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Theorem basellem9 23085
Description: Lemma for basel 23086. Since by basellem8 23084 
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 13414. 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 11108 . . 3  |-  NN  =  ( ZZ>= `  1 )
2 1zzd 10886 . . 3  |-  ( T. 
->  1  e.  ZZ )
3 oveq1 6284 . . . . 5  |-  ( n  =  k  ->  (
n ^ -u 2
)  =  ( k ^ -u 2 ) )
4 eqid 2462 . . . . 5  |-  ( n  e.  NN  |->  ( n ^ -u 2 ) )  =  ( n  e.  NN  |->  ( n ^ -u 2 ) )
5 ovex 6302 . . . . 5  |-  ( k ^ -u 2 )  e.  _V
63, 4, 5fvmpt 5943 . . . 4  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  ( n ^ -u 2
) ) `  k
)  =  ( k ^ -u 2 ) )
76adantl 466 . . 3  |-  ( ( T.  /\  k  e.  NN )  ->  (
( n  e.  NN  |->  ( n ^ -u 2
) ) `  k
)  =  ( k ^ -u 2 ) )
8 nnre 10534 . . . . . . . . 9  |-  ( n  e.  NN  ->  n  e.  RR )
9 nnne0 10559 . . . . . . . . 9  |-  ( n  e.  NN  ->  n  =/=  0 )
10 2z 10887 . . . . . . . . . . 11  |-  2  e.  ZZ
11 znegcl 10889 . . . . . . . . . . 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 12292 . . . . . . . 8  |-  ( n  e.  NN  ->  (
n ^ -u 2
)  e.  RR )
1514adantl 466 . . . . . . 7  |-  ( ( T.  /\  n  e.  NN )  ->  (
n ^ -u 2
)  e.  RR )
1615, 4fmptd 6038 . . . . . 6  |-  ( T. 
->  ( n  e.  NN  |->  ( n ^ -u 2
) ) : NN --> RR )
1716ffvelrnda 6014 . . . . 5  |-  ( ( T.  /\  k  e.  NN )  ->  (
( n  e.  NN  |->  ( n ^ -u 2
) ) `  k
)  e.  RR )
187, 17eqeltrrd 2551 . . . 4  |-  ( ( T.  /\  k  e.  NN )  ->  (
k ^ -u 2
)  e.  RR )
1918recnd 9613 . . 3  |-  ( ( T.  /\  k  e.  NN )  ->  (
k ^ -u 2
)  e.  CC )
201, 2, 17serfre 12094 . . . . . . . . . . 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 5716 . . . . . . . . . . 11  |-  ( F : NN --> RR  <->  seq 1
(  +  ,  ( n  e.  NN  |->  ( n ^ -u 2
) ) ) : NN --> RR )
2320, 22sylibr 212 . . . . . . . . . 10  |-  ( T. 
->  F : NN --> RR )
2423ffvelrnda 6014 . . . . . . . . 9  |-  ( ( T.  /\  n  e.  NN )  ->  ( F `  n )  e.  RR )
2524recnd 9613 . . . . . . . 8  |-  ( ( T.  /\  n  e.  NN )  ->  ( F `  n )  e.  CC )
26 remulcl 9568 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  x.  y
)  e.  RR )
2726adantl 466 . . . . . . . . . . . 12  |-  ( ( T.  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  x.  y
)  e.  RR )
28 ovex 6302 . . . . . . . . . . . . . . . 16  |-  ( ( pi ^ 2 )  /  6 )  e. 
_V
2928fconst 5764 . . . . . . . . . . . . . . 15  |-  ( NN 
X.  { ( ( pi ^ 2 )  /  6 ) } ) : NN --> { ( ( pi ^ 2 )  /  6 ) }
30 pire 22580 . . . . . . . . . . . . . . . . . . 19  |-  pi  e.  RR
3130resqcli 12210 . . . . . . . . . . . . . . . . . 18  |-  ( pi
^ 2 )  e.  RR
32 6re 10607 . . . . . . . . . . . . . . . . . 18  |-  6  e.  RR
33 6nn 10688 . . . . . . . . . . . . . . . . . . 19  |-  6  e.  NN
3433nnne0i 10561 . . . . . . . . . . . . . . . . . 18  |-  6  =/=  0
3531, 32, 34redivcli 10302 . . . . . . . . . . . . . . . . 17  |-  ( ( pi ^ 2 )  /  6 )  e.  RR
3635a1i 11 . . . . . . . . . . . . . . . 16  |-  ( T. 
->  ( ( pi ^
2 )  /  6
)  e.  RR )
3736snssd 4167 . . . . . . . . . . . . . . 15  |-  ( T. 
->  { ( ( pi
^ 2 )  / 
6 ) }  C_  RR )
38 fss 5732 . . . . . . . . . . . . . . 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 663 . . . . . . . . . . . . . 14  |-  ( T. 
->  ( NN  X.  {
( ( pi ^
2 )  /  6
) } ) : NN --> RR )
40 resubcl 9874 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  -  y
)  e.  RR )
4140adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( T.  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  -  y
)  e.  RR )
42 1ex 9582 . . . . . . . . . . . . . . . . 17  |-  1  e.  _V
4342fconst 5764 . . . . . . . . . . . . . . . 16  |-  ( NN 
X.  { 1 } ) : NN --> { 1 }
44 1red 9602 . . . . . . . . . . . . . . . . 17  |-  ( T. 
->  1  e.  RR )
4544snssd 4167 . . . . . . . . . . . . . . . 16  |-  ( T. 
