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Theorem heiborlem4 26514
Description: Lemma for heibor 26521. Using the function  T constructed in heiborlem3 26513, construct an infinite path in  G. (Contributed by Jeff Madsen, 23-Jan-2014.)
Hypotheses
Ref Expression
heibor.1  |-  J  =  ( MetOpen `  D )
heibor.3  |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin )
u  C_  U. v }
heibor.4  |-  G  =  { <. y ,  n >.  |  ( n  e. 
NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K
) }
heibor.5  |-  B  =  ( z  e.  X ,  m  e.  NN0  |->  ( z ( ball `  D ) ( 1  /  ( 2 ^ m ) ) ) )
heibor.6  |-  ( ph  ->  D  e.  ( CMet `  X ) )
heibor.7  |-  ( ph  ->  F : NN0 --> ( ~P X  i^i  Fin )
)
heibor.8  |-  ( ph  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n
) ( y B n ) )
heibor.9  |-  ( ph  ->  A. x  e.  G  ( ( T `  x ) G ( ( 2nd `  x
)  +  1 )  /\  ( ( B `
 x )  i^i  ( ( T `  x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )
heibor.10  |-  ( ph  ->  C G 0 )
heibor.11  |-  S  =  seq  0 ( T ,  ( m  e. 
NN0  |->  if ( m  =  0 ,  C ,  ( m  - 
1 ) ) ) )
Assertion
Ref Expression
heiborlem4  |-  ( (
ph  /\  A  e.  NN0 )  ->  ( S `  A ) G A )
Distinct variable groups:    x, n, y, A    u, n, F, x, y    x, G    ph, x    m, n, u, v, x, y, z, D    T, m, n, x, y, z    B, n, u, v, y    m, J, n, u, v, x, y, z    U, n, u, v, x, y, z    S, m, n, u, v, x, y, z   
m, X, n, u, v, x, y, z    C, m, n, u, v, y    n, K, x, y, z    x, B
Allowed substitution hints:    ph( y, z, v, u, m, n)    A( z, v, u, m)    B( z, m)    C( x, z)    T( v, u)    U( m)    F( z, v, m)    G( y, z, v, u, m, n)    K( v, u, m)

Proof of Theorem heiborlem4
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 fveq2 5720 . . . . 5  |-  ( x  =  0  ->  ( S `  x )  =  ( S ` 
0 ) )
2 id 20 . . . . 5  |-  ( x  =  0  ->  x  =  0 )
31, 2breq12d 4217 . . . 4  |-  ( x  =  0  ->  (
( S `  x
) G x  <->  ( S `  0 ) G 0 ) )
43imbi2d 308 . . 3  |-  ( x  =  0  ->  (
( ph  ->  ( S `
 x ) G x )  <->  ( ph  ->  ( S `  0
) G 0 ) ) )
5 fveq2 5720 . . . . 5  |-  ( x  =  k  ->  ( S `  x )  =  ( S `  k ) )
6 id 20 . . . . 5  |-  ( x  =  k  ->  x  =  k )
75, 6breq12d 4217 . . . 4  |-  ( x  =  k  ->  (
( S `  x
) G x  <->  ( S `  k ) G k ) )
87imbi2d 308 . . 3  |-  ( x  =  k  ->  (
( ph  ->  ( S `
 x ) G x )  <->  ( ph  ->  ( S `  k
) G k ) ) )
9 fveq2 5720 . . . . 5  |-  ( x  =  ( k  +  1 )  ->  ( S `  x )  =  ( S `  ( k  +  1 ) ) )
10 id 20 . . . . 5  |-  ( x  =  ( k  +  1 )  ->  x  =  ( k  +  1 ) )
119, 10breq12d 4217 . . . 4  |-  ( x  =  ( k  +  1 )  ->  (
( S `  x
) G x  <->  ( S `  ( k  +  1 ) ) G ( k  +  1 ) ) )
1211imbi2d 308 . . 3  |-  ( x  =  ( k  +  1 )  ->  (
( ph  ->  ( S `
 x ) G x )  <->  ( ph  ->  ( S `  (
k  +  1 ) ) G ( k  +  1 ) ) ) )
13 fveq2 5720 . . . . 5  |-  ( x  =  A  ->  ( S `  x )  =  ( S `  A ) )
14 id 20 . . . . 5  |-  ( x  =  A  ->  x  =  A )
1513, 14breq12d 4217 . . . 4  |-  ( x  =  A  ->  (
( S `  x
) G x  <->  ( S `  A ) G A ) )
1615imbi2d 308 . . 3  |-  ( x  =  A  ->  (
( ph  ->  ( S `
 x ) G x )  <->  ( ph  ->  ( S `  A
) G A ) ) )
17 heibor.11 . . . . . . 7  |-  S  =  seq  0 ( T ,  ( m  e. 
