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Theorem efgsp1 16734
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 16727 . . . . . . 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 1012 . . . . . 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 3473 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  F  e. Word  W )
111, 2, 3, 4, 5, 6efgsf 16726 . . . . . . . . . . . 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 5726 . . . . . . . . . . . . 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 5714 . . . . . . . . . . . 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 6015 . . . . . . . . . 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 16719 . . . . . . . . 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 5727 . . . . . . 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 3490 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  A  e.  W )
2423s1cld 12597 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  <" A ">  e. Word  W )
25 ccatcl 12575 . . . 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 12576 . . . . . . 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 12599 . . . . . . 7  |-  ( # `  <" A "> )  =  1
3029oveq2i 6292 . . . . . 6  |-  ( (
# `  F )  +  ( # `  <" A "> )
)  =  ( (
# `  F )  +  1 )
3128, 30syl6eq 2500 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  ( F concat  <" A "> ) )  =  ( ( # `  F
)  +  1 ) )
32 lencl 12544 . . . . . 6  |-  ( F  e. Word  W  ->  ( # `
 F )  e. 
NN0 )
33 nn0p1nn 10842 . . . . . 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 2531 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  ( F concat  <" A "> ) )  e.  NN )
36 wrdfin 12543 . . . . 5  |-  ( ( F concat  <" A "> )  e. Word  W  -> 
( F concat  <" A "> )  e.  Fin )
37 hashnncl 12418 . . . . 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 4140 . . 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 4141 . . . . . . 7  |-  ( F  e.  (Word  W  \  { (/) } )  ->  F  =/=  (/) )
439, 42syl 16 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  F  =/=  (/) )
44 wrdfin 12543 . . . . . . 7  |-  ( F  e. Word  W  ->  F  e.  Fin )
45 hashnncl 12418 . . . . . . 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 11844 . . . . 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 12577 . . . 4  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  0  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( F concat  <" A "> ) `  0 )  =  ( F ` 
0 ) )
5110, 24, 49, 50syl3anc 1229 . . 3  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  0 )  =  ( F ` 
0 ) )
527simp2bi 1013 . . . 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 2531 . 2  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  0 )  e.  D )
557simp3bi 1014 . . . . . 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 fzo0ss1 11837 . . . . . . . . . . 11  |-  ( 1..^ ( # `  F
) )  C_  (
0..^ ( # `  F
) )
5857sseli 3485 . . . . . . . . . 10  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  i  e.  ( 0..^ ( # `  F
) ) )
59 ccatval1 12577 . . . . . . . . . 10  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  i  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( F concat  <" A "> ) `  i )  =  ( F `  i ) )
6058, 59syl3an3 1264 . . . . . . . . 9  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  i  e.  ( 1..^ ( # `  F ) ) )  ->  ( ( F concat  <" A "> ) `  i )  =  ( F `  i ) )
61 elfzoel2 11810 . . . . . . . . . . . . . . . 16  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( # `  F
)  e.  ZZ )
62 peano2zm 10914 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  e.  ZZ  ->  ( ( # `
 F )  - 
1 )  e.  ZZ )
6361, 62syl 16 . . . . . . . . . . . . . . 15  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( ( # `
 F )  - 
1 )  e.  ZZ )
6461zred 10976 . . . . . . . . . . . . . . . 16  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( # `  F
)  e.  RR )
6564lem1d 10486 . . . . . . . . . . . . . . 15  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( ( # `
 F )  - 
1 )  <_  ( # `
 F ) )
66 eluz2 11098 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  e.  ( ZZ>= `  ( ( # `
 F )  - 
1 ) )  <->  ( (
( # `  F )  -  1 )  e.  ZZ  /\  ( # `  F )  e.  ZZ  /\  ( ( # `  F
)  -  1 )  <_  ( # `  F
) ) )
6763, 61, 65, 66syl3anbrc 1181 . . . . . . . . . . . . . 14  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( # `  F
)  e.  ( ZZ>= `  ( ( # `  F
)  -  1 ) ) )
68 fzoss2 11835 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  e.  ( ZZ>= `  ( ( # `
 F )  - 
1 ) )  -> 
( 0..^ ( (
# `  F )  -  1 ) ) 
C_  ( 0..^ (
# `  F )
) )
6967, 68syl 16 . . . . . . . . . . . . 13  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( 0..^ ( ( # `  F
)  -  1 ) )  C_  ( 0..^ ( # `  F
) ) )
70 elfzoelz 11811 . . . . . . . . . . . . . . 15  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  i  e.  ZZ )
71 elfzom1b 11893 . . . . . . . . . . . . . . 15  |-  ( ( i  e.  ZZ  /\  ( # `  F )  e.  ZZ )  -> 
( i  e.  ( 1..^ ( # `  F