->  { 1 }  C_  RR )
46 fss 5732 . . . . . . . . . . . . . . . 16  |-  ( ( ( NN  X.  {
1 } ) : NN --> { 1 }  /\  { 1 } 
C_  RR )  -> 
( NN  X.  {
1 } ) : NN --> RR )
4743, 45, 46sylancr 663 . . . . . . . . . . . . . . 15  |-  ( T. 
->  ( NN  X.  {
1 } ) : NN --> RR )
48 2nn 10684 . . . . . . . . . . . . . . . . . . . 20  |-  2  e.  NN
4948a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( T. 
->  2  e.  NN )
50 nnmulcl 10550 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 2  e.  NN  /\  n  e.  NN )  ->  ( 2  x.  n
)  e.  NN )
5149, 50sylan 471 . . . . . . . . . . . . . . . . . 18  |-  ( ( T.  /\  n  e.  NN )  ->  (
2  x.  n )  e.  NN )
5251peano2nnd 10544 . . . . . . . . . . . . . . . . 17  |-  ( ( T.  /\  n  e.  NN )  ->  (
( 2  x.  n
)  +  1 )  e.  NN )
5352nnrecred 10572 . . . . . . . . . . . . . . . 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 6038 . . . . . . . . . . . . . . 15  |-  ( T. 
->  G : NN --> RR )
56 nnex 10533 . . . . . . . . . . . . . . . 16  |-  NN  e.  _V
5756a1i 11 . . . . . . . . . . . . . . 15  |-  ( T. 
->  NN  e.  _V )
58 inidm 3702 . . . . . . . . . . . . . . 15  |-  ( NN 
i^i  NN )  =  NN
5941, 47, 55, 57, 57, 58off 6531 . . . . . . . . . . . . . 14  |-  ( T. 
->  ( ( NN  X.  { 1 } )  oF  -  G
) : NN --> RR )
6027, 39, 59, 57, 57, 58off 6531 . . . . . . . . . . . . 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 5716 . . . . . . . . . . . . 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 9566 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
6564adantl 466 . . . . . . . . . . . . 13  |-  ( ( T.  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( x  +  y )  e.  RR )
6612elexi 3118 . . . . . . . . . . . . . . . 16  |-  -u 2  e.  _V
6766fconst 5764 . . . . . . . . . . . . . . 15  |-  ( NN 
X.  { -u 2 } ) : NN --> { -u 2 }
6812zrei 10861 . . . . . . . . . . . . . . . . 17  |-  -u 2  e.  RR
6968a1i 11 . . . . . . . . . . . . . . . 16  |-  ( T. 
->  -u 2  e.  RR )
7069snssd 4167 . . . . . . . . . . . . . . 15  |-  ( T. 
->  { -u 2 } 
C_  RR )
71 fss 5732 . . . . . . . . . . . . . . 15  |-  ( ( ( NN  X.  { -u 2 } ) : NN --> { -u 2 }  /\  { -u 2 }  C_  RR )  -> 
( NN  X.  { -u 2 } ) : NN --> RR )
7267, 70, 71sylancr 663 . . . . . . . . . . . . . 14  |-  ( T. 
->  ( NN  X.  { -u 2 } ) : NN --> RR )
7327, 72, 55, 57, 57, 58off 6531 . . . . . . . . . . . . 13  |-  ( T. 
->  ( ( NN  X.  { -u 2 } )  oF  x.  G
) : NN --> RR )
7465, 47, 73, 57, 57, 58off 6531 . . . . . . . . . . . 12  |-  ( T. 
->  ( ( NN  X.  { 1 } )  oF  +  ( ( NN  X.  { -u 2 } )  oF  x.  G ) ) : NN --> RR )
7527, 63, 74, 57, 57, 58off 6531 . . . . . . . . . . 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 5716 . . . . . . . . . . 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 6014 . . . . . . . . 9  |-  ( ( T.  /\  n  e.  NN )  ->  ( J `  n )  e.  RR )
8079recnd 9613 . . . . . . . 8  |-  ( ( T.  /\  n  e.  NN )  ->  ( J `  n )  e.  CC )
8125, 80npcand 9925 . . . . . . 7  |-  ( ( T.  /\  n  e.  NN )  ->  (
( ( F `  n )  -  ( J `  n )
)  +  ( J `
 n ) )  =  ( F `  n ) )
8281mpteq2dva 4528 . . . . . 6  |-  ( T. 
->  ( n  e.  NN  |->  ( ( ( F `
 n )  -  ( J `  n ) )  +  ( J `
 n ) ) )  =  ( n  e.  NN  |->  ( F `
 n ) ) )
83 ovex 6302 . . . . . . . 8  |-  ( ( F `  n )  -  ( J `  n ) )  e. 
_V
8483a1i 11 . . . . . . 7  |-  ( ( T.  /\  n  e.  NN )  ->  (
( F `  n
)  -  ( J `
 n ) )  e.  _V )
8523feqmptd 5913 . . . . . . . 8  |-  ( T. 
->  F  =  (
n  e.  NN  |->  ( F `  n ) ) )
8678feqmptd 5913 . . . . . . . 8  |-  ( T. 
->  J  =  (
n  e.  NN  |->  ( J `  n ) ) )
8757, 24, 79, 85, 86offval2 6533 . . . . . . 7  |-  ( T. 
->  ( F  oF  -  J )  =  ( n  e.  NN  |->  ( ( F `  n )  -  ( J `  n )
) ) )
8857, 84, 79, 87, 86offval2 6533 . . . . . 6  |-  ( T. 
->  ( ( F  oF  -  J )  oF  +  J
)  =  ( n  e.  NN  |->  ( ( ( F `  n
)  -  ( J `
 n ) )  +  ( J `  n ) ) ) )
8982, 88, 853eqtr4d 2513 . . . . 5  |-  ( T. 
