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Theorem efgrelexlemb 15059
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 15033 . 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 4811 . . . . . . 7  |-  Rel  L
87a1i 10 . . . . . 6  |-  (  T. 
->  Rel  L )
9 simpr 447 . . . . . . 7  |-  ( (  T.  /\  f L g )  ->  f L g )
10 eqcom 2285 . . . . . . . . . 10  |-  ( ( a `  0 )  =  ( b ` 
0 )  <->  ( b `  0 )  =  ( a `  0
) )
11102rexbii 2570 . . . . . . . . 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 2701 . . . . . . . . 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 240 . . . . . . . 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 15058 . . . . . . . 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 15058 . . . . . . . 8  |-  ( g L f  <->  E. b  e.  ( `' S " { g } ) E. a  e.  ( `' S " { f } ) ( b `
 0 )  =  ( a `  0
) )
1813, 16, 173bitr4i 268 . . . . . . 7  |-  ( f L g  <->  g L
f )
199, 18sylib 188 . . . . . 6  |-  ( (  T.  /\  f L g )  ->  g L f )
201, 2, 3, 4, 14, 15, 6efgrelexlema 15058 . . . . . . . . 9  |-  ( g L h  <->  E. r  e.  ( `' S " { g } ) E. s  e.  ( `' S " { h } ) ( r `
 0 )  =  ( s `  0
) )
21 reeanv 2707 . . . . . . . . . 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 15048 . . . . . . . . . . . . . . . . . . . 20  |-  S : dom  S -onto-> W
23 fofn 5453 . . . . . . . . . . . . . . . . . . . 20  |-  ( S : dom  S -onto-> W  ->  S  Fn  dom  S
)
2422, 23ax-mp 8 . . . . . . . . . . . . . . . . . . 19  |-  S  Fn  dom  S
25 fniniseg 5646 . . . . . . . . . . . . . . . . . . 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 5646 . . . . . . . . . . . . . . . . . . 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 2302 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( S `  r
)  =  g  /\  ( S `  b )  =  g )  -> 
( S `  r
)  =  ( S `
 b ) )
301, 2, 3, 4, 14, 15efgred 15057 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( r  e.  dom  S  /\  b  e.  dom  S  /\  ( S `  r )  =  ( S `  b ) )  ->  ( r `  0 )  =  ( b `  0
) )
3130eqcomd 2288 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( r  e.  dom  S  /\  b  e.  dom  S  /\  ( S `  r )  =  ( S `  b ) )  ->  ( b `  0 )  =  ( r `  0
) )
32313expa 1151 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( r  e.  dom  S  /\  b  e.  dom  S )  /\  ( S `
 r )  =  ( S `  b
) )  ->  (
b `  0 )  =  ( r ` 
0 ) )
3329, 32sylan2 460 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( r  e.  dom  S  /\  b  e.  dom  S )  /\  ( ( S `  r )  =  g  /\  ( S `  b )  =  g ) )  ->  ( b ` 
0 )  =  ( r `  0 ) )
3433an4s 799 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( r  e.  dom  S  /\  ( S `  r )  =  g )  /\  ( b  e.  dom  S  /\  ( S `  b )  =  g ) )  ->  ( b ` 
0 )  =  ( r `  0 ) )
3526, 28, 34syl2anb 465 . . . . . . . . . . . . . . . . 17  |-  ( ( r  e.  ( `' S " { g } )  /\  b  e.  ( `' S " { g } ) )  ->  ( b `  0 )  =  ( r `  0
) )
36 eqeq2 2292 . . . . . . . . . . . . . . . . 17  |-  ( ( r `  0 )  =  ( s ` 
0 )  ->  (
( b `  0
)  =  ( r `
 0 )  <->  ( b `  0 )  =  ( s `  0
) ) )
3735, 36syl5ibcom 211 . . . . . . . . . . . . . . . 16  |-  ( ( r  e.  ( `' S " { g } )  /\  b  e.  ( `' S " { g } ) )  ->  ( (
