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Theorem efgsp1 16253
Description: If  F is an extension sequence and  A is an extension of the last element of  F, then  F  +  <" A "> is an extension sequence. (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
efgsp1  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( F concat  <" A "> )  e.  dom  S )
Distinct variable groups:    y, z    t, n, v, w, y, z, m, x    m, M    x, n, M, t, v, w    k, m, t, x, T    k, n, v, w, y, z, W, m, t, x    .~ , m, t, x, y, z    m, I, n, t, v, w, x, y, z    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)    F( x, y, z, w, v, t, k, m, n)    I( k)    M( y, z, k)

Proof of Theorem efgsp1
Dummy variables  a 
i are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efgval.w . . . . . . . 8  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
2 efgval.r . . . . . . . 8  |-  .~  =  ( ~FG  `  I )
3 efgval2.m . . . . . . . 8  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
4 efgval2.t . . . . . . . 8  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
5 efgred.d . . . . . . . 8  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
6 efgred.s . . . . . . . 8  |-  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, 6efgsdm 16246 . . . . . . 7  |-  ( F  e.  dom  S  <->  ( F  e.  (Word  W  \  { (/)
} )  /\  ( F `  0 )  e.  D  /\  A. i  e.  ( 1..^ ( # `  F ) ) ( F `  i )  e.  ran  ( T `
 ( F `  ( i  -  1 ) ) ) ) )
87simp1bi 1003 . . . . . 6  |-  ( F  e.  dom  S  ->  F  e.  (Word  W  \  { (/) } ) )
98adantr 465 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  F  e.  (Word  W  \  { (/) } ) )
109eldifad 3359 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  F  e. Word  W )
111, 2, 3, 4, 5, 6efgsf 16245 . . . . . . . . . . . 12  |-  S : { t  e.  (Word 
W  \  { (/) } )  |  ( ( t `
 0 )  e.  D  /\  A. k  e.  ( 1..^ ( # `  t ) ) ( t `  k )  e.  ran  ( T `
 ( t `  ( k  -  1 ) ) ) ) } --> W
1211fdmi 5583 . . . . . . . . . . . . 13  |-  dom  S  =  { t  e.  (Word 
W  \  { (/) } )  |  ( ( t `
 0 )  e.  D  /\  A. k  e.  ( 1..^ ( # `  t ) ) ( t `  k )  e.  ran  ( T `
 ( t `  ( k  -  1 ) ) ) ) }
1312feq2i 5571 . . . . . . . . . . . 12  |-  ( S : dom  S --> W  <->  S : { t  e.  (Word 
W  \  { (/) } )  |  ( ( t `
 0 )  e.  D  /\  A. k  e.  ( 1..^ ( # `  t ) ) ( t `  k )  e.  ran  ( T `
 ( t `  ( k  -  1 ) ) ) ) } --> W )
1411, 13mpbir 209 . . . . . . . . . . 11  |-  S : dom  S --> W
1514ffvelrni 5861 . . . . . . . . . 10  |-  ( F  e.  dom  S  -> 
( S `  F
)  e.  W )
1615adantr 465 . . . . . . . . 9  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( S `  F )  e.  W
)
171, 2, 3, 4efgtf 16238 . . . . . . . . 9  |-  ( ( S `  F )  e.  W  ->  (
( T `  ( S `  F )
)  =  ( a  e.  ( 0 ... ( # `  ( S `  F )
) ) ,  i  e.  ( I  X.  2o )  |->  ( ( S `  F ) splice  <. a ,  a , 
<" i ( M `
 i ) "> >. ) )  /\  ( T `  ( S `
 F ) ) : ( ( 0 ... ( # `  ( S `  F )
) )  X.  (
I  X.  2o ) ) --> W ) )
1816, 17syl 16 . . . . . . . 8  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( T `  ( S `  F ) )  =  ( a  e.  ( 0 ... ( # `  ( S `  F
) ) ) ,  i  e.  ( I  X.  2o )  |->  ( ( S `  F
) splice  <. a ,  a ,  <" i ( M `  i ) "> >. )
)  /\  ( T `  ( S `  F
) ) : ( ( 0 ... ( # `
 ( S `  F ) ) )  X.  ( I  X.  2o ) ) --> W ) )
1918simprd 463 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( T `  ( S `  F
) ) : ( ( 0 ... ( # `
 ( S `  F ) ) )  X.  ( I  X.  2o ) ) --> W )
20 frn 5584 . . . . . . 7  |-  ( ( T `  ( S `
 F ) ) : ( ( 0 ... ( # `  ( S `  F )
) )  X.  (
I  X.  2o ) ) --> W  ->  ran  ( T `  ( S `
 F ) ) 
C_  W )
2119, 20syl 16 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ran  ( T `
 ( S `  F ) )  C_  W )
