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Theorem efgrelexlemb 15337
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 15311 . 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 4959 . . . . . . 7  |-  Rel  L
87a1i 11 . . . . . 6  |-  (  T. 
->  Rel  L )
9 simpr 448 . . . . . . 7  |-  ( (  T.  /\  f L g )  ->  f L g )
10 eqcom 2406 . . . . . . . . . 10  |-  ( ( a `  0 )  =  ( b ` 
0 )  <->  ( b `  0 )  =  ( a `  0
) )
11102rexbii 2693 . . . . . . . . 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 2829 . . . . . . . . 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 241 . . . . . . . 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 15336 . . . . . . . 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 15336 . . . . . . . 8  |-  ( g L f  <->  E. b  e.  ( `' S " { g } ) E. a  e.  ( `' S " { f } ) ( b `
 0 )  =  ( a `  0
) )
1813, 16, 173bitr4i 269 . . . . . . 7  |-  ( f L g  <->  g L
f )
199, 18sylib 189 . . . . . 6  |-  ( (  T.  /\  f L g )  ->  g L f )
201, 2, 3, 4, 14, 15, 6efgrelexlema 15336 . . . . . . . . 9  |-  ( g L h  <->  E. r  e.  ( `' S " { g } ) E. s  e.  ( `' S " { h } ) ( r `
 0 )  =  ( s `  0
) )
21 reeanv 2835 . . . . . . . . . 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 15326 . . . . . . . . . . . . . . . . . . . 20  |-  S : dom  S -onto-> W
23 fofn 5614 . . . . . . . . . . . . . . . . . . . 20  |-  ( S : dom  S -onto-> W  ->  S  Fn  dom  S
)
2422, 23ax-mp 8 . . . . . . . . . . . . . . . . . . 19  |-  S  Fn  dom  S
25 fniniseg 5810 . . . . . . . . . . . . . . . . . . 19  |-  ( S  Fn  dom  S  -> 
( r  e.  ( `' S " { g } )  <->  ( r  e.  dom  S  /\  ( S `  r )  =  g ) ) )
2624, 25ax-mp 8 . . . . . . . . . . . . . . . . . 18  |-  ( r  e.  ( `' S " { g } )  <-> 
( r  e.  dom  S  /\  ( S `  r )  =  g ) )
27 fniniseg 5810 . . . . . . . . . . . . . . . . . . 19  |-  ( S  Fn  dom  S  -> 
( b  e.  ( `' S " { g } )  <->  ( b  e.  dom  S  /\  ( S `  b )  =  g ) ) )
2824, 27ax-mp 8 . . . . . . . . . . . . . . . . . 18  |-  ( b  e.  ( `' S " { g } )  <-> 
( b  e.  dom  S  /\  ( S `  b )  =  g ) )
29 eqtr3 2423 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( S `  r
)  =  g  /\  ( S `  b )  =  g )  -> 
( S `  r
)  =  ( S `
 b ) )
301, 2, 3, 4, 14, 15efgred 15335 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( r  e.  dom  S  /\  b  e.  dom  S  /\  ( S `  r )  =  ( S `  b ) )  ->  ( r `  0 )  =  ( b `  0
) )
3130eqcomd 2409 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( r  e.  dom  S  /\  b  e.  dom  S  /\  ( S `  r )  =  ( S `  b ) )  ->  ( b `  0 )  =  ( r `  0
) )
32313expa 1153 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( r  e.  dom  S  /\  b  e.  dom  S )  /\  ( S `
 r )  =  ( S `  b
) )  ->  (
b `  0 )  =  ( r ` 
0 ) )
3329, 32sylan2 461 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( r  e.  dom  S  /\  b  e.  dom  S )  /\  ( ( S `  r )  =  g  /\  ( S `  b )  =  g ) )  ->  ( b ` 
0 )  =  ( r `  0 ) )
3433an4s 800 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( r  e.  dom  S  /\  ( S `  r )  =  g )  /\  ( b  e.  dom  S  /\  ( S `  b )  =  g ) )  ->  ( b ` 
