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Theorem leibpilem2 20253
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 9832 . . . . . . . . . . . 12  |-  2  e.  CC
3 nn0cn 9991 . . . . . . . . . . . 12  |-  ( n  e.  NN0  ->  n  e.  CC )
4 mulcl 8837 . . . . . . . . . . . 12  |-  ( ( 2  e.  CC  /\  n  e.  CC )  ->  ( 2  x.  n
)  e.  CC )
52, 3, 4sylancr 644 . . . . . . . . . . 11  |-  ( n  e.  NN0  ->  ( 2  x.  n )  e.  CC )
6 ax-1cn 8811 . . . . . . . . . . 11  |-  1  e.  CC
7 pncan 9073 . . . . . . . . . . 11  |-  ( ( ( 2  x.  n
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( 2  x.  n )  +  1 )  -  1 )  =  ( 2  x.  n ) )
85, 6, 7sylancl 643 . . . . . . . . . 10  |-  ( n  e.  NN0  ->  ( ( ( 2  x.  n
)  +  1 )  -  1 )  =  ( 2  x.  n
) )
98oveq1d 5889 . . . . . . . . 9  |-  ( n  e.  NN0  ->  ( ( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 )  =  ( ( 2  x.  n )  /  2
) )
10 2ne0 9845 . . . . . . . . . . 11  |-  2  =/=  0
11 divcan3 9464 . . . . . . . . . . 11  |-  ( ( n  e.  CC  /\  2  e.  CC  /\  2  =/=  0 )  ->  (
( 2  x.  n
)  /  2 )  =  n )
122, 10, 11mp3an23 1269 . . . . . . . . . 10  |-  ( n  e.  CC  ->  (
( 2  x.  n
)  /  2 )  =  n )
133, 12syl 15 . . . . . . . . 9  |-  ( n  e.  NN0  ->  ( ( 2  x.  n )  /  2 )  =  n )
149, 13eqtrd 2328 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( ( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 )  =  n )
1514oveq2d 5890 . . . . . . 7  |-  ( n  e.  NN0  ->  ( -u
1 ^ ( ( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 ) )  =  ( -u 1 ^ n ) )
1615oveq1d 5889 . . . . . 6  |-  ( n  e.  NN0  ->  ( (
-u 1 ^ (
( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 ) )  /  ( ( 2  x.  n )  +  1 ) )  =  ( ( -u
1 ^ n )  /  ( ( 2  x.  n )  +  1 ) ) )
1716mpteq2ia 4118 . . . . 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 2319 . . . 4  |-  F  =  ( n  e.  NN0  |->  ( ( -u 1 ^ ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 ) )  / 
( ( 2  x.  n )  +  1 ) ) )
19 seqeq3 11067 . . . 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 4046 . 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 8853 . . . . . . . . . 10  |-  1  e.  RR
2322renegcli 9124 . . . . . . . . 9  |-  -u 1  e.  RR
24 reexpcl 11136 . . . . . . . . 9  |-  ( (
-u 1  e.  RR  /\  n  e.  NN0 )  ->  ( -u 1 ^ n )  e.  RR )
2523, 24mpan 651 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( -u
1 ^ n )  e.  RR )
26 2nn0 9998 . . . . . . . . . 10  |-  2  e.  NN0
27 nn0mulcl 10016 . . . . . . . . . 10  |-  ( ( 2  e.  NN0  /\  n  e.  NN0 )  -> 
( 2  x.  n
)  e.  NN0 )
2826, 27mpan 651 . . . . . . . . 9  |-  ( n  e.  NN0  ->  ( 2  x.  n )  e. 