->  ( ( F  oF  -  J )  oF  +  J
)  =  F )
9065, 47, 55, 57, 57, 58off 6531 . . . . . . . . . 10  |-  ( T. 
->  ( ( NN  X.  { 1 } )  oF  +  G
) : NN --> RR )
91 recn 9573 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  x  e.  CC )
92 recn 9573 . . . . . . . . . . . 12  |-  ( y  e.  RR  ->  y  e.  CC )
93 recn 9573 . . . . . . . . . . . 12  |-  ( z  e.  RR  ->  z  e.  CC )
94 subdi 9981 . . . . . . . . . . . 12  |-  ( ( x  e.  CC  /\  y  e.  CC  /\  z  e.  CC )  ->  (
x  x.  ( y  -  z ) )  =  ( ( x  x.  y )  -  ( x  x.  z
) ) )
9591, 92, 93, 94syl3an 1265 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  z  e.  RR )  ->  (
x  x.  ( y  -  z ) )  =  ( ( x  x.  y )  -  ( x  x.  z
) ) )
9695adantl 466 . . . . . . . . . 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 6553 . . . . . . . . 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 2521 . . . . . . . 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 9599 . . . . . . . . . . . . . 14  |-  ( ( pi ^ 2 )  /  6 )  e.  CC
1021eqimss2i 3554 . . . . . . . . . . . . . . 15  |-  ( ZZ>= ` 
1 )  C_  NN
103102, 56climconst2 13322 . . . . . . . . . . . . . 14  |-  ( ( ( ( pi ^
2 )  /  6
)  e.  CC  /\  1  e.  ZZ )  ->  ( NN  X.  {
( ( pi ^
2 )  /  6
) } )  ~~>  ( ( pi ^ 2 )  /  6 ) )
104101, 2, 103sylancr 663 . . . . . . . . . . . . 13  |-  ( T. 
->  ( NN  X.  {
( ( pi ^
2 )  /  6
) } )  ~~>  ( ( pi ^ 2 )  /  6 ) )
105 ovex 6302 . . . . . . . . . . . . . 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 9540 . . . . . . . . . . . . . . . 16  |-  RR  C_  CC
108 fss 5732 . . . . . . . . . . . . . . . 16  |-  ( ( ( NN  X.  {
1 } ) : NN --> RR  /\  RR  C_  CC )  ->  ( NN  X.  { 1 } ) : NN --> CC )
10947, 107, 108sylancl 662 . . . . . . . . . . . . . . 15  |-  ( T. 
->  ( NN  X.  {
1 } ) : NN --> CC )
110 fss 5732 . . . . . . . . . . . . . . . 16  |-  ( ( G : NN --> RR  /\  RR  C_  CC )  ->  G : NN --> CC )
11155, 107, 110sylancl 662 . . . . . . . . . . . . . . 15  |-  ( T. 
->  G : NN --> CC )
112 ofnegsub 10525 . . . . . . . . . . . . . . 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 1223 . . . . . . . . . . . . . 14  |-  ( T. 
->  ( ( NN  X.  { 1 } )  oF  +  ( ( NN  X.  { -u 1 } )  oF  x.  G ) )  =  ( ( NN  X.  { 1 } )  oF  -  G ) )
114 neg1cn 10630 . . . . . . . . . . . . . . 15  |-  -u 1  e.  CC
11554, 114basellem7 23083 . . . . . . . . . . . . . 14  |-  ( ( NN  X.  { 1 } )  oF  +  ( ( NN 
X.  { -u 1 } )  oF  x.  G ) )  ~~>  1
116113, 115syl6eqbrr 4480 . . . . . . . . . . . . 13  |-  ( T. 
->  ( ( NN  X.  { 1 } )  oF  -  G
)  ~~>  1 )
11739ffvelrnda 6014 . . . . . . . . . . . . . 14  |-  ( ( T.  /\  k  e.  NN )  ->  (
( NN  X.  {
( ( pi ^
2 )  /  6
) } ) `  k )  e.  RR )
118117recnd 9613 . . . . . . . . . . . . 13  |-  ( ( T.  /\  k  e.  NN )  ->  (
( NN  X.  {
( ( pi ^
2 )  /  6
) } ) `  k )  e.  CC )
11959ffvelrnda 6014 . . . . . . . . . . . . . 14  |-  ( ( T.  /\  k  e.  NN )  ->  (
( ( NN  X.  { 1 } )  oF  -  G
) `  k )  e.  RR )
120119recnd 9613 . . . . . . . . . . . . 13  |-  ( ( T.  /\  k  e.  NN )  ->  (
( ( NN  X.  { 1 } )  oF  -  G
) `  k )  e.  CC )
121 ffn 5724 . . . . . . . . . . . . . . 15  |-  ( ( NN  X.  { ( ( pi ^ 2 )  /  6 ) } ) : NN --> RR  ->  ( NN  X.  { ( ( pi
^ 2 )  / 
6 ) } )  Fn  NN )
12239, 121syl 16 . . . . . . . . . . . . . 14  |-  ( T. 
->  ( NN  X.  {
( ( pi ^
2 )  /  6
) } )  Fn  NN )
123 fnconstg 5766 . . . . . . . . . . . . . . . 16  |-  ( 1  e.  ZZ  ->  ( NN  X.  { 1 } )  Fn  NN )
1242, 123syl 16 . . . . . . . . . . . . . . 15  |-  ( T. 
->  ( NN  X.  {
1 } )  Fn  NN )
125 ffn 5724 . . . . . . . . . . . . . . . 16  |-  ( G : NN --> RR  ->  G  Fn  NN )
12655, 125syl 16 . . . . . . . . . . . . . . 15  |-  ( T. 
