF  -  J ) `  k )  e.  RR )
22441, 23, 78, 57, 57, 58off 6535 . . . . . . . 8  |-  ( T. 
->  ( F  oF  -  J ) : NN --> RR )
225224ffvelrnda 6008 . . . . . . 7  |-  ( ( T.  /\  k  e.  NN )  ->  (
( F  oF  -  J ) `  k )  e.  RR )
22623ffvelrnda 6008 . . . . . . . . 9  |-  ( ( T.  /\  k  e.  NN )  ->  ( F `  k )  e.  RR )
227221ffvelrnda 6008 . . . . . . . . 9  |-  ( ( T.  /\  k  e.  NN )  ->  ( K `  k )  e.  RR )
22878ffvelrnda 6008 . . . . . . . . 9  |-  ( ( T.  /\  k  e.  NN )  ->  ( J `  k )  e.  RR )
229 eqid 2402 . . . . . . . . . . . 12  |-  ( ( 2  x.  k )  +  1 )  =  ( ( 2  x.  k )  +  1 )
23054, 21, 61, 76, 98, 229basellem8 23740 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  (
( J `  k
)  <_  ( F `  k )  /\  ( F `  k )  <_  ( K `  k
) ) )
231230adantl 464 . . . . . . . . . 10  |-  ( ( T.  /\  k  e.  NN )  ->  (
( J `  k
)  <_  ( F `  k )  /\  ( F `  k )  <_  ( K `  k
) ) )
232231simprd 461 . . . . . . . . 9  |-  ( ( T.  /\  k  e.  NN )  ->  ( F `  k )  <_  ( K `  k
) )
233226, 227, 228, 232lesub1dd 10207 . . . . . . . 8  |-  ( ( T.  /\  k  e.  NN )  ->  (
( F `  k
)  -  ( J `
 k ) )  <_  ( ( K `
 k )  -  ( J `  k ) ) )
234 ffn 5713 . . . . . . . . . 10  |-  ( F : NN --> RR  ->  F  Fn  NN )
23523, 234syl 17 . . . . . . . . 9  |-  ( T. 
->  F  Fn  NN )
236 ffn 5713 . . . . . . . . . 10  |-  ( J : NN --> RR  ->  J  Fn  NN )
23778, 236syl 17 . . . . . . . . 9  |-  ( T. 
->  J  Fn  NN )
238 eqidd 2403 . . . . . . . . 9  |-  ( ( T.  /\  k  e.  NN )  ->  ( F `  k )  =  ( F `  k ) )
239 eqidd 2403 . . . . . . . . 9  |-  ( ( T.  /\  k  e.  NN )  ->  ( J `  k )  =  ( J `  k ) )
240235, 237, 57, 57, 58, 238, 239ofval 6529 . . . . . . . 8  |-  ( ( T.  /\  k  e.  NN )  ->  (
( F  oF  -  J ) `  k )  =  ( ( F `  k
)  -  ( J `
 k ) ) )
241 ffn 5713 . . . . . . . . . 10  |-  ( K : NN --> RR  ->  K  Fn  NN )
242221, 241syl 17 . . . . . . . . 9  |-  ( T. 
->  K  Fn  NN )
243 eqidd 2403 . . . . . . . . 9  |-  ( ( T.  /\  k  e.  NN )  ->  ( K `  k )  =  ( K `  k ) )
244242, 237, 57, 57, 58, 243, 239ofval 6529 . . . . . . . 8  |-  ( ( T.  /\  k  e.  NN )  ->  (
( K  oF  -  J ) `  k )  =  ( ( K `  k
)  -  ( J `
 k ) ) )
245233, 240, 2443brtr4d 4424 . . . . . . 7  |-  ( ( T.  /\  k  e.  NN )  ->  (
( F  oF  -  J ) `  k )  <_  (
( K  oF  -  J ) `  k ) )
246231simpld 457 . . . . . . . . 9  |-  ( ( T.  /\  k  e.  NN )  ->  ( J `  k )  <_  ( F `  k
) )
247226, 228subge0d 10181 . . . . . . . . 9  |-  ( ( T.  /\  k  e.  NN )  ->  (
0  <_  ( ( F `  k )  -  ( J `  k ) )  <->  ( J `  k )  <_  ( F `  k )
) )
248246, 247mpbird 232 . . . . . . . 8  |-  ( ( T.  /\  k  e.  NN )  ->  0  <_  ( ( F `  k )  -  ( J `  k )
) )
249248, 240breqtrrd 4420 . . . . . . 7  |-  ( ( T.  /\  k  e.  NN )  ->  0  <_  ( ( F  oF  -  J ) `  k ) )
2501, 2, 216, 218, 223, 225, 245, 249climsqz2 13611 . . . . . 6  |-  ( T. 
