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Theorem efgrelexlemb 15387
Description: If two words  A ,  B are related under the free group equivalence, then there exist two extension sequences  a ,  b such that  a ends at  A,  b ends at  B, and  a and  B have the same starting point. (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 ) ) )
efgrelexlem.1  |-  L  =  { <. i ,  j
>.  |  E. c  e.  ( `' S " { i } ) E. d  e.  ( `' S " { j } ) ( c `
 0 )  =  ( d `  0
) }
Assertion
Ref Expression
efgrelexlemb  |-  .~  C_  L
Distinct variable groups:    c, d,
i, j    y, z    n, c, t, v, w, y, z, m, x    M, c    i, m, n, t, v, w, x, M, j    k, c, T, i, j, m, t, x    W, c   
k, d, m, n, t, v, w, x, y, z, W, i, j    .~ , c, d, i, j, m, t, x, y, z    S, c, d, i, j    I,
c, i, j, m, n, t, v, w, x, y, z    D, c, d, i, j, m, t
Allowed substitution hints:    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)    L( x, y, z, w, v, t, i, j, k, m, n, c, d)    M( y, z, k, d)

Proof of Theorem efgrelexlemb
Dummy variables  a 
b  f  g  h  r  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efgval.w . . 3  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
2 efgval.r . . 3  |-  .~  =  ( ~FG  `  I )
3 efgval2.m . . 3  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
4 efgval2.t . . 3  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
51, 2, 3, 4efgval2 15361 . 2  |-  .~  =  |^| { r  |  ( r  Er  W  /\  A. a  e.  W  ran  ( T `  a ) 
C_  [ a ] r ) }
6 efgrelexlem.1 . . . . . . . 8  |-  L  =  { <. i ,  j
>.  |  E. c  e.  ( `' S " { i } ) E. d  e.  ( `' S " { j } ) ( c `
 0 )  =  ( d `  0
) }
76relopabi 5003 . . . . . . 7  |-  Rel  L
87a1i 11 . . . . . 6  |-  (  T. 
->  Rel  L )
9 simpr 449 . . . . . . 7  |-  ( (  T.  /\  f L g )  ->  f L g )
10 eqcom 2440 . . . . . . . . . 10  |-  ( ( a `  0 )  =  ( b ` 
0 )  <->  ( b `  0 )  =  ( a `  0
) )
11102rexbii 2734 . . . . . . . . 9  |-  ( E. a  e.  ( `' S " { f } ) E. b  e.  ( `' S " { g } ) ( a `  0
)  =  ( b `
 0 )  <->  E. a  e.  ( `' S " { f } ) E. b  e.  ( `' S " { g } ) ( b `
 0 )  =  ( a `  0
) )
12 rexcom 2871 . . . . . . . . 9  |-  ( E. a  e.  ( `' S " { f } ) E. b  e.  ( `' S " { g } ) ( b `  0
)  =  ( a `
 0 )  <->  E. b  e.  ( `' S " { g } ) E. a  e.  ( `' S " { f } ) ( b `
 0 )  =  ( a `  0
) )
1311, 12bitri 242 . . . . . . . 8  |-  ( E. a  e.  ( `' S " { f } ) E. b  e.  ( `' S " { g } ) ( a `  0
)  =  ( b `
 0 )  <->  E. b  e.  ( `' S " { g } ) E. a  e.  ( `' S " { f } ) ( b `
 0 )  =  ( a `  0
) )
14 efgred.d . . . . . . . . 9  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