->  G  Fn  NN )
127124, 126, 57, 57, 58offn 6528 . . . . . . . . . . . . . 14  |-  ( T. 
->  ( ( NN  X.  { 1 } )  oF  -  G
)  Fn  NN )
128 eqidd 2463 . . . . . . . . . . . . . 14  |-  ( ( T.  /\  k  e.  NN )  ->  (
( NN  X.  {
( ( pi ^
2 )  /  6
) } ) `  k )  =  ( ( NN  X.  {
( ( pi ^
2 )  /  6
) } ) `  k ) )
129 eqidd 2463 . . . . . . . . . . . . . 14  |-  ( ( T.  /\  k  e.  NN )  ->  (
( ( NN  X.  { 1 } )  oF  -  G
) `  k )  =  ( ( ( NN  X.  { 1 } )  oF  -  G ) `  k ) )
130122, 127, 57, 57, 58, 128, 129ofval 6526 . . . . . . . . . . . . 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 13406 . . . . . . . . . . . 12  |-  ( T. 
->  ( ( NN  X.  { ( ( pi
^ 2 )  / 
6 ) } )  oF  x.  (
( NN  X.  {
1 } )  oF  -  G ) )  ~~>  ( ( ( pi ^ 2 )  /  6 )  x.  1 ) )
132101mulid1i 9589 . . . . . . . . . . . 12  |-  ( ( ( pi ^ 2 )  /  6 )  x.  1 )  =  ( ( pi ^
2 )  /  6
)
133131, 132syl6breq 4481 . . . . . . . . . . 11  |-  ( T. 
->  ( ( NN  X.  { ( ( pi
^ 2 )  / 
6 ) } )  oF  x.  (
( NN  X.  {
1 } )  oF  -  G ) )  ~~>  ( ( pi
^ 2 )  / 
6 ) )
13461, 133syl5eqbr 4475 . . . . . . . . . 10  |-  ( T. 
->  H  ~~>  ( (
pi ^ 2 )  /  6 ) )
135 ovex 6302 . . . . . . . . . . 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 10601 . . . . . . . . . . . . 13  |-  3  e.  CC
138102, 56climconst2 13322 . . . . . . . . . . . . 13  |-  ( ( 3  e.  CC  /\  1  e.  ZZ )  ->  ( NN  X.  {
3 } )  ~~>  3 )
139137, 2, 138sylancr 663 . . . . . . . . . . . 12  |-  ( T. 
->  ( NN  X.  {
3 } )  ~~>  3 )
140 ovex 6302 . . . . . . . . . . . . 13  |-  ( ( NN  X.  { 3 } )  oF  x.  G )  e. 
_V
141140a1i 11 . . . . . . . . . . . 12  |-  ( T. 
->  ( ( NN  X.  { 3 } )  oF  x.  G
)  e.  _V )
14254basellem6 23082 . . . . . . . . . . . . 13  |-  G  ~~>  0
143142a1i 11 . . . . . . . . . . . 12  |-  ( T. 
->  G  ~~>  0 )
144 3ex 10602 . . . . . . . . . . . . . . . 16  |-  3  e.  _V
145144fconst 5764 . . . . . . . . . . . . . . 15  |-  ( NN 
X.  { 3 } ) : NN --> { 3 }
146 3re 10600 . . . . . . . . . . . . . . . . 17  |-  3  e.  RR
147146a1i 11 . . . . . . . . . . . . . . . 16  |-  ( T. 
->  3  e.  RR )
148147snssd 4167 . . . . . . . . . . . . . . 15  |-  ( T. 
->  { 3 }  C_  RR )
149 fss 5732 . . . . . . . . . . . . . . 15  |-  ( ( ( NN  X.  {
3 } ) : NN --> { 3 }  /\  { 3 } 
C_  RR )  -> 
( NN  X.  {
3 } ) : NN --> RR )
150145, 148, 149sylancr 663 . . . . . . . . . . . . . 14  |-  ( T. 
->  ( NN  X.  {
3 } ) : NN --> RR )
151150ffvelrnda 6014 . . . . . . . . . . . . 13  |-  ( ( T.  /\  k  e.  NN )  ->  (
( NN  X.  {
3 } ) `  k )  e.  RR )
152151recnd 9613 . . . . . . . . . . . 12  |-  ( ( T.  /\  k  e.  NN )  ->  (
( NN  X.  {
3 } ) `  k )  e.  CC )
15355ffvelrnda 6014 . . . . . . . . . . . . 13  |-  ( ( T.  /\  k  e.  NN )  ->  ( G `  k )  e.  RR )
154153recnd 9613 . . . . . . . . . . . 12  |-  ( ( T.  /\  k  e.  NN )  ->  ( G `  k )  e.  CC )
155 ffn 5724 . . . . . . . . . . . . . 14  |-  ( ( NN  X.  { 3 } ) : NN --> RR  ->  ( NN  X.  { 3 } )  Fn  NN )
156150, 155syl 16 . . . . . . . . . . . . 13  |-  ( T. 
->  ( NN  X.  {
3 } )  Fn  NN )
157 eqidd 2463 . . . . . . . . . . . . 13  |-  ( ( T.  /\  k  e.  NN )  ->  (
( NN  X.  {
3 } ) `  k )  =  ( ( NN  X.  {
3 } ) `  k ) )
158 eqidd 2463 . . . . . . . . . . . . 13  |-  ( ( T.  /\  k  e.  NN )  ->  ( G `  k )  =  ( G `  k ) )
159156, 126, 57, 57, 58, 157, 158ofval 6526 . . . . . . . . . . . 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 13406 . . . . . . . . . . 11  |-  ( T. 