22 simpr 461 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  A  e.  ran  ( T `  ( S `  F )
) )
2321, 22sseldd 3376 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  A  e.  W )
2423s1cld 12313 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  <" A ">  e. Word  W )
25 ccatcl 12293 . . . 4  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W )  ->  ( F concat  <" A "> )  e. Word  W )
2610, 24, 25syl2anc 661 . . 3  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( F concat  <" A "> )  e. Word  W )
27 ccatlen 12294 . . . . . . 7  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W )  ->  ( # `
 ( F concat  <" A "> ) )  =  ( ( # `  F
)  +  ( # `  <" A "> ) ) )
2810, 24, 27syl2anc 661 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  ( F concat  <" A "> ) )  =  ( ( # `  F
)  +  ( # `  <" A "> ) ) )
29 s1len 12315 . . . . . . 7  |-  ( # `  <" A "> )  =  1
3029oveq2i 6121 . . . . . 6  |-  ( (
# `  F )  +  ( # `  <" A "> )
)  =  ( (
# `  F )  +  1 )
3128, 30syl6eq 2491 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  ( F concat  <" A "> ) )  =  ( ( # `  F
)  +  1 ) )
32 lencl 12268 . . . . . 6  |-  ( F  e. Word  W  ->  ( # `
 F )  e. 
NN0 )
33 nn0p1nn 10638 . . . . . 6  |-  ( (
# `  F )  e.  NN0  ->  ( ( # `
 F )  +  1 )  e.  NN )
3410, 32, 333syl 20 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( # `
 F )  +  1 )  e.  NN )
3531, 34eqeltrd 2517 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  ( F concat  <" A "> ) )  e.  NN )
36 wrdfin 12267 . . . . 5  |-  ( ( F concat  <" A "> )  e. Word  W  -> 
( F concat  <" A "> )  e.  Fin )
37 hashnncl 12153 . . . . 5  |-  ( ( F concat  <" A "> )  e.  Fin  ->  ( ( # `  ( F concat  <" A "> ) )  e.  NN  <->  ( F concat  <" A "> )  =/=  (/) ) )
3826, 36, 373syl 20 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( # `
 ( F concat  <" A "> ) )  e.  NN  <->  ( F concat  <" A "> )  =/=  (/) ) )
3935, 38mpbid 210 . . 3  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( F concat  <" A "> )  =/=  (/) )
40 eldifsn 4019 . . 3  |-  ( ( F concat  <" A "> )  e.  (Word  W  \  { (/) } )  <-> 
( ( F concat  <" A "> )  e. Word  W  /\  ( F concat  <" A "> )  =/=  (/) ) )
4126, 39, 40sylanbrc 664 . 2  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( F concat  <" A "> )  e.  (Word  W  \  { (/) } ) )
42 eldifsni 4020 . . . . . . 7  |-  ( F  e.  (Word  W  \  { (/) } )  ->  F  =/=  (/) )
439, 42syl 16 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  F  =/=  (/) )
44 wrdfin 12267 . . . . . . 7  |-  ( F  e. Word  W  ->  F  e.  Fin )
45 hashnncl 12153 . . . . . . 7  |-  ( F  e.  Fin  ->  (
( # `  F )  e.  NN  <->  F  =/=  (/) ) )
4610, 44, 453syl 20 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( # `
 F )  e.  NN  <->  F  =/=  (/) ) )
4743, 46mpbird 232 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  F
)  e.  NN )
48 lbfzo0 11605 . . . . 5  |-  ( 0  e.  ( 0..^ (
# `  F )
)  <->  ( # `  F
)  e.  NN )
4947, 48sylibr 212 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  0  e.  ( 0..^ ( # `  F
) ) )
50 ccatval1 12295 . . . 4  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  0  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( F concat  <" A "> ) `  0 )  =  ( F ` 
0 ) )
5110, 24, 49, 50syl3anc 1218 . . 3  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  0 )  =  ( F ` 
0 ) )
527simp2bi 1004 . . . 4  |-  ( F  e.  dom  S  -> 
( F `  0
)  e.  D )
5352adantr 465 . . 3  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( F `  0 )  e.  D )
5451, 53eqeltrd 2517 . 2  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  0 )  e.  D )
557simp3bi 1005 . . . . . 6  |-  ( F  e.  dom  S  ->  A. i  e.  (
1..^ ( # `  F
) ) ( F `
 i )  e. 