0 )  =  ( r `  0 ) )
3526, 28, 34syl2anb 466 . . . . . . . . . . . . . . . . 17  |-  ( ( r  e.  ( `' S " { g } )  /\  b  e.  ( `' S " { g } ) )  ->  ( b `  0 )  =  ( r `  0
) )
36 eqeq2 2413 . . . . . . . . . . . . . . . . 17  |-  ( ( r `  0 )  =  ( s ` 
0 )  ->  (
( b `  0
)  =  ( r `
 0 )  <->  ( b `  0 )  =  ( s `  0
) ) )
3735, 36syl5ibcom 212 . . . . . . . . . . . . . . . 16  |-  ( ( r  e.  ( `' S " { g } )  /\  b  e.  ( `' S " { g } ) )  ->  ( (
r `  0 )  =  ( s ` 
0 )  ->  (
b `  0 )  =  ( s ` 
0 ) ) )
3837reximdv 2777 . . . . . . . . . . . . . . 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 2410 . . . . . . . . . . . . . . . . 17  |-  ( ( a `  0 )  =  ( b ` 
0 )  ->  (
( a `  0
)  =  ( s `
 0 )  <->  ( b `  0 )  =  ( s `  0
) ) )
4039rexbidv 2687 . . . . . . . . . . . . . . . 16  |-  ( ( a `  0 )  =  ( b ` 
0 )  ->  ( E. s  e.  ( `' S " { h } ) ( a `
 0 )  =  ( s `  0
)  <->  E. s  e.  ( `' S " { h } ) ( b `
 0 )  =  ( s `  0
) ) )
4140imbi2d 308 . . . . . . . . . . . . . . 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 214 . . . . . . . . . . . . . 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 2790 . . . . . . . . . . . . 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 421 . . . . . . . . . . . 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 2784 . . . . . . . . . . 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 2773 . . . . . . . . . 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 205 . . . . . . . . 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 466 . . . . . . . 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 15336 . . . . . . . 8  |-  ( f L h  <->  E. a  e.  ( `' S " { f } ) E. s  e.  ( `' S " { h } ) ( a `
 0 )  =  ( s `  0
) )
5048, 49sylibr 204 . . . . . . 7  |-  ( ( f L g  /\  g L h )  -> 
f L h )
5150adantl 453 . . . . . 6  |-  ( (  T.  /\  ( f L g  /\  g L h ) )  ->  f L h )
52 eqid 2404 . . . . . . . . . . . 12  |-  ( a `
 0 )  =  ( a `  0
)
53 fveq1 5686 . . . . . . . . . . . . . 14  |-  ( b  =  a  ->  (
b `  0 )  =  ( a ` 
0 ) )
5453eqeq2d 2415 . . . . . . . . . . . . 13  |-  ( b  =  a  ->  (
( a `  0
)  =  ( b `
 0 )  <->  ( a `  0 )  =  ( a `  0
) ) )
5554rspcev 3012 . . . . . . . . . . . 12  |-  ( ( a  e.  ( `' S " { f } )  /\  (
a `  0 )  =  ( a ` 
0 ) )  ->  E. b  e.  ( `' S " { f } ) ( a `
 0 )  =  ( b `  0
) )
5652, 55mpan2 653 . . . . . . . . . . 11  |-  ( a  e.  ( `' S " { f } )  ->  E. b  e.  ( `' S " { f } ) ( a `
 0 )  =  ( b `  0
) )
5756pm4.71i 614 . . . . . . . . . 10  |-  ( a  e.  ( `' S " { f } )  <-> 
( a  e.  ( `' S " { f } )  /\  E. b  e.  ( `' S " { f } ) ( a ` 
0 )  =  ( b `  0 ) ) )
58 fniniseg 5810 . . . . . . . . . . 11  |-  ( S  Fn  dom  S  -> 
( a  e.  ( `' S " { f } )  <->  ( a  e.  dom  S  /\  ( S `  a )  =  f ) ) )
5924, 58ax-mp 8 . . . . . . . . . 10  |-  ( a  e.  ( `' S " { f } )  <-> 
( a  e.  dom  S  /\  ( S `  a )  =  f ) )
6057, 59bitr3i 243 . . . . . . . . 9  |-  ( ( a  e.  ( `' S " { f } )  /\  E. b  e.  ( `' S " { f } ) ( a ` 
0 )  =  ( b `  0 ) )  <->  ( a  e. 
