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Theorem heiborlem4 28739
Description: Lemma for heibor 28746. Using the function  T constructed in heiborlem3 28738, 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 5712 . . . . 5  |-  ( x  =  0  ->  ( S `  x )  =  ( S ` 
0 ) )
2 id 22 . . . . 5  |-  ( x  =  0  ->  x  =  0 )
31, 2breq12d 4326 . . . 4  |-  ( x  =  0  ->  (
( S `  x
) G x  <->  ( S `  0 ) G 0 ) )
43imbi2d 316 . . 3  |-  ( x  =  0  ->  (
( ph  ->  ( S `
 x ) G x )  <->  ( ph  ->  ( S `  0
) G 0 ) ) )
5 fveq2 5712 . . . . 5  |-  ( x  =  k  ->  ( S `  x )  =  ( S `  k ) )
6 id 22 . . . . 5  |-  ( x  =  k  ->  x  =  k )
75, 6breq12d 4326 . . . 4  |-  ( x  =  k  ->  (
( S `  x
) G x  <->  ( S `  k ) G k ) )
87imbi2d 316 . . 3  |-  ( x  =  k  ->  (
( ph  ->  ( S `
 x ) G x )  <->  ( ph  ->  ( S `  k
) G k ) ) )
9 fveq2 5712 . . . . 5  |-  ( x  =  ( k  +  1 )  ->  ( S `  x )  =  ( S `  ( k  +  1 ) ) )
10 id 22 . . . . 5  |-  ( x  =  ( k  +  1 )  ->  x  =  ( k  +  1 ) )
119, 10breq12d 4326 . . . 4  |-  ( x  =  ( k  +  1 )  ->  (
( S `  x
) G x  <->  ( S `  ( k  +  1 ) ) G ( k  +  1 ) ) )
1211imbi2d 316 . . 3  |-  ( x  =  ( k  +  1 )  ->  (
( ph  ->  ( S `
 x ) G x )  <->  ( ph  ->  ( S `  (
k  +  1 ) ) G ( k  +  1 ) ) ) )
13 fveq2 5712 . . . . 5  |-  ( x  =  A  ->  ( S `  x )  =  ( S `  A ) )
14 id 22 . . . . 5  |-  ( x  =  A  ->  x  =  A )
1513, 14breq12d 4326 . . . 4  |-  ( x  =  A  ->  (
( S `  x
) G x  <->  ( S `  A ) G A ) )
1615imbi2d 316 . . 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 5713 . . . . . 6  |-  ( S `
 0 )  =  (  seq 0 ( T ,  ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) ) `  0
)
19 0z 10678 . . . . . . 7  |-  0  e.  ZZ
20 seq1 11840 . . . . . . 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 5 . . . . . 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 2463 . . . . 5  |-  ( S `
 0 )  =  ( ( m  e. 
NN0  |->  if ( m  =  0 ,  C ,  ( m  - 
1 ) ) ) `
 0 )
23 0nn0 10615 . . . . . 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 4986 . . . . . . . 8  |-  Rel  G
2726brrelexi 4900 . . . . . . 7  |-  ( C G 0  ->  C  e.  _V )
2824, 27syl 16 . . . . . 6  |-  ( ph  ->  C  e.  _V )
29 iftrue 3818 . . . . . . 7  |-  ( m  =  0  ->  if ( m  =  0 ,  C ,  ( m  -  1 ) )  =  C )
30 eqid 2443 . . . . . . 7  |-  ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) )  =  ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) )
3129, 30fvmptg 5793 . . . . . 6  |-  ( ( 0  e.  NN0  /\  C  e.  _V )  ->  ( ( m  e. 
NN0  |->  if ( m  =  0 ,  C ,  ( m  - 
1 ) ) ) `
 0 )  =  C )
3223, 28, 31sylancr 663 . . . . 5  |-  ( ph  ->  ( ( m  e. 