ran  ( T `  ( F `  ( i  -  1 ) ) ) )
5655adantr 465 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  A. i  e.  ( 1..^ ( # `  F ) ) ( F `  i )  e.  ran  ( T `
 ( F `  ( i  -  1 ) ) ) )
57 1nn0 10614 . . . . . . . . . . . . 13  |-  1  e.  NN0
58 nn0uz 10914 . . . . . . . . . . . . 13  |-  NN0  =  ( ZZ>= `  0 )
5957, 58eleqtri 2515 . . . . . . . . . . . 12  |-  1  e.  ( ZZ>= `  0 )
60 fzoss1 11595 . . . . . . . . . . . 12  |-  ( 1  e.  ( ZZ>= `  0
)  ->  ( 1..^ ( # `  F
) )  C_  (
0..^ ( # `  F
) ) )
6159, 60ax-mp 5 . . . . . . . . . . 11  |-  ( 1..^ ( # `  F
) )  C_  (
0..^ ( # `  F
) )
6261sseli 3371 . . . . . . . . . 10  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  i  e.  ( 0..^ ( # `  F
) ) )
63 ccatval1 12295 . . . . . . . . . 10  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  i  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( F concat  <" A "> ) `  i )  =  ( F `  i ) )
6462, 63syl3an3 1253 . . . . . . . . 9  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  i  e.  ( 1..^ ( # `  F ) ) )  ->  ( ( F concat  <" A "> ) `  i )  =  ( F `  i ) )
65 elfzoel2 11571 . . . . . . . . . . . . . . . 16  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( # `  F
)  e.  ZZ )
66 peano2zm 10707 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  e.  ZZ  ->  ( ( # `
 F )  - 
1 )  e.  ZZ )
6765, 66syl 16 . . . . . . . . . . . . . . 15  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( ( # `
 F )  - 
1 )  e.  ZZ )
6865zred 10766 . . . . . . . . . . . . . . . 16  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( # `  F
)  e.  RR )
6968lem1d 10285 . . . . . . . . . . . . . . 15  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( ( # `
 F )  - 
1 )  <_  ( # `
 F ) )
70 eluz2 10886 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  e.  ( ZZ>= `  ( ( # `
 F )  - 
1 ) )  <->  ( (
( # `  F )  -  1 )  e.  ZZ  /\  ( # `  F )  e.  ZZ  /\  ( ( # `  F
)  -  1 )  <_  ( # `  F
) ) )
7167, 65, 69, 70syl3anbrc 1172 . . . . . . . . . . . . . 14  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( # `  F
)  e.  ( ZZ>= `  ( ( # `  F
)  -  1 ) ) )
72 fzoss2 11596 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  e.  ( ZZ>= `  ( ( # `
 F )  - 
1 ) )  -> 
( 0..^ ( (
# `  F )  -  1 ) ) 
C_  ( 0..^ (
# `  F )
) )
7371, 72syl 16 . . . . . . . . . . . . 13  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( 0..^ ( ( # `  F
)  -  1 ) )  C_  ( 0..^ ( # `  F
) ) )
74 elfzoelz 11572 . . . . . . . . . . . . . . 15  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  i  e.  ZZ )