dom  S  /\  ( S `  a )  =  f ) )
6160rexbii2 2695 . . . . . . . 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 15336 . . . . . . . 8  |-  ( f L f  <->  E. a  e.  ( `' S " { f } ) E. b  e.  ( `' S " { f } ) ( a `
 0 )  =  ( b `  0
) )
63 forn 5615 . . . . . . . . . . 11  |-  ( S : dom  S -onto-> W  ->  ran  S  =  W )
6422, 63ax-mp 8 . . . . . . . . . 10  |-  ran  S  =  W
6564eleq2i 2468 . . . . . . . . 9  |-  ( f  e.  ran  S  <->  f  e.  W )
66 fvelrnb 5733 . . . . . . . . . 10  |-  ( S  Fn  dom  S  -> 
( f  e.  ran  S  <->  E. a  e.  dom  S ( S `  a
)  =  f ) )
6724, 66ax-mp 8 . . . . . . . . 9  |-  ( f  e.  ran  S  <->  E. a  e.  dom  S ( S `
 a )  =  f )
6865, 67bitr3i 243 . . . . . . . 8  |-  ( f  e.  W  <->  E. a  e.  dom  S ( S `
 a )  =  f )
6961, 62, 683bitr4ri 270 . . . . . . 7  |-  ( f  e.  W  <->  f L
f )
7069a1i 11 . . . . . 6  |-  (  T. 
->  ( f  e.  W  <->  f L f ) )
718, 19, 51, 70iserd 6890 . . . . 5  |-  (  T. 
->  L  Er  W
)
7271trud 1329 . . . 4  |-  L  Er  W
73 simpl 444 . . . . . . . . . . 11  |-  ( ( a  e.  W  /\  b  e.  ran  ( T `
 a ) )  ->  a  e.  W
)
74 foelrn 5847 . . . . . . . . . . 11  |-  ( ( S : dom  S -onto-> W  /\  a  e.  W
)  ->  E. r  e.  dom  S  a  =  ( S `  r
) )
7522, 73, 74sylancr 645 . . . . . . . . . 10  |-  ( ( a  e.  W  /\  b  e.  ran  ( T `
 a ) )  ->  E. r  e.  dom  S  a  =  ( S `
 r ) )
76 simprl 733 . . . . . . . . . . . . . 14  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  r  e.  dom  S )
77 simprr 734 . . . . . . . . . . . . . . 15  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  a  =  ( S `  r ) )
7877eqcomd 2409 . . . . . . . . . . . . . 14  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  ( S `  r )  =  a )
79 fniniseg 5810 . . . . . . . . . . . . . . 15  |-  ( S  Fn  dom  S  -> 
( r  e.  ( `' S " { a } )  <->  ( r  e.  dom  S  /\  ( S `  r )  =  a ) ) )
8024, 79ax-mp 8 . . . . . . . . . . . . . 14  |-  ( r  e.  ( `' S " { a } )  <-> 
( r  e.  dom  S  /\  ( S `  r )  =  a ) )
8176, 78, 80sylanbrc 646 . . . . . . . . . . . . 13  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  r  e.  ( `' S " { a } ) )
82 simplr 732 . . . . . . . . . . . . . . . . 17  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  b  e.  ran  ( T `  a ) )
8377fveq2d 5691 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  ( T `  a )  =  ( T `  ( S `
 r ) ) )
8483rneqd 5056 . . . . . . . . . . . . . . . . 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 2480 . . . . . . . . . . . . . . . 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 15324 . . . . . . . . . . . . . . . 16  |-  ( ( r  e.  dom  S  /\  b  e.  ran  ( T `  ( S `
 r ) ) )  ->  ( r concat  <" b "> )  e.  dom  S )
8776, 85, 86syl2anc 643 . . . . . . . . . . . . . . 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 15317 . . . . . . . . . . . . . . . . . . 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 972 . . . . . . . . . . . . . . . . . 18  |-  ( r  e.  dom  S  -> 
r  e.  (Word  W  \  { (/) } ) )
9089ad2antrl 709 . . . . . . . . . . . . . . . . 17  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  r  e.  (Word 
W  \  { (/) } ) )
9190eldifad 3292 . . . . . . . . . . . . . . . 16  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  r  e. Word  W
)
921, 2, 3, 4efgtf 15309 . . . . . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . . . . . . . 19  |-  ( a  e.  W  ->  ( T `  a ) : ( ( 0 ... ( # `  a
