MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  leibpilem2 Structured version   Unicode version

Theorem leibpilem2 20773
Description: The Leibniz formula for  pi. (Contributed by Mario Carneiro, 7-Apr-2015.)
Hypotheses
Ref Expression
leibpi.1  |-  F  =  ( n  e.  NN0  |->  ( ( -u 1 ^ n )  / 
( ( 2  x.  n )  +  1 ) ) )
leibpilem2.2  |-  G  =  ( k  e.  NN0  |->  if ( ( k  =  0  \/  2  ||  k ) ,  0 ,  ( ( -u
1 ^ ( ( k  -  1 )  /  2 ) )  /  k ) ) )
leibpilem2.3  |-  A  e. 
_V
Assertion
Ref Expression
leibpilem2  |-  (  seq  0 (  +  ,  F )  ~~>  A  <->  seq  0
(  +  ,  G
)  ~~>  A )
Distinct variable groups:    k, n    n, G
Allowed substitution hints:    A( k, n)    F( k, n)    G( k)

Proof of Theorem leibpilem2
StepHypRef Expression
1 leibpi.1 . . . . 5  |-  F  =  ( n  e.  NN0  |->  ( ( -u 1 ^ n )  / 
( ( 2  x.  n )  +  1 ) ) )
2 2cn 10062 . . . . . . . . . . . 12  |-  2  e.  CC
3 nn0cn 10223 . . . . . . . . . . . 12  |-  ( n  e.  NN0  ->  n  e.  CC )
4 mulcl 9066 . . . . . . . . . . . 12  |-  ( ( 2  e.  CC  /\  n  e.  CC )  ->  ( 2  x.  n
)  e.  CC )
52, 3, 4sylancr 645 . . . . . . . . . . 11  |-  ( n  e.  NN0  ->  ( 2  x.  n )  e.  CC )
6 ax-1cn 9040 . . . . . . . . . . 11  |-  1  e.  CC
7 pncan 9303 . . . . . . . . . . 11  |-  ( ( ( 2  x.  n
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( 2  x.  n )  +  1 )  -  1 )  =  ( 2  x.  n ) )
85, 6, 7sylancl 644 . . . . . . . . . 10  |-  ( n  e.  NN0  ->  ( ( ( 2  x.  n
)  +  1 )  -  1 )  =  ( 2  x.  n
) )
98oveq1d 6088 . . . . . . . . 9  |-  ( n  e.  NN0  ->  ( ( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 )  =  ( ( 2  x.  n )  /  2
) )
10 2ne0 10075 . . . . . . . . . . 11  |-  2  =/=  0
11 divcan3 9694 . . . . . . . . . . 11  |-  ( ( n  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
( 2  x.  n
)  /  2 )  =  n )
122, 10, 11mp3an23 1271 . . . . . . . . . 10  |-  ( n  e.  CC  ->  (
( 2  x.  n
)  /  2 )  =  n )
133, 12syl 16 . . . . . . . . 9  |-  ( n  e.  NN0  ->  ( ( 2  x.  n )  /  2 )  =  n )
149, 13eqtrd 2467 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( ( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 )  =  n )
1514oveq2d 6089 . . . . . . 7  |-  ( n  e.  NN0  ->  ( -u
1 ^ ( ( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 ) )  =  ( -u 1 ^ n ) )
1615oveq1d 6088 . . . . . 6  |-  ( n  e.  NN0  ->  ( (
-u 1 ^ (
( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 ) )  /  ( ( 2  x.  n )  +  1 ) )  =  ( ( -u
1 ^ n )  /  ( ( 2  x.  n )  +  1 ) ) )
1716mpteq2ia 4283 . . . . 5  |-  ( n  e.  NN0  |->  ( (
-u 1 ^ (
( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 ) )  /  ( ( 2  x.  n )  +  1 ) ) )  =  ( n  e.  NN0  |->  ( (
-u 1 ^ n
)  /  ( ( 2  x.  n )  +  1 ) ) )
181, 17eqtr4i 2458 . . . 4  |-  F  =  ( n  e.  NN0  |->  ( ( -u 1 ^ ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 ) )  / 
( ( 2  x.  n )  +  1 ) ) )
19 seqeq3 11320 . . . 4  |-  ( F  =  ( n  e. 