75 elfzom1b 11645 . . . . . . . . . . . . . . 15  |-  ( ( i  e.  ZZ  /\  ( # `  F )  e.  ZZ )  -> 
( i  e.  ( 1..^ ( # `  F
) )  <->  ( i  -  1 )  e.  ( 0..^ ( (
# `  F )  -  1 ) ) ) )
7674, 65, 75syl2anc 661 . . . . . . . . . . . . . 14  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( i  e.  ( 1..^ ( # `  F ) )  <->  ( i  -  1 )  e.  ( 0..^ ( (
# `  F )  -  1 ) ) ) )
7776ibi 241 . . . . . . . . . . . . 13  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( i  -  1 )  e.  ( 0..^ ( (
# `  F )  -  1 ) ) )
7873, 77sseldd 3376 . . . . . . . . . . . 12  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( i  -  1 )  e.  ( 0..^ ( # `  F ) ) )
79 ccatval1 12295 . . . . . . . . . . . 12  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  ( i  -  1 )  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( F concat  <" A "> ) `  ( i  -  1 ) )  =  ( F `  ( i  -  1 ) ) )
8078, 79syl3an3 1253 . . . . . . . . . . 11  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  i  e.  ( 1..^ ( # `  F ) ) )  ->  ( ( F concat  <" A "> ) `  ( i  -  1 ) )  =  ( F `  ( i  -  1 ) ) )
8180fveq2d 5714 . . . . . . . . . 10  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  i  e.  ( 1..^ ( # `  F ) ) )  ->  ( T `  ( ( F concat  <" A "> ) `  (
i  -  1 ) ) )  =  ( T `  ( F `
 ( i  - 
1 ) ) ) )
8281rneqd 5086 . . . . . . . . 9  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  i  e.  ( 1..^ ( # `  F ) ) )  ->  ran  ( T `  ( ( F concat  <" A "> ) `  (
i  -  1 ) ) )  =  ran  ( T `  ( F `
 ( i  - 
1 ) ) ) )
8364, 82eleq12d 2511 . . . . . . . 8  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  i  e.  ( 1..^ ( # `  F ) ) )  ->  ( ( ( F concat  <" A "> ) `  i )  e.  ran  ( T `
 ( ( F concat  <" A "> ) `  ( i  -  1 ) ) )  <->  ( F `  i )  e.  ran  ( T `  ( F `
 ( i  - 
1 ) ) ) ) )
84833expa 1187 . . . . . . 7  |-  ( ( ( F  e. Word  W  /\  <" A ">  e. Word  W )  /\  i  e.  ( 1..^ ( # `  F
) ) )  -> 
( ( ( F concat  <" A "> ) `  i )  e.  ran  ( T `  ( ( F concat  <" A "> ) `  (
i  -  1 ) ) )  <->  ( F `  i )  e.  ran  ( T `  ( F `
 ( i  - 
1 ) ) ) ) )
8584ralbidva 2750 . . . . . 6  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W )  ->  ( A. i  e.  (
1..^ ( # `  F
) ) ( ( F concat  <" A "> ) `  i )  e.  ran  ( T `
 ( ( F concat  <" A "> ) `  ( i  -  1 ) ) )  <->  A. i  e.  ( 1..^ ( # `  F
) ) ( F `
 i )  e. 
ran  ( T `  ( F `  ( i  -  1 ) ) ) ) )
8610, 24, 85syl2anc 661 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( A. i  e.  ( 1..^ ( # `  F
) ) ( ( F concat  <" A "> ) `  i )  e.  ran  ( T `
 ( ( F concat  <" A "> ) `  ( i  -  1 ) ) )  <->  A. i  e.  ( 1..^ ( # `  F
) ) ( F `
 i )  e. 