) )  X.  (
I  X.  2o ) ) --> W )
94 frn 5556 . . . . . . . . . . . . . . . . . . 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 3308 . . . . . . . . . . . . . . . . 17  |-  ( ( a  e.  W  /\  b  e.  ran  ( T `
 a ) )  ->  b  e.  W
)
9796adantr 452 . . . . . . . . . . . . . . . 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 15320 . . . . . . . . . . . . . . . 16  |-  ( ( r  e. Word  W  /\  b  e.  W  /\  ( r concat  <" b "> )  e.  dom  S )  ->  ( S `  ( r concat  <" b "> ) )  =  b )
9991, 97, 87, 98syl3anc 1184 . . . . . . . . . . . . . . 15  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  ( S `  ( r concat  <" b "> ) )  =  b )
100 fniniseg 5810 . . . . . . . . . . . . . . . 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 8 . . . . . . . . . . . . . . 15  |-  ( ( r concat  <" b "> )  e.  ( `' S " { b } )  <->  ( (
r concat  <" b "> )  e.  dom  S  /\  ( S `  ( r concat  <" b "> ) )  =  b ) )
10287, 99, 101sylanbrc 646 . . . . . . . . . . . . . 14  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  ( r concat  <" b "> )  e.  ( `' S " { b } ) )
10397s1cld 11711 . . . . . . . . . . . . . . . 16  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  <" b ">  e. Word  W )
104 eldifsn 3887 . . . . . . . . . . . . . . . . . . 19  |-  ( r  e.  (Word  W  \  { (/) } )  <->  ( r  e. Word  W  /\  r  =/=  (/) ) )
105 lennncl 11691 . . . . . . . . . . . . . . . . . . 19  |-  ( ( r  e. Word  W  /\  r  =/=  (/) )  ->  ( # `
 r )  e.  NN )
106104, 105sylbi 188 . . . . . . . . . . . . . . . . . 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 11125 . . . . . . . . . . . . . . . . 17  |-  ( 0  e.  ( 0..^ (
# `  r )
)  <->  ( # `  r
)  e.  NN )
109107, 108sylibr 204 . . . . . . . . . . . . . . . 16  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  0  e.  ( 0..^ ( # `  r
) ) )
110 ccatval1 11700 . . . . . . . . . . . . . . . 16  |-  ( ( r  e. Word  W  /\  <" b ">  e. Word  W  /\  0  e.  ( 0..^ ( # `  r ) ) )  ->  ( ( r concat  <" b "> ) `  0 )  =  ( r ` 
0 ) )
11191, 103, 109, 110syl3anc 1184 . . . . . . . . . . . . . . 15  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  ( ( r concat  <" b "> ) `  0 )  =  ( r ` 
0 ) )
112111eqcomd 2409 . . . . . . . . . . . . . 14  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  ( r ` 
0 )  =  ( ( r concat  <" b "> ) `  0
) )
113 fveq1 5686 . . . . . . . . . . . . . . . 16  |-  ( s  =  ( r concat  <" b "> )  ->  ( s `  0
)  =  ( ( r concat  <" b "> ) `  0
) )
114113eqeq2d 2415 . . . . . . . . . . . . . . 15  |-  ( s  =  ( r concat  <" b "> )  ->  ( ( r ` 
0 )  =  ( s `  0 )  <-> 
( r `  0
)  =  ( ( r concat  <" b "> ) `  0
) ) )
115114rspcev 3012 . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . 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 519 . . . . . . . . . . . 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 424 . . . . . . . . . . 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 2775 . . . . . . . . . 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 15336 . . . . . . . . 9  |-  ( a L b  <->  E. r  e.  ( `' S " { a } ) E. s  e.  ( `' S " { b } ) ( r `
 0 )  =  ( s `  0
) )
122120, 121sylibr 204 . . . . . . . 8  |-  ( ( a  e.  W  /\  b  e.  ran  ( T `
 a ) )  ->  a L b )
123 vex 2919 . . . . . . . . 9  |-  b  e. 
