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Theorem leibpilem2 20734
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 10026 . . . . . . . . . . . 12  |-  2  e.  CC
3 nn0cn 10187 . . . . . . . . . . . 12  |-  ( n  e.  NN0  ->  n  e.  CC )
4 mulcl 9030 . . . . . . . . . . . 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 9004 . . . . . . . . . . 11  |-  1  e.  CC
7 pncan 9267 . . . . . . . . . . 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 6055 . . . . . . . . 9  |-  ( n  e.  NN0  ->  ( ( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 )  =  ( ( 2  x.  n )  /  2
) )
10 2ne0 10039 . . . . . . . . . . 11  |-  2  =/=  0
11 divcan3 9658 . . . . . . . . . . 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 2436 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( ( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 )  =  n )
1514oveq2d 6056 . . . . . . 7  |-  ( n  e.  NN0  ->  ( -u
1 ^ ( ( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 ) )  =  ( -u 1 ^ n ) )
1615oveq1d 6055 . . . . . 6  |-  ( n  e.  NN0  ->  ( (
-u 1 ^ (
( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 ) )  /  ( ( 2  x.  n )  +  1 ) )  =  ( ( -u
1 ^ n )  /  ( ( 2  x.  n )  +  1 ) ) )
1716mpteq2ia 4251 . . . . 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 2427 . . . 4  |-  F  =  ( n  e.  NN0  |->  ( ( -u 1 ^ ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 ) )  / 
( ( 2  x.  n )  +  1 ) ) )
19 seqeq3 11283 . . . 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 4179 . 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 9046 . . . . . . . . . 10  |-  1  e.  RR
2322renegcli 9318 . . . . . . . . 9  |-  -u 1  e.  RR
24 reexpcl 11353 . . . . . . . . 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 10194 . . . . . . . . . 10  |-  2  e.  NN0
27 nn0mulcl 10212 . . . . . . . . . 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 10215 . . . . . . . . 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 10004 . . . . . . 7  |-  ( n  e.  NN0  ->  ( (
-u 1 ^ n
)  /  ( ( 2  x.  n )  +  1 ) )  e.  RR )
3231recnd 9070 . . . . . 6  |-  ( n  e.  NN0  ->  ( (
-u 1 ^ n
)  /  ( ( 2  x.  n )  +  1 ) )  e.  CC )
3316, 32eqeltrd 2478 . . . . 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 6047 . . . . . . 7  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  (
k  -  1 )  =  ( ( ( 2  x.  n )  +  1 )  - 
1 ) )
3635oveq1d 6055 . . . . . 6  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  (
( k  -  1 )  /  2 )  =  ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 ) )
3736oveq2d 6056 . . . . 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 6058 . . . 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 13164 . . 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 1329 . 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 9205 . . . . . . . 8  |-  ( n  e.  CC  ->  (
0  +  n )  =  n )
4342adantl 453 . . . . . . 7  |-  ( (  T.  /\  n  e.  CC )  ->  (
0  +  n )  =  n )
44 0cn 9040 . . . . . . . 8  |-  0  e.  CC
4544a1i 11 . . . . . . 7  |-  (  T. 
->  0  e.  CC )
46 1nn0 10193 . . . . . . . . 9  |-  1  e.  NN0
47 nn0uz 10476 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
4846, 47eleqtri 2476 . . . . . . . 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 20733 . . . . . . . . . . . . . . . 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 11353 . . . . . . . . . . . . . . 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 10004 . . . . . . . . . . . . 13  |-  ( ( k  e.  NN0  /\  ( -.  k  = 
0  /\  -.  2  ||  k ) )  -> 
( ( -u 1 ^ ( ( k  -  1 )  / 
2 ) )  / 
k )  e.  RR )
5958recnd 9070 . . . . . . . . . . . 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 3726 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  if ( ( k  =  0  \/  2  ||  k
) ,  0 ,  ( ( -u 1 ^ ( ( k  -  1 )  / 
2 ) )  / 
k ) )  e.  CC )
6250, 61fmpti 5851 . . . . . . . . 9  |-  G : NN0
--> CC
6362ffvelrni 5828 . . . . . . . 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 10024 . . . . . . . . . . . 12  |-  ( 1  -  1 )  =  0
6766oveq2i 6051 . . . . . . . . . . 11  |-  ( 0 ... ( 1  -  1 ) )  =  ( 0 ... 0
)
6865, 67syl6eleq 2494 . . . . . . . . . 10  |-  ( (  T.  /\  n  e.  ( 0 ... (
1  -  1 ) ) )  ->  n  e.  ( 0 ... 0
) )
69 elfz1eq 11024 . . . . . . . . . 10  |-  ( n  e.  ( 0 ... 0 )  ->  n  =  0 )
7068, 69syl 16 . . . . . . . . 9  |-  ( (  T.  /\  n  e.  ( 0 ... (
1  -  1 ) ) )  ->  n  =  0 )
7170fveq2d 5691 . . . . . . . 8  |-  ( (  T.  /\  n  e.  ( 0 ... (
1  -  1 ) ) )  ->  ( G `  n )  =  ( G ` 
0 ) )
72 0nn0 10192 . . . . . . . . 9  |-  0  e.  NN0
73 iftrue 3705 . . . . . . . . . . 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 9041 . . . . . . . . . 10  |-  0  e.  _V
7674, 50, 75fvmpt 5765 . . . . . . . . 9  |-  ( 0  e.  NN0  ->  ( G `
 0 )  =  0 )
7772, 76ax-mp 8 . . . . . . . 8  |-  ( G `
 0 )  =  0
7871, 77syl6eq 2452 . . . . . . 7  |-  ( (  T.  /\  n  e.  ( 0 ... (
1  -  1 ) ) )  ->  ( G `  n )  =  0 )
7943, 45, 49, 64, 78seqid 11323 . . . . . 6  |-  (  T. 