NN0  |->  ( ( -u
1 ^ ( ( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 ) )  /  ( ( 2  x.  n )  +  1 ) ) )  ->  seq  0 (  +  ,  F )  =  seq  0 (  +  ,  ( n  e.  NN0  |->  ( (
-u 1 ^ (
( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 ) )  /  ( ( 2  x.  n )  +  1 ) ) ) ) )
2018, 19ax-mp 8 . . 3  |-  seq  0
(  +  ,  F
)  =  seq  0
(  +  ,  ( n  e.  NN0  |->  ( (
-u 1 ^ (
( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 ) )  /  ( ( 2  x.  n )  +  1 ) ) ) )
2120breq1i 4211 . 2  |-  (  seq  0 (  +  ,  F )  ~~>  A  <->  seq  0
(  +  ,  ( n  e.  NN0  |->  ( (
-u 1 ^ (
( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 ) )  /  ( ( 2  x.  n )  +  1 ) ) ) )  ~~>  A )
22 1re 9082 . . . . . . . . . 10  |-  1  e.  RR
2322renegcli 9354 . . . . . . . . 9  |-  -u 1  e.  RR
24 reexpcl 11390 . . . . . . . . 9  |-  ( (
-u 1  e.  RR  /\  n  e.  NN0 )  ->  ( -u 1 ^ n )  e.  RR )
2523, 24mpan 652 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( -u
1 ^ n )  e.  RR )
26 2nn0 10230 . . . . . . . . . 10  |-  2  e.  NN0
27 nn0mulcl 10248 . . . . . . . . . 10  |-  ( ( 2  e.  NN0  /\  n  e.  NN0 )  -> 
( 2  x.  n
)  e.  NN0 )
2826, 27mpan 652 . . . . . . . . 9  |-  ( n  e.  NN0  ->  ( 2  x.  n )  e. 
NN0 )
29 nn0p1nn 10251 . . . . . . . . 9  |-  ( ( 2  x.  n )  e.  NN0  ->  ( ( 2  x.  n )  +  1 )  e.  NN )
3028, 29syl 16 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( ( 2  x.  n )  +  1 )  e.  NN )
3125, 30nndivred 10040 . . . . . . 7  |-  ( n  e.  NN0  ->  ( (
-u 1 ^ n
)  /  ( ( 2  x.  n )  +  1 ) )  e.  RR )
3231recnd 9106 . . . . . 6  |-  ( n  e.  NN0  ->  ( (
-u 1 ^ n
)  /  ( ( 2  x.  n )  +  1 ) )  e.  CC )
3316, 32eqeltrd 2509 . . . . 5  |-  ( n  e.  NN0  ->  ( (
-u 1 ^ (
( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 ) )  /  ( ( 2  x.  n )  +  1 ) )  e.  CC )
3433adantl 453 . . . 4  |-  ( (  T.  /\  n  e. 
NN0 )  ->  (
( -u 1 ^ (
( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 ) )  /  ( ( 2  x.  n )  +  1 ) )  e.  CC )
35 oveq1 6080 . . . . . . 7  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  (
k  -  1 )  =  ( ( ( 2  x.  n )  +  1 )  - 
1 ) )
3635oveq1d 6088 . . . . . 6  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  (
( k  -  1 )  /  2 )  =  ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 ) )
3736oveq2d 6089 . . . . 5  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  ( -u 1 ^ ( ( k  -  1 )  /  2 ) )  =  ( -u 1 ^ ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 ) ) )
38 id 20 . . . . 5  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  k  =  ( ( 2  x.  n )  +  1 ) )
3937, 38oveq12d 6091 . . . 4  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  (
( -u 1 ^ (
( k  -  1 )  /  2 ) )  /  k )  =  ( ( -u
1 ^ ( ( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 ) )  /  ( ( 2  x.  n )  +  1 ) ) )
4034, 39iserodd 13201 . . 3  |-  (  T. 