) )  <->  ( i  -  1 )  e.  ( 0..^ ( (
# `  F )  -  1 ) ) ) )
7270, 61, 71syl2anc 661 . . . . . . . . . . . . . 14  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( i  e.  ( 1..^ ( # `  F ) )  <->  ( i  -  1 )  e.  ( 0..^ ( (
# `  F )  -  1 ) ) ) )
7372ibi 241 . . . . . . . . . . . . 13  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( i  -  1 )  e.  ( 0..^ ( (
# `  F )  -  1 ) ) )
7469, 73sseldd 3490 . . . . . . . . . . . 12  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( i  -  1 )  e.  ( 0..^ ( # `  F ) ) )
75 ccatval1 12577 . . . . . . . . . . . 12  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  ( i  -  1 )  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( F concat  <" A "> ) `  ( i  -  1 ) )  =  ( F `  ( i  -  1 ) ) )
7674, 75syl3an3 1264 . . . . . . . . . . 11  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  i  e.  ( 1..^ ( # `  F ) ) )  ->  ( ( F concat  <" A "> ) `  ( i  -  1 ) )  =  ( F `  ( i  -  1 ) ) )
7776fveq2d 5860 . . . . . . . . . 10  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  i  e.  ( 1..^ ( # `  F ) ) )  ->  ( T `  ( ( F concat  <" A "> ) `  (
i  -  1 ) ) )  =  ( T `  ( F `
 ( i  - 
1 ) ) ) )
7877rneqd 5220 . . . . . . . . 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 ) ) ) )
7960, 78eleq12d 2525 . . . . . . . 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 ) ) ) ) )
80793expa 1197 . . . . . . 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 ) ) ) ) )
8180ralbidva 2879 . . . . . 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 ) ) ) ) )
8210, 24, 81syl2anc 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 ) ) ) ) )
8356, 82mpbird 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 ) ) ) )
8410, 32syl 16 . . . . . . . . . 10  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  F
)  e.  NN0 )
8584nn0cnd 10861 . . . . . . . . 9  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  F
)  e.  CC )
8685addid2d 9784 . . . . . . . 8  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( 0  +  ( # `  F
) )  =  (
# `  F )
)
8786fveq2d 5860 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  ( 0  +  ( # `  F
) ) )  =  ( ( F concat  <" A "> ) `  ( # `
 F ) ) )
88 1nn 10554 . . . . . . . . . . 11  |-  1  e.  NN
8929, 88eqeltri 2527 . . . . . . . . . 10  |-  ( # `  <" A "> )  e.  NN
9089a1i 11 . . . . . . . . 9  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  <" A "> )  e.  NN )
91 lbfzo0 11844 . . . . . . . . 9  |-  ( 0  e.  ( 0..^ (
# `  <" A "> ) )  <->  ( # `  <" A "> )  e.  NN )
9290, 91sylibr 212 . . . . . . . 8  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  0  e.  ( 0..^ ( # `  <" A "> )
) )
93 ccatval3 12579 . . . . . . . 8  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  0  e.  ( 0..^ ( # `  <" A "> ) ) )  -> 
( ( F concat  <" A "> ) `  (
0  +  ( # `  F ) ) )  =  ( <" A "> `  0 )
)
9410, 24, 92, 93syl3anc 1229 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  ( 0  +  ( # `  F
) ) )  =  ( <" A "> `  0 )
)
9587, 94eqtr3d 2486 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  ( # `  F ) )  =  ( <" A "> `  0 )
)
96 s1fv 12601 . . . . . . . 8  |-  ( A  e.  ran  ( T `
 ( S `  F ) )  -> 
( <" A "> `  0 )  =  A )
9796adantl 466 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( <" A "> `  0
)  =  A )
98 fzo0end 11886 . . . . . . . . . . . 12  |-  ( (
# `  F )  e.  NN  ->  ( ( # `
 F )  - 
1 )  e.  ( 0..^ ( # `  F
) ) )
9947, 98syl 16 . . . . . . . . . . 11  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( # `
 F )  - 
1 )  e.  ( 0..^ ( # `  F
) ) )
100 ccatval1 12577 . . . . . . . . . . 11  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  ( (
# `  F )  -  1 )  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( F concat  <" A "> ) `  ( ( # `
 F )  - 
1 ) )  =  ( F `  (
( # `  F )  -  1 ) ) )
10110, 24, 99, 100syl3anc 1229 . . . . . . . . . 10  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  ( (
# `  F )  -  1 ) )  =  ( F `  ( ( # `  F
)  -  1 ) ) )
1021, 2, 3, 4, 5, 6efgsval 16728 . . . . . . . . . . 11  |-  ( F  e.  dom  S  -> 
( S `  F
)  =  ( F `
 ( ( # `  F )  -  1 ) ) )
103102adantr 465 . . . . . . . . . 10  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( S `  F )  =  ( F `  ( (
# `  F )  -  1 ) ) )
104101, 103eqtr4d 2487 . . . . . . . . 9  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  ( (
# `  F )  -  1 ) )  =  ( S `  F ) )
105104fveq2d 5860 . . . . . . . 8  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( T `  ( ( F concat  <" A "> ) `  (
( # `  F )  -  1 ) ) )  =  ( T `
 ( S `  F ) ) )
106105rneqd 5220 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ran  ( T `
 ( ( F concat  <" A "> ) `  ( ( # `
 F )  - 
1 ) ) )  =  ran  ( T `
 ( S `  F ) ) )
10722, 97, 1063eltr4d 2546 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( <" A "> `  0
)  e.  ran  ( T `  ( ( F concat  <" A "> ) `  ( (
# `  F )  -  1 ) ) ) )
10895, 107eqeltrd 2531 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  ( # `  F ) )  e. 