ran  ( T `  ( F `  ( i  -  1 ) ) ) ) )
8756, 86mpbird 232 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  A. i  e.  ( 1..^ ( # `  F ) ) ( ( F concat  <" A "> ) `  i
)  e.  ran  ( T `  ( ( F concat  <" A "> ) `  ( i  -  1 ) ) ) )
8810, 32syl 16 . . . . . . . . . 10  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  F
)  e.  NN0 )
8988nn0cnd 10657 . . . . . . . . 9  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  F
)  e.  CC )
9089addid2d 9589 . . . . . . . 8  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( 0  +  ( # `  F
) )  =  (
# `  F )
)
9190fveq2d 5714 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  ( 0  +  ( # `  F
) ) )  =  ( ( F concat  <" A "> ) `  ( # `
 F ) ) )
92 1nn 10352 . . . . . . . . . . 11  |-  1  e.  NN
9329, 92eqeltri 2513 . . . . . . . . . 10  |-  ( # `  <" A "> )  e.  NN
9493a1i 11 . . . . . . . . 9  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  <" A "> )  e.  NN )
95 lbfzo0 11605 . . . . . . . . 9  |-  ( 0  e.  ( 0..^ (
# `  <" A "> ) )  <->  ( # `  <" A "> )  e.  NN )
9694, 95sylibr 212 . . . . . . . 8  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  0  e.  ( 0..^ ( # `  <" A "> )
) )
97 ccatval3 12297 . . . . . . . 8  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  0  e.  ( 0..^ ( # `  <" A "> ) ) )  -> 
( ( F concat  <" A "> ) `  (
0  +  ( # `  F ) ) )  =  ( <" A "> `  0 )
)
9810, 24, 96, 97syl3anc 1218 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  ( 0  +  ( # `  F
) ) )  =  ( <" A "> `  0 )
)
9991, 98eqtr3d 2477 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  ( # `  F ) )  =  ( <" A "> `  0 )
)
100 s1fv 12317 . . . . . . . 8  |-  ( A  e.  ran  ( T `
 ( S `  F ) )  -> 
( <" A "> `  0 )  =  A )
101100adantl 466 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( <" A "> `  0
)  =  A )
102 fzo0end 11638 . . . . . . . . . . . 12  |-  ( (
# `  F )  e.  NN  ->  ( ( # `
 F )  - 
1 )  e.  ( 0..^ ( # `  F
) ) )
10347, 102syl 16 . . . . . . . . . . 11  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( # `
 F )  - 
1 )  e.  ( 0..^ ( # `  F
) ) )
104 ccatval1 12295 . . . . . . . . . . 11  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  ( (
# `  F )  -  1 )  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( F concat  <" A "> ) `  ( ( # `
 F )  - 
1 ) )  =  ( F `  (
( # `  F )  -  1 ) ) )
10510, 24, 103, 104syl3anc 1218 . . . . . . . . . 10  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  ( (
# `  F )  -  1 ) )  =  ( F `  ( ( # `  F
)  -  1 ) ) )
1061, 2, 3, 4, 5, 6efgsval 16247 . . . . . . . . . . 11  |-  ( F  e.  dom  S  -> 
( S `  F
)  =  ( F `
 ( ( # `  F )  -  1 ) ) )
107106adantr 465 . . . . . . . . . 10  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( S `  F )  =  ( F `  ( (
# `  F )  -  1 ) ) )
108105, 107eqtr4d 2478 . . . . . . . . 9  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  ( (
# `  F )  -  1 ) )  =  ( S `  F ) )
109108fveq2d 5714 . . . . . . . 8  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( T `  ( ( F concat  <" A "> ) `  (
( # `  F )  -  1 ) ) )  =  ( T `
 ( S `  F ) ) )
110109rneqd 5086 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ran  ( T `
 ( ( F concat  <" A "> ) `  ( ( # `
 F )  - 
1 ) ) )  =  ran  ( T `
 ( S `  F ) ) )
11122, 101, 1103eltr4d 2524 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( <" A "> `  0
)  e.  ran  ( T `  ( ( F concat  <" A "> ) `  ( (
# `  F )  -  1 ) ) ) )
11299, 111eqeltrd 2517 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  ( # `  F ) )  e. 