->  (  seq  0
(  +  ,  ( n  e.  NN0  |->  ( (
-u 1 ^ (
( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 ) )  /  ( ( 2  x.  n )  +  1 ) ) ) )  ~~>  A  <->  seq  1
(  +  ,  ( k  e.  NN  |->  if ( 2  ||  k ,  0 ,  ( ( -u 1 ^ ( ( k  - 
1 )  /  2
) )  /  k
) ) ) )  ~~>  A ) )
4140trud 1332 . 2  |-  (  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( -u 1 ^ ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 ) )  / 
( ( 2  x.  n )  +  1 ) ) ) )  ~~>  A  <->  seq  1 (  +  ,  ( k  e.  NN  |->  if ( 2 
||  k ,  0 ,  ( ( -u
1 ^ ( ( k  -  1 )  /  2 ) )  /  k ) ) ) )  ~~>  A )
42 addid2 9241 . . . . . . . 8  |-  ( n  e.  CC  ->  (
0  +  n )  =  n )
4342adantl 453 . . . . . . 7  |-  ( (  T.  /\  n  e.  CC )  ->  (
0  +  n )  =  n )
44 0cn 9076 . . . . . . . 8  |-  0  e.  CC
4544a1i 11 . . . . . . 7  |-  (  T. 
->  0  e.  CC )
46 1nn0 10229 . . . . . . . . 9  |-  1  e.  NN0
47 nn0uz 10512 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
4846, 47eleqtri 2507 . . . . . . . 8  |-  1  e.  ( ZZ>= `  0 )
4948a1i 11 . . . . . . 7  |-  (  T. 
->  1  e.  ( ZZ>=
`  0 ) )
50 leibpilem2.2 . . . . . . . . . 10  |-  G  =  ( k  e.  NN0  |->  if ( ( k  =  0  \/  2  ||  k ) ,  0 ,  ( ( -u
1 ^ ( ( k  -  1 )  /  2 ) )  /  k ) ) )
5144a1i 11 . . . . . . . . . . 11  |-  ( ( k  e.  NN0  /\  ( k  =  0  \/  2  ||  k
) )  ->  0  e.  CC )
52 ioran 477 . . . . . . . . . . . 12  |-  ( -.  ( k  =  0  \/  2  ||  k
)  <->  ( -.  k  =  0  /\  -.  2  ||  k ) )
53 leibpilem1 20772 . . . . . . . . . . . . . . . 16  |-  ( ( k  e.  NN0  /\  ( -.  k  = 
0  /\  -.  2  ||  k ) )  -> 
( k  e.  NN  /\  ( ( k  - 
1 )  /  2
)  e.  NN0 )
)
5453simprd 450 . . . . . . . . . . . . . . 15  |-  ( ( k  e.  NN0  /\  ( -.  k  = 
0  /\  -.  2  ||  k ) )  -> 
( ( k  - 
1 )  /  2
)  e.  NN0 )
55 reexpcl 11390 . . . . . . . . . . . . . . 15  |-  ( (
-u 1  e.  RR  /\  ( ( k  - 
1 )  /  2
)  e.  NN0 )  ->  ( -u 1 ^ ( ( k  - 
1 )  /  2
) )  e.  RR )
5623, 54, 55sylancr 645 . . . . . . . . . . . . . 14  |-  ( ( k  e.  NN0  /\  ( -.  k  = 
0  /\  -.  2  ||  k ) )  -> 
( -u 1 ^ (
( k  -  1 )  /  2 ) )  e.  RR )
5753simpld 446 . . . . . . . . . . . . . 14  |-  ( ( k  e.  NN0  /\  ( -.  k  = 
0  /\  -.  2  ||  k ) )  -> 
k  e.  NN )
5856, 57nndivred 10040 . . . . . . . . . . . . 13  |-  ( ( k  e.  NN0  /\  ( -.  k  = 
0  /\  -.  2  ||  k ) )  -> 
( ( -u 1 ^ ( ( k  -  1 )  / 
2 ) )  / 
k )  e.  RR )
5958recnd 9106 . . . . . . . . . . . 12  |-  ( ( k  e.  NN0  /\  ( -.  k  = 
0  /\  -.  2  ||  k ) )  -> 
( ( -u 1 ^ ( ( k  -  1 )  / 
2 ) )  / 
k )  e.  CC )
6052, 59sylan2b 462 . . . . . . . . . . 11  |-  ( ( k  e.  NN0  /\  -.  ( k  =  0  \/  2  ||  k
) )  ->  (
( -u 1 ^ (
( k  -  1 )  /  2 ) )  /  k )  e.  CC )
6151, 60ifclda 3758 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  if ( ( k  =  0  \/  2  ||  k
) ,  0 ,  ( ( -u 1 ^ ( ( k  -  1 )  / 
2 ) )  / 
k ) )  e.  CC )
6250, 61fmpti 5884 . . . . . . . . 9  |-  G : NN0
--> CC
6362ffvelrni 5861 . . . . . . . 8  |-  ( 1  e.  NN0  ->  ( G `
 1 )  e.  CC )
6446, 63mp1i 12 . . . . . . 7  |-  (  T. 