15 efgred.s . . . . . . . . 9  |-  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 ) ) )
161, 2, 3, 4, 14, 15, 6efgrelexlema 15386 . . . . . . . 8  |-  ( f L g  <->  E. a  e.  ( `' S " { f } ) E. b  e.  ( `' S " { g } ) ( a `
 0 )  =  ( b `  0
) )
171, 2, 3, 4, 14, 15, 6efgrelexlema 15386 . . . . . . . 8  |-  ( g L f  <->  E. b  e.  ( `' S " { g } ) E. a  e.  ( `' S " { f } ) ( b `
 0 )  =  ( a `  0
) )
1813, 16, 173bitr4i 270 . . . . . . 7  |-  ( f L g  <->  g L
f )
199, 18sylib 190 . . . . . 6  |-  ( (  T.  /\  f L g )  ->  g L f )
201, 2, 3, 4, 14, 15, 6efgrelexlema 15386 . . . . . . . . 9  |-  ( g L h  <->  E. r  e.  ( `' S " { g } ) E. s  e.  ( `' S " { h } ) ( r `
 0 )  =  ( s `  0
) )
21 reeanv 2877 . . . . . . . . . 10  |-  ( E. a  e.  ( `' S " { f } ) E. r  e.  ( `' S " { g } ) ( E. b  e.  ( `' S " { g } ) ( a `  0
)  =  ( b `
 0 )  /\  E. s  e.  ( `' S " { h } ) ( r `
 0 )  =  ( s `  0
) )  <->  ( E. a  e.  ( `' S " { f } ) E. b  e.  ( `' S " { g } ) ( a `  0
)  =  ( b `
 0 )  /\  E. r  e.  ( `' S " { g } ) E. s  e.  ( `' S " { h } ) ( r `  0
)  =  ( s `
 0 ) ) )
221, 2, 3, 4, 14, 15efgsfo 15376 . . . . . . . . . . . . . . . . . . . 20  |-  S : dom  S -onto-> W
23 fofn 5658 . . . . . . . . . . . . . . . . . . . 20  |-  ( S : dom  S -onto-> W  ->  S  Fn  dom  S
)
2422, 23ax-mp 5 . . . . . . . . . . . . . . . . . . 19  |-  S  Fn  dom  S
25 fniniseg 5854 . . . . . . . . . . . . . . . . . . 19  |-  ( S  Fn  dom  S  -> 
( r  e.  ( `' S " { g } )  <->  ( r  e.  dom  S  /\  ( S `  r )  =  g ) ) )
2624, 25ax-mp 5 . . . . . . . . . . . . . . . . . 18  |-  ( r  e.  ( `' S " { g } )  <-> 
( r  e.  dom  S  /\  ( S `  r )  =  g ) )
27 fniniseg 5854 . . . . . . . . . . . . . . . . . . 19  |-  ( S  Fn  dom  S  -> 
( b  e.  ( `' S " { g } )  <->  ( b  e.  dom  S  /\  ( S `  b )  =  g ) ) )
2824, 27ax-mp 5 . . . . . . . . . . . . . . . . . 18  |-  ( b  e.  ( `' S " { g } )  <-> 
( b  e.  dom  S  /\  ( S `  b )  =  g ) )
29 eqtr3 2457 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( S `  r
)  =  g  /\  ( S `  b )  =  g )  -> 
( S `  r
)  =  ( S `
 b ) )
301, 2, 3, 4, 14, 15efgred 15385 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( r  e.  dom  S  /\  b  e.  dom  S  /\  ( S `  r )  =  ( S `  b ) )  ->  ( r `  0 )  =  ( b `  0
) )
3130eqcomd 2443 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( r  e.  dom  S  /\  b  e.  dom  S  /\  ( S `  r )  =  ( S `  b ) )  ->  ( b `  0 )  =  ( r `  0
) )
32313expa 1154 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( r  e.  dom  S  /\  b  e.  dom  S )  /\  ( S `