->  ( ( NN  X.  { 3 } )  oF  x.  G
)  ~~>  ( 3  x.  0 ) )
161137mul01i 9760 . . . . . . . . . . 11  |-  ( 3  x.  0 )  =  0
162160, 161syl6breq 4481 . . . . . . . . . 10  |-  ( T. 
->  ( ( NN  X.  { 3 } )  oF  x.  G
)  ~~>  0 )
16363ffvelrnda 6014 . . . . . . . . . . 11  |-  ( ( T.  /\  k  e.  NN )  ->  ( H `  k )  e.  RR )
164163recnd 9613 . . . . . . . . . 10  |-  ( ( T.  /\  k  e.  NN )  ->  ( H `  k )  e.  CC )
16527, 150, 55, 57, 57, 58off 6531 . . . . . . . . . . . 12  |-  ( T. 
->  ( ( NN  X.  { 3 } )  oF  x.  G
) : NN --> RR )
166165ffvelrnda 6014 . . . . . . . . . . 11  |-  ( ( T.  /\  k  e.  NN )  ->  (
( ( NN  X.  { 3 } )  oF  x.  G
) `  k )  e.  RR )
167166recnd 9613 . . . . . . . . . 10  |-  ( ( T.  /\  k  e.  NN )  ->  (
( ( NN  X.  { 3 } )  oF  x.  G
) `  k )  e.  CC )
168 ffn 5724 . . . . . . . . . . . 12  |-  ( H : NN --> RR  ->  H  Fn  NN )
16963, 168syl 16 . . . . . . . . . . 11  |-  ( T. 
->  H  Fn  NN )
17041, 90, 74, 57, 57, 58off 6531 . . . . . . . . . . . 12  |-  ( T. 
->  ( ( ( NN 
X.  { 1 } )  oF  +  G )  oF  -  ( ( NN 
X.  { 1 } )  oF  +  ( ( NN  X.  { -u 2 } )  oF  x.  G
) ) ) : NN --> RR )
171 ffn 5724 . . . . . . . . . . . 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 16 . . . . . . . . . . 11  |-  ( T. 
->  ( ( ( NN 
X.  { 1 } )  oF  +  G )  oF  -  ( ( NN 
X.  { 1 } )  oF  +  ( ( NN  X.  { -u 2 } )  oF  x.  G
) ) )  Fn  NN )
173 eqidd 2463 . . . . . . . . . . 11  |-  ( ( T.  /\  k  e.  NN )  ->  ( H `  k )  =  ( H `  k ) )
174154mulid2d 9605 . . . . . . . . . . . . . . 15  |-  ( ( T.  /\  k  e.  NN )  ->  (
1  x.  ( G `
 k ) )  =  ( G `  k ) )
175 2cn 10597 . . . . . . . . . . . . . . . . . 18  |-  2  e.  CC
176 mulneg1 9984 . . . . . . . . . . . . . . . . . 18  |-  ( ( 2  e.  CC  /\  ( G `  k )  e.  CC )  -> 
( -u 2  x.  ( G `  k )
)  =  -u (
2  x.  ( G `
 k ) ) )
177175, 154, 176sylancr 663 . . . . . . . . . . . . . . . . 17  |-  ( ( T.  /\  k  e.  NN )  ->  ( -u 2  x.  ( G `
 k ) )  =  -u ( 2  x.  ( G `  k
) ) )
178177negeqd 9805 . . . . . . . . . . . . . . . 16  |-  ( ( T.  /\  k  e.  NN )  ->  -u ( -u 2  x.  ( G `
 k ) )  =  -u -u ( 2  x.  ( G `  k
) ) )
179 mulcl 9567 . . . . . . . . . . . . . . . . . 18  |-  ( ( 2  e.  CC  /\  ( G `  k )  e.  CC )  -> 
( 2  x.  ( G `  k )
)  e.  CC )
180175, 154, 179sylancr 663 . . . . . . . . . . . . . . . . 17  |-  ( ( T.  /\  k  e.  NN )  ->  (
2  x.  ( G `
 k ) )  e.  CC )
181180negnegd 9912 . . . . . . . . . . . . . . . 16  |-  ( ( T.  /\  k  e.  NN )  ->  -u -u (
2  x.  ( G `
 k ) )  =  ( 2  x.  ( G `  k
) ) )
182178, 181eqtr2d 2504 . . . . . . . . . . . . . . 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 9568 . . . . . . . . . . . . . . . . 17  |-  ( (
-u 2  e.  RR  /\  ( G `  k
)  e.  RR )  ->  ( -u 2  x.  ( G `  k
) )  e.  RR )
18568, 153, 184sylancr 663 . . . . . . . . . . . . . . . 16  |-  ( ( T.  /\  k  e.  NN )  ->  ( -u 2  x.  ( G `
 k ) )  e.  RR )
186185recnd 9613 . . . . . . . . . . . . . . 15  |-  ( ( T.  /\  k  e.  NN )  ->  ( -u 2  x.  ( G `
 k ) )  e.  CC )
187154, 186negsubd 9927 . . . . . . . . . . . . . 14  |-  ( ( T.  /\  k  e.  NN )  ->  (
( G `  k
)  +  -u ( -u 2  x.  ( G `
 k ) ) )  =  ( ( G `  k )  -  ( -u 2  x.  ( G `  k
) ) ) )
188183, 187eqtrd 2503 . . . . . . . . . . . . 13  |-  ( ( T.  /\  k  e.  NN )  ->  (
( 1  x.  ( G `  k )
)  +  ( 2  x.  ( G `  k ) ) )  =  ( ( G `
 k )  -  ( -u 2  x.  ( G `  k )
) ) )
189 df-3 10586 . . . . . . . . . . . . . . . 16  |-  3  =  ( 2  +  1 )
190 ax-1cn 9541 . . . . . . . . . . . . . . . . 17  |-  1  e.  CC