r `  0 )  =  ( s ` 
0 )  ->  (
b `  0 )  =  ( s ` 
0 ) ) )
3837reximdv 2654 . . . . . . . . . . . . . . 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 2289 . . . . . . . . . . . . . . . . 17  |-  ( ( a `  0 )  =  ( b ` 
0 )  ->  (
( a `  0
)  =  ( s `
 0 )  <->  ( b `  0 )  =  ( s `  0
) ) )
4039rexbidv 2564 . . . . . . . . . . . . . . . 16  |-  ( ( a `  0 )  =  ( b ` 
0 )  ->  ( E. s  e.  ( `' S " { h } ) ( a `
 0 )  =  ( s `  0
)  <->  E. s  e.  ( `' S " { h } ) ( b `
 0 )  =  ( s `  0
) ) )
4140imbi2d 307 . . . . . . . . . . . . . . 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 213 . . . . . . . . . . . . . 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 2667 . . . . . . . . . . . . 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 420 . . . . . . . . . . . 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 2661 . . . . . . . . . . 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 2650 . . . . . . . . . 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 204 . . . . . . . . 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 465 . . . . . . . 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 15058 . . . . . . . 8  |-  ( f L h  <->  E. a  e.  ( `' S " { f } ) E. s  e.  ( `' S " { h } ) ( a `
 0 )  =  ( s `  0
) )
5048, 49sylibr 203 . . . . . . 7  |-  ( ( f L g  /\  g L h )  -> 
f L h )
5150adantl 452 . . . . . 6  |-  ( (  T.  /\  ( f L g  /\  g L h ) )  ->  f L h )
52 eqid 2283 . . . . . . . . . . . 12  |-  ( a `
 0 )  =  ( a `  0
)
53 fveq1 5524 . . . . . . . . . . . . . 14  |-  ( b  =  a  ->  (
b `  0 )  =  ( a ` 
0 ) )
5453eqeq2d 2294 . . . . . . . . . . . . 13  |-  ( b  =  a  ->  (
( a `  0
)  =  ( b `
 0 )  <->  ( a `  0 )  =  ( a `  0
) ) )
5554rspcev 2884 . . . . . . . . . . . 12  |-  ( ( a  e.  ( `' S " { f } )  /\  (
a `  0 )  =  ( a ` 
0 ) )  ->  E. b  e.  ( `' S " { f } ) ( a `
 0 )  =  ( b `  0
) )
5652, 55mpan2 652 . . . . . . . . . . 11  |-  ( a  e.  ( `' S " { f } )  ->  E. b  e.  ( `' S " { f } ) ( a `
 0 )  =  ( b `  0
) )
5756pm4.71i 613 . . . . . . . . . 10  |-  ( a  e.  ( `' S " { f } )  <-> 
( a  e.  ( `' S " { f } )  /\  E. b  e.  ( `' S " { f } ) ( a ` 
0 )  =  ( b `  0 ) ) )
58 fniniseg 5646 . . . . . . . . . . 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 242 . . . . . . . . 9  |-  ( ( a  e.  ( `' S " { f } )  /\  E. b  e.  ( `' S " { f } ) ( a ` 
0 )  =  ( b `  0 ) )  <->  ( a  e. 
dom  S  /\  ( S `  a )  =  f ) )
6160rexbii2 2572 . . . . . . . 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 15058 . . . . . . . 8  |-  ( f L f  <->  E. a  e.  ( `' S " { f } ) E. b  e.  ( `' S " { f } ) ( a `
 0 )  =  ( b `  0
) )
63 forn 5454 . . . . . . . . . . 11  |-  ( S : dom  S -onto-> W  ->  ran  S  =  W )
6422, 63ax-mp 8 . . . . . . . . . 10  |-  ran  S  =  W
6564eleq2i 2347 . . . . . . . . 9  |-  ( f  e.  ran  S  <->  f  e.  W )
66 fvelrnb 5570 . . . . . . . . . 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 242 . . . . . . . 8  |-  ( f  e.  W  <->  E. a  e.  dom  S ( S `
 a )  =  f )
6961, 62, 683bitr4ri 269 . . . . . . 7  |-  ( f  e.  W  <->  f L
f )
7069a1i 10 . . . . . 6  |-  (  T. 