->  ( G `  1
)  e.  CC )
65 simpr 448 . . . . . . . . . . 11  |-  ( (  T.  /\  n  e.  ( 0 ... (
1  -  1 ) ) )  ->  n  e.  ( 0 ... (
1  -  1 ) ) )
66 1m1e0 10060 . . . . . . . . . . . 12  |-  ( 1  -  1 )  =  0
6766oveq2i 6084 . . . . . . . . . . 11  |-  ( 0 ... ( 1  -  1 ) )  =  ( 0 ... 0
)
6865, 67syl6eleq 2525 . . . . . . . . . 10  |-  ( (  T.  /\  n  e.  ( 0 ... (
1  -  1 ) ) )  ->  n  e.  ( 0 ... 0
) )
69 elfz1eq 11060 . . . . . . . . . 10  |-  ( n  e.  ( 0 ... 0 )  ->  n  =  0 )
7068, 69syl 16 . . . . . . . . 9  |-  ( (  T.  /\  n  e.  ( 0 ... (
1  -  1 ) ) )  ->  n  =  0 )
7170fveq2d 5724 . . . . . . . 8  |-  ( (  T.  /\  n  e.  ( 0 ... (
1  -  1 ) ) )  ->  ( G `  n )  =  ( G ` 
0 ) )
72 0nn0 10228 . . . . . . . . 9  |-  0  e.  NN0
73 iftrue 3737 . . . . . . . . . . 11  |-  ( ( k  =  0  \/  2  ||  k )  ->  if ( ( k  =  0  \/  2  ||  k ) ,  0 ,  ( ( -u 1 ^ ( ( k  - 
1 )  /  2
) )  /  k
) )  =  0 )
7473orcs 384 . . . . . . . . . 10  |-  ( k  =  0  ->  if ( ( k  =  0  \/  2  ||  k ) ,  0 ,  ( ( -u
1 ^ ( ( k  -  1 )  /  2 ) )  /  k ) )  =  0 )
75 c0ex 9077 . . . . . . . . . 10  |-  0  e.  _V
7674, 50, 75fvmpt 5798 . . . . . . . . 9  |-  ( 0  e.  NN0  ->  ( G `
 0 )  =  0 )
7772, 76ax-mp 8 . . . . . . . 8  |-  ( G `
 0 )  =  0
7871, 77syl6eq 2483 . . . . . . 7  |-  ( (  T.  /\  n  e.  ( 0 ... (
1  -  1 ) ) )  ->  ( G `  n )  =  0 )
7943, 45, 49, 64, 78seqid 11360 . . . . . 6  |-  (  T. 
->  (  seq  0
(  +  ,  G
)  |`  ( ZZ>= `  1
) )  =  seq  1 (  +  ,  G ) )
80 1z 10303 . . . . . . . 8  |-  1  e.  ZZ
8180a1i 11 . . . . . . 7  |-  (  T. 
->  1  e.  ZZ )
82 simpr 448 . . . . . . . . 9  |-  ( (  T.  /\  n  e.  ( ZZ>= `  1 )
)  ->  n  e.  ( ZZ>= `  1 )
)
83 nnuz 10513 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
8482, 83syl6eleqr 2526 . . . . . . . 8  |-  ( (  T.  /\  n  e.  ( ZZ>= `  1 )
)  ->  n  e.  NN )
85 nnne0 10024 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  n  =/=  0 )
8685neneqd 2614 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  -.  n  =  0 )
87 biorf 395 . . . . . . . . . . 11  |-  ( -.  n  =  0  -> 
( 2  ||  n  <->  ( n  =  0  \/  2  ||  n ) ) )
8886, 87syl 16 . . . . . . . . . 10  |-  ( n  e.  NN  ->  (
2  ||  n  <->  ( n  =  0  \/  2 
||  n ) ) )
8988ifbid 3749 . . . . . . . . 9  |-  ( n  e.  NN  ->  if ( 2  ||  n ,  0 ,  ( ( -u 1 ^ ( ( n  - 
