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Theorem efgredeu 16563
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 16550 . . . 4  |-  S : dom  S -onto-> W
8 foelrn 6038 . . . 4  |-  ( ( S : dom  S -onto-> W  /\  A  e.  W
)  ->  E. a  e.  dom  S  A  =  ( S `  a
) )
97, 8mpan 670 . . 3  |-  ( A  e.  W  ->  E. a  e.  dom  S  A  =  ( S `  a
) )
101, 2, 3, 4, 5, 6efgsdm 16541 . . . . . . . 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 1012 . . . . . . 7  |-  ( a  e.  dom  S  -> 
( a `  0
)  e.  D )
1211adantl 466 . . . . . 6  |-  ( ( A  e.  W  /\  a  e.  dom  S )  ->  ( a ` 
0 )  e.  D
)
131, 2, 3, 4, 5, 6efgsrel 16545 . . . . . . 7  |-  ( a  e.  dom  S  -> 
( a `  0
)  .~  ( S `  a ) )
1413adantl 466 . . . . . 6  |-  ( ( A  e.  W  /\  a  e.  dom  S )  ->  ( a ` 
0 )  .~  ( S `  a )
)
15 breq1 4450 . . . . . . 7  |-  ( d  =  ( a ` 
0 )  ->  (
d  .~  ( S `  a )  <->  ( a `  0 )  .~  ( S `  a ) ) )
1615rspcev 3214 . . . . . 6  |-  ( ( ( a `  0
)  e.  D  /\  ( a `  0
)  .~  ( S `  a ) )  ->  E. d  e.  D  d  .~  ( S `  a ) )
1712, 14, 16syl2anc 661 . . . . 5  |-  ( ( A  e.  W  /\  a  e.  dom  S )  ->  E. d  e.  D  d  .~  ( S `  a ) )
18 breq2 4451 . . . . . 6  |-  ( A  =  ( S `  a )  ->  (
d  .~  A  <->  d  .~  ( S `  a ) ) )
1918rexbidv 2973 . . . . 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 2955 . . 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 16529 . . . . . . 7  |-  .~  Er  W
2423a1i 11 . . . . . 6  |-  ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D
) )  /\  (
d  .~  A  /\  c  .~  A ) )  ->  .~  Er  W
)
25 simprl 755 . . . . . 6  |-  ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D
) )  /\  (
d  .~  A  /\  c  .~  A ) )  ->  d  .~  A
)
26 simprr 756 . . . . . 6  |-  ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D
) )  /\  (
d  .~  A  /\  c  .~  A ) )  ->  c  .~  A
)
2724, 25, 26ertr4d 7327 . . . . 5  |-  ( ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D
) )  /\  (
d  .~  A  /\  c  .~  A ) )  ->  d  .~  c
)
281, 2, 3, 4, 5, 6efgrelex 16562 . . . . . 6  |-  ( d  .~  c  ->  E. a  e.  ( `' S " { d } ) E. b  e.  ( `' S " { c } ) ( a `
 0 )  =  ( b `  0
) )
29 fofn 5795 . . . . . . . . . . . . . 14  |-  ( S : dom  S -onto-> W  ->  S  Fn  dom  S
)
30 fniniseg 6000 . . . . . . . . . . . . . 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 460 . . . . . . . . . . . 12  |-  ( a  e.  ( `' S " { d } )  ->  a  e.  dom  S )
3332ad2antrl 727 . . . . . . . . . . 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 16542 . . . . . . . . . . 11  |-  ( a  e.  dom  S  -> 
( S `  a
)  =  ( a `
 ( ( # `  a )  -  1 ) ) )
3533, 34syl 16 . . . . . . . . . 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 464 . . . . . . . . . . 11  |-  ( a  e.  ( `' S " { d } )  ->  ( S `  a )  =  d )
3736ad2antrl 727 . . . . . . . . . 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 758 . . . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . . . 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 2555 . . . . . . . . . . . . . 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 16547 . . . . . . . . . . . . . . 15  |-  ( a  e.  dom  S  -> 
( ( S `  a )  e.  D  <->  (
# `  a )  =  1 ) )
4233, 41syl 16 . . . . . . . . . . . . . 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 6297 . . . . . . . . . . . 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 10600 . . . . . . . . . . . 12  |-  ( 1  -  1 )  =  0
4644, 45syl6eq 2524 . . . . . . . . . . 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 5868 . . . . . . . . . 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 2517 . . . . . . . . 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 6000 . . . . . . . . . . . . . 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 460 . . . . . . . . . . . 12  |-  ( b  e.  ( `' S " { c } )  ->  b  e.  dom  S )
5251ad2antll 728 . . . . . . . . . . 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 16542 . . . . . . . . . . 11  |-  ( b  e.  dom  S  -> 
( S `  b
)  =  ( b `
 ( ( # `  b )  -  1 ) ) )
5452, 53syl 16 . . . . . . . . . 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 464 . . . . . . . . . . 11  |-  ( b  e.  ( `' S " { c } )  ->  ( S `  b )  =  c )
5655ad2antll 728 . . . . . . . . . 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 463 . . . . . . . . . . . . . . 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 2555 . . . . . . . . . . . . . 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 16547 . . . . . . . . . . . . . . 15  |-  ( b  e.  dom  S  -> 
( ( S `  b )  e.  D  <->  (
# `  b )  =  1 ) )
6052, 59syl 16 . . . . . . . . . . . . . 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 6297 . . . . . . . . . . . 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 2524 . . . . . . . . . . 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 5868 . . . . . . . . . 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 2517 . . . . . . . . 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 2489 . . . . . . . 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 2962 . . . . . 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 32 . . . . 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 434 . . 3  |-  ( ( A  e.  W  /\  ( d  e.  D  /\  c  e.  D
) )  ->  (
( d  .~  A  /\  c  .~  A )  ->  d  =  c ) )
7271ralrimivva 2885 . 2  |-  ( A  e.  W  ->  A. d  e.  D  A. c  e.  D  ( (
d  .~  A  /\  c  .~  A )  -> 
d  =  c ) )
73 breq1 4450 . . 3  |-  ( d  =  c  ->  (
d  .~  A  <->  c  .~  A ) )
7473reu4 3297 . 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 664 1  |-  ( A  e.  W  ->  E! d  e.  D  d  .~  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   E!wreu 2816   {crab 2818    \ cdif 3473   (/)c0 3785   {csn 4027   <.cop 4033   <.cotp 4035   U_ciun 4325   class class class wbr 4447    |-> cmpt 4505    _I cid 4790    X. cxp 4997   `'ccnv 4998   dom cdm 4999   ran crn 5000   "cima 5002    Fn wfn 5581   -onto->wfo 5584   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   1oc1o 7120   2oc2o 7121    Er wer 7305   0cc0 9488   1c1 9489    - cmin 9801   ...cfz 11668  ..^cfzo 11788   #chash 12367  Word cword 12494   splice csplice 12499   <"cs2 12763   ~FG cefg 16517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-ec 7310  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fz 11669  df-fzo 11789  df-hash 12368  df-word 12502  df-concat 12504  df-s1 12505  df-substr 12506  df-splice 12507  df-s2 12770  df-efg 16520
This theorem is referenced by:  efgred2  16564  frgpnabllem2  16666
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