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Theorem efgredeu 15077
Description: There is a unique reduced word equivalent to a given word. (Contributed by Mario Carneiro, 1-Oct-2015.)
Hypotheses
Ref Expression
efgval.w  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
efgval.r  |-  .~  =  ( ~FG  `  I )
efgval2.m  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
efgval2.t  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
efgred.d  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
efgred.s  |-  S  =  ( m  e.  {
t  e.  (Word  W  \  { (/) } )  |  ( ( t ` 
0 )  e.  D  /\  A. k  e.  ( 1..^ ( # `  t
) ) ( t `
 k )  e. 
ran  ( T `  ( t `  (
k  -  1 ) ) ) ) } 
|->  ( m `  (
( # `  m )  -  1 ) ) )
Assertion
Ref Expression
efgredeu  |-  ( A  e.  W  ->  E! d  e.  D  d  .~  A )
Distinct variable groups:    A, d    y, z    t, n, v, w, y, z, m, x    m, M    x, n, M, t, v, w   
k, m, t, x, T    k, d, m, n, t, v, w, x, y, z, W    .~ , d, m, t, x, y, z    S, d   
m, I, n, t, v, w, x, y, z    D, d, m, t
Allowed substitution hints:    A( x, y, z, w, v, t, k, m, n)    D( x, y, z, w, v, k, n)    .~ ( w, v, k, n)    S( x, y, z, w, v, t, k, m, n)    T( y, z, w, v, n, d)    I( k, d)    M( y, z, k, d)

Proof of Theorem efgredeu
Dummy variables  a 
b  c  i are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efgval.w . . . . 5  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
2 efgval.r . . . . 5  |-  .~  =  ( ~FG  `  I )
3 efgval2.m . . . . 5  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
4 efgval2.t . . . . 5  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
5 efgred.d . . . . 5  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
6 efgred.s . . . . 5  |-  S  =  ( m  e.  {
t  e.  (Word  W  \  { (/) } )  |  ( ( t ` 
0 )  e.  D  /\  A. k  e.  ( 1..^ ( # `  t
) ) ( t `
 k )  e. 
ran  ( T `  ( t `  (
k  -  1 ) ) ) ) } 
|->  ( m `  (
( # `  m )  -  1 ) ) )
71, 2, 3, 4, 5, 6efgsfo 15064 . . . 4  |-  S : dom  S -onto-> W
8 foelrn 5695 . . . 4  |-  ( ( S : dom  S -onto-> W  /\  A  e.  W
)  ->  E. a  e.  dom  S  A  =  ( S `  a
) )
97, 8mpan 651 . . 3  |-  ( A  e.  W  ->  E. a  e.  dom  S  A  =  ( S `  a
) )
101, 2, 3, 4, 5, 6efgsdm 15055 . . . . . . . 8  |-  ( a  e.  dom  S  <->  ( a  e.  (Word  W  \  { (/)
} )  /\  (
a `  0 )  e.  D  /\  A. i  e.  ( 1..^ ( # `  a ) ) ( a `  i )  e.  ran  ( T `
 ( a `  ( i  -  1 ) ) ) ) )
1110simp2bi 971 . . . . . . 7  |-  ( a  e.  dom  S  -> 
( a `  0
)  e.  D )
1211adantl 452 . . . . . 6  |-  ( ( A  e.  W  /\  a  e.  dom  S )  ->  ( a ` 
0 )  e.  D
)
131, 2, 3, 4, 5, 6efgsrel 15059 . . . . . . 7  |-  ( a  e.  dom  S  -> 
( a `  0
)  .~  ( S `  a ) )
1413adantl 452 . . . . . 6  |-  ( ( A  e.  W  /\  a  e.  dom  S )  ->  ( a ` 
0 )  .~  ( S `  a )
)
15 breq1 4042 . . . . . . 7  |-  ( d  =  ( a ` 
0 )  ->  (
d  .~  ( S `  a )  <->  ( a `  0 )  .~  ( S `  a ) ) )
1615rspcev 2897 . . . . . 6  |-  ( ( ( a `  0
)  e.  D  /\  ( a `  0
)  .~  ( S `  a ) )  ->  E. d  e.  D  d  .~  ( S `  a ) )
1712, 14, 16syl2anc 642 . . . . 5  |-  ( ( A  e.  W  /\  a  e.  dom  S )  ->  E. d  e.  D  d  .~  ( S `  a ) )
18 breq2 4043 . . . . . 6  |-  ( A  =  ( S `  a )  ->  (
d  .~  A  <->  d  .~  ( S `  a ) ) )
1918rexbidv 2577 . . . . 5  |-  ( A  =  ( S `  a )  ->  ( E. d  e.  D  d  .~  A  <->  E. d  e.  D  d  .~  ( S `  a ) ) )
2017, 19syl5ibrcom 213 . . . 4  |-  ( ( A  e.  W  /\  a  e.  dom  S )  ->  ( A  =  ( S `  a
)  ->  E. d  e.  D  d  .~  A ) )
2120rexlimdva 2680 . . 3  |-  ( A  e.  W  ->  ( E. a  e.  dom  S  A  =  ( S `
 a )  ->  E. d  e.  D  d  .~  A ) )
229, 21mpd 14 . 2  |-  ( A  e.  W  ->  E. d  e.  D  d  .~  A )
231, 2efger 15043 . . . . . . 7  |-  .~  Er  W
2423a1i 10 . . . . . 6  |-  ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D