1 )  /  2
) )  /  n
) )  =  if ( ( n  =  0  \/  2  ||  n ) ,  0 ,  ( ( -u
1 ^ ( ( n  -  1 )  /  2 ) )  /  n ) ) )
90 breq2 4208 . . . . . . . . . . 11  |-  ( k  =  n  ->  (
2  ||  k  <->  2  ||  n ) )
91 oveq1 6080 . . . . . . . . . . . . . 14  |-  ( k  =  n  ->  (
k  -  1 )  =  ( n  - 
1 ) )
9291oveq1d 6088 . . . . . . . . . . . . 13  |-  ( k  =  n  ->  (
( k  -  1 )  /  2 )  =  ( ( n  -  1 )  / 
2 ) )
9392oveq2d 6089 . . . . . . . . . . . 12  |-  ( k  =  n  ->  ( -u 1 ^ ( ( k  -  1 )  /  2 ) )  =  ( -u 1 ^ ( ( n  -  1 )  / 
2 ) ) )
94 id 20 . . . . . . . . . . . 12  |-  ( k  =  n  ->  k  =  n )
9593, 94oveq12d 6091 . . . . . . . . . . 11  |-  ( k  =  n  ->  (
( -u 1 ^ (
( k  -  1 )  /  2 ) )  /  k )  =  ( ( -u
1 ^ ( ( n  -  1 )  /  2 ) )  /  n ) )
9690, 95ifbieq2d 3751 . . . . . . . . . 10  |-  ( k  =  n  ->  if ( 2  ||  k ,  0 ,  ( ( -u 1 ^ ( ( k  - 
1 )  /  2
) )  /  k
) )  =  if ( 2  ||  n ,  0 ,  ( ( -u 1 ^ ( ( n  - 
1 )  /  2
) )  /  n
) ) )
97 eqid 2435 . . . . . . . . . 10  |-  ( k  e.  NN  |->  if ( 2  ||  k ,  0 ,  ( (
-u 1 ^ (
( k  -  1 )  /  2 ) )  /  k ) ) )  =  ( k  e.  NN  |->  if ( 2  ||  k ,  0 ,  ( ( -u 1 ^ ( ( k  - 
1 )  /  2
) )  /  k
) ) )
98 ovex 6098 . . . . . . . . . . 11  |-  ( (
-u 1 ^ (
( n  -  1 )  /  2 ) )  /  n )  e.  _V
9975, 98ifex 3789 . . . . . . . . . 10  |-  if ( 2  ||  n ,  0 ,  ( (
-u 1 ^ (
( n  -  1 )  /  2 ) )  /  n ) )  e.  _V
10096, 97, 99fvmpt 5798 . . . . . . . . 9  |-  ( n  e.  NN  ->  (
( k  e.  NN  |->  if ( 2  ||  k ,  0 ,  ( ( -u 1 ^ ( ( k  - 
1 )  /  2
) )  /  k
) ) ) `  n )  =  if ( 2  ||  n ,  0 ,  ( ( -u 1 ^ ( ( n  - 
1 )  /  2
) )  /  n
) ) )
101 nnnn0 10220 . . . . . . . . . 10  |-  ( n  e.  NN  ->  n  e.  NN0 )
102 eqeq1 2441 . . . . . . . . . . . . 13  |-  ( k  =  n  ->  (
k  =  0  <->  n  =  0 ) )
103102, 90orbi12d 691 . . . . . . . . . . . 12  |-  ( k  =  n  ->  (
( k  =  0  \/  2  ||  k
)  <->  ( n  =  0  \/  2  ||  n ) ) )
104103, 95ifbieq2d 3751 . . . . . . . . . . 11  |-  ( k  =  n  ->  if ( ( k  =  0  \/  2  ||  k ) ,  0 ,  ( ( -u
1 ^ ( ( k  -  1 )  /  2 ) )  /  k ) )  =  if ( ( n  =  0  \/  2  ||  n ) ,  0 ,  ( ( -u 1 ^ ( ( n  - 
1 )  /  2
) )  /  n
) ) )
10575, 98ifex 3789 . . . . . . . . . . 11  |-  if ( ( n  =  0  \/  2  ||  n
) ,  0 ,  ( ( -u 1 ^ ( ( n  -  1 )  / 
2 ) )  /  n ) )  e. 