ran  ( T `  ( ( F concat  <" A "> ) `  (
( # `  F )  -  1 ) ) ) )
109 fvex 5866 . . . . . 6  |-  ( # `  F )  e.  _V
110 fveq2 5856 . . . . . . 7  |-  ( i  =  ( # `  F
)  ->  ( ( F concat  <" A "> ) `  i )  =  ( ( F concat  <" A "> ) `  ( # `  F
) ) )
111 oveq1 6288 . . . . . . . . . 10  |-  ( i  =  ( # `  F
)  ->  ( i  -  1 )  =  ( ( # `  F
)  -  1 ) )
112111fveq2d 5860 . . . . . . . . 9  |-  ( i  =  ( # `  F
)  ->  ( ( F concat  <" A "> ) `  ( i  -  1 ) )  =  ( ( F concat  <" A "> ) `  ( ( # `
 F )  - 
1 ) ) )
113112fveq2d 5860 . . . . . . . 8  |-  ( i  =  ( # `  F
)  ->  ( T `  ( ( F concat  <" A "> ) `  (
i  -  1 ) ) )  =  ( T `  ( ( F concat  <" A "> ) `  ( (
# `  F )  -  1 ) ) ) )
114113rneqd 5220 . . . . . . 7  |-  ( i  =  ( # `  F
)  ->  ran  ( T `
 ( ( F concat  <" A "> ) `  ( i  -  1 ) ) )  =  ran  ( T `  ( ( F concat  <" A "> ) `  ( (
# `  F )  -  1 ) ) ) )
115110, 114eleq12d 2525 . . . . . 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 ) ) ) ) )
116109, 115ralsn 4053 . . . . 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 ) ) ) )
117108, 116sylibr 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 ) ) ) )
118 ralunb 3670 . . . 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 ) ) ) ) )
11983, 117, 118sylanbrc 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 ) ) ) )
12031oveq2d 6297 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( 1..^ ( # `  ( F concat  <" A "> ) ) )  =  ( 1..^ ( (
# `  F )  +  1 ) ) )
121 nnuz 11127 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
12247, 121syl6eleq 2541 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  F
)  e.  ( ZZ>= ` 
1 ) )
123 fzosplitsn 11900 . . . . . 6  |-  ( (
# `  F )  e.  ( ZZ>= `  1 )  ->  ( 1..^ ( (
# `  F )  +  1 ) )  =  ( ( 1..^ ( # `  F
) )  u.  {
( # `  F ) } ) )
124122, 123syl 16 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( 1..^ ( ( # `  F
)  +  1 ) )  =  ( ( 1..^ ( # `  F
) )  u.  {
( # `  F ) } ) )
125120, 124eqtrd 2484 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( 1..^ ( # `  ( F concat  <" A "> ) ) )  =  ( ( 1..^ (
# `  F )
)  u.  { (
# `  F ) } ) )
126125raleqdv 3046 . . 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 ) ) ) ) )
127119, 126mpbird 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 ) ) ) )
1281, 2, 3, 4, 5, 6efgsdm 16727 . 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 ) ) ) ) )
12941, 54, 127, 128syl3anbrc 1181 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 974    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   {crab 2797    \ cdif 3458    u. cun 3459    C_ wss 3461   (/)c0 3770   {csn 4014   <.cop 4020   <.cotp 4022   U_ciun 4315   class class class wbr 4437    |-> cmpt 4495    _I cid 4780    X. cxp 4987   dom cdm 4989   ran crn 4990   -->wf 5574   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283   1oc1o 7125   2oc2o 7126   Fincfn 7518   0cc0 9495   1c1 9496    + caddc 9498    <_ cle 9632    - cmin 9810   NNcn 10543   NN0cn0 10802   ZZcz 10871   ZZ>=cuz 11092   ...cfz 11683  ..^cfzo 11806   #chash 12387  Word cword 12516   concat cconcat 12518   <"cs1 12519   splice csplice 12521   <"cs2 12788   ~FG cefg 16703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-ot 4023  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-n0 10803  df-z 10872  df-uz 11093  df-fz 11684  df-fzo 11807  df-hash 12388  df-word 12524  df-concat 12526  df-s1 12527  df-substr 12528  df-splice 12529  df-s2 12795
This theorem is referenced by:  efgsfo  16736  efgredlemd  16741  efgrelexlemb  16747
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