->  ( F  oF  -  J )  ~~>  0 )
251 ovex 6305 . . . . . . 7  |-  ( ( F  oF  -  J )  oF  +  J )  e. 
_V
252251a1i 11 . . . . . 6  |-  ( T. 
->  ( ( F  oF  -  J )  oF  +  J
)  e.  _V )
253 ovex 6305 . . . . . . . . . 10  |-  ( H  oF  x.  (
( NN  X.  {
1 } )  oF  +  ( ( NN  X.  { -u
2 } )  oF  x.  G ) ) )  e.  _V
254253a1i 11 . . . . . . . . 9  |-  ( T. 
->  ( H  oF  x.  ( ( NN 
X.  { 1 } )  oF  +  ( ( NN  X.  { -u 2 } )  oF  x.  G
) ) )  e. 
_V )
25568recni 9637 . . . . . . . . . . 11  |-  -u 2  e.  CC
25654, 255basellem7 23739 . . . . . . . . . 10  |-  ( ( NN  X.  { 1 } )  oF  +  ( ( NN 
X.  { -u 2 } )  oF  x.  G ) )  ~~>  1
257256a1i 11 . . . . . . . . 9  |-  ( T. 
->  ( ( NN  X.  { 1 } )  oF  +  ( ( NN  X.  { -u 2 } )  oF  x.  G ) )  ~~>  1 )
25874ffvelrnda 6008 . . . . . . . . . 10  |-  ( ( T.  /\  k  e.  NN )  ->  (
( ( NN  X.  { 1 } )  oF  +  ( ( NN  X.  { -u 2 } )  oF  x.  G ) ) `  k )  e.  RR )
259258recnd 9651 . . . . . . . . 9  |-  ( ( T.  /\  k  e.  NN )  ->  (
( ( NN  X.  { 1 } )  oF  +  ( ( NN  X.  { -u 2 } )  oF  x.  G ) ) `  k )  e.  CC )
260 eqidd 2403 . . . . . . . . . 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 ) )
261169, 205, 57, 57, 58, 173, 260ofval 6529 . . . . . . . . 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 ) ) )
2621, 2, 134, 254, 257, 164, 259, 261climmul 13602 . . . . . . . 8  |-  ( T. 
->  ( H  oF  x.  ( ( NN 
X.  { 1 } )  oF  +  ( ( NN  X.  { -u 2 } )  oF  x.  G
) ) )  ~~>  ( ( ( pi ^ 2 )  /  6 )  x.  1 ) )
263262, 132syl6breq 4433 . . . . . . 7  |-  ( T. 
->  ( H  oF  x.  ( ( NN 
X.  { 1 } )  oF  +  ( ( NN  X.  { -u 2 } )  oF  x.  G
) ) )  ~~>  ( ( pi ^ 2 )  /  6 ) )
26476, 263syl5eqbr 4427 . . . . . 6  |-  ( T. 
->  J  ~~>  ( (
pi ^ 2 )  /  6 ) )
265225recnd 9651 . . . . . 6  |-  ( ( T.  /\  k  e.  NN )  ->  (
( F  oF  -  J ) `  k )  e.  CC )
266228recnd 9651 . . . . . 6  |-  ( ( T.  /\  k  e.  NN )  ->  ( J `  k )  e.  CC )
267 ffn 5713 . . . . . . . 8  |-  ( ( F  oF  -  J ) : NN --> RR  ->  ( F  oF  -  J )  Fn  NN )
268224, 267syl 17 . . . . . . 7  |-  ( T. 
->  ( F  oF  -  J )  Fn  NN )
269 eqidd 2403 . . . . . . 7  |-  ( ( T.  /\  k  e.  NN )  ->  (
( F  oF  -  J ) `  k )  =  ( ( F  oF  -  J ) `  k ) )
270268, 237, 57, 57, 58, 269, 239ofval 6529 . . . . . 6  |-  ( ( T.  /\  k  e.  NN )  ->  (
( ( F  oF  -  J )  oF  +  J
) `  k )  =  ( ( ( F  oF  -  J ) `  k
)  +  ( J `
 k ) ) )
2711, 2, 250, 252, 264, 265, 266, 270climadd 13601 . . . . 5  |-  ( T. 
