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Theorem efgredeu 17094
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 17081 . . . 4  |-  S : dom  S -onto-> W
8 foelrn 6028 . . . 4  |-  ( ( S : dom  S -onto-> W  /\  A  e.  W
)  ->  E. a  e.  dom  S  A  =  ( S `  a
) )
97, 8mpan 668 . . 3  |-  ( A  e.  W  ->  E. a  e.  dom  S  A  =  ( S `  a
) )
101, 2, 3, 4, 5, 6efgsdm 17072 . . . . . . . 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 1013 . . . . . . 7  |-  ( a  e.  dom  S  -> 
( a `  0
)  e.  D )
1211adantl 464 . . . . . 6  |-  ( ( A  e.  W  /\  a  e.  dom  S )  ->  ( a ` 
0 )  e.  D
)
131, 2, 3, 4, 5, 6efgsrel 17076 . . . . . . 7  |-  ( a  e.  dom  S  -> 
( a `  0
)  .~  ( S `  a ) )
1413adantl 464 . . . . . 6  |-  ( ( A  e.  W  /\  a  e.  dom  S )  ->  ( a ` 
0 )  .~  ( S `  a )
)
15 breq1 4398 . . . . . . 7  |-  ( d  =  ( a ` 
0 )  ->  (
d  .~  ( S `  a )  <->  ( a `  0 )  .~  ( S `  a ) ) )
1615rspcev 3160 . . . . . 6  |-  ( ( ( a `  0
)  e.  D  /\  ( a `  0
)  .~  ( S `  a ) )  ->  E. d  e.  D  d  .~  ( S `  a ) )
1712, 14, 16syl2anc 659 . . . . 5  |-  ( ( A  e.  W  /\  a  e.  dom  S )  ->  E. d  e.  D  d  .~  ( S `  a ) )
18 breq2 4399 . . . . . 6  |-  ( A  =  ( S `  a )  ->  (
d  .~  A  <->  d  .~  ( S `  a ) ) )
1918rexbidv 2918 . . . . 5  |-  ( A  =  ( S `  a )  ->  ( E. d  e.  D  d  .~  A  <->  E. d  e.  D  d  .~  ( S `  a ) ) )
2017, 19syl5ibrcom 222 . . . 4  |-  ( ( A  e.  W  /\  a  e.  dom  S )  ->  ( A  =  ( S `  a
)  ->  E. d  e.  D  d  .~  A ) )
2120rexlimdva 2896 . . 3  |-  ( A  e.  W  ->  ( E. a  e.  dom  S  A  =  ( S `
 a )  ->  E. d  e.  D  d  .~  A ) )
229, 21mpd 15 . 2  |-  ( A  e.  W  ->  E. d  e.  D  d  .~  A )
231, 2efger 17060 . . . . . . 7  |-  .~  Er  W
2423a1i 11 . . . . . 6  |-  ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D
) )  /\  (
d  .~  A  /\  c  .~  A ) )  ->  .~  Er  W
)
25 simprl 756 . . . . . 6  |-  ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D
) )  /\  (
d  .~  A  /\  c  .~  A ) )  ->  d  .~  A
)
26 simprr 758 . . . . . 6  |-  ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D
) )  /\  (
d  .~  A  /\  c  .~  A ) )  ->  c  .~  A
)
2724, 25, 26ertr4d 7367 . . . . 5  |-  ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D
) )  /\  (
d  .~  A  /\  c  .~  A ) )  ->  d  .~  c
)
281, 2, 3, 4, 5, 6efgrelex 17093 . . . . . 6  |-  ( d  .~  c  ->  E. a  e.  ( `' S " { d } ) E. b  e.  ( `' S " { c } ) ( a `
 0 )  =  ( b `  0
) )
29 fofn 5780 . . . . . . . . . . . . . 14  |-  ( S : dom  S -onto-> W  ->  S  Fn  dom  S
)
30 fniniseg 5986 . . . . . . . . . . . . . 14  |-  ( S  Fn  dom  S  -> 
( a  e.  ( `' S " { d } )  <->  ( a  e.  dom  S  /\  ( S `  a )  =  d ) ) )
317, 29, 30mp2b 10 . . . . . . . . . . . . 13  |-  ( a  e.  ( `' S " { d } )  <-> 
( a  e.  dom  S  /\  ( S `  a )  =  d ) )
3231simplbi 458 . . . . . . . . . . . 12  |-  ( a  e.  ( `' S " { d } )  ->  a  e.  dom  S )
3332ad2antrl 726 . . . . . . . . . . 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 17073 . . . . . . . . . . 11  |-  ( a  e.  dom  S  -> 
( S `  a
)  =  ( a `
 ( ( # `  a )  -  1 ) ) )
3533, 34syl 17 . . . . . . . . . 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 462 . . . . . . . . . . 11  |-  ( a  e.  ( `' S " { d } )  ->  ( S `  a )  =  d )
3736ad2antrl 726 . . . . . . . . . 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 761 . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . 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 2490 . . . . . . . . . . . . . 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 17078 . . . . . . . . . . . . . . 15  |-  ( a  e.  dom  S  -> 
( ( S `  a )  e.  D  <->  (
# `  a )  =  1 ) )
4233, 41syl 17 . . . . . . . . . . . . . 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 210 . . . . . . . . . . . . 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 6293 . . . . . . . . . . . 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 10645 . . . . . . . . . . . 12  |-  ( 1  -  1 )  =  0