 r )  =  ( S `  b
) )  ->  (
b `  0 )  =  ( r ` 
0 ) )
3329, 32sylan2 462 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( r  e.  dom  S  /\  b  e.  dom  S )  /\  ( ( S `  r )  =  g  /\  ( S `  b )  =  g ) )  ->  ( b ` 
0 )  =  ( r `  0 ) )
3433an4s 801 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( r  e.  dom  S  /\  ( S `  r )  =  g )  /\  ( b  e.  dom  S  /\  ( S `  b )  =  g ) )  ->  ( b ` 
0 )  =  ( r `  0 ) )
3526, 28, 34syl2anb 467 . . . . . . . . . . . . . . . . 17  |-  ( ( r  e.  ( `' S " { g } )  /\  b  e.  ( `' S " { g } ) )  ->  ( b `  0 )  =  ( r `  0
) )
36 eqeq2 2447 . . . . . . . . . . . . . . . . 17  |-  ( ( r `  0 )  =  ( s ` 
0 )  ->  (
( b `  0
)  =  ( r `
 0 )  <->  ( b `  0 )  =  ( s `  0
) ) )
3735, 36syl5ibcom 213 . . . . . . . . . . . . . . . 16  |-  ( ( r  e.  ( `' S " { g } )  /\  b  e.  ( `' S " { g } ) )  ->  ( (
r `  0 )  =  ( s ` 
0 )  ->  (
b `  0 )  =  ( s ` 
0 ) ) )
3837reximdv 2819 . . . . . . . . . . . . . . 15  |-  ( ( r  e.  ( `' S " { g } )  /\  b  e.  ( `' S " { g } ) )  ->  ( E. s  e.  ( `' S " { h }
) ( r ` 
0 )  =  ( s `  0 )  ->  E. s  e.  ( `' S " { h } ) ( b `
 0 )  =  ( s `  0
) ) )
39 eqeq1 2444 . . . . . . . . . . . . . . . . 17  |-  ( ( a `  0 )  =  ( b ` 
0 )  ->  (
( a `  0
)  =  ( s `
 0 )  <->  ( b `  0 )  =  ( s `  0
) ) )
4039rexbidv 2728 . . . . . . . . . . . . . . . 16  |-  ( ( a `  0 )  =  ( b ` 
0 )  ->  ( E. s  e.  ( `' S " { h } ) ( a `
 0 )  =  ( s `  0
)  <->  E. s  e.  ( `' S " { h } ) ( b `
 0 )  =  ( s `  0
) ) )
4140imbi2d 309 . . . . . . . . . . . . . . 15  |-  ( ( a `  0 )  =  ( b ` 
0 )  ->  (
( E. s  e.  ( `' S " { h } ) ( r `  0
)  =  ( s `
 0 )  ->  E. s  e.  ( `' S " { h } ) ( a `
 0 )  =  ( s `  0
) )  <->  ( E. s  e.  ( `' S " { h }
) ( r ` 
0 )  =  ( s `  0 )  ->  E. s  e.  ( `' S " { h } ) ( b `
 0 )  =  ( s `  0
) ) ) )
4238, 41syl5ibrcom 215 . . . . . . . . . . . . . 14  |-  ( ( r  e.  ( `' S " { g } )  /\  b  e.  ( `' S " { g } ) )  ->  ( (
a `  0 )  =  ( b ` 
0 )  ->  ( E. s  e.  ( `' S " { h } ) ( r `
 0 )  =  ( s `  0
)  ->  E. s  e.  ( `' S " { h } ) ( a `  0
)  =  ( s `
 0 ) ) ) )
4342rexlimdva 2832 . . . . . . . . . . . . 13  |-  ( r  e.  ( `' S " { g } )  ->  ( E. b  e.  ( `' S " { g } ) ( a `  0
)  =  ( b `
 0 )  -> 
( E. s  e.  ( `' S " { h } ) ( r `  0
)  =  ( s `
 0 )  ->  E. s  e.  ( `' S " { h } ) ( a `
 0 )  =  ( s `  0
) ) ) )
4443imp3a 422 . . . . . . . . . . . 12  |-  ( r  e.  ( `' S " { g } )  ->  ( ( E. b  e.  ( `' S " { g } ) ( a `
 0 )  =  ( b `  0
)  /\  E. s  e.  ( `' S " { h } ) ( r `  0