NN0  |->  if ( m  =  0 ,  C ,  ( m  - 
1 ) ) ) `
 0 )  =  C )
3322, 32syl5eq 2487 . . . 4  |-  ( ph  ->  ( S `  0
)  =  C )
3433, 24eqbrtrd 4333 . . 3  |-  ( ph  ->  ( S `  0
) G 0 )
35 df-br 4314 . . . . . 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 5712 . . . . . . . . . . 11  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( T `  x )  =  ( T `  <. ( S `  k ) ,  k >. )
)
38 df-ov 6115 . . . . . . . . . . 11  |-  ( ( S `  k ) T k )  =  ( T `  <. ( S `  k ) ,  k >. )
3937, 38syl6eqr 2493 . . . . . . . . . 10  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( T `  x )  =  ( ( S `  k
) T k ) )
40 fvex 5722 . . . . . . . . . . . 12  |-  ( S `
 k )  e. 
_V
41 vex 2996 . . . . . . . . . . . 12  |-  k  e. 
_V
4240, 41op2ndd 6609 . . . . . . . . . . 11  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( 2nd `  x
)  =  k )
4342oveq1d 6127 . . . . . . . . . 10  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( ( 2nd `  x )  +  1 )  =  ( k  +  1 ) )
4439, 43breq12d 4326 . . . . . . . . 9  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( ( T `
 x ) G ( ( 2nd `  x
)  +  1 )  <-> 
( ( S `  k ) T k ) G ( k  +  1 ) ) )
45 fveq2 5712 . . . . . . . . . . . 12  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( B `  x )  =  ( B `  <. ( S `  k ) ,  k >. )
)
46 df-ov 6115 . . . . . . . . . . . 12  |-  ( ( S `  k ) B k )  =  ( B `  <. ( S `  k ) ,  k >. )
4745, 46syl6eqr 2493 . . . . . . . . . . 11  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( B `  x )  =  ( ( S `  k
) B k ) )
4839, 43oveq12d 6130 . . . . . . . . . . 11  |-  ( x  =  <. ( S `  k ) ,  k
>.  ->  ( ( T `
 x ) B ( ( 2nd `  x
)  +  1 ) )  =  ( ( ( S `  k
) T k ) B ( k  +  1 ) ) )
4947, 48ineq12d 3574 . . . . . . . . . 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 2509 . . . . . . . . 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 710 . . . . . . . 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 3091 . . . . . . 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 217 . . . . 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 11842 . . . . . . . . . . 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 10916 . . . . . . . . . . 11  |-  NN0  =  ( ZZ>= `  0 )
5755, 56eleq2s 2535 . . . . . . . . . 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 5713 . . . . . . . . . 10  |-  ( S `
 ( k  +  1 ) )  =  (  seq 0 ( T ,  ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) ) `  (
k  +  1 ) )
5917fveq1i 5713 . . . . . . . . . . 11  |-  ( S `
 k )  =  (  seq 0 ( T ,  ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) ) `  k
)
6059oveq1i 6122 . . . . . . . . . 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 2500 . . . . . . . . 9  |-  ( k  e.  NN0  ->  ( S `
 ( k  +  1 ) )  =  ( ( S `  k ) T ( ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) `  (
k  +  1 ) ) ) )
62 peano2nn0 10641 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  ( k  +  1 )  e. 
NN0 )
63 nn0p1nn 10640 . . . . . . . . . . . . . 14  |-  ( k  e.  NN0  ->  ( k  +  1 )  e.  NN )
64 nnne0 10375 . . . . . . . . . . . . . . 15  |-  ( ( k  +  1 )  e.  NN  ->  (
k  +  1 )  =/=  0 )
6564neneqd 2627 . . . . . . . . . . . . . 14  |-  ( ( k  +  1 )  e.  NN  ->  -.  ( k  +  1 )  =  0 )
66 iffalse 3820 . . . . . . . . . . . . . 14  |-  ( -.  ( k  +  1 )  =  0  ->  if ( ( k  +  1 )  =  0 ,  C ,  ( ( k  +  1 )  -  1 ) )  =  ( ( k  +  1 )  -  1 ) )
6763, 65, 663syl 20 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  ->  if ( ( k  +  1 )  =  0 ,  C ,  ( ( k  +  1 )  -  1 ) )  =  ( ( k  +  1 )  - 
1 ) )
68 ovex 6137 . . . . . . . . . . . . 13  |-  ( ( k  +  1 )  -  1 )  e. 