191175, 190addcomi 9761 . . . . . . . . . . . . . . . 16  |-  ( 2  +  1 )  =  ( 1  +  2 )
192189, 191eqtri 2491 . . . . . . . . . . . . . . 15  |-  3  =  ( 1  +  2 )
193192oveq1i 6287 . . . . . . . . . . . . . 14  |-  ( 3  x.  ( G `  k ) )  =  ( ( 1  +  2 )  x.  ( G `  k )
)
194 1cnd 9603 . . . . . . . . . . . . . . 15  |-  ( ( T.  /\  k  e.  NN )  ->  1  e.  CC )
195175a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( T.  /\  k  e.  NN )  ->  2  e.  CC )
196194, 195, 154adddird 9612 . . . . . . . . . . . . . 14  |-  ( ( T.  /\  k  e.  NN )  ->  (
( 1  +  2 )  x.  ( G `
 k ) )  =  ( ( 1  x.  ( G `  k ) )  +  ( 2  x.  ( G `  k )
) ) )
197193, 196syl5eq 2515 . . . . . . . . . . . . 13  |-  ( ( T.  /\  k  e.  NN )  ->  (
3  x.  ( G `
 k ) )  =  ( ( 1  x.  ( G `  k ) )  +  ( 2  x.  ( G `  k )
) ) )
198194, 154, 186pnpcand 9958 . . . . . . . . . . . . 13  |-  ( ( T.  /\  k  e.  NN )  ->  (
( 1  +  ( G `  k ) )  -  ( 1  +  ( -u 2  x.  ( G `  k
) ) ) )  =  ( ( G `
 k )  -  ( -u 2  x.  ( G `  k )
) ) )
199188, 197, 1983eqtr4rd 2514 . . . . . . . . . . . 12  |-  ( ( T.  /\  k  e.  NN )  ->  (
( 1  +  ( G `  k ) )  -  ( 1  +  ( -u 2  x.  ( G `  k
) ) ) )  =  ( 3  x.  ( G `  k
) ) )
200124, 126, 57, 57, 58offn 6528 . . . . . . . . . . . . 13  |-  ( T. 
->  ( ( NN  X.  { 1 } )  oF  +  G
)  Fn  NN )
20112a1i 11 . . . . . . . . . . . . . . . 16  |-  ( T. 
->  -u 2  e.  ZZ )
202 fnconstg 5766 . . . . . . . . . . . . . . . 16  |-  ( -u
2  e.  ZZ  ->  ( NN  X.  { -u
2 } )  Fn  NN )
203201, 202syl 16 . . . . . . . . . . . . . . 15  |-  ( T. 
->  ( NN  X.  { -u 2 } )  Fn  NN )
204203, 126, 57, 57, 58offn 6528 . . . . . . . . . . . . . 14  |-  ( T. 
->  ( ( NN  X.  { -u 2 } )  oF  x.  G
)  Fn  NN )
205124, 204, 57, 57, 58offn 6528 . . . . . . . . . . . . 13  |-  ( T. 
->  ( ( NN  X.  { 1 } )  oF  +  ( ( NN  X.  { -u 2 } )  oF  x.  G ) )  Fn  NN )
20657, 44, 126, 158ofc1 6540 . . . . . . . . . . . . 13  |-  ( ( T.  /\  k  e.  NN )  ->  (
( ( NN  X.  { 1 } )  oF  +  G
) `  k )  =  ( 1  +  ( G `  k
) ) )
20757, 69, 126, 158ofc1 6540 . . . . . . . . . . . . . 14  |-  ( ( T.  /\  k  e.  NN )  ->  (
( ( NN  X.  { -u 2 } )  oF  x.  G
) `  k )  =  ( -u 2  x.  ( G `  k
) ) )
20857, 44, 204, 207ofc1 6540 . . . . . . . . . . . . 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 6526 . . . . . . . . . . . 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 6540 . . . . . . . . . . . 12  |-  ( ( T.  /\  k  e.  NN )  ->  (
( ( NN  X.  { 3 } )  oF  x.  G
) `  k )  =  ( 3  x.  ( G `  k
) ) )
211199, 209, 2103eqtr4d 2513 . . . . . . . . . . 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 6526 . . . . . . . . . 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 13406 . . . . . . . . 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 9760 . . . . . . . . 9  |-  ( ( ( pi ^ 2 )  /  6 )  x.  0 )  =  0
215213, 214syl6breq 4481 . . . . . . . 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 4464 . . . . . . 7  |-  ( T. 
->  ( K  oF  -  J )  ~~>  0 )
217 ovex 6302 . . . . . . . 8  |-  ( F  oF  -  J
)  e.  _V
218217a1i 11 . . . . . . 7  |-  ( T. 
->  ( F  oF  -  J )  e. 
_V )
21927, 63, 90, 57, 57, 58off 6531 . . . . . . . . . 10  |-  ( T. 
->  ( H  oF  x.  ( ( NN 
X.  { 1 } )  oF  +  G ) ) : NN --> RR )
22098feq1i 5716 . . . . . . . . . 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 6531 . . . . . . . 8  |-  ( T. 
->  ( K  oF  -  J ) : NN --> RR )
223222ffvelrnda 6014 . . . . . . 7  |-  ( ( T.  /\  k  e.  NN )  ->  (
( K  oF  -  J ) `  k )  e.  RR )
22441, 23, 78, 57, 57, 58off 6531 . . . . . . . 8  |-  ( T. 