NN0 )
29 nn0p1nn 10019 . . . . . . . . 9  |-  ( ( 2  x.  n )  e.  NN0  ->  ( ( 2  x.  n )  +  1 )  e.  NN )
3028, 29syl 15 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( ( 2  x.  n )  +  1 )  e.  NN )
3125, 30nndivred 9810 . . . . . . 7  |-  ( n  e.  NN0  ->  ( (
-u 1 ^ n
)  /  ( ( 2  x.  n )  +  1 ) )  e.  RR )
3231recnd 8877 . . . . . 6  |-  ( n  e.  NN0  ->  ( (
-u 1 ^ n
)  /  ( ( 2  x.  n )  +  1 ) )  e.  CC )
3316, 32eqeltrd 2370 . . . . 5  |-  ( n  e.  NN0  ->  ( (
-u 1 ^ (
( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 ) )  /  ( ( 2  x.  n )  +  1 ) )  e.  CC )
3433adantl 452 . . . 4  |-  ( (  T.  /\  n  e. 
NN0 )  ->  (
( -u 1 ^ (
( ( ( 2  x.  n )  +  1 )  -  1 )  /  2 ) )  /  ( ( 2  x.  n )  +  1 ) )  e.  CC )
35 oveq1 5881 . . . . . . 7  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  (
k  -  1 )  =  ( ( ( 2  x.  n )  +  1 )  - 
1 ) )
3635oveq1d 5889 . . . . . 6  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  (
( k  -  1 )  /  2 )  =  ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 ) )
3736oveq2d 5890 . . . . 5  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  ( -u 1 ^ ( ( k  -  1 )  /  2 ) )  =  ( -u 1 ^ ( ( ( ( 2  x.  n
)  +  1 )  -  1 )  / 
2 ) ) )
38 id 19 . . . . 5  |-  ( k  =  ( ( 2  x.  n )  +  1 )  ->  k  =  ( ( 2  x.  n )  +  1 ) )
3937, 38oveq12d 5892 . . . 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 12904 . . 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 1314 . 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 9011 . . . . . . . 8  |-  ( n  e.  CC  ->  (
0  +  n )  =  n )
4342adantl 452 . . . . . . 7  |-  ( (  T.  /\  n  e.  CC )  ->  (
0  +  n )  =  n )
44 0cn 8847 . . . . . . . 8  |-  0  e.  CC
4544a1i 10 . . . . . . 7  |-  (  T. 
->  0  e.  CC )
46 1nn0 9997 . . . . . . . . 9  |-  1  e.  NN0
47 nn0uz 10278 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
4846, 47eleqtri 2368 . . . . . . . 8  |-  1  e.  ( ZZ>= `  0 )
4948a1i 10 . . . . . . 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 10 . . . . . . . . . . 11  |-  ( ( k  e.  NN0  /\  ( k  =  0  \/  2  ||  k
) )  ->  0  e.  CC )
52 ioran 476 . . . . . . . . . . . 12  |-  ( -.  ( k  =  0  \/  2  ||  k
)  <->  ( -.  k  =  0  /\  -.  2  ||  k ) )
53 leibpilem1 20252 . . . . . . . . . . . . . . . 16  |-  ( ( k  e.  NN0  /\  ( -.  k  = 
0  /\  -.  2  ||  k ) )  -> 
( k  e.  NN  /\  ( ( k  - 
1 )  /  2
)  e.  NN0 )
)
5453simprd 449 . . . . . . . . . . . . . . 15  |-  ( ( k  e.  NN0  /\  ( -.  k  = 
0  /\  -.  2  ||  k ) )  -> 
( ( k  - 
1 )  /  2
)  e.  NN0 )
55 reexpcl 11136 . . . . . . . . . . . . . . 15  |-  ( (