_V
6967, 68syl6eqel 2531 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  if ( ( k  +  1 )  =  0 ,  C ,  ( ( k  +  1 )  -  1 ) )  e.  _V )
70 eqeq1 2449 . . . . . . . . . . . . . 14  |-  ( m  =  ( k  +  1 )  ->  (
m  =  0  <->  (
k  +  1 )  =  0 ) )
71 oveq1 6119 . . . . . . . . . . . . . 14  |-  ( m  =  ( k  +  1 )  ->  (
m  -  1 )  =  ( ( k  +  1 )  - 
1 ) )
7270, 71ifbieq2d 3835 . . . . . . . . . . . . 13  |-  ( m  =  ( k  +  1 )  ->  if ( m  =  0 ,  C ,  ( m  -  1 ) )  =  if ( ( k  +  1 )  =  0 ,  C ,  ( ( k  +  1 )  - 
1 ) ) )
7372, 30fvmptg 5793 . . . . . . . . . . . 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 661 . . . . . . . . . . 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 10610 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  k  e.  CC )
76 ax-1cn 9361 . . . . . . . . . . . 12  |-  1  e.  CC
77 pncan 9637 . . . . . . . . . . . 12  |-  ( ( k  e.  CC  /\  1  e.  CC )  ->  ( ( k  +  1 )  -  1 )  =  k )
7875, 76, 77sylancl 662 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( ( k  +  1 )  -  1 )  =  k )
7974, 67, 783eqtrd 2479 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  ( ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) `  ( k  +  1 ) )  =  k )
8079oveq2d 6128 . . . . . . . . 9  |-  ( k  e.  NN0  ->  ( ( S `  k ) T ( ( m  e.  NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1 ) ) ) `  ( k  +  1 ) ) )  =  ( ( S `  k ) T k ) )
8161, 80eqtrd 2475 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( S `
 ( k  +  1 ) )  =  ( ( S `  k ) T k ) )
8281breq1d 4323 . . . . . . 7  |-  ( k  e.  NN0  ->  ( ( S `  ( k  +  1 ) ) G ( k  +  1 )  <->  ( ( S `  k ) T k ) G ( k  +  1 ) ) )
8382biimprd 223 . . . . . 6  |-  ( k  e.  NN0  ->  ( ( ( S `  k
) T k ) G ( k  +  1 )  ->  ( S `  ( k  +  1 ) ) G ( k  +  1 ) ) )
8483adantrd 468 . . . . 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 72 . . . 4  |-  ( k  e.  NN0  ->  ( ph  ->  ( ( S `  k ) G k  ->  ( S `  ( k  +  1 ) ) G ( k  +  1 ) ) ) )
8685a2d 26 . . 3  |-  ( k  e.  NN0  ->  ( (
ph  ->  ( S `  k ) G k )  ->  ( ph  ->  ( S `  (
k  +  1 ) ) G ( k  +  1 ) ) ) )
874, 8, 12, 16, 34, 86nn0ind 10759 . 2  |-  ( A  e.  NN0  ->  ( ph  ->  ( S `  A
) G A ) )
8887impcom 430 1  |-  ( (
ph  /\  A  e.  NN0 )  ->  ( S `  A ) G A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   {cab 2429   A.wral 2736   E.wrex 2737   _Vcvv 2993    i^i cin 3348    C_ wss 3349   ifcif 3812   ~Pcpw 3881   <.cop 3904   U.cuni 4112   U_ciun 4192   class class class wbr 4313   {copab 4370    e. cmpt 4371   -->wf 5435   ` cfv 5439  (class class class)co 6112    e. cmpt2 6114   2ndc2nd 6597   Fincfn 7331   CCcc 9301   0cc0 9303   1c1 9304    + caddc 9306    - cmin 9616    / cdiv 10014   NNcn 10343   2c2 10392   NN0cn0 10600   ZZcz 10667   ZZ>=cuz 10882    seqcseq 11827   ^cexp 11886   ballcbl 17825   MetOpencmopn 17828   CMetcms 20787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-2nd 6599  df-recs 6853  df-rdg 6887  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-n0 10601  df-z 10668  df-uz 10883  df-seq 11828
This theorem is referenced by:  heiborlem5  28740  heiborlem6  28741  heiborlem8  28743
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