->  ( F  oF  -  J ) : NN --> RR )
225224ffvelrnda 6014 . . . . . . 7  |-  ( ( T.  /\  k  e.  NN )  ->  (
( F  oF  -  J ) `  k )  e.  RR )
22623ffvelrnda 6014 . . . . . . . . 9  |-  ( ( T.  /\  k  e.  NN )  ->  ( F `  k )  e.  RR )
227221ffvelrnda 6014 . . . . . . . . 9  |-  ( ( T.  /\  k  e.  NN )  ->  ( K `  k )  e.  RR )
22878ffvelrnda 6014 . . . . . . . . 9  |-  ( ( T.  /\  k  e.  NN )  ->  ( J `  k )  e.  RR )
229 eqid 2462 . . . . . . . . . . . 12  |-  ( ( 2  x.  k )  +  1 )  =  ( ( 2  x.  k )  +  1 )
23054, 21, 61, 76, 98, 229basellem8 23084 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  (
( J `  k
)  <_  ( F `  k )  /\  ( F `  k )  <_  ( K `  k
) ) )
231230adantl 466 . . . . . . . . . 10  |-  ( ( T.  /\  k  e.  NN )  ->  (
( J `  k
)  <_  ( F `  k )  /\  ( F `  k )  <_  ( K `  k
) ) )
232231simprd 463 . . . . . . . . 9  |-  ( ( T.  /\  k  e.  NN )  ->  ( F `  k )  <_  ( K `  k
) )
233226, 227, 228, 232lesub1dd 10159 . . . . . . . 8  |-  ( ( T.  /\  k  e.  NN )  ->  (
( F `  k
)  -  ( J `
 k ) )  <_  ( ( K `
 k )  -  ( J `  k ) ) )
234 ffn 5724 . . . . . . . . . 10  |-  ( F : NN --> RR  ->  F  Fn  NN )
23523, 234syl 16 . . . . . . . . 9  |-  ( T. 
->  F  Fn  NN )
236 ffn 5724 . . . . . . . . . 10  |-  ( J : NN --> RR  ->  J  Fn  NN )
23778, 236syl 16 . . . . . . . . 9  |-  ( T. 
->  J  Fn  NN )
238 eqidd 2463 . . . . . . . . 9  |-  ( ( T.  /\  k  e.  NN )  ->  ( F `  k )  =  ( F `  k ) )
239 eqidd 2463 . . . . . . . . 9  |-  ( ( T.  /\  k  e.  NN )  ->  ( J `  k )  =  ( J `  k ) )
240235, 237, 57, 57, 58, 238, 239ofval 6526 . . . . . . . 8  |-  ( ( T.  /\  k  e.  NN )  ->  (
( F  oF  -  J ) `  k )  =  ( ( F `  k
)  -  ( J `
 k ) ) )
241 ffn 5724 . . . . . . . . . 10  |-  ( K : NN --> RR  ->  K  Fn  NN )
242221, 241syl 16 . . . . . . . . 9  |-  ( T. 
->  K  Fn  NN )
243 eqidd 2463 . . . . . . . . 9  |-  ( ( T.  /\  k  e.  NN )  ->  ( K `  k )  =  ( K `  k ) )
244242, 237, 57, 57, 58, 243, 239ofval 6526 . . . . . . . 8  |-  ( ( T.  /\  k  e.  NN )  ->  (
( K  oF  -  J ) `  k )  =  ( ( K `  k
)  -  ( J `
 k ) ) )
245233, 240, 2443brtr4d 4472 . . . . . . 7  |-  ( ( T.  /\  k  e.  NN )  ->  (
( F  oF  -  J ) `  k )  <_  (
( K  oF  -  J ) `  k ) )
246231simpld 459 . . . . . . . . 9  |-  ( ( T.  /\  k  e.  NN )  ->  ( J `  k )  <_  ( F `  k
) )
247226, 228subge0d 10133 . . . . . . . . 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 4468 . . . . . . 7  |-  ( ( T.  /\  k  e.  NN )  ->  0  <_  ( ( F  oF  -  J ) `  k ) )
2501, 2, 216, 218, 223, 225, 245, 249climsqz2 13415 . . . . . 6  |-  ( T. 
->  ( F  oF  -  J )  ~~>  0 )
251 ovex 6302 . . . . . . 7  |-  ( ( F  oF  -  J )  oF  +  J )  e. 
_V
252251a1i 11 . . . . . 6  |-  ( T. 
->  ( ( F  oF  -  J )  oF  +  J
)  e.  _V )
253 ovex 6302 . . . . . . . . . 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 9599 . . . . . . . . . . 11  |-  -u 2  e.  CC
25654, 255basellem7 23083 . . . . . . . . . 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 6014 . . . . . . . . . 10  |-  ( ( T.  /\  k  e.  NN )  ->  (
( ( NN  X.  { 1 } )  oF  +  ( ( NN  X.  { -u 2 } )  oF  x.  G ) ) `  k )  e.  RR )
259258recnd 9613 . . . . . . . . 9  |-  ( ( T.  /\  k  e.  NN )  ->  (
( ( NN  X.  { 1 } )  oF  +  ( ( NN  X.  { -u 2 } )  oF  x.  G ) ) `  k )  e.  CC )
260 eqidd 2463 . . . . . . . . . 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 6526 . . . . . . . . 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 13406 . . . . . . . 8  |-  ( T. 
->  ( H  oF  x.  ( ( NN 
X.  { 1 } )  oF  +  ( ( NN  X.  { -u 2 } )  oF  x.  G
) ) )  ~~>  ( ( ( pi ^ 2 )  /  6 )  x.  1 ) )
263262, 132syl6breq 4481 . . . . . . 7  |-  ( T. 