) )  /\  (
d  .~  A  /\  c  .~  A ) )  ->  .~  Er  W
)
25 simprl 732 . . . . . 6  |-  ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D
) )  /\  (
d  .~  A  /\  c  .~  A ) )  ->  d  .~  A
)
26 simprr 733 . . . . . 6  |-  ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D
) )  /\  (
d  .~  A  /\  c  .~  A ) )  ->  c  .~  A
)
2724, 25, 26ertr4d 6695 . . . . 5  |-  ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D
) )  /\  (
d  .~  A  /\  c  .~  A ) )  ->  d  .~  c
)
281, 2, 3, 4, 5, 6efgrelex 15076 . . . . . 6  |-  ( d  .~  c  ->  E. a  e.  ( `' S " { d } ) E. b  e.  ( `' S " { c } ) ( a `
 0 )  =  ( b `  0
) )
29 fofn 5469 . . . . . . . . . . . . . 14  |-  ( S : dom  S -onto-> W  ->  S  Fn  dom  S
)
30 fniniseg 5662 . . . . . . . . . . . . . 14  |-  ( S  Fn  dom  S  -> 
( a  e.  ( `' S " { d } )  <->  ( a  e.  dom  S  /\  ( S `  a )  =  d ) ) )
317, 29, 30mp2b 9 . . . . . . . . . . . . 13  |-  ( a  e.  ( `' S " { d } )  <-> 
( a  e.  dom  S  /\  ( S `  a )  =  d ) )
3231simplbi 446 . . . . . . . . . . . 12  |-  ( a  e.  ( `' S " { d } )  ->  a  e.  dom  S )
3332ad2antrl 708 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  a  e.  dom  S )
341, 2, 3, 4, 5, 6efgsval 15056 . . . . . . . . . . 11  |-  ( a  e.  dom  S  -> 
( S `  a
)  =  ( a `
 ( ( # `  a )  -  1 ) ) )
3533, 34syl 15 . . . . . . . . . 10  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( S `  a )  =  ( a `  ( (
# `  a )  -  1 ) ) )
3631simprbi 450 . . . . . . . . . . 11  |-  ( a  e.  ( `' S " { d } )  ->  ( S `  a )  =  d )
3736ad2antrl 708 . . . . . . . . . 10  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( S `  a )  =  d )
38 simpllr 735 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( d  e.  D  /\  c  e.  D ) )
3938simpld 445 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  d  e.  D
)
4037, 39eqeltrd 2370 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( S `  a )  e.  D
)
411, 2, 3, 4, 5, 6efgs1b 15061 . . . . . . . . . . . . . . 15  |-  ( a  e.  dom  S  -> 
( ( S `  a )  e.  D  <->  (
# `  a )  =  1 ) )
4233, 41syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( ( S `
 a )  e.  D  <->  ( # `  a
)  =  1 ) )
4340, 42mpbid 201 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( # `  a
)  =  1 )
4443oveq1d 5889 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( ( # `  a )  -  1 )  =  ( 1  -  1 ) )
45 1m1e0 9830 . . . . . . . . . . . 12  |-  ( 1  -  1 )  =  0
4644, 45syl6eq 2344 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( ( # `  a )  -  1 )  =  0 )
4746fveq2d 5545 . . . . . . . . . 10  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( a `  ( ( # `  a
)  -  1 ) )  =  ( a `
 0 ) )
4835, 37, 473eqtr3rd 2337 . . . . . . . . 9  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( a ` 
0 )  =  d )
49 fniniseg 5662 . . . . . . . . . . . . . 14  |-  ( S  Fn  dom  S  -> 
( b  e.  ( `' S " { c } )  <->  ( b  e.  dom  S  /\  ( S `  b )  =  c ) ) )
507, 29, 49mp2b 9 . . . . . . . . . . . . 13  |-  ( b  e.  ( `' S " { c } )  <-> 
( b  e.  dom  S  /\  ( S `  b )  =  c ) )
5150simplbi 446 . . . . . . . . . . . 12  |-  ( b  e.  ( `' S " { c } )  ->  b  e.  dom  S )
5251ad2antll 709 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  b  e.  dom  S )
531, 2, 3, 4, 5, 6efgsval 15056 . . . . . . . . . . 11  |-  ( b  e.  dom  S  -> 
( S `  b
)  =  ( b `
 ( ( # `  b )  -  1 ) ) )
5452, 53syl 15 . . . . . . . . . 10  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( S `  b )  =  ( b `  ( (
# `  b )  -  1 ) ) )
5550simprbi 450 . . . . . . . . . . 11  |-  ( b  e.  ( `' S " { c } )  ->  ( S `  b )  =  c )
5655ad2antll 709 . . . . . . . . . 10  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( S `  b )  =  c )