)  =  ( s `
 0 ) )  ->  E. s  e.  ( `' S " { h } ) ( a `
 0 )  =  ( s `  0
) ) )
4544rexlimiv 2826 . . . . . . . . . . 11  |-  ( E. r  e.  ( `' S " { g } ) ( E. b  e.  ( `' S " { g } ) ( a `
 0 )  =  ( b `  0
)  /\  E. s  e.  ( `' S " { h } ) ( r `  0
)  =  ( s `
 0 ) )  ->  E. s  e.  ( `' S " { h } ) ( a `
 0 )  =  ( s `  0
) )
4645reximi 2815 . . . . . . . . . 10  |-  ( E. a  e.  ( `' S " { f } ) E. r  e.  ( `' S " { g } ) ( E. b  e.  ( `' S " { g } ) ( a `  0
)  =  ( b `
 0 )  /\  E. s  e.  ( `' S " { h } ) ( r `
 0 )  =  ( s `  0
) )  ->  E. a  e.  ( `' S " { f } ) E. s  e.  ( `' S " { h } ) ( a `
 0 )  =  ( s `  0
) )
4721, 46sylbir 206 . . . . . . . . 9  |-  ( ( E. a  e.  ( `' S " { f } ) E. b  e.  ( `' S " { g } ) ( a `  0
)  =  ( b `
 0 )  /\  E. r  e.  ( `' S " { g } ) E. s  e.  ( `' S " { h } ) ( r `  0
)  =  ( s `
 0 ) )  ->  E. a  e.  ( `' S " { f } ) E. s  e.  ( `' S " { h } ) ( a `  0
)  =  ( s `
 0 ) )
4816, 20, 47syl2anb 467 . . . . . . . 8  |-  ( ( f L g  /\  g L h )  ->  E. a  e.  ( `' S " { f } ) E. s  e.  ( `' S " { h } ) ( a `  0
)  =  ( s `
 0 ) )
491, 2, 3, 4, 14, 15, 6efgrelexlema 15386 . . . . . . . 8  |-  ( f L h  <->  E. a  e.  ( `' S " { f } ) E. s  e.  ( `' S " { h } ) ( a `
 0 )  =  ( s `  0
) )
5048, 49sylibr 205 . . . . . . 7  |-  ( ( f L g  /\  g L h )  -> 
f L h )
5150adantl 454 . . . . . 6  |-  ( (  T.  /\  ( f L g  /\  g L h ) )  ->  f L h )
52 eqid 2438 . . . . . . . . . . . 12  |-  ( a `
 0 )  =  ( a `  0
)
53 fveq1 5730 . . . . . . . . . . . . . 14  |-  ( b  =  a  ->  (
b `  0 )  =  ( a ` 
0 ) )
5453eqeq2d 2449 . . . . . . . . . . . . 13  |-  ( b  =  a  ->  (
( a `  0
)  =  ( b `
 0 )  <->  ( a `  0 )  =  ( a `  0
) ) )
5554rspcev 3054 . . . . . . . . . . . 12  |-  ( ( a  e.  ( `' S " { f } )  /\  (
a `  0 )  =  ( a ` 
0 ) )  ->  E. b  e.  ( `' S " { f } ) ( a `
 0 )  =  ( b `  0
) )
5652, 55mpan2 654 . . . . . . . . . . 11  |-  ( a  e.  ( `' S " { f } )  ->  E. b  e.  ( `' S " { f } ) ( a `
 0 )  =  ( b `  0
) )
5756pm4.71i 615 . . . . . . . . . 10  |-  ( a  e.  ( `' S " { f } )  <-> 
( a  e.  ( `' S " { f } )  /\  E. b  e.  ( `' S " { f } ) ( a ` 
0 )  =  ( b `  0 ) ) )
58 fniniseg 5854 . . . . . . . . . . 11  |-  ( S  Fn  dom  S  -> 
( a  e.  ( `' S " { f } )  <->  ( a  e.  dom  S  /\  ( S `  a )  =  f ) ) )
5924, 58ax-mp 5 . . . . . . . . . 10  |-  ( a  e.  ( `' S " { f } )  <-> 
( a  e.  dom  S  /\  ( S `  a )  =  f ) )
6057, 59bitr3i 244 . . . . . . . . 9  |-  ( ( a  e.  ( `' S " { f } )  /\  E. b  e.  ( `' S " { f } ) ( a ` 
0 )  =  ( b `  0 ) )  <->  ( a  e. 