->  (  seq  0
(  +  ,  G
)  |`  ( ZZ>= `  1
) )  =  seq  1 (  +  ,  G ) )
80 1z 10267 . . . . . . . 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 10477 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
8482, 83syl6eleqr 2495 . . . . . . . 8  |-  ( (  T.  /\  n  e.  ( ZZ>= `  1 )
)  ->  n  e.  NN )
85 nnne0 9988 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  n  =/=  0 )
8685neneqd 2583 . . . . . . . . . . 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 3717 . . . . . . . . 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 4176 . . . . . . . . . . 11  |-  ( k  =  n  ->  (
2  ||  k  <->  2  ||  n ) )
91 oveq1 6047 . . . . . . . . . . . . . 14  |-  ( k  =  n  ->  (
k  -  1 )  =  ( n  - 
1 ) )
9291oveq1d 6055 . . . . . . . . . . . . 13  |-  ( k  =  n  ->  (
( k  -  1 )  /  2 )  =  ( ( n  -  1 )  / 
2 ) )
9392oveq2d 6056 . . . . . . . . . . . 12  |-  ( k  =  n  ->  ( -u 1 ^ ( ( k  -  1 )  /  2 ) )  =  ( -u 1 ^ ( ( n  -  1 )  / 
2 ) ) )
94 id 20 . . . . . . . . . . . 12  |-  ( k  =  n  ->  k  =  n )
9593, 94oveq12d 6058 . . . . . . . . . . 11  |-  ( k  =  n  ->  (
( -u 1 ^ (
( k  -  1 )  /  2 ) )  /  k )  =  ( ( -u
1 ^ ( ( n  -  1 )  /  2 ) )  /  n ) )
9690, 95ifbieq2d 3719 . . . . . . . . . 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 2404 . . . . . . . . . 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 6065 . . . . . . . . . . 11  |-  ( (
-u 1 ^ (
( n  -  1 )  /  2 ) )  /  n )  e.  _V
9975, 98ifex 3757 . . . . . . . . . 10  |-  if ( 2  ||  n ,  0 ,  ( (
-u 1 ^ (
( n  -  1 )  /  2 ) )  /  n ) )  e.  _V
10096, 97, 99fvmpt 5765 . . . . . . . . 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 10184 . . . . . . . . . 10  |-  ( n  e.  NN  ->  n  e.  NN0 )
102 eqeq1 2410 . . . . . . . . . . . . 13  |-  ( k  =  n  ->  (
k  =  0  <->  n  =  0 ) )
103102, 90orbi12d 691 . . . . . . . . . . . 12  |-  ( k  =  n  ->  (
( k  =  0  \/  2  ||  k
)  <->  ( n  =  0  \/  2  ||  n ) ) )
104103, 95ifbieq2d 3719 . . . . . . . . . . 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 3757 . . . . . . . . . . 11  |-  if ( ( n  =  0  \/  2  ||  n
) ,  0 ,  ( ( -u 1 ^ ( ( n  -  1 )  / 
2 ) )  /  n ) )  e. 
_V
106104, 50, 105fvmpt 5765 . . . . . . . . . 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 2446 . . . . . . . 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 11303 . . . . . 6  |-  (  T. 
->  seq  1 (  +  ,  ( k  e.  NN  |->  if ( 2 
||  k ,  0 ,  ( ( -u
1 ^ ( ( k  -  1 )  /  2 ) )  /  k ) ) ) )  =  seq  1 (  +  ,  G ) )
11179, 110eqtr4d 2439 . . . . 5  |-  (  T. 
->  (  seq  0
(  +  ,  G
)  |`  ( ZZ>= `  1
) )  =  seq  1 (  +  , 
( k  e.  NN  |->  if ( 2  ||  k ,  0 ,  ( ( -u 1 ^ ( ( k  - 
1 )  /  2
) )  /  k
) ) ) ) )
112111trud 1329 . . . 4  |-  (  seq  0 (  +  ,  G )  |`  ( ZZ>=
`  1 ) )  =  seq  1 (  +  ,  ( k  e.  NN  |->  if ( 2  ||  k ,  0 ,  ( (
-u 1 ^ (
( k  -  1 )  /  2 ) )  /  k ) ) ) )
113112breq1i 4179 . . 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 11280 . . . 4  |-  seq  0
(  +  ,  G
)  e.  _V
115 climres 12324 . . . 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 1322    = wceq 1649    e. wcel 1721    =/= wne 2567   _Vcvv 2916   ifcif 3699   class class class wbr 4172    e. cmpt 4226    |` cres 4839   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    - cmin 9247   -ucneg 9248    / cdiv 9633   NNcn 9956   2c2 10005   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999    seq cseq 11278   ^cexp 11337    ~~> cli 12233    || cdivides 12807
This theorem is referenced by:  leibpi  20735
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-seq 11279  df-exp 11338  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-dvds 12808
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