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Theorem leibpilem2 20641
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 9995 . . . . . . . . . . . 12  |-  2  e.  CC
3 nn0cn 10156 . . . . . . . . . . . 12  |-  ( n  e.  NN0  ->  n  e.  CC )
4 mulcl 9000 . . . . . . . . . . . 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 8974 . . . . . . . . . . 11  |-  1  e.  CC
7 pncan 9236 . . . . . . . . . . 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 6028 . . . . . . . . 9  |-  ( n  e.  NN0  ->  ( ( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 )  =  ( ( 2  x.  n )  /  2
) )
10 2ne0 10008 . . . . . . . . . . 11  |-  2  =/=  0
11 divcan3 9627 . . . . . . . . . . 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 2412 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( ( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 )  =  n )
1514oveq2d 6029 . . . . . . 7  |-  ( n  e.  NN0  ->  ( -u
1 ^ ( ( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 ) )  =  ( -u 1 ^ n ) )
1615oveq1d 6028 . . . . . 6  |-  ( n  e.  NN0  ->  ( (
-u 1 ^ (
( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 ) )  /  ( ( 2  x.  n )  +  1 ) )  =  ( ( -u
1 ^ n )  /  ( ( 2  x.  n )  +  1 ) ) )
1716mpteq2ia 4225 . . . . 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 2403 . . . 4  |-  F  =  ( n  e.  NN0  |->  ( ( -u 1 ^ ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 ) )  / 
( ( 2  x.  n )  +  1 ) ) )
19 seqeq3 11248 . . . 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 4153 . 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 9016 . . . . . . . . . 10  |-  1  e.  RR
2322renegcli 9287 . . . . . . . . 9  |-  -u 1  e.  RR
24 reexpcl 11318 . . . . . . . . 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 10163 . . . . . . . . . 10  |-  2  e.  NN0
27 nn0mulcl 10181 . . . . . . . . . 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 10184 . . . . . . . . 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 9973 . . . . . . 7  |-  ( n  e.  NN0  ->  ( (
-u 1 ^ n
)  /  ( ( 2  x.  n )  +  1 ) )  e.  RR )
3231recnd 9040 . . . . . 6  |-  ( n  e.  NN0  ->  ( (
-u 1 ^ n
)  /  ( ( 2  x.  n )  +  1 ) )  e.  CC )
3316, 32eqeltrd 2454 . . . . 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 6020 . . . . . . 7  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  (
k  -  1 )  =  ( ( ( 2  x.  n )  +  1 )  - 
1 ) )
3635oveq1d 6028 . . . . . 6  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  (
( k  -  1 )  /  2 )  =  ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 ) )
3736oveq2d 6029 . . . . 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 6031 . . . 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 13129 . . 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 9174 . . . . . . . 8  |-  ( n  e.  CC  ->  (
0  +  n )  =  n )
4342adantl 453 . . . . . . 7  |-  ( (  T.  /\  n  e.  CC )  ->  (
0  +  n )  =  n )
44 0cn 9010 . . . . . . . 8  |-  0  e.  CC
4544a1i 11 . . . . . . 7  |-  (  T. 
->  0  e.  CC )
46 1nn0 10162 . . . . . . . . 9  |-  1  e.  NN0
47 nn0uz 10445 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
4846, 47eleqtri 2452 . . . . . . . 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 20640 . . . . . . . . . . . . . . . 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 11318 . . . . . . . . . . . . . . 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 9973 . . . . . . . . . . . . 13  |-  ( ( k  e.  NN0  /\  ( -.  k  = 
0  /\  -.  2  ||  k ) )  -> 
( ( -u 1 ^ ( ( k  -  1 )  / 
2 ) )  / 
k )  e.  RR )
5958recnd 9040 . . . . . . . . . . . 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 3702 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  if ( ( k  =  0  \/  2  ||  k
) ,  0 ,  ( ( -u 1 ^ ( ( k  -  1 )  / 
2 ) )  / 
k ) )  e.  CC )
6250, 61fmpti 5824 . . . . . . . . 9  |-  G : NN0
--> CC
6362ffvelrni 5801 . . . . . . . 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 9993 . . . . . . . . . . . 12  |-  ( 1  -  1 )  =  0
6766oveq2i 6024 . . . . . . . . . . 11  |-  ( 0 ... ( 1  -  1 ) )  =  ( 0 ... 0
)
6865, 67syl6eleq 2470 . . . . . . . . . 10  |-  ( (  T.  /\  n  e.  ( 0 ... (
1  -  1 ) ) )  ->  n  e.  ( 0 ... 0
) )
69 elfz1eq 10993 . . . . . . . . . 10  |-  ( n  e.  ( 0 ... 0 )  ->  n  =  0 )
7068, 69syl 16 . . . . . . . . 9  |-  ( (  T.  /\  n  e.  ( 0 ... (
1  -  1 ) ) )  ->  n  =  0 )
7170fveq2d 5665 . . . . . . . 8  |-  ( (  T.  /\  n  e.  ( 0 ... (
1  -  1 ) ) )  ->  ( G `  n )  =  ( G ` 
0 ) )
72 0nn0 10161 . . . . . . . . 9  |-  0  e.  NN0
73 iftrue 3681 . . . . . . . . . . 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 9011 . . . . . . . . . 10  |-  0  e.  _V
7674, 50, 75fvmpt 5738 . . . . . . . . 9  |-  ( 0  e.  NN0  ->  ( G `
 0 )  =  0 )
7772, 76ax-mp 8 . . . . . . . 8  |-  ( G `
 0 )  =  0
7871, 77syl6eq 2428 . . . . . . 7  |-  ( (  T.  /\  n  e.  ( 0 ... (
1  -  1 ) ) )  ->  ( G `  n )  =  0 )
7943, 45, 49, 64, 78seqid 11288 . . . . . 6  |-  (  T. 