->  ( ( F  oF  -  J )  oF  +  J
)  ~~>  ( 0  +  ( ( pi ^
2 )  /  6
) ) )
27289, 271eqbrtrrd 4416 . . . 4  |-  ( T. 
->  F  ~~>  ( 0  +  ( ( pi
^ 2 )  / 
6 ) ) )
273101addid2i 9801 . . . 4  |-  ( 0  +  ( ( pi
^ 2 )  / 
6 ) )  =  ( ( pi ^
2 )  /  6
)
274272, 21, 2733brtr3g 4425 . . 3  |-  ( T. 
->  seq 1 (  +  ,  ( n  e.  NN  |->  ( n ^ -u 2 ) ) )  ~~>  ( ( pi ^
2 )  /  6
) )
2751, 2, 7, 19, 274isumclim 13721 . 2  |-  ( T. 
->  sum_ k  e.  NN  ( k ^ -u 2
)  =  ( ( pi ^ 2 )  /  6 ) )
276275trud 1414 1  |-  sum_ k  e.  NN  ( k ^ -u 2 )  =  ( ( pi ^ 2 )  /  6 )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    /\ w3a 974    = wceq 1405   T. wtru 1406    e. wcel 1842   _Vcvv 3058    C_ wss 3413   {csn 3971   class class class wbr 4394    |-> cmpt 4452    X. cxp 4820    Fn wfn 5563   -->wf 5564   ` cfv 5568  (class class class)co 6277    oFcof 6518   CCcc 9519   RRcr 9520   0cc0 9521   1c1 9522    + caddc 9524    x. cmul 9526    <_ cle 9658    - cmin 9840   -ucneg 9841    / cdiv 10246   NNcn 10575   2c2 10625   3c3 10626   6c6 10629   ZZcz 10904   ZZ>=cuz 11126    seqcseq 12149   ^cexp 12208    ~~> cli 13454   sum_csu 13655   picpi 14009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-inf2 8090  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599  ax-addf 9600  ax-mulf 9601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6520  df-om 6683  df-1st 6783  df-2nd 6784  df-supp 6902  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-2o 7167  df-oadd 7170  df-er 7347  df-map 7458  df-pm 7459  df-ixp 7507  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-fsupp 7863  df-fi 7904  df-sup 7934  df-oi 7968  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-7 10639  df-8 10640  df-9 10641  df-10 10642  df-n0 10836  df-z 10905  df-dec 11019  df-uz 11127  df-q 11227  df-rp 11265  df-xneg 11370  df-xadd 11371  df-xmul 11372  df-ioo 11585  df-ioc 11586  df-ico 11587  df-icc 11588  df-fz 11725  df-fzo 11853  df-fl 11964  df-mod 12033  df-seq 12150  df-exp 12209  df-fac 12396  df-bc 12423  df-hash 12451  df-shft 13047  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-limsup 13441  df-clim 13458  df-rlim 13459  df-sum 13656  df-ef 14010  df-sin 14012  df-cos 14013  df-tan 14014  df-pi 14015  df-struct 14841  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-ress 14846  df-plusg 14920  df-mulr 14921  df-starv 14922  df-sca 14923  df-vsca 14924  df-ip 14925  df-tset 14926  df-ple 14927  df-ds 14929  df-unif 14930  df-hom 14931  df-cco 14932  df-rest 15035  df-topn 15036  df-0g 15054  df-gsum 15055  df-topgen 15056  df-pt 15057  df-prds 15060  df-xrs 15114  df-qtop 15119  df-imas 15120  df-xps 15122  df-mre 15198  df-mrc 15199  df-acs 15201  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-submnd 16289  df-mulg 16382  df-cntz 16677  df-cmn 17122  df-psmet 18729  df-xmet 18730  df-met 18731  df-bl 18732  df-mopn 18733  df-fbas 18734  df-fg 18735  df-cnfld 18739  df-top 19689  df-bases 19691  df-topon 19692  df-topsp 19693  df-cld 19810  df-ntr 19811  df-cls 19812  df-nei 19890  df-lp 19928  df-perf 19929  df-cn 20019  df-cnp 20020  df-haus 20107  df-tx 20353  df-hmeo 20546  df-fil 20637  df-fm 20729  df-flim 20730  df-flf 20731  df-xms 21113  df-ms 21114  df-tms 21115  df-cncf 21672  df-0p 22367  df-limc 22560  df-dv 22561  df-ply 22875  df-idp 22876  df-coe 22877  df-dgr 22878  df-quot 22977
This theorem is referenced by:  basel  23742
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