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Theorem wwlktovf1 12949
Description: Lemma 2 for wrd2f1tovbij 12952. (Contributed by Alexander van der Vekens, 27-Jul-2018.)
Hypotheses
Ref Expression
wrd2f1tovbij.d  |-  D  =  { w  e. Word  V  |  ( ( # `  w )  =  2  /\  ( w ` 
0 )  =  P  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  X
) }
wrd2f1tovbij.r  |-  R  =  { n  e.  V  |  { P ,  n }  e.  X }
wrd2f1tovbij.f  |-  F  =  ( t  e.  D  |->  ( t `  1
) )
Assertion
Ref Expression
wwlktovf1  |-  F : D -1-1-> R
Distinct variable groups:    t, D    P, n, t, w    t, R    n, V, t, w   
n, X, w
Allowed substitution hints:    D( w, n)    R( w, n)    F( w, t, n)    X( t)

Proof of Theorem wwlktovf1
Dummy variables  x  y  i are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wrd2f1tovbij.d . . 3  |-  D  =  { w  e. Word  V  |  ( ( # `  w )  =  2  /\  ( w ` 
0 )  =  P  /\  { ( w `
 0 ) ,  ( w `  1
) }  e.  X
) }
2 wrd2f1tovbij.r . . 3  |-  R  =  { n  e.  V  |  { P ,  n }  e.  X }
3 wrd2f1tovbij.f . . 3  |-  F  =  ( t  e.  D  |->  ( t `  1
) )
41, 2, 3wwlktovf 12948 . 2  |-  F : D
--> R
5 fveq1 5847 . . . . . 6  |-  ( t  =  x  ->  (
t `  1 )  =  ( x ` 
1 ) )
6 fvex 5858 . . . . . 6  |-  ( x `
 1 )  e. 
_V
75, 3, 6fvmpt 5931 . . . . 5  |-  ( x  e.  D  ->  ( F `  x )  =  ( x ` 
1 ) )
8 fveq1 5847 . . . . . 6  |-  ( t  =  y  ->  (
t `  1 )  =  ( y ` 
1 ) )
9 fvex 5858 . . . . . 6  |-  ( y `
 1 )  e. 
_V
108, 3, 9fvmpt 5931 . . . . 5  |-  ( y  e.  D  ->  ( F `  y )  =  ( y ` 
1 ) )
117, 10eqeqan12d 2425 . . . 4  |-  ( ( x  e.  D  /\  y  e.  D )  ->  ( ( F `  x )  =  ( F `  y )  <-> 
( x `  1
)  =  ( y `
 1 ) ) )
12 fveq2 5848 . . . . . . . 8  |-  ( w  =  x  ->  ( # `
 w )  =  ( # `  x
) )
1312eqeq1d 2404 . . . . . . 7  |-  ( w  =  x  ->  (
( # `  w )  =  2  <->  ( # `  x
)  =  2 ) )
14 fveq1 5847 . . . . . . . 8  |-  ( w  =  x  ->  (
w `  0 )  =  ( x ` 
0 ) )
1514eqeq1d 2404 . . . . . . 7  |-  ( w  =  x  ->  (
( w `  0
)  =  P  <->  ( x `  0 )  =  P ) )
16 fveq1 5847 . . . . . . . . 9  |-  ( w  =  x  ->  (
w `  1 )  =  ( x ` 
1 ) )
1714, 16preq12d 4058 . . . . . . . 8  |-  ( w  =  x  ->  { ( w `  0 ) ,  ( w ` 
1 ) }  =  { ( x ` 
0 ) ,  ( x `  1 ) } )
1817eleq1d 2471 . . . . . . 7  |-  ( w  =  x  ->  ( { ( w ` 
0 ) ,  ( w `  1 ) }  e.  X  <->  { (
x `  0 ) ,  ( x ` 
1 ) }  e.  X ) )
1913, 15, 183anbi123d 1301 . . . . . 6  |-  ( w  =  x  ->  (
( ( # `  w
)  =  2  /\  ( w `  0
)  =  P  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  X )  <->  ( ( # `