-u 1  e.  RR  /\  ( ( k  - 
1 )  /  2
)  e.  NN0 )  ->  ( -u 1 ^ ( ( k  - 
1 )  /  2
) )  e.  RR )
5623, 54, 55sylancr 644 . . . . . . . . . . . . . 14  |-  ( ( k  e.  NN0  /\  ( -.  k  = 
0  /\  -.  2  ||  k ) )  -> 
( -u 1 ^ (
( k  -  1 )  /  2 ) )  e.  RR )
5753simpld 445 . . . . . . . . . . . . . 14  |-  ( ( k  e.  NN0  /\  ( -.  k  = 
0  /\  -.  2  ||  k ) )  -> 
k  e.  NN )
5856, 57nndivred 9810 . . . . . . . . . . . . 13  |-  ( ( k  e.  NN0  /\  ( -.  k  = 
0  /\  -.  2  ||  k ) )  -> 
( ( -u 1 ^ ( ( k  -  1 )  / 
2 ) )  / 
k )  e.  RR )
5958recnd 8877 . . . . . . . . . . . 12  |-  ( ( k  e.  NN0  /\  ( -.  k  = 
0  /\  -.  2  ||  k ) )  -> 
( ( -u 1 ^ ( ( k  -  1 )  / 
2 ) )  / 
k )  e.  CC )
6052, 59sylan2b 461 . . . . . . . . . . 11  |-  ( ( k  e.  NN0  /\  -.  ( k  =  0  \/  2  ||  k
) )  ->  (
( -u 1 ^ (
( k  -  1 )  /  2 ) )  /  k )  e.  CC )
6151, 60ifclda 3605 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  if ( ( k  =  0  \/  2  ||  k
) ,  0 ,  ( ( -u 1 ^ ( ( k  -  1 )  / 
2 ) )  / 
k ) )  e.  CC )
6250, 61fmpti 5699 . . . . . . . . 9  |-  G : NN0
--> CC
6362ffvelrni 5680 . . . . . . . 8  |-  ( 1  e.  NN0  ->  ( G `
 1 )  e.  CC )
6446, 63mp1i 11 . . . . . . 7  |-  (  T. 
->  ( G `  1
)  e.  CC )
65 simpr 447 . . . . . . . . . . 11  |-  ( (  T.  /\  n  e.  ( 0 ... (
1  -  1 ) ) )  ->  n  e.  ( 0 ... (
1  -  1 ) ) )
66 1m1e0 9830 . . . . . . . . . . . 12  |-  ( 1  -  1 )  =  0
6766oveq2i 5885 . . . . . . . . . . 11  |-  ( 0 ... ( 1  -  1 ) )  =  ( 0 ... 0
)
6865, 67syl6eleq 2386 . . . . . . . . . 10  |-  ( (  T.  /\  n  e.  ( 0 ... (
1  -  1 ) ) )  ->  n  e.  ( 0 ... 0
) )
69 elfz1eq 10823 . . . . . . . . . 10  |-  ( n  e.  ( 0 ... 0 )  ->  n  =  0 )
7068, 69syl 15 . . . . . . . . 9  |-  ( (  T.  /\  n  e.  ( 0 ... (
1  -  1 ) ) )  ->  n  =  0 )
7170fveq2d 5545 . . . . . . . 8  |-  ( (  T.  /\  n  e.  ( 0 ... (
1  -  1 ) ) )  ->  ( G `  n )  =  ( G ` 
0 ) )
72 0nn0 9996 . . . . . . . . 9  |-  0  e.  NN0
73 iftrue 3584 . . . . . . . . . . 11  |-  ( ( k  =  0  \/  2  ||  k )  ->  if ( ( k  =  0  \/  2  ||  k ) ,  0 ,  ( ( -u 1 ^ ( ( k  - 
1 )  /  2
) )  /  k
) )  =  0 )
7473orcs 383 . . . . . . . . . 10  |-  ( k  =  0  ->  if ( ( k  =  0  \/  2  ||  k ) ,  0 ,  ( ( -u
1 ^ ( ( k  -  1 )  /  2 ) )  /  k ) )  =  0 )
75 c0ex 8848 . . . . . . . . . 10  |-  0  e.  _V
7674, 50, 75fvmpt 5618 . . . . . . . . 9  |-  ( 0  e.  NN0  ->  ( G `
 0 )  =  0 )
7772, 76ax-mp 8 . . . . . . . 8  |-  ( G `
 0 )  =  0
7871, 77syl6eq 2344 . . . . . . 7  |-  ( (  T.  /\  n  e.  ( 0 ... (
1  -  1 ) ) )  ->  ( G `  n )  =  0 )
7943, 45, 49, 64, 78seqid 11107 . . . . . 6  |-  (  T. 