_V
106104, 50, 105fvmpt 5798 . . . . . . . . . 10  |-  ( n  e.  NN0  ->  ( G `
 n )  =  if ( ( n  =  0  \/  2 
||  n ) ,  0 ,  ( (
-u 1 ^ (
( n  -  1 )  /  2 ) )  /  n ) ) )
107101, 106syl 16 . . . . . . . . 9  |-  ( n  e.  NN  ->  ( G `  n )  =  if ( ( n  =  0  \/  2 
||  n ) ,  0 ,  ( (
-u 1 ^ (
( n  -  1 )  /  2 ) )  /  n ) ) )
10889, 100, 1073eqtr4d 2477 . . . . . . . 8  |-  ( n  e.  NN  ->  (
( k  e.  NN  |->  if ( 2  ||  k ,  0 ,  ( ( -u 1 ^ ( ( k  - 
1 )  /  2
) )  /  k
) ) ) `  n )  =  ( G `  n ) )
10984, 108syl 16 . . . . . . 7  |-  ( (  T.  /\  n  e.  ( ZZ>= `  1 )
)  ->  ( (
k  e.  NN  |->  if ( 2  ||  k ,  0 ,  ( ( -u 1 ^ ( ( k  - 
1 )  /  2
) )  /  k
) ) ) `  n )  =  ( G `  n ) )
11081, 109seqfeq 11340 . . . . . 6  |-  (  T. 
->  seq  1 (  +  ,  ( k  e.  NN  |->  if ( 2 
||  k ,  0 ,  ( ( -u
1 ^ ( ( k  -  1 )  /  2 ) )  /  k ) ) ) )  =  seq  1 (  +  ,  G ) )
11179, 110eqtr4d 2470 . . . . 5  |-  (  T. 
->  (  seq  0
(  +  ,  G
)  |`  ( ZZ>= `  1
) )  =  seq  1 (  +  , 
( k  e.  NN  |->  if ( 2  ||  k ,  0 ,  ( ( -u 1 ^ ( ( k  - 
1 )  /  2
) )  /  k
) ) ) ) )
112111trud 1332 . . . 4  |-  (  seq  0 (  +  ,  G )  |`  ( ZZ>=
`  1 ) )  =  seq  1 (  +  ,  ( k  e.  NN  |->  if ( 2  ||  k ,  0 ,  ( (
-u 1 ^ (
( k  -  1 )  /  2 ) )  /  k ) ) ) )
113112breq1i 4211 . . 3  |-  ( (  seq  0 (  +  ,  G )  |`  ( ZZ>= `  1 )
)  ~~>  A  <->  seq  1
(  +  ,  ( k  e.  NN  |->  if ( 2  ||  k ,  0 ,  ( ( -u 1 ^ ( ( k  - 
1 )  /  2
) )  /  k
) ) ) )  ~~>  A )
114 seqex 11317 . . . 4  |-  seq  0
(  +  ,  G
)  e.  _V
115 climres 12361 . . . 4  |-  ( ( 1  e.  ZZ  /\  seq  0 (  +  ,  G )  e.  _V )  ->  ( (  seq  0 (  +  ,  G )  |`  ( ZZ>=
`  1 ) )  ~~>  A  <->  seq  0 (  +  ,  G )  ~~>  A ) )
11680, 114, 115mp2an 654 . . 3  |-  ( (  seq  0 (  +  ,  G )  |`  ( ZZ>= `  1 )
)  ~~>  A  <->  seq  0
(  +  ,  G
)  ~~>  A )
117113, 116bitr3i 243 . 2  |-  (  seq  1 (  +  , 
( k  e.  NN  |->  if ( 2  ||  k ,  0 ,  ( ( -u 1 ^ ( ( k  - 
1 )  /  2
) )  /  k
) ) ) )  ~~>  A  <->  seq  0 (  +  ,  G )  ~~>  A )
11821, 41, 1173bitri 263 1  |-  (  seq  0 (  +  ,  F )  ~~>  A  <->  seq  0
(  +  ,  G
)  ~~>  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    \/ wo 358    /\ wa 359    T. wtru 1325    = wceq 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948   ifcif 3731   class class class wbr 4204    e. cmpt 4258    |` cres 4872   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    - cmin 9283   -ucneg 9284    / cdiv 9669   NNcn 9992   2c2 10041   NN0cn0 10213   ZZcz 10274   ZZ>=cuz 10480   ...cfz 11035    seq cseq 11315   ^cexp 11374    ~~> cli 12270    || cdivides 12844
This theorem is referenced by:  leibpi  20774
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-seq 11316  df-exp 11375  df-hash 11611  df-shft 11874  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-dvds 12845
  Copyright terms: Public domain W3C validator