5738simprd 449 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  c  e.  D
)
5856, 57eqeltrd 2370 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( S `  b )  e.  D
)
591, 2, 3, 4, 5, 6efgs1b 15061 . . . . . . . . . . . . . . 15  |-  ( b  e.  dom  S  -> 
( ( S `  b )  e.  D  <->  (
# `  b )  =  1 ) )
6052, 59syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( ( S `
 b )  e.  D  <->  ( # `  b
)  =  1 ) )
6158, 60mpbid 201 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( # `  b
)  =  1 )
6261oveq1d 5889 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( ( # `  b )  -  1 )  =  ( 1  -  1 ) )
6362, 45syl6eq 2344 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( ( # `  b )  -  1 )  =  0 )
6463fveq2d 5545 . . . . . . . . . 10  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( b `  ( ( # `  b
)  -  1 ) )  =  ( b `
 0 ) )
6554, 56, 643eqtr3rd 2337 . . . . . . . . 9  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( b ` 
0 )  =  c )
6648, 65eqeq12d 2310 . . . . . . . 8  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( ( a `
 0 )  =  ( b `  0
)  <->  d  =  c ) )
6766biimpd 198 . . . . . . 7  |-  ( ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D ) )  /\  ( d  .~  A  /\  c  .~  A ) )  /\  ( a  e.  ( `' S " { d } )  /\  b  e.  ( `' S " { c } ) ) )  ->  ( ( a `
 0 )  =  ( b `  0
)  ->  d  =  c ) )
6867rexlimdvva 2687 . . . . . 6  |-  ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D
) )  /\  (
d  .~  A  /\  c  .~  A ) )  ->  ( E. a  e.  ( `' S " { d } ) E. b  e.  ( `' S " { c } ) ( a `
 0 )  =  ( b `  0
)  ->  d  =  c ) )
6928, 68syl5 28 . . . . 5  |-  ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D
) )  /\  (
d  .~  A  /\  c  .~  A ) )  ->  ( d  .~  c  ->  d  =  c ) )
7027, 69mpd 14 . . . 4  |-  ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D
) )  /\  (
d  .~  A  /\  c  .~  A ) )  ->  d  =  c )
7170ex 423 . . 3  |-  ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D
) )  ->  (
( d  .~  A  /\  c  .~  A )  ->  d  =  c ) )
7271ralrimivva 2648 . 2  |-  ( A  e.  W  ->  A. d  e.  D  A. c  e.  D  ( (
d  .~  A  /\  c  .~  A )  -> 
d  =  c ) )
73 breq1 4042 . . 3  |-  ( d  =  c  ->  (
d  .~  A  <->  c  .~  A ) )
7473reu4 2972 . 2  |-  ( E! d  e.  D  d  .~  A  <->  ( E. d  e.  D  d  .~  A  /\  A. d  e.  D  A. c  e.  D  ( (
d  .~  A  /\  c  .~  A )  -> 
d  =  c ) ) )
7522, 72, 74sylanbrc 645 1  |-  ( A  e.  W  ->  E! d  e.  D  d  .~  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   E!wreu 2558   {crab 2560    \ cdif 3162   (/)c0 3468   {csn 3653   <.cop 3656   <.cotp 3657   U_ciun 3921   class class class wbr 4039    e. cmpt 4093    _I cid 4320    X. cxp 4703   `'ccnv 4704   dom cdm 4705   ran crn 4706   "cima 4708    Fn wfn 5266   -onto->wfo 5269   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1oc1o 6488   2oc2o 6489    Er wer 6673   0cc0 8753   1c1 8754    - cmin 9053   ...cfz 10798  ..^cfzo 10886   #chash 11353  Word cword 11419   splice csplice 11423   <"cs2 11507   ~FG cefg 15031
This theorem is referenced by:  efgred2  15078  frgpnabllem2  15178
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-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
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-ot 3663  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  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-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-2o 6496  df-oadd 6499  df-er 6676  df-ec 6678  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  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-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-hash 11354  df-word 11425  df-concat 11426  df-s1 11427  df-substr 11428  df-splice 11429  df-s2 11514  df-efg 15034
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