NN0  |->  if ( m  =  0 ,  C ,  ( m  - 
1 ) ) ) )
1817fveq1i 5721 . . . . . 6  |-  ( S `
 0 )  =  (  seq  0 ( T ,  ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) ) `  0
)
19 0z 10285 . . . . . . 7  |-  0  e.  ZZ
20 seq1 11328 . . . . . . 7  |-  ( 0  e.  ZZ  ->  (  seq  0 ( T , 
( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) ) ` 
0 )  =  ( ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) `  0
) )
2119, 20ax-mp 8 . . . . . 6  |-  (  seq  0 ( T , 
( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) ) ` 
0 )  =  ( ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) `  0
)
2218, 21eqtri 2455 . . . . 5  |-  ( S `
 0 )  =  ( ( m  e. 
NN0  |->  if ( m  =  0 ,  C ,  ( m  - 
1 ) ) ) `
 0 )
23 0nn0 10228 . . . . . 6  |-  0  e.  NN0
24 heibor.10 . . . . . . 7  |-  ( ph  ->  C G 0 )
25 heibor.4 . . . . . . . . 9  |-  G  =  { <. y ,  n >.  |  ( n  e. 
NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K
) }
2625relopabi 4992 . . . . . . . 8  |-  Rel  G
2726brrelexi 4910 . . . . . . 7  |-  ( C G 0  ->  C  e.  _V )
2824, 27syl 16 . . . . . 6  |-  ( ph  ->  C  e.  _V )
29 iftrue 3737 . . . . . . 7  |-  ( m  =  0  ->  if ( m  =  0 ,  C ,  ( m  -  1 ) )  =  C )
30 eqid 2435 . . . . . . 7  |-  ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) )  =  ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) )
3129, 30fvmptg 5796 . . . . . 6  |-  ( ( 0  e.  NN0  /\  C  e.  _V )  ->  ( ( m  e. 
NN0  |->  if ( m  =  0 ,  C ,  ( m  - 
1 ) ) ) `
 0 )  =  C )
3223, 28, 31sylancr 645 . . . . 5  |-  ( ph  ->  ( ( m  e. 
NN0  |->  if ( m  =  0 ,  C ,  ( m  - 
1 ) ) ) `
 0 )  =  C )
3322, 32syl5eq 2479 . . . 4  |-  ( ph  ->  ( S `  0
)  =  C )
3433, 24eqbrtrd 4224 . . 3  |-  ( ph  ->  ( S `  0
) G 0 )
35 df-br 4205 . . . . . 6  |-  ( ( S `  k ) G k  <->  <. ( S `
 k ) ,  k >.  e.  G
)
36 heibor.9 . . . . . . 7  |-  ( ph  ->  A. x  e.  G  ( ( T `  x ) G ( ( 2nd `  x
)  +  1 )  /\  ( ( B `
 x )  i^i  ( ( T `  x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )
37 fveq2 5720 . . . . . . . . . . 11  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( T `  x )  =  ( T `  <. ( S `  k ) ,  k >. )
)
38 df-ov 6076 . . . . . . . . . . 11  |-  ( ( S `  k ) T k )  =  ( T `  <. ( S `  k ) ,  k >. )
3937, 38syl6eqr 2485 . . . . . . . . . 10  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( T `  x )  =  ( ( S `  k
) T k ) )
40 fvex 5734 . . . . . . . . . . . 12  |-  ( S `
 k )  e. 