 x )  =  2  /\  ( x `
 0 )  =  P  /\  { ( x `  0 ) ,  ( x ` 
1 ) }  e.  X ) ) )
2019, 1elrab2 3208 . . . . 5  |-  ( x  e.  D  <->  ( x  e. Word  V  /\  ( (
# `  x )  =  2  /\  (
x `  0 )  =  P  /\  { ( x `  0 ) ,  ( x ` 
1 ) }  e.  X ) ) )
21 fveq2 5848 . . . . . . . 8  |-  ( w  =  y  ->  ( # `
 w )  =  ( # `  y
) )
2221eqeq1d 2404 . . . . . . 7  |-  ( w  =  y  ->  (
( # `  w )  =  2  <->  ( # `  y
)  =  2 ) )
23 fveq1 5847 . . . . . . . 8  |-  ( w  =  y  ->  (
w `  0 )  =  ( y ` 
0 ) )
2423eqeq1d 2404 . . . . . . 7  |-  ( w  =  y  ->  (
( w `  0
)  =  P  <->  ( y `  0 )  =  P ) )
25 fveq1 5847 . . . . . . . . 9  |-  ( w  =  y  ->  (
w `  1 )  =  ( y ` 
1 ) )
2623, 25preq12d 4058 . . . . . . . 8  |-  ( w  =  y  ->  { ( w `  0 ) ,  ( w ` 
1 ) }  =  { ( y ` 
0 ) ,  ( y `  1 ) } )
2726eleq1d 2471 . . . . . . 7  |-  ( w  =  y  ->  ( { ( w ` 
0 ) ,  ( w `  1 ) }  e.  X  <->  { (
y `  0 ) ,  ( y ` 
1 ) }  e.  X ) )
2822, 24, 273anbi123d 1301 . . . . . 6  |-  ( w  =  y  ->  (
( ( # `  w
)  =  2  /\  ( w `  0
)  =  P  /\  { ( w `  0
) ,  ( w `
 1 ) }  e.  X )  <->  ( ( # `
 y )  =  2  /\  ( y `
 0 )  =  P  /\  { ( y `  0 ) ,  ( y ` 
1 ) }  e.  X ) ) )
2928, 1elrab2 3208 . . . . 5  |-  ( y  e.  D  <->  ( y  e. Word  V  /\  ( (
# `  y )  =  2  /\  (
y `  0 )  =  P  /\  { ( y `  0 ) ,  ( y ` 
1 ) }  e.  X ) ) )
30 simp1 997 . . . . . . . . . 10  |-  ( ( ( # `  x
)  =  2  /\  ( x `  0
)  =  P  /\  { ( x `  0
) ,  ( x `
 1 ) }  e.  X )  -> 
( # `  x )  =  2 )
3130adantl 464 . . . . . . . . 9  |-  ( ( x  e. Word  V  /\  ( ( # `  x
)  =  2  /\  ( x `  0
)  =  P  /\  { ( x `  0
) ,  ( x `
 1 ) }  e.  X ) )  ->  ( # `  x
)  =  2 )
32 simp1 997 . . . . . . . . . . 11  |-  ( ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X )  -> 
( # `  y )  =  2 )
3332eqcomd 2410 . . . . . . . . . 10  |-  ( ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X )  -> 
2  =  ( # `  y ) )
3433adantl 464 . . . . . . . . 9  |-  ( ( y  e. Word  V  /\  ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X ) )  ->  2  =  (
# `  y )
)
3531, 34sylan9eq 2463 . . . . . . . 8  |-  ( ( ( x  e. Word  V  /\  ( ( # `  x
)  =  2  /\  ( x `  0
)  =  P  /\  { ( x `  0
) ,  ( x `
 1 ) }  e.  X ) )  /\  ( y  e. Word  V  /\  ( ( # `  y )  =  2  /\  ( y ` 
0 )  =  P  /\  { ( y `
 0 ) ,  ( y `  1
) }  e.  X
) ) )  -> 
( # `  x )  =  ( # `  y
) )
3635adantr 463 . . . . . . 7  |-  ( ( ( ( x  e. Word  V  /\  ( ( # `  x )  =  2  /\  ( x ` 
0 )  =  P  /\  { ( x `
 0 ) ,  ( x `  1
) }  e.  X
) )  /\  (