->  (  seq  0
(  +  ,  G
)  |`  ( ZZ>= `  1
) )  =  seq  1 (  +  ,  G ) )
80 1z 10069 . . . . . . . 8  |-  1  e.  ZZ
8180a1i 10 . . . . . . 7  |-  (  T. 
->  1  e.  ZZ )
82 simpr 447 . . . . . . . . 9  |-  ( (  T.  /\  n  e.  ( ZZ>= `  1 )
)  ->  n  e.  ( ZZ>= `  1 )
)
83 nnuz 10279 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
8482, 83syl6eleqr 2387 . . . . . . . 8  |-  ( (  T.  /\  n  e.  ( ZZ>= `  1 )
)  ->  n  e.  NN )
85 nnne0 9794 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  n  =/=  0 )
8685neneqd 2475 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  -.  n  =  0 )
87 biorf 394 . . . . . . . . . . 11  |-  ( -.  n  =  0  -> 
( 2  ||  n  <->  ( n  =  0  \/  2  ||  n ) ) )
8886, 87syl 15 . . . . . . . . . 10  |-  ( n  e.  NN  ->  (
2  ||  n  <->  ( n  =  0  \/  2 
||  n ) ) )
8988ifbid 3596 . . . . . . . . 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 4043 . . . . . . . . . . 11  |-  ( k  =  n  ->  (
2  ||  k  <->  2  ||  n ) )
91 oveq1 5881 . . . . . . . . . . . . . 14  |-  ( k  =  n  ->  (
k  -  1 )  =  ( n  - 
1 ) )
9291oveq1d 5889 . . . . . . . . . . . . 13  |-  ( k  =  n  ->  (
( k  -  1 )  /  2 )  =  ( ( n  -  1 )  / 
2 ) )
9392oveq2d 5890 . . . . . . . . . . . 12  |-  ( k  =  n  ->  ( -u 1 ^ ( ( k  -  1 )  /  2 ) )  =  ( -u 1 ^ ( ( n  -  1 )  / 
2 ) ) )
94 id 19 . . . . . . . . . . . 12  |-  ( k  =  n  ->  k  =  n )
9593, 94oveq12d 5892 . . . . . . . . . . 11  |-  ( k  =  n  ->  (
( -u 1 ^ (
( k  -  1 )  /  2 ) )  /  k )  =  ( ( -u
1 ^ ( ( n  -  1 )  /  2 ) )  /  n ) )
9690, 95ifbieq2d 3598 . . . . . . . . . 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 2296 . . . . . . . . . 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 5899 . . . . . . . . . . 11  |-  ( (
-u 1 ^ (
( n  -  1 )  /  2 ) )  /  n )  e.  _V
9975, 98ifex 3636 . . . . . . . . . 10  |-  if ( 2  ||  n ,  0 ,  ( (
-u 1 ^ (
( n  -  1 )  /  2 ) )  /  n ) )  e.  _V
10096, 97, 99fvmpt 5618 . . . . . . . . 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 9988 . . . . . . . . . 10  |-  ( n  e.  NN  ->  n  e.  NN0 )
102 eqeq1 2302 . . . . . . . . . . . . 13  |-  ( k  =  n  ->  (
k  =  0  <->  n  =  0 ) )
103102, 90orbi12d 690 . . . . . . . . . . . 12  |-  ( k  =  n  ->  (
( k  =  0  \/  2  ||  k
)  <->  ( n  =  0  \/  2  ||  n ) ) )
104103, 95ifbieq2d 3598 . . . . . . . . . . 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 3636 . . . . . . . . . . 11  |-  if ( ( n  =  0  \/  2  ||  n
) ,  0 ,  ( ( -u 1 ^ ( ( n  -  1 )  / 
2 ) )  /  n ) )  e. 