_V
41 vex 2951 . . . . . . . . . . . 12  |-  k  e. 
_V
4240, 41op2ndd 6350 . . . . . . . . . . 11  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( 2nd `  x
)  =  k )
4342oveq1d 6088 . . . . . . . . . 10  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( ( 2nd `  x )  +  1 )  =  ( k  +  1 ) )
4439, 43breq12d 4217 . . . . . . . . 9  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( ( T `
 x ) G ( ( 2nd `  x
)  +  1 )  <-> 
( ( S `  k ) T k ) G ( k  +  1 ) ) )
45 fveq2 5720 . . . . . . . . . . . 12  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( B `  x )  =  ( B `  <. ( S `  k ) ,  k >. )
)
46 df-ov 6076 . . . . . . . . . . . 12  |-  ( ( S `  k ) B k )  =  ( B `  <. ( S `  k ) ,  k >. )
4745, 46syl6eqr 2485 . . . . . . . . . . 11  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( B `  x )  =  ( ( S `  k
) B k ) )
4839, 43oveq12d 6091 . . . . . . . . . . 11  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( ( T `
 x ) B ( ( 2nd `  x
)  +  1 ) )  =  ( ( ( S `  k
) T k ) B ( k  +  1 ) ) )
4947, 48ineq12d 3535 . . . . . . . . . 10  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( ( B `
 x )  i^i  ( ( T `  x ) B ( ( 2nd `  x
)  +  1 ) ) )  =  ( ( ( S `  k ) B k )  i^i  ( ( ( S `  k
) T k ) B ( k  +  1 ) ) ) )
5049eleq1d 2501 . . . . . . . . 9  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( ( ( B `  x )  i^i  ( ( T `
 x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K  <->  ( ( ( S `  k ) B k )  i^i  ( ( ( S `  k
) T k ) B ( k  +  1 ) ) )  e.  K ) )
5144, 50anbi12d 692 . . . . . . . 8  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( ( ( T `  x ) G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( ( T `
 x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
)  <->  ( ( ( S `  k ) T k ) G ( k  +  1 )  /\  ( ( ( S `  k
) B k )  i^i  ( ( ( S `  k ) T k ) B ( k  +  1 ) ) )  e.  K ) ) )
5251rspccv 3041 . . . . . . 7  |-  ( A. x  e.  G  (
( T `  x
) G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  (
( T `  x
) B ( ( 2nd `  x )  +  1 ) ) )  e.  K )  ->  ( <. ( S `  k ) ,  k >.  e.  G  ->  ( ( ( S `
 k ) T k ) G ( k  +  1 )  /\  ( ( ( S `  k ) B k )  i^i  ( ( ( S `
 k ) T k ) B ( k  +  1 ) ) )  e.  K
) ) )
5336, 52syl 16 . . . . . 6  |-  ( ph  ->  ( <. ( S `  k ) ,  k
>.  e.  G  ->  (
( ( S `  k ) T k ) G ( k  +  1 )  /\  ( ( ( S `
 k ) B k )  i^i  (
( ( S `  k ) T k ) B ( k  +  1 ) ) )  e.  K ) ) )
5435, 53syl5bi 209 . . . . 5  |-  ( ph  ->  ( ( S `  k ) G k  ->  ( ( ( S `  k ) T k ) G ( k  +  1 )  /\  ( ( ( S `  k
) B k )  i^i  ( ( ( S `  k ) T k ) B ( k  +  1 ) ) )  e.  K ) ) )
55 seqp1 11330 . . . . . . . . . . 11  |-  ( k  e.  ( ZZ>= `  0
)  ->  (  seq  0 ( T , 
( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) ) `  ( k  +  1 ) )  =  ( (  seq  0 ( T ,  ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) ) `  k
) T ( ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) `  ( k  +  1 ) ) ) )
56 nn0uz 10512 . . . . . . . . . . 11  |-  NN0  =  ( ZZ>= `  0 )
5755, 56eleq2s 2527 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  (  seq  0 ( T , 
( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) ) `  ( k  +  1 ) )  =  ( (  seq  0 ( T ,  ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) ) `  k
) T ( ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) `  ( k  +  1 ) ) ) )
5817fveq1i 5721 . . . . . . . . . 10  |-  ( S `
 ( k  +  1 ) )  =  (  seq  0 ( T ,  ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) ) `  (
k  +  1 ) )
5917fveq1i 5721 . . . . . . . . . . 11  |-  ( S `
 k )  =  (  seq  0 ( T ,  ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) ) `  k
)
6059oveq1i 6083 . . . . . . . . . 10  |-  ( ( S `  k ) T ( ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) `  ( k  +  1 ) ) )  =  ( (  seq  0 ( T ,  ( m  e. 
