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Theorem efgsp1 15324
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 15317 . . . . . . 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 972 . . . . . 6  |-  ( F  e.  dom  S  ->  F  e.  (Word  W  \  { (/) } ) )
98adantr 452 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  F  e.  (Word  W  \  { (/) } ) )
109eldifad 3292 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  F  e. Word  W )
111, 2, 3, 4, 5, 6efgsf 15316 . . . . . . . . . . . 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 5555 . . . . . . . . . . . . 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 5545 . . . . . . . . . . . 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 201 . . . . . . . . . . 11  |-  S : dom  S --> W
1514ffvelrni 5828 . . . . . . . . . 10  |-  ( F  e.  dom  S  -> 
( S `  F
)  e.  W )
1615adantr 452 . . . . . . . . 9  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( S `  F )  e.  W
)
171, 2, 3, 4efgtf 15309 . . . . . . . . 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 450 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( T `  ( S `  F
) ) : ( ( 0 ... ( # `
 ( S `  F ) ) )  X.  ( I  X.  2o ) ) --> W )
20 frn 5556 . . . . . . 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 448 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  A  e.  ran  ( T `  ( S `  F )
) )
2321, 22sseldd 3309 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  A  e.  W )
2423s1cld 11711 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  <" A ">  e. Word  W )
25 ccatcl 11698 . . . 4  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W )  ->  