y  e. Word  V  /\  ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X ) ) )  /\  ( x `
 1 )  =  ( y `  1
) )  ->  ( # `
 x )  =  ( # `  y
) )
37 simp2 998 . . . . . . . . . . 11  |-  ( ( ( # `  x
)  =  2  /\  ( x `  0
)  =  P  /\  { ( x `  0
) ,  ( x `
 1 ) }  e.  X )  -> 
( x `  0
)  =  P )
3837adantl 464 . . . . . . . . . 10  |-  ( ( x  e. Word  V  /\  ( ( # `  x
)  =  2  /\  ( x `  0
)  =  P  /\  { ( x `  0
) ,  ( x `
 1 ) }  e.  X ) )  ->  ( x ` 
0 )  =  P )
39 simp2 998 . . . . . . . . . . . 12  |-  ( ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X )  -> 
( y `  0
)  =  P )
4039eqcomd 2410 . . . . . . . . . . 11  |-  ( ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X )  ->  P  =  ( y `  0 ) )
4140adantl 464 . . . . . . . . . 10  |-  ( ( y  e. Word  V  /\  ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X ) )  ->  P  =  ( y `  0 ) )
4238, 41sylan9eq 2463 . . . . . . . . 9  |-  ( ( ( x  e. Word  V  /\  ( ( # `  x
)  =  2  /\  ( x `  0
)  =  P  /\  { ( x `  0
) ,  ( x `
 1 ) }  e.  X ) )  /\  ( y  e. Word  V  /\  ( ( # `  y )  =  2  /\  ( y ` 
0 )  =  P  /\  { ( y `
 0 ) ,  ( y `  1
) }  e.  X
) ) )  -> 
( x `  0
)  =  ( y `
 0 ) )
4342adantr 463 . . . . . . . 8  |-  ( ( ( ( x  e. Word  V  /\  ( ( # `  x )  =  2  /\  ( x ` 
0 )  =  P  /\  { ( x `
 0 ) ,  ( x `  1
) }  e.  X
) )  /\  (
y  e. Word  V  /\  ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X ) ) )  /\  ( x `
 1 )  =  ( y `  1
) )  ->  (
x `  0 )  =  ( y ` 
0 ) )
44 simpr 459 . . . . . . . 8  |-  ( ( ( ( x  e. Word  V  /\  ( ( # `  x )  =  2  /\  ( x ` 
0 )  =  P  /\  { ( x `
 0 ) ,  ( x `  1
) }  e.  X
) )  /\  (
y  e. Word  V  /\  ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X ) ) )  /\  ( x `
 1 )  =  ( y `  1
) )  ->  (
x `  1 )  =  ( y ` 
1 ) )
45 oveq2 6285 . . . . . . . . . . . . 13  |-  ( (
# `  x )  =  2  ->  (
0..^ ( # `  x
) )  =  ( 0..^ 2 ) )
46 fzo0to2pr 11934 . . . . . . . . . . . . 13  |-  ( 0..^ 2 )  =  {
0 ,  1 }
4745, 46syl6eq 2459 . . . . . . . . . . . 12  |-  ( (
# `  x )  =  2  ->  (
0..^ ( # `  x
) )  =  {
0 ,  1 } )
4847raleqdv 3009 . . . . . . . . . . 11  |-  ( (
# `  x )  =  2  ->  ( A. i  e.  (
0..^ ( # `  x
) ) ( x `
 i )  =  ( y `  i
)  <->  A. i  e.  {
0 ,  1 }  ( x `  i
)  =  ( y `
 i ) ) )
49 c0ex 9619 . . . . . . . . . . . 12  |-  0  e.  _V
50 1ex 9620 . . . . . . . . . . . 12  |-  1  e.  _V
51 fveq2 5848 . . . . . . . . . . . . 13  |-  ( i  =  0  ->  (
x `  i )  =  ( x ` 
0 ) )
52 fveq2 5848 . . . . . . . . . . . . 13  |-  ( i  =  0  ->  (