->  ( H  oF  x.  ( ( NN 
X.  { 1 } )  oF  +  ( ( NN  X.  { -u 2 } )  oF  x.  G
) ) )  ~~>  ( ( pi ^ 2 )  /  6 ) )
26476, 263syl5eqbr 4475 . . . . . 6  |-  ( T. 
->  J  ~~>  ( (
pi ^ 2 )  /  6 ) )
265225recnd 9613 . . . . . 6  |-  ( ( T.  /\  k  e.  NN )  ->  (
( F  oF  -  J ) `  k )  e.  CC )
266228recnd 9613 . . . . . 6  |-  ( ( T.  /\  k  e.  NN )  ->  ( J `  k )  e.  CC )
267 ffn 5724 . . . . . . . 8  |-  ( ( F  oF  -  J ) : NN --> RR  ->  ( F  oF  -  J )  Fn  NN )
268224, 267syl 16 . . . . . . 7  |-  ( T. 
->  ( F  oF  -  J )  Fn  NN )
269 eqidd 2463 . . . . . . 7  |-  ( ( T.  /\  k  e.  NN )  ->  (
( F  oF  -  J ) `  k )  =  ( ( F  oF  -  J ) `  k ) )
270268, 237, 57, 57, 58, 269, 239ofval 6526 . . . . . 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 13405 . . . . 5  |-  ( T. 
->  ( ( F  oF  -  J )  oF  +  J
)  ~~>  ( 0  +  ( ( pi ^
2 )  /  6
) ) )
27289, 271eqbrtrrd 4464 . . . 4  |-  ( T. 
->  F  ~~>  ( 0  +  ( ( pi
^ 2 )  / 
6 ) ) )
273101addid2i 9758 . . . 4  |-  ( 0  +  ( ( pi
^ 2 )  / 
6 ) )  =  ( ( pi ^
2 )  /  6
)
274272, 21, 2733brtr3g 4473 . . 3  |-  ( T. 
->  seq 1 (  +  ,  ( n  e.  NN  |->  ( n ^ -u 2 ) ) )  ~~>  ( ( pi ^
2 )  /  6
) )
2751, 2, 7, 19, 274isumclim 13523 . 2  |-  ( T. 
->  sum_ k  e.  NN  ( k ^ -u 2
)  =  ( ( pi ^ 2 )  /  6 ) )
276275trud 1383 1  |-  sum_ k  e.  NN  ( k ^ -u 2 )  =  ( ( pi ^ 2 )  /  6 )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    /\ w3a 968    = wceq 1374   T. wtru 1375    e. wcel 1762   _Vcvv 3108    C_ wss 3471   {csn 4022   class class class wbr 4442    |-> cmpt 4500    X. cxp 4992    Fn wfn 5576   -->wf 5577   ` cfv 5581  (class class class)co 6277    oFcof 6515   CCcc 9481   RRcr 9482   0cc0 9483   1c1 9484    + caddc 9486    x. cmul 9488    <_ cle 9620    - cmin 9796   -ucneg 9797    / cdiv 10197   NNcn 10527   2c2 10576   3c3 10577   6c6 10580   ZZcz 10855   ZZ>=cuz 11073    seqcseq 12065   ^cexp 12124    ~~> cli 13258   sum_csu 13459   picpi 13655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561  ax-addf 9562  ax-mulf 9563
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517  df-om 6674  df-1st 6776  df-2nd 6777  df-supp 6894  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7462  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-fsupp 7821  df-fi 7862  df-sup 7892  df-oi 7926  df-card 8311  df-cda 8539  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-uz 11074  df-q 11174  df-rp 11212  df-xneg 11309  df-xadd 11310  df-xmul 11311  df-ioo 11524  df-ioc 11525  df-ico 11526  df-icc 11527  df-fz 11664  df-fzo 11784  df-fl 11888  df-mod 11955  df-seq 12066  df-exp 12125  df-fac 12311  df-bc 12338  df-hash 12363  df-shft 12852  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-limsup 13245  df-clim 13262  df-rlim 13263  df-sum 13460  df-ef 13656  df-sin 13658  df-cos 13659  df-tan 13660  df-pi 13661  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-mulr 14560  df-starv 14561  df-sca 14562  df-vsca 14563  df-ip 14564  df-tset 14565  df-ple 14566  df-ds 14568  df-unif 14569  df-hom 14570  df-cco 14571  df-rest 14669  df-topn 14670  df-0g 14688  df-gsum 14689  df-topgen 14690  df-pt 14691  df-prds 14694  df-xrs 14748  df-qtop 14753  df-imas 14754  df-xps 14756  df-mre 14832  df-mrc 14833  df-acs 14835  df-mnd 15723  df-submnd 15773  df-mulg 15856  df-cntz 16145  df-cmn 16591  df-psmet 18177  df-xmet 18178  df-met 18179  df-bl 18180  df-mopn 18181  df-fbas 18182  df-fg 18183  df-cnfld 18187  df-top 19161  df-bases 19163  df-topon 19164  df-topsp 19165  df-cld 19281  df-ntr 19282  df-cls 19283  df-nei 19360  df-lp 19398  df-perf 19399  df-cn 19489  df-cnp 19490  df-haus 19577  df-tx 19793  df-hmeo 19986  df-fil 20077  df-fm 20169  df-flim 20170  df-flf 20171  df-xms 20553  df-ms 20554  df-tms 20555  df-cncf 21112  df-0p 21807  df-limc 22000  df-dv 22001  df-ply 22315  df-idp 22316  df-coe 22317  df-dgr 22318  df-quot 22416
This theorem is referenced by:  basel  23086
  Copyright terms: Public domain W3C validator