dom  S  /\  ( S `  a )  =  f ) )
6160rexbii2 2736 . . . . . . . 8  |-  ( E. a  e.  ( `' S " { f } ) E. b  e.  ( `' S " { f } ) ( a `  0
)  =  ( b `
 0 )  <->  E. a  e.  dom  S ( S `
 a )  =  f )
621, 2, 3, 4, 14, 15, 6efgrelexlema 15386 . . . . . . . 8  |-  ( f L f  <->  E. a  e.  ( `' S " { f } ) E. b  e.  ( `' S " { f } ) ( a `
 0 )  =  ( b `  0
) )
63 forn 5659 . . . . . . . . . . 11  |-  ( S : dom  S -onto-> W  ->  ran  S  =  W )
6422, 63ax-mp 5 . . . . . . . . . 10  |-  ran  S  =  W
6564eleq2i 2502 . . . . . . . . 9  |-  ( f  e.  ran  S  <->  f  e.  W )
66 fvelrnb 5777 . . . . . . . . . 10  |-  ( S  Fn  dom  S  -> 
( f  e.  ran  S  <->  E. a  e.  dom  S ( S `  a
)  =  f ) )
6724, 66ax-mp 5 . . . . . . . . 9  |-  ( f  e.  ran  S  <->  E. a  e.  dom  S ( S `
 a )  =  f )
6865, 67bitr3i 244 . . . . . . . 8  |-  ( f  e.  W  <->  E. a  e.  dom  S ( S `
 a )  =  f )
6961, 62, 683bitr4ri 271 . . . . . . 7  |-  ( f  e.  W  <->  f L
f )
7069a1i 11 . . . . . 6  |-  (  T. 
->  ( f  e.  W  <->  f L f ) )
718, 19, 51, 70iserd 6934 . . . . 5  |-  (  T. 
->  L  Er  W
)
7271trud 1333 . . . 4  |-  L  Er  W
73 simpl 445 . . . . . . . . . . 11  |-  ( ( a  e.  W  /\  b  e.  ran  ( T `
 a ) )  ->  a  e.  W
)
74 foelrn 5891 . . . . . . . . . . 11  |-  ( ( S : dom  S -onto-> W  /\  a  e.  W
)  ->  E. r  e.  dom  S  a  =  ( S `  r
) )
7522, 73, 74sylancr 646 . . . . . . . . . 10  |-  ( ( a  e.  W  /\  b  e.  ran  ( T `
 a ) )  ->  E. r  e.  dom  S  a  =  ( S `
 r ) )
76 simprl 734 . . . . . . . . . . . . . 14  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  r  e.  dom  S )
77 simprr 735 . . . . . . . . . . . . . . 15  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  a  =  ( S `  r ) )
7877eqcomd 2443 . . . . . . . . . . . . . 14  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  ( S `  r )  =  a )
79 fniniseg 5854 . . . . . . . . . . . . . . 15  |-  ( S  Fn  dom  S  -> 
( r  e.  ( `' S " { a } )  <->  ( r  e.  dom  S  /\  ( S `  r )  =  a ) ) )
8024, 79ax-mp 5 . . . . . . . . . . . . . 14  |-  ( r  e.  ( `' S " { a } )  <-> 
( r  e.  dom  S  /\  ( S `  r )  =  a ) )
8176, 78, 80sylanbrc 647 . . . . . . . . . . . . 13  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  r  e.  ( `' S " { a } ) )
82 simplr 733 . . . . . . . . . . . . . . . . 17  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  b  e.  ran  ( T `  a ) )
8377fveq2d 5735 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  ( T `  a )  =  ( T `  ( S `
 r ) ) )
8483rneqd 5100 . . . . . . . . . . . . . . . . 17  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  ran  ( T `  a )  =  ran  ( T `  ( S `
 r ) ) )
8582, 84eleqtrd 2514 . . . . . . . . . . . . . . . 16  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  b  e.  ran  ( T `  ( S `
 r ) ) )
861, 2, 3, 4, 14, 15efgsp1 15374 . . . . . . . . . . . . . . . 16  |-  ( ( r  e.  dom  S  /\  b  e.  ran  ( T `  ( S `
 r ) ) )  ->  ( r concat  <" b "> )  e.  dom  S )
8776, 85, 86syl2anc 644 . . . . . . . . . . . . . . 15  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  ( r concat  <" b "> )  e.  dom  S )