->  ( f  e.  W  <->  f L f ) )
718, 19, 51, 70iserd 6686 . . . . 5  |-  (  T. 
->  L  Er  W
)
7271trud 1314 . . . 4  |-  L  Er  W
73 simpl 443 . . . . . . . . . . 11  |-  ( ( a  e.  W  /\  b  e.  ran  ( T `
 a ) )  ->  a  e.  W
)
74 foelrn 5679 . . . . . . . . . . 11  |-  ( ( S : dom  S -onto-> W  /\  a  e.  W
)  ->  E. r  e.  dom  S  a  =  ( S `  r
) )
7522, 73, 74sylancr 644 . . . . . . . . . 10  |-  ( ( a  e.  W  /\  b  e.  ran  ( T `
 a ) )  ->  E. r  e.  dom  S  a  =  ( S `
 r ) )
76 simprl 732 . . . . . . . . . . . . . 14  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  r  e.  dom  S )
77 simprr 733 . . . . . . . . . . . . . . 15  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  a  =  ( S `  r ) )
7877eqcomd 2288 . . . . . . . . . . . . . 14  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  ( S `  r )  =  a )
79 fniniseg 5646 . . . . . . . . . . . . . . 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 645 . . . . . . . . . . . . 13  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  r  e.  ( `' S " { a } ) )
82 simplr 731 . . . . . . . . . . . . . . . . 17  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  b  e.  ran  ( T `  a ) )
8377fveq2d 5529 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  ( T `  a )  =  ( T `  ( S `
 r ) ) )
8483rneqd 4906 . . . . . . . . . . . . . . . . 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 2359 . . . . . . . . . . . . . . . 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 15046 . . . . . . . . . . . . . . . 16  |-  ( ( r  e.  dom  S  /\  b  e.  ran  ( T `  ( S `
 r ) ) )  ->  ( r concat  <" b "> )  e.  dom  S )
8776, 85, 86syl2anc 642 . . . . . . . . . . . . . . 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 15039 . . . . . . . . . . . . . . . . . . 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 970 . . . . . . . . . . . . . . . . . 18  |-  ( r  e.  dom  S  -> 
r  e.  (Word  W  \  { (/) } ) )
9089ad2antrl 708 . . . . . . . . . . . . . . . . 17  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  r  e.  (Word 
W  \  { (/) } ) )
91 eldifi 3298 . . . . . . . . . . . . . . . . 17  |-  ( r  e.  (Word  W  \  { (/) } )  -> 
r  e. Word  W )
9290, 91syl 15 . . . . . . . . . . . . . . . 16  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  r  e. Word  W
)
931, 2, 3, 4efgtf 15031 . . . . . . . . . . . . . . . . . . . 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 ) )
9493simprd 449 . . . . . . . . . . . . . . . . . . 19  |-  ( a  e.  W  ->  ( T `  a ) : ( ( 0 ... ( # `  a
) )  X.  (
I  X.  2o ) ) --> W )
95 frn 5395 . . . . . . . . . . . . . . . . . . 19  |-  ( ( T `  a ) : ( ( 0 ... ( # `  a
) )  X.  (
I  X.  2o ) ) --> W  ->  ran  ( T `  a ) 
C_  W )
9694, 95syl 15 . . . . . . . . . . . . . . . . . 18  |-  ( a  e.  W  ->  ran  ( T `  a ) 
C_  W )
9796sselda 3180 . . . . . . . . . . . . . . . . 17  |-  ( ( a  e.  W  /\  b  e.  ran  ( T `
 a ) )  ->  b  e.  W
)
9897adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  b  e.  W
)
991, 2, 3, 4, 14, 15efgsval2 15042 . . . . . . . . . . . . . . . 16  |-  ( ( r  e. Word  W  /\  b  e.  W  /\  ( r concat  <" b "> )  e.  dom  S )  ->  ( S `  ( r concat  <" b "> ) )  =  b )
10092, 98, 87, 99syl3anc 1182 . . . . . . . . . . . . . . 15  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  ( S `  ( r concat  <" b "> ) )  =  b )