->  (  seq  0
(  +  ,  G
)  |`  ( ZZ>= `  1
) )  =  seq  1 (  +  ,  G ) )
80 1z 10236 . . . . . . . 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 10446 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
8482, 83syl6eleqr 2471 . . . . . . . 8  |-  ( (  T.  /\  n  e.  ( ZZ>= `  1 )
)  ->  n  e.  NN )
85 nnne0 9957 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  n  =/=  0 )
8685neneqd 2559 . . . . . . . . . . 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 3693 . . . . . . . . 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 4150 . . . . . . . . . . 11  |-  ( k  =  n  ->  (
2  ||  k  <->  2  ||  n ) )
91 oveq1 6020 . . . . . . . . . . . . . 14  |-  ( k  =  n  ->  (
k  -  1 )  =  ( n  - 
1 ) )
9291oveq1d 6028 . . . . . . . . . . . . 13  |-  ( k  =  n  ->  (
( k  -  1 )  /  2 )  =  ( ( n  -  1 )  / 
2 ) )
9392oveq2d 6029 . . . . . . . . . . . 12  |-  ( k  =  n  ->  ( -u 1 ^ ( ( k  -  1 )  /  2 ) )  =  ( -u 1 ^ ( ( n  -  1 )  / 
2 ) ) )
94 id 20 . . . . . . . . . . . 12  |-  ( k  =  n  ->  k  =  n )
9593, 94oveq12d 6031 . . . . . . . . . . 11  |-  ( k  =  n  ->  (
( -u 1 ^ (
( k  -  1 )  /  2 ) )  /  k )  =  ( ( -u
1 ^ ( ( n  -  1 )  /  2 ) )  /  n ) )
9690, 95ifbieq2d 3695 . . . . . . . . . 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 2380 . . . . . . . . . 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 6038 . . . . . . . . . . 11  |-  ( (
-u 1 ^ (
( n  -  1 )  /  2 ) )  /  n )  e.  _V
9975, 98ifex 3733 . . . . . . . . . 10  |-  if ( 2  ||  n ,  0 ,  ( (
-u 1 ^ (
( n  -  1 )  /  2 ) )  /  n ) )  e.  _V
10096, 97, 99fvmpt 5738 . . . . . . . . 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 10153 . . . . . . . . . 10  |-  ( n  e.  NN  ->  n  e.  NN0 )
102 eqeq1 2386 . . . . . . . . . . . . 13  |-  ( k  =  n  ->  (
k  =  0  <->  n  =  0 ) )
103102, 90orbi12d 691 . . . . . . . . . . . 12  |-  ( k  =  n  ->  (
( k  =  0  \/  2  ||  k
)  <->  ( n  =  0  \/  2  ||  n ) ) )
104103, 95ifbieq2d 3695 . . . . . . . . . . 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 3733 . . . . . . . . . . 11  |-  if ( ( n  =  0  \/  2  ||  n
) ,  0 ,  ( ( -u 1 ^ ( ( n  -  1 )  / 
2 ) )  /  n ) )  e. 
_V
106104, 50, 105fvmpt 5738 . . . . . . . . . 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 2422 . . . . . . . 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 11268 . . . . . 6  |-  (  T. 
->  seq  1 (  +  ,  ( k  e.  NN  |->  if ( 2 
||  k ,  0 ,  ( ( -u
1 ^ ( ( k  -  1 )  /  2 ) )  /  k ) ) ) )  =  seq  1 (  +  ,  G ) )
11179, 110eqtr4d 2415 . . . . 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 4153 . . 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 11245 . . . 4  |-  seq  0
(  +  ,  G
)  e.  _V
115 climres 12289 . . . 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 1717    =/= wne 2543   _Vcvv 2892   ifcif 3675   class class class wbr 4146    e. cmpt 4200    |` cres 4813   ` cfv 5387  (class class class)co 6013   CCcc 8914   RRcr 8915   0cc0 8916   1c1 8917    + caddc 8919    x. cmul 8921    - cmin 9216   -ucneg 9217    / cdiv 9602   NNcn 9925   2c2 9974   NN0cn0 10146   ZZcz 10207   ZZ>=cuz 10413   ...cfz 10968    seq cseq 11243   ^cexp 11302    ~~> cli 12198    || cdivides 12772
This theorem is referenced by:  leibpi  20642
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-card 7752  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-fz 10969  df-seq 11244  df-exp 11303  df-hash 11539  df-shft 11802  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-clim 12202  df-dvds 12773
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