ran  ( T `  ( ( F concat  <" A "> ) `  (
( # `  F )  -  1 ) ) ) )
113 fvex 5720 . . . . . 6  |-  ( # `  F )  e.  _V
114 fveq2 5710 . . . . . . 7  |-  ( i  =  ( # `  F
)  ->  ( ( F concat  <" A "> ) `  i )  =  ( ( F concat  <" A "> ) `  ( # `  F
) ) )
115 oveq1 6117 . . . . . . . . . 10  |-  ( i  =  ( # `  F
)  ->  ( i  -  1 )  =  ( ( # `  F
)  -  1 ) )
116115fveq2d 5714 . . . . . . . . 9  |-  ( i  =  ( # `  F
)  ->  ( ( F concat  <" A "> ) `  ( i  -  1 ) )  =  ( ( F concat  <" A "> ) `  ( ( # `
 F )  - 
1 ) ) )
117116fveq2d 5714 . . . . . . . 8  |-  ( i  =  ( # `  F
)  ->  ( T `  ( ( F concat  <" A "> ) `  (
i  -  1 ) ) )  =  ( T `  ( ( F concat  <" A "> ) `  ( (
# `  F )  -  1 ) ) ) )
118117rneqd 5086 . . . . . . 7  |-  ( i  =  ( # `  F
)  ->  ran  ( T `
 ( ( F concat  <" A "> ) `  ( i  -  1 ) ) )  =  ran  ( T `  ( ( F concat  <" A "> ) `  ( (
# `  F )  -  1 ) ) ) )
119114, 118eleq12d 2511 . . . . . 6  |-  ( i  =  ( # `  F
)  ->  ( (
( F concat  <" A "> ) `  i
)  e.  ran  ( T `  ( ( F concat  <" A "> ) `  ( i  -  1 ) ) )  <->  ( ( F concat  <" A "> ) `  ( # `  F
) )  e.  ran  ( T `  ( ( F concat  <" A "> ) `  ( (
# `  F )  -  1 ) ) ) ) )
120113, 119ralsn 3934 . . . . 5  |-  ( A. i  e.  { ( # `
 F ) }  ( ( F concat  <" A "> ) `  i
)  e.  ran  ( T `  ( ( F concat  <" A "> ) `  ( i  -  1 ) ) )  <->  ( ( F concat  <" A "> ) `  ( # `  F
) )  e.  ran  ( T `  ( ( F concat  <" A "> ) `  ( (
# `  F )  -  1 ) ) ) )
121112, 120sylibr 212 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  A. i  e.  { ( # `  F
) }  ( ( F concat  <" A "> ) `  i )  e.  ran  ( T `
 ( ( F concat  <" A "> ) `  ( i  -  1 ) ) ) )
122 ralunb 3556 . . . 4  |-  ( A. i  e.  ( (
1..^ ( # `  F
) )  u.  {
( # `  F ) } ) ( ( F concat  <" A "> ) `  i )  e.  ran  ( T `
 ( ( F concat  <" A "> ) `  ( i  -  1 ) ) )  <->  ( A. i  e.  ( 1..^ ( # `  F ) ) ( ( F concat  <" A "> ) `  i
)  e.  ran  ( T `  ( ( F concat  <" A "> ) `  ( i  -  1 ) ) )  /\  A. i  e.  { ( # `  F
) }  ( ( F concat  <" A "> ) `  i )  e.  ran  ( T `
 ( ( F concat  <" A "> ) `  ( i  -  1 ) ) ) ) )
12387, 121, 122sylanbrc 664 . . 3  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  A. i  e.  ( ( 1..^ (
# `  F )
)  u.  { (
# `  F ) } ) ( ( F concat  <" A "> ) `  i )  e.  ran  ( T `
 ( ( F concat  <" A "> ) `  ( i  -  1 ) ) ) )
12431oveq2d 6126 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( 1..^ ( # `  ( F concat  <" A "> ) ) )  =  ( 1..^ ( (
# `  F )  +  1 ) ) )
125 nnuz 10915 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
12647, 125syl6eleq 2533 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  F
)  e.  ( ZZ>= ` 
1 ) )
127 fzosplitsn 11652 . . . . . 6  |-  ( (
# `  F )  e.  ( ZZ>= `  1 )  ->  ( 1..^ ( (
# `  F )  +  1 ) )  =  ( ( 1..^ ( # `  F
) )  u.  {
( # `  F ) } ) )
128126, 127syl 16 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( 1..^ ( ( # `  F
)  +  1 ) )  =  ( ( 1..^ ( # `  F
) )  u.  {
( # `  F ) } ) )
129124, 128eqtrd 2475 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( 1..^ ( # `  ( F concat  <" A "> ) ) )  =  ( ( 1..^ (
# `  F )
)  u.  { (
# `  F ) } ) )
130129raleqdv 2942 . . 3  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( A. i  e.  ( 1..^ ( # `  ( F concat  <" A "> ) ) ) ( ( F concat  <" A "> ) `  i
)  e.  ran  ( T `  ( ( F concat  <" A "> ) `  ( i  -  1 ) ) )  <->  A. i  e.  ( ( 1..^ ( # `  F ) )  u. 