y `  i )  =  ( y ` 
0 ) )
5351, 52eqeq12d 2424 . . . . . . . . . . . 12  |-  ( i  =  0  ->  (
( x `  i
)  =  ( y `
 i )  <->  ( x `  0 )  =  ( y `  0
) ) )
54 fveq2 5848 . . . . . . . . . . . . 13  |-  ( i  =  1  ->  (
x `  i )  =  ( x ` 
1 ) )
55 fveq2 5848 . . . . . . . . . . . . 13  |-  ( i  =  1  ->  (
y `  i )  =  ( y ` 
1 ) )
5654, 55eqeq12d 2424 . . . . . . . . . . . 12  |-  ( i  =  1  ->  (
( x `  i
)  =  ( y `
 i )  <->  ( x `  1 )  =  ( y `  1
) ) )
5749, 50, 53, 56ralpr 4024 . . . . . . . . . . 11  |-  ( A. i  e.  { 0 ,  1 }  (
x `  i )  =  ( y `  i )  <->  ( (
x `  0 )  =  ( y ` 
0 )  /\  (
x `  1 )  =  ( y ` 
1 ) ) )
5848, 57syl6bb 261 . . . . . . . . . 10  |-  ( (
# `  x )  =  2  ->  ( A. i  e.  (
0..^ ( # `  x
) ) ( x `
 i )  =  ( y `  i
)  <->  ( ( x `
 0 )  =  ( y `  0
)  /\  ( x `  1 )  =  ( y `  1
) ) ) )
59583ad2ant1 1018 . . . . . . . . 9  |-  ( ( ( # `  x
)  =  2  /\  ( x `  0
)  =  P  /\  { ( x `  0
) ,  ( x `
 1 ) }  e.  X )  -> 
( A. i  e.  ( 0..^ ( # `  x ) ) ( x `  i )  =  ( y `  i )  <->  ( (
x `  0 )  =  ( y ` 
0 )  /\  (
x `  1 )  =  ( y ` 
1 ) ) ) )
6059ad3antlr 729 . . . . . . . 8  |-  ( ( ( ( x  e. Word  V  /\  ( ( # `  x )  =  2  /\  ( x ` 
0 )  =  P  /\  { ( x `
 0 ) ,  ( x `  1
) }  e.  X
) )  /\  (
y  e. Word  V  /\  ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X ) ) )  /\  ( x `
 1 )  =  ( y `  1
) )  ->  ( A. i  e.  (
0..^ ( # `  x
) ) ( x `
 i )  =  ( y `  i
)  <->  ( ( x `
 0 )  =  ( y `  0
)  /\  ( x `  1 )  =  ( y `  1
) ) ) )
6143, 44, 60mpbir2and 923 . . . . . . 7  |-  ( ( ( ( x  e. Word  V  /\  ( ( # `  x )  =  2  /\  ( x ` 
0 )  =  P  /\  { ( x `
 0 ) ,  ( x `  1
) }  e.  X
) )  /\  (
y  e. Word  V  /\  ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X ) ) )  /\  ( x `
 1 )  =  ( y `  1
) )  ->  A. i  e.  ( 0..^ ( # `  x ) ) ( x `  i )  =  ( y `  i ) )
62 simpl 455 . . . . . . . . . 10  |-  ( ( x  e. Word  V  /\  ( ( # `  x
)  =  2  /\  ( x `  0
)  =  P  /\  { ( x `  0
) ,  ( x `
 1 ) }  e.  X ) )  ->  x  e. Word  V
)
63 simpl 455 . . . . . . . . . 10  |-  ( ( y  e. Word  V  /\  ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X ) )  ->  y  e. Word  V
)
6462, 63anim12i 564 . . . . . . . . 9  |-  ( ( ( x  e. Word  V  /\  ( ( # `  x
)  =  2  /\  ( x `  0
)  =  P  /\  { ( x `  0
) ,  ( x `
 1 ) }  e.  X ) )  /\  ( y  e. Word  V  /\  ( ( # `  y )  =  2  /\  ( y ` 
0 )  =  P  /\  { ( y `
 0 ) ,  ( y `  1
) }  e.  X
) ) )  -> 
( x  e. Word  V  /\  y  e. Word  V ) )