_V
106104, 50, 105fvmpt 5618 . . . . . . . . . 10  |-  ( n  e.  NN0  ->  ( G `
 n )  =  if ( ( n  =  0  \/  2 
||  n ) ,  0 ,  ( (
-u 1 ^ (
( n  -  1 )  /  2 ) )  /  n ) ) )
107101, 106syl 15 . . . . . . . . 9  |-  ( n  e.  NN  ->  ( G `  n )  =  if ( ( n  =  0  \/  2 
||  n ) ,  0 ,  ( (
-u 1 ^ (
( n  -  1 )  /  2 ) )  /  n ) ) )
10889, 100, 1073eqtr4d 2338 . . . . . . . 8  |-  ( n  e.  NN  ->  (
( k  e.  NN  |->  if ( 2  ||  k ,  0 ,  ( ( -u 1 ^ ( ( k  - 
1 )  /  2
) )  /  k
) ) ) `  n )  =  ( G `  n ) )
10984, 108syl 15 . . . . . . 7  |-  ( (  T.  /\  n  e.  ( ZZ>= `  1 )
)  ->  ( (
k  e.  NN  |->  if ( 2  ||  k ,  0 ,  ( ( -u 1 ^ ( ( k  - 
1 )  /  2
) )  /  k
) ) ) `  n )  =  ( G `  n ) )
11081, 109seqfeq 11087 . . . . . 6  |-  (  T. 
->  seq  1 (  +  ,  ( k  e.  NN  |->  if ( 2 
||  k ,  0 ,  ( ( -u
1 ^ ( ( k  -  1 )  /  2 ) )  /  k ) ) ) )  =  seq  1 (  +  ,  G ) )
11179, 110eqtr4d 2331 . . . . 5  |-  (  T. 
->  (  seq  0
(  +  ,  G
)  |`  ( ZZ>= `  1
) )  =  seq  1 (  +  , 
( k  e.  NN  |->  if ( 2  ||  k ,  0 ,  ( ( -u 1 ^ ( ( k  - 
1 )  /  2
) )  /  k
) ) ) ) )
112111trud 1314 . . . 4  |-  (  seq  0 (  +  ,  G )  |`  ( ZZ>=
`  1 ) )  =  seq  1 (  +  ,  ( k  e.  NN  |->  if ( 2  ||  k ,  0 ,  ( (
-u 1 ^ (
( k  -  1 )  /  2 ) )  /  k ) ) ) )
113112breq1i 4046 . . 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 11064 . . . 4  |-  seq  0
(  +  ,  G
)  e.  _V
115 climres 12065 . . . 4  |-  ( ( 1  e.  ZZ  /\  seq  0 (  +  ,  G )  e.  _V )  ->  ( (  seq  0 (  +  ,  G )  |`  ( ZZ>=
`  1 ) )  ~~>  A  <->  seq  0 (  +  ,  G )  ~~>  A ) )
11680, 114, 115mp2an 653 . . 3  |-  ( (  seq  0 (  +  ,  G )  |`  ( ZZ>= `  1 )
)  ~~>  A  <->  seq  0
(  +  ,  G
)  ~~>  A )
117113, 116bitr3i 242 . 2  |-  (  seq  1 (  +  , 
( k  e.  NN  |->  if ( 2  ||  k ,  0 ,  ( ( -u 1 ^ ( ( k  - 
1 )  /  2
) )  /  k
) ) ) )  ~~>  A  <->  seq  0 (  +  ,  G )  ~~>  A )
11821, 41, 1173bitri 262 1  |-  (  seq  0 (  +  ,  F )  ~~>  A  <->  seq  0
(  +  ,  G
)  ~~>  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/ wo 357    /\ wa 358    T. wtru 1307    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801   ifcif 3578   class class class wbr 4039    e. cmpt 4093    |` cres 4707   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    - cmin 9053   -ucneg 9054    / cdiv 9439   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798    seq cseq 11062   ^cexp 11120    ~~> cli 11974    || cdivides 12547
This theorem is referenced by:  leibpi  20254
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-seq 11063  df-exp 11121  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-dvds 12548
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