NN0  |->  if ( m  =  0 ,  C ,  ( m  - 
1 ) ) ) ) `  k ) T ( ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) `  ( k  +  1 ) ) )
6157, 58, 603eqtr4g 2492 . . . . . . . . 9  |-  ( k  e.  NN0  ->  ( S `
 ( k  +  1 ) )  =  ( ( S `  k ) T ( ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) `  (
k  +  1 ) ) ) )
62 peano2nn0 10252 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  ( k  +  1 )  e. 
NN0 )
63 nn0p1nn 10251 . . . . . . . . . . . . . 14  |-  ( k  e.  NN0  ->  ( k  +  1 )  e.  NN )
64 nnne0 10024 . . . . . . . . . . . . . . 15  |-  ( ( k  +  1 )  e.  NN  ->  (
k  +  1 )  =/=  0 )
6564neneqd 2614 . . . . . . . . . . . . . 14  |-  ( ( k  +  1 )  e.  NN  ->  -.  ( k  +  1 )  =  0 )
66 iffalse 3738 . . . . . . . . . . . . . 14  |-  ( -.  ( k  +  1 )  =  0  ->  if ( ( k  +  1 )  =  0 ,  C ,  ( ( k  +  1 )  -  1 ) )  =  ( ( k  +  1 )  -  1 ) )
6763, 65, 663syl 19 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  ->  if ( ( k  +  1 )  =  0 ,  C ,  ( ( k  +  1 )  -  1 ) )  =  ( ( k  +  1 )  - 
1 ) )
68 ovex 6098 . . . . . . . . . . . . 13  |-  ( ( k  +  1 )  -  1 )  e. 
_V
6967, 68syl6eqel 2523 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  if ( ( k  +  1 )  =  0 ,  C ,  ( ( k  +  1 )  -  1 ) )  e.  _V )
70 eqeq1 2441 . . . . . . . . . . . . . 14  |-  ( m  =  ( k  +  1 )  ->  (
m  =  0  <->  (
k  +  1 )  =  0 ) )
71 oveq1 6080 . . . . . . . . . . . . . 14  |-  ( m  =  ( k  +  1 )  ->  (
m  -  1 )  =  ( ( k  +  1 )  - 
1 ) )
7270, 71ifbieq2d 3751 . . . . . . . . . . . . 13  |-  ( m  =  ( k  +  1 )  ->  if ( m  =  0 ,  C ,  ( m  -  1 ) )  =  if ( ( k  +  1 )  =  0 ,  C ,  ( ( k  +  1 )  - 
1 ) ) )
7372, 30fvmptg 5796 . . . . . . . . . . . 12  |-  ( ( ( k  +  1 )  e.  NN0  /\  if ( ( k  +  1 )  =  0 ,  C ,  ( ( k  +  1 )  -  1 ) )  e.  _V )  ->  ( ( m  e. 