{ ( # `  F
) } ) ( ( F concat  <" A "> ) `  i
)  e.  ran  ( T `  ( ( F concat  <" A "> ) `  ( i  -  1 ) ) ) ) )
131123, 130mpbird 232 . 2  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  A. i  e.  ( 1..^ ( # `  ( F concat  <" A "> ) ) ) ( ( F concat  <" A "> ) `  i
)  e.  ran  ( T `  ( ( F concat  <" A "> ) `  ( i  -  1 ) ) ) )
1321, 2, 3, 4, 5, 6efgsdm 16246 . 2  |-  ( ( F concat  <" A "> )  e.  dom  S  <-> 
( ( F concat  <" A "> )  e.  (Word 
W  \  { (/) } )  /\  ( ( F concat  <" A "> ) `  0 )  e.  D  /\  A. i  e.  ( 1..^ ( # `  ( F concat  <" A "> ) ) ) ( ( F concat  <" A "> ) `  i
)  e.  ran  ( T `  ( ( F concat  <" A "> ) `  ( i  -  1 ) ) ) ) )
13341, 54, 131, 132syl3anbrc 1172 1  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( F concat  <" A "> )  e.  dom  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2620   A.wral 2734   {crab 2738    \ cdif 3344    u. cun 3345    C_ wss 3347   (/)c0 3656   {csn 3896   <.cop 3902   <.cotp 3904   U_ciun 4190   class class class wbr 4311    e. cmpt 4369    _I cid 4650    X. cxp 4857   dom cdm 4859   ran crn 4860   -->wf 5433   ` cfv 5437  (class class class)co 6110    e. cmpt2 6112   1oc1o 6932   2oc2o 6933   Fincfn 7329   0cc0 9301   1c1 9302    + caddc 9304    <_ cle 9438    - cmin 9614   NNcn 10341   NN0cn0 10598   ZZcz 10665   ZZ>=cuz 10880   ...cfz 11456  ..^cfzo 11567   #chash 12122  Word cword 12240   concat cconcat 12242   <"cs1 12243   splice csplice 12245   <"cs2 12487   ~FG cefg 16222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-ot 3905  df-uni 4111  df-int 4148  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-om 6496  df-1st 6596  df-2nd 6597  df-recs 6851  df-rdg 6885  df-1o 6939  df-2o 6940  df-oadd 6943  df-er 7120  df-map 7235  df-pm 7236  df-en 7330  df-dom 7331  df-sdom 7332  df-fin 7333  df-card 8128  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-nn 10342  df-n0 10599  df-z 10666  df-uz 10881  df-fz 11457  df-fzo 11568  df-hash 12123  df-word 12248  df-concat 12250  df-s1 12251  df-substr 12252  df-splice 12253  df-s2 12494
This theorem is referenced by:  efgsfo  16255  efgredlemd  16260  efgrelexlemb  16266
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