881, 2, 3, 4, 14, 15efgsdm 15367 . . . . . . . . . . . . . . . . . . 19  |-  ( r  e.  dom  S  <->  ( r  e.  (Word  W  \  { (/)
} )  /\  (
r `  0 )  e.  D  /\  A. i  e.  ( 1..^ ( # `  r ) ) ( r `  i )  e.  ran  ( T `
 ( r `  ( i  -  1 ) ) ) ) )
8988simp1bi 973 . . . . . . . . . . . . . . . . . 18  |-  ( r  e.  dom  S  -> 
r  e.  (Word  W  \  { (/) } ) )
9089ad2antrl 710 . . . . . . . . . . . . . . . . 17  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  r  e.  (Word 
W  \  { (/) } ) )
9190eldifad 3334 . . . . . . . . . . . . . . . 16  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  r  e. Word  W
)
921, 2, 3, 4efgtf 15359 . . . . . . . . . . . . . . . . . . . 20  |-  ( a  e.  W  ->  (
( T `  a
)  =  ( f  e.  ( 0 ... ( # `  a
) ) ,  g  e.  ( I  X.  2o )  |->  ( a splice  <. f ,  f , 
<" g ( M `
 g ) "> >. ) )  /\  ( T `  a ) : ( ( 0 ... ( # `  a
) )  X.  (
I  X.  2o ) ) --> W ) )
9392simprd 451 . . . . . . . . . . . . . . . . . . 19  |-  ( a  e.  W  ->  ( T `  a ) : ( ( 0 ... ( # `  a
) )  X.  (
I  X.  2o ) ) --> W )
94 frn 5600 . . . . . . . . . . . . . . . . . . 19  |-  ( ( T `  a ) : ( ( 0 ... ( # `  a
) )  X.  (
I  X.  2o ) ) --> W  ->  ran  ( T `  a ) 
C_  W )
9593, 94syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( a  e.  W  ->  ran  ( T `  a ) 
C_  W )
9695sselda 3350 . . . . . . . . . . . . . . . . 17  |-  ( ( a  e.  W  /\  b  e.  ran  ( T `
 a ) )  ->  b  e.  W
)
9796adantr 453 . . . . . . . . . . . . . . . 16  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  b  e.  W
)
981, 2, 3, 4, 14, 15efgsval2 15370 . . . . . . . . . . . . . . . 16  |-  ( ( r  e. Word  W  /\  b  e.  W  /\  ( r concat  <" b "> )  e.  dom  S )  ->  ( S `  ( r concat  <" b "> ) )  =  b )
9991, 97, 87, 98syl3anc 1185 . . . . . . . . . . . . . . 15  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  ( S `  ( r concat  <" b "> ) )  =  b )
100 fniniseg 5854 . . . . . . . . . . . . . . . 16  |-  ( S  Fn  dom  S  -> 
( ( r concat  <" b "> )  e.  ( `' S " { b } )  <-> 
( ( r concat  <" b "> )  e.  dom  S  /\  ( S `  ( r concat  <" b "> ) )  =  b ) ) )
10124, 100ax-mp 5 . . . . . . . . . . . . . . 15  |-  ( ( r concat  <" b "> )  e.  ( `' S " { b } )  <->  ( (
r concat  <" b "> )  e.  dom  S  /\  ( S `  ( r concat  <" b "> ) )  =  b ) )
10287, 99, 101sylanbrc 647 . . . . . . . . . . . . . 14  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  ( r concat  <" b "> )  e.  ( `' S " { b } ) )
10397s1cld 11761 . . . . . . . . . . . . . . . 16  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  <" b ">  e. Word  W )
104 eldifsn 3929 . . . . . . . . . . . . . . . . . . 19  |-  ( r  e.  (Word  W  \  { (/) } )  <->  ( r  e. Word  W  /\  r  =/=  (/) ) )
105 lennncl 11741 . . . . . . . . . . . . . . . . . . 19  |-  ( ( r  e. Word  W  /\  r  =/=  (/) )  ->  ( # `
 r )  e.  NN )
106104, 105sylbi 189 . . . . . . . . . . . . . . . . . 18  |-  ( r  e.  (Word  W  \  { (/) } )  -> 
( # `  r )  e.  NN )
10790, 106syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  ( # `  r
)  e.  NN )
108 lbfzo0 11175 . . . . . . . . . . . . . . . . 17  |-  ( 0  e.  ( 0..^ (