_V
124 vex 2919 . . . . . . . . 9  |-  a  e. 
_V
125123, 124elec 6903 . . . . . . . 8  |-  ( b  e.  [ a ] L  <->  a L b )
126122, 125sylibr 204 . . . . . . 7  |-  ( ( a  e.  W  /\  b  e.  ran  ( T `
 a ) )  ->  b  e.  [
a ] L )
127126ex 424 . . . . . 6  |-  ( a  e.  W  ->  (
b  e.  ran  ( T `  a )  ->  b  e.  [ a ] L ) )
128127ssrdv 3314 . . . . 5  |-  ( a  e.  W  ->  ran  ( T `  a ) 
C_  [ a ] L )
129128rgen 2731 . . . 4  |-  A. a  e.  W  ran  ( T `
 a )  C_  [ a ] L
130 fvex 5701 . . . . . . 7  |-  (  _I 
` Word  ( I  X.  2o ) )  e.  _V
1311, 130eqeltri 2474 . . . . . 6  |-  W  e. 
_V
132 erex 6888 . . . . . 6  |-  ( L  Er  W  ->  ( W  e.  _V  ->  L  e.  _V ) )
13372, 131, 132mp2 9 . . . . 5  |-  L  e. 
_V
134 ereq1 6871 . . . . . 6  |-  ( r  =  L  ->  (
r  Er  W  <->  L  Er  W ) )
135 eceq2 6901 . . . . . . . 8  |-  ( r  =  L  ->  [ a ] r  =  [
a ] L )
136135sseq2d 3336 . . . . . . 7  |-  ( r  =  L  ->  ( ran  ( T `  a
)  C_  [ a ] r  <->  ran  ( T `
 a )  C_  [ a ] L ) )
137136ralbidv 2686 . . . . . 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 692 . . . . 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 3042 . . . 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 887 . . 3  |-  L  e. 
{ r  |  ( r  Er  W  /\  A. a  e.  W  ran  ( T `  a ) 
C_  [ a ] r ) }
141 intss1 4025 . . 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 8 . 2  |-  |^| { r  |  ( r  Er  W  /\  A. a  e.  W  ran  ( T `
 a )  C_  [ a ] r ) }  C_  L
1435, 142eqsstri 3338 1  |-  .~  C_  L
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    T. wtru 1322    = wceq 1649    e. wcel 1721   {cab 2390    =/= wne 2567   A.wral 2666   E.wrex 2667   {crab 2670   _Vcvv 2916    \ cdif 3277    C_ wss 3280   (/)c0 3588   {csn 3774   <.cop 3777   <.cotp 3778   |^|cint 4010   U_ciun 4053   class class class wbr 4172   {copab 4225    e. cmpt 4226    _I cid 4453    X. cxp 4835   `'ccnv 4836   dom cdm 4837   ran crn 4838   "cima 4840   Rel wrel 4842    Fn wfn 5408   -->wf 5409   -onto->wfo 5411   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1oc1o 6676   2oc2o 6677    Er wer 6861   [cec 6862   0cc0 8946   1c1 8947    - cmin 9247   NNcn 9956   ...cfz 10999  ..^cfzo 11090   #chash 11573  Word cword 11672   concat cconcat 11673   <"cs1 11674   splice csplice 11676   <"cs2 11760   ~FG cefg 15293
This theorem is referenced by:  efgrelex  15338
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-ot 3784  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-ec 6866  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-hash 11574  df-word 11678  df-concat 11679  df-s1 11680  df-substr 11681  df-splice 11682  df-s2 11767  df-efg 15296
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