101 fniniseg 5646 . . . . . . . . . . . . . . . 16  |-  ( S  Fn  dom  S  -> 
( ( r concat  <" b "> )  e.  ( `' S " { b } )  <-> 
( ( r concat  <" b "> )  e.  dom  S  /\  ( S `  ( r concat  <" b "> ) )  =  b ) ) )
10224, 101ax-mp 8 . . . . . . . . . . . . . . 15  |-  ( ( r concat  <" b "> )  e.  ( `' S " { b } )  <->  ( (
r concat  <" b "> )  e.  dom  S  /\  ( S `  ( r concat  <" b "> ) )  =  b ) )
10387, 100, 102sylanbrc 645 . . . . . . . . . . . . . 14  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  ( r concat  <" b "> )  e.  ( `' S " { b } ) )
10498s1cld 11442 . . . . . . . . . . . . . . . 16  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  <" b ">  e. Word  W )
105 eldifsn 3749 . . . . . . . . . . . . . . . . . . 19  |-  ( r  e.  (Word  W  \  { (/) } )  <->  ( r  e. Word  W  /\  r  =/=  (/) ) )
106 lennncl 11422 . . . . . . . . . . . . . . . . . . 19  |-  ( ( r  e. Word  W  /\  r  =/=  (/) )  ->  ( # `
 r )  e.  NN )
107105, 106sylbi 187 . . . . . . . . . . . . . . . . . 18  |-  ( r  e.  (Word  W  \  { (/) } )  -> 
( # `  r )  e.  NN )
10890, 107syl 15 . . . . . . . . . . . . . . . . 17  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  ( # `  r
)  e.  NN )
109 lbfzo0 10903 . . . . . . . . . . . . . . . . 17  |-  ( 0  e.  ( 0..^ (
# `  r )
)  <->  ( # `  r
)  e.  NN )
110108, 109sylibr 203 . . . . . . . . . . . . . . . 16  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  0  e.  ( 0..^ ( # `  r
) ) )
111 ccatval1 11431 . . . . . . . . . . . . . . . 16  |-  ( ( r  e. Word  W  /\  <" b ">  e. Word  W  /\  0  e.  ( 0..^ ( # `  r ) ) )  ->  ( ( r concat  <" b "> ) `  0 )  =  ( r ` 
0 ) )
11292, 104, 110, 111syl3anc 1182 . . . . . . . . . . . . . . 15  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  ( ( r concat  <" b "> ) `  0 )  =  ( r ` 
0 ) )
113112eqcomd 2288 . . . . . . . . . . . . . 14  |-  ( ( ( a  e.  W  /\  b  e.  ran  ( T `  a ) )  /\  ( r  e.  dom  S  /\  a  =  ( S `  r ) ) )  ->  ( r ` 
0 )  =  ( ( r concat  <" b "> ) `  0
) )
114 fveq1 5524 . . . . . . . . . . . . . . . 16  |-  ( s  =  ( r concat  <" b "> )  ->  ( s `  0
)  =  ( ( r concat  <" b "> ) `  0
) )
115114eqeq2d 2294 . . . . . . . . . . . . . . 15  |-  ( s  =  ( r concat  <" b "> )  ->  ( ( r ` 
0 )  =  ( s `  0 )  <-> 
( r `  0
)  =  ( ( r concat  <" b "> ) `  0
) ) )
116115rspcev 2884 . . . . . . . . . . . . . 14  |-  ( ( ( r concat  <" b "> )  e.  ( `' S " { b } )  /\  (
r `  0 )  =  ( ( r concat  <" b "> ) `  0 )
)  ->  E. s  e.  ( `' S " { b } ) ( r `  0
)  =  ( s `
 0 ) )
117103, 113, 116syl2anc 642 . . . . . . . . . . . . 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
) )
11881, 117jca 518 . . . . . . . . . . . 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 ) ) )
119118ex 423 . . . . . . . . . . 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 ) ) ) )
120119reximdv2 2652 . . . . . . . . . 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
) ) )
12175, 120mpd 14 . . . . . . . . 9  |-  ( ( a  e.  W  /\  b  e.  ran  ( T `
 a ) )  ->  E. r  e.  ( `' S " { a } ) E. s  e.  ( `' S " { b } ) ( r `  0
)  =  ( s `
 0 ) )
1221, 2, 3, 4, 14, 15, 6efgrelexlema 15058 . . . . . . . . 9  |-  ( a L b  <->  E. r  e.  ( `' S " { a } ) E. s  e.  ( `' S " { b } ) ( r `
 0 )  =  ( s `  0
) )
123121, 122sylibr 203 . . . . . . . 8  |-  ( ( a  e.  W  /\  b  e.  ran  ( T `
 a ) )  ->  a L b )
124 vex 2791 . . . . . . . . 9  |-  b  e. 