# `  r )
)  <->  ( # `  r
)  e.  NN )
109107, 108sylibr 205 . . . . . . . . . . . . . . . 16  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  0  e.  ( 0..^ ( # `  r
) ) )
110 ccatval1 11750 . . . . . . . . . . . . . . . 16  |-  ( ( r  e. Word  W  /\  <" b ">  e. Word  W  /\  0  e.  ( 0..^ ( # `  r ) ) )  ->  ( ( r concat  <" b "> ) `  0 )  =  ( r ` 
0 ) )
11191, 103, 109, 110syl3anc 1185 . . . . . . . . . . . . . . 15  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  ( ( r concat  <" b "> ) `  0 )  =  ( r ` 
0 ) )
112111eqcomd 2443 . . . . . . . . . . . . . 14  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  ( r ` 
0 )  =  ( ( r concat  <" b "> ) `  0
) )
113 fveq1 5730 . . . . . . . . . . . . . . . 16  |-  ( s  =  ( r concat  <" b "> )  ->  ( s `  0
)  =  ( ( r concat  <" b "> ) `  0
) )
114113eqeq2d 2449 . . . . . . . . . . . . . . 15  |-  ( s  =  ( r concat  <" b "> )  ->  ( ( r ` 
0 )  =  ( s `  0 )  <-> 
( r `  0
)  =  ( ( r concat  <" b "> ) `  0
) ) )
115114rspcev 3054 . . . . . . . . . . . . . 14  |-  ( ( ( r concat  <" b "> )  e.  ( `' S " { b } )  /\  (
r `  0 )  =  ( ( r concat  <" b "> ) `  0 )
)  ->  E. s  e.  ( `' S " { b } ) ( r `  0
)  =  ( s `
 0 ) )
116102, 112, 115syl2anc 644 . . . . . . . . . . . . 13  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  E. s  e.  ( `' S " { b } ) ( r `
 0 )  =  ( s `  0
) )
11781, 116jca 520 . . . . . . . . . . . 12  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  ( r  e.  ( `' S " { a } )  /\  E. s  e.  ( `' S " { b } ) ( r `  0
)  =  ( s `
 0 ) ) )
118117ex 425 . . . . . . . . . . 11  |-  ( ( a  e.  W  /\  b  e.  ran  ( T `
 a ) )  ->  ( ( r  e.  dom  S  /\  a  =  ( S `  r ) )  -> 
( r  e.  ( `' S " { a } )  /\  E. s  e.  ( `' S " { b } ) ( r ` 
0 )  =  ( s `  0 ) ) ) )
119118reximdv2 2817 . . . . . . . . . 10  |-  ( ( a  e.  W  /\  b  e.  ran  ( T `
 a ) )  ->  ( E. r  e.  dom  S  a  =  ( S `  r
)  ->  E. r  e.  ( `' S " { a } ) E. s  e.  ( `' S " { b } ) ( r `
 0 )  =  ( s `  0
) ) )
12075, 119mpd 15 . . . . . . . . 9  |-  ( ( a  e.  W  /\  b  e.  ran  ( T `
 a ) )  ->  E. r  e.  ( `' S " { a } ) E. s  e.  ( `' S " { b } ) ( r `  0
)  =  ( s `
 0 ) )
1211, 2, 3, 4, 14, 15, 6efgrelexlema 15386 . . . . . . . . 9  |-  ( a L b  <->  E. r  e.  ( `' S " { a } ) E. s  e.  ( `' S " { b } ) ( r `
 0 )  =  ( s `  0
) )
122120, 121sylibr 205 . . . . . . . 8  |-  ( ( a  e.  W  /\  b  e.  ran  ( T `
 a ) )  ->  a L b )
123 vex 2961 . . . . . . . . 9  |-  b  e. 
_V
124 vex 2961 . . . . . . . . 9  |-  a  e. 
_V
125123, 124elec 6947 . . . . . . . 8  |-  ( b  e.  [ a ] L  <->  a L b )
126122, 125sylibr 205 . . . . . . 7  |-  ( ( a  e.  W  /\  b  e.  ran  ( T `
 a ) )  ->  b  e.  [
a ] L )
127126ex 425 . . . . . 6  |-  ( a  e.  W  ->  (
b  e.  ran  ( T `  a )  ->  b  e.  [ a ] L ) )
128127ssrdv 3356 . . . . 5  |-  ( a  e.  W  ->  ran  ( T `  a ) 
C_  [ a ] L )
129128rgen 2773 . . . 4  |-  A. a  e.  W  ran  ( T `
 a )  C_  [ a ] L
130 fvex 5745 . . . . . . 7  |-  (  _I 
` Word  ( I  X.  2o ) )  e.  _V
1311, 130eqeltri 2508 . . . . . 6  |-  W  e. 