4644, 45syl6eq 2459 . . . . . . . . . . 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 5853 . . . . . . . . . 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 2452 . . . . . . . . 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 5986 . . . . . . . . . . . . . 14  |-  ( S  Fn  dom  S  -> 
( b  e.  ( `' S " { c } )  <->  ( b  e.  dom  S  /\  ( S `  b )  =  c ) ) )
507, 29, 49mp2b 10 . . . . . . . . . . . . 13  |-  ( b  e.  ( `' S " { c } )  <-> 
( b  e.  dom  S  /\  ( S `  b )  =  c ) )
5150simplbi 458 . . . . . . . . . . . 12  |-  ( b  e.  ( `' S " { c } )  ->  b  e.  dom  S )
5251ad2antll 727 . . . . . . . . . . 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 17073 . . . . . . . . . . 11  |-  ( b  e.  dom  S  -> 
( S `  b
)  =  ( b `
 ( ( # `  b )  -  1 ) ) )
5452, 53syl 17 . . . . . . . . . 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 462 . . . . . . . . . . 11  |-  ( b  e.  ( `' S " { c } )  ->  ( S `  b )  =  c )
5655ad2antll 727 . . . . . . . . . 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 461 . . . . . . . . . . . . . . 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 2490 . . . . . . . . . . . . . 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 17078 . . . . . . . . . . . . . . 15  |-  ( b  e.  dom  S  -> 
( ( S `  b )  e.  D  <->  (
# `  b )  =  1 ) )
6052, 59syl 17 . . . . . . . . . . . . . 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 210 . . . . . . . . . . . . 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 6293 . . . . . . . . . . . 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 2459 . . . . . . . . . . 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 5853 . . . . . . . . . 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 2452 . . . . . . . . 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 2424 . . . . . . . 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 207 . . . . . . 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 2903 . . . . . 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 30 . . . . 5  |-  ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D
) )  /\  (
d  .~  A  /\  c  .~  A ) )  ->  ( d  .~  c  ->  d  =  c ) )
7027, 69mpd 15 . . . 4  |-  ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D
) )  /\  (
d  .~  A  /\  c  .~  A ) )  ->  d  =  c )
7170ex 432 . . 3  |-  ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D
) )  ->  (
( d  .~  A  /\  c  .~  A )  ->  d  =  c ) )
7271ralrimivva 2825 . 2  |-  ( A  e.  W  ->  A. d  e.  D  A. c  e.  D  ( (
d  .~  A  /\  c  .~  A )  -> 
d  =  c ) )
73 breq1 4398 . . 3  |-  ( d  =  c  ->  (
d  .~  A  <->  c  .~  A ) )
7473reu4 3243 . 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 662 1  |-  ( A  e.  W  ->  E! d  e.  D  d  .~  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2754   E.wrex 2755   E!wreu 2756   {crab 2758    \ cdif 3411   (/)c0 3738   {csn 3972   <.cop 3978   <.cotp 3980   U_ciun 4271   class class class wbr 4395    |-> cmpt 4453    _I cid 4733    X. cxp 4821   `'ccnv 4822   dom cdm 4823   ran crn 4824   "cima 4826    Fn wfn 5564   -onto->wfo 5567   ` cfv 5569  (class class class)co 6278    |-> cmpt2 6280   1oc1o 7160   2oc2o 7161    Er wer 7345   0cc0 9522   1c1 9523    - cmin 9841   ...cfz 11726  ..^cfzo 11854   #chash 12452  Word cword 12583   splice csplice 12588   <"cs2 12862   ~FG cefg 17048
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 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
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 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-ot 3981  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-ec 7350  df-map 7459  df-pm 7460  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-fz 11727  df-fzo 11855  df-hash 12453  df-word 12591  df-concat 12593  df-s1 12594  df-substr 12595  df-splice 12596  df-s2 12869  df-efg 17051
This theorem is referenced by:  efgred2  17095  frgpnabllem2  17202
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