_V
125 vex 2791 . . . . . . . . 9  |-  a  e. 
_V
126124, 125elec 6699 . . . . . . . 8  |-  ( b  e.  [ a ] L  <->  a L b )
127123, 126sylibr 203 . . . . . . 7  |-  ( ( a  e.  W  /\  b  e.  ran  ( T `
 a ) )  ->  b  e.  [
a ] L )
128127ex 423 . . . . . 6  |-  ( a  e.  W  ->  (
b  e.  ran  ( T `  a )  ->  b  e.  [ a ] L ) )
129128ssrdv 3185 . . . . 5  |-  ( a  e.  W  ->  ran  ( T `  a ) 
C_  [ a ] L )
130129rgen 2608 . . . 4  |-  A. a  e.  W  ran  ( T `
 a )  C_  [ a ] L
131 fvex 5539 . . . . . . 7  |-  (  _I 
` Word  ( I  X.  2o ) )  e.  _V
1321, 131eqeltri 2353 . . . . . 6  |-  W  e. 
_V
133 erex 6684 . . . . . 6  |-  ( L  Er  W  ->  ( W  e.  _V  ->  L  e.  _V ) )
13472, 132, 133mp2 17 . . . . 5  |-  L  e. 
_V
135 ereq1 6667 . . . . . 6  |-  ( r  =  L  ->  (
r  Er  W  <->  L  Er  W ) )
136 eceq2 6697 . . . . . . . 8  |-  ( r  =  L  ->  [ a ] r  =  [
a ] L )
137136sseq2d 3206 . . . . . . 7  |-  ( r  =  L  ->  ( ran  ( T `  a
)  C_  [ a ] r  <->  ran  ( T `
 a )  C_  [ a ] L ) )
138137ralbidv 2563 . . . . . 6  |-  ( r  =  L  ->  ( A. a  e.  W  ran  ( T `  a
)  C_  [ a ] r  <->  A. a  e.  W  ran  ( T `
 a )  C_  [ a ] L ) )
139135, 138anbi12d 691 . . . . 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 ) ) )
140134, 139elab 2914 . . . 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 ) )
14172, 130, 140mpbir2an 886 . . 3  |-  L  e. 
{ r  |  ( r  Er  W  /\  A. a  e.  W  ran  ( T `  a ) 
C_  [ a ] r ) }
142 intss1 3877 . . 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 )
143141, 142ax-mp 8 . 2  |-  |^| { r  |  ( r  Er  W  /\  A. a  e.  W  ran  ( T `
 a )  C_  [ a ] r ) }  C_  L
1445, 143eqsstri 3208 1  |-  .~  C_  L
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    T. wtru 1307    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    \ cdif 3149    C_ wss 3152   (/)c0 3455   {csn 3640   <.cop 3643   <.cotp 3644   |^|cint 3862   U_ciun 3905   class class class wbr 4023   {copab 4076    e. cmpt 4077    _I cid 4304    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692   Rel wrel 4694    Fn wfn 5250   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1oc1o 6472   2oc2o 6473    Er wer 6657   [cec 6658   0cc0 8737   1c1 8738    - cmin 9037   NNcn 9746   ...cfz 10782  ..^cfzo 10870   #chash 11337  Word cword 11403   concat cconcat 11404   <"cs1 11405   splice csplice 11407   <"cs2 11491   ~FG cefg 15015
This theorem is referenced by:  efgrelex  15060
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-ot 3650  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-ec 6662  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-hash 11338  df-word 11409  df-concat 11410  df-s1 11411  df-substr 11412  df-splice 11413  df-s2 11498  df-efg 15018
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