_V
132 erex 6932 . . . . . 6  |-  ( L  Er  W  ->  ( W  e.  _V  ->  L  e.  _V ) )
13372, 131, 132mp2 9 . . . . 5  |-  L  e. 
_V
134 ereq1 6915 . . . . . 6  |-  ( r  =  L  ->  (
r  Er  W  <->  L  Er  W ) )
135 eceq2 6945 . . . . . . . 8  |-  ( r  =  L  ->  [ a ] r  =  [
a ] L )
136135sseq2d 3378 . . . . . . 7  |-  ( r  =  L  ->  ( ran  ( T `  a
)  C_  [ a ] r  <->  ran  ( T `
 a )  C_  [ a ] L ) )
137136ralbidv 2727 . . . . . 6  |-  ( r  =  L  ->  ( A. a  e.  W  ran  ( T `  a
)  C_  [ a ] r  <->  A. a  e.  W  ran  ( T `
 a )  C_  [ a ] L ) )
138134, 137anbi12d 693 . . . . 5  |-  ( r  =  L  ->  (
( r  Er  W  /\  A. a  e.  W  ran  ( T `  a
)  C_  [ a ] r )  <->  ( L  Er  W  /\  A. a  e.  W  ran  ( T `
 a )  C_  [ a ] L ) ) )
139133, 138elab 3084 . . . 4  |-  ( L  e.  { r  |  ( r  Er  W  /\  A. a  e.  W  ran  ( T `  a
)  C_  [ a ] r ) }  <-> 
( L  Er  W  /\  A. a  e.  W  ran  ( T `  a
)  C_  [ a ] L ) )
14072, 129, 139mpbir2an 888 . . 3  |-  L  e. 
{ r  |  ( r  Er  W  /\  A. a  e.  W  ran  ( T `  a ) 
C_  [ a ] r ) }
141 intss1 4067 . . 3  |-  ( L  e.  { r  |  ( r  Er  W  /\  A. a  e.  W  ran  ( T `  a
)  C_  [ a ] r ) }  ->  |^| { r  |  ( r  Er  W  /\  A. a  e.  W  ran  ( T `  a
)  C_  [ a ] r ) } 
C_  L )
142140, 141ax-mp 5 . 2  |-  |^| { r  |  ( r  Er  W  /\  A. a  e.  W  ran  ( T `
 a )  C_  [ a ] r ) }  C_  L
1435, 142eqsstri 3380 1  |-  .~  C_  L
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    T. wtru 1326    = wceq 1653    e. wcel 1726   {cab 2424    =/= wne 2601   A.wral 2707   E.wrex 2708   {crab 2711   _Vcvv 2958    \ cdif 3319    C_ wss 3322   (/)c0 3630   {csn 3816   <.cop 3819   <.cotp 3820   |^|cint 4052   U_ciun 4095   class class class wbr 4215   {copab 4268    e. cmpt 4269    _I cid 4496    X. cxp 4879   `'ccnv 4880   dom cdm 4881   ran crn 4882   "cima 4884   Rel wrel 4886    Fn wfn 5452   -->wf 5453   -onto->wfo 5455   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   1oc1o 6720   2oc2o 6721    Er wer 6905   [cec 6906   0cc0 8995   1c1 8996    - cmin 9296   NNcn 10005   ...cfz 11048  ..^cfzo 11140   #chash 11623  Word cword 11722   concat cconcat 11723   <"cs1 11724   splice csplice 11726   <"cs2 11810   ~FG cefg 15343
This theorem is referenced by:  efgrelex  15388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-ot 3826  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-ec 6910  df-map 7023  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fz 11049  df-fzo 11141  df-hash 11624  df-word 11728  df-concat 11729  df-s1 11730  df-substr 11731  df-splice 11732  df-s2 11817  df-efg 15346
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