NN0  |->  if ( m  =  0 ,  C ,  ( m  - 
1 ) ) ) `
 ( k  +  1 ) )  =  if ( ( k  +  1 )  =  0 ,  C , 
( ( k  +  1 )  -  1 ) ) )
7462, 69, 73syl2anc 643 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) `  ( k  +  1 ) )  =  if ( ( k  +  1 )  =  0 ,  C ,  ( ( k  +  1 )  - 
1 ) ) )
75 nn0cn 10223 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  k  e.  CC )
76 ax-1cn 9040 . . . . . . . . . . . 12  |-  1  e.  CC
77 pncan 9303 . . . . . . . . . . . 12  |-  ( ( k  e.  CC  /\  1  e.  CC )  ->  ( ( k  +  1 )  -  1 )  =  k )
7875, 76, 77sylancl 644 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( ( k  +  1 )  -  1 )  =  k )
7974, 67, 783eqtrd 2471 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  ( ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) `  ( k  +  1 ) )  =  k )
8079oveq2d 6089 . . . . . . . . 9  |-  ( k  e.  NN0  ->  ( ( S `  k ) T ( ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) `  ( k  +  1 ) ) )  =  ( ( S `  k ) T k ) )
8161, 80eqtrd 2467 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( S `
 ( k  +  1 ) )  =  ( ( S `  k ) T k ) )
8281breq1d 4214 . . . . . . 7  |-  ( k  e.  NN0  ->  ( ( S `  ( k  +  1 ) ) G ( k  +  1 )  <->  ( ( S `  k ) T k ) G ( k  +  1 ) ) )
8382biimprd 215 . . . . . 6  |-  ( k  e.  NN0  ->  ( ( ( S `  k
) T k ) G ( k  +  1 )  ->  ( S `  ( k  +  1 ) ) G ( k  +  1 ) ) )
8483adantrd 455 . . . . 5  |-  ( k  e.  NN0  ->  ( ( ( ( S `  k ) T k ) G ( k  +  1 )  /\  ( ( ( S `
 k ) B k )  i^i  (
( ( S `  k ) T k ) B ( k  +  1 ) ) )  e.  K )  ->  ( S `  ( k  +  1 ) ) G ( k  +  1 ) ) )
8554, 84syl9r 69 . . . 4  |-  ( k  e.  NN0  ->  ( ph  ->  ( ( S `  k ) G k  ->  ( S `  ( k  +  1 ) ) G ( k  +  1 ) ) ) )
8685a2d 24 . . 3  |-  ( k  e.  NN0  ->  ( (
ph  ->  ( S `  k ) G k )  ->  ( ph  ->  ( S `  (
k  +  1 ) ) G ( k  +  1 ) ) ) )
874, 8, 12, 16, 34, 86nn0ind 10358 . 2  |-  ( A  e.  NN0  ->  ( ph  ->  ( S `  A
) G A ) )
8887impcom 420 1  |-  ( (
ph  /\  A  e.  NN0 )  ->  ( S `  A ) G A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {cab 2421   A.wral 2697   E.wrex 2698   _Vcvv 2948    i^i cin 3311    C_ wss 3312   ifcif 3731   ~Pcpw 3791   <.cop 3809   U.cuni 4007   U_ciun 4085   class class class wbr 4204   {copab 4257    e. cmpt 4258   -->wf 5442   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   2ndc2nd 6340   Fincfn 7101   CCcc 8980   0cc0 8982   1c1 8983    + caddc 8985    - cmin 9283    / cdiv 9669   NNcn 9992   2c2 10041   NN0cn0 10213   ZZcz 10274   ZZ>=cuz 10480    seq cseq 11315   ^cexp 11374   ballcbl 16680   MetOpencmopn 16683   CMetcms 19199
This theorem is referenced by:  heiborlem5  26515  heiborlem6  26516  heiborlem8  26518
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-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  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
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-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-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-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-seq 11316
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