6564adantr 463 . . . . . . . 8  |-  ( ( ( ( x  e. Word  V  /\  ( ( # `  x )  =  2  /\  ( x ` 
0 )  =  P  /\  { ( x `
 0 ) ,  ( x `  1
) }  e.  X
) )  /\  (
y  e. Word  V  /\  ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X ) ) )  /\  ( x `
 1 )  =  ( y `  1
) )  ->  (
x  e. Word  V  /\  y  e. Word  V )
)
66 eqwrd 12633 . . . . . . . 8  |-  ( ( x  e. Word  V  /\  y  e. Word  V )  ->  ( x  =  y  <-> 
( ( # `  x
)  =  ( # `  y )  /\  A. i  e.  ( 0..^ ( # `  x
) ) ( x `
 i )  =  ( y `  i
) ) ) )
6765, 66syl 17 . . . . . . 7  |-  ( ( ( ( x  e. Word  V  /\  ( ( # `  x )  =  2  /\  ( x ` 
0 )  =  P  /\  { ( x `
 0 ) ,  ( x `  1
) }  e.  X
) )  /\  (
y  e. Word  V  /\  ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X ) ) )  /\  ( x `
 1 )  =  ( y `  1
) )  ->  (
x  =  y  <->  ( ( # `
 x )  =  ( # `  y
)  /\  A. i  e.  ( 0..^ ( # `  x ) ) ( x `  i )  =  ( y `  i ) ) ) )
6836, 61, 67mpbir2and 923 . . . . . 6  |-  ( ( ( ( x  e. Word  V  /\  ( ( # `  x )  =  2  /\  ( x ` 
0 )  =  P  /\  { ( x `
 0 ) ,  ( x `  1
) }  e.  X
) )  /\  (
y  e. Word  V  /\  ( ( # `  y
)  =  2  /\  ( y `  0
)  =  P  /\  { ( y `  0
) ,  ( y `
 1 ) }  e.  X ) ) )  /\  ( x `
 1 )  =  ( y `  1
) )  ->  x  =  y )
6968ex 432 . . . . 5  |-  ( ( ( x  e. Word  V  /\  ( ( # `  x
)  =  2  /\  ( x `  0
)  =  P  /\  { ( x `  0
) ,  ( x `
 1 ) }  e.  X ) )  /\  ( y  e. Word  V  /\  ( ( # `  y )  =  2  /\  ( y ` 
0 )  =  P  /\  { ( y `
 0 ) ,  ( y `  1
) }  e.  X
) ) )  -> 
( ( x ` 
1 )  =  ( y `  1 )  ->  x  =  y ) )
7020, 29, 69syl2anb 477 . . . 4  |-  ( ( x  e.  D  /\  y  e.  D )  ->  ( ( x ` 
1 )  =  ( y `  1 )  ->  x  =  y ) )
7111, 70sylbid 215 . . 3  |-  ( ( x  e.  D  /\  y  e.  D )  ->  ( ( F `  x )  =  ( F `  y )  ->  x  =  y ) )
7271rgen2a 2830 . 2  |-  A. x  e.  D  A. y  e.  D  ( ( F `  x )  =  ( F `  y )  ->  x  =  y )
73 dff13 6146 . 2  |-  ( F : D -1-1-> R  <->  ( F : D --> R  /\  A. x  e.  D  A. y  e.  D  (
( F `  x
)  =  ( F `
 y )  ->  x  =  y )
) )
744, 72, 73mpbir2an 921 1  |-  F : D -1-1-> R
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2753   {crab 2757   {cpr 3973    |-> cmpt 4452   -->wf 5564   -1-1->wf1 5565   ` cfv 5568  (class class class)co 6277   0cc0 9521   1c1 9522   2c2 10625  ..^cfzo 11852   #chash 12450  Word cword 12581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-n0 10836  df-z 10905  df-uz 11127  df-fz 11725  df-fzo 11853  df-hash 12451  df-word 12589
This theorem is referenced by:  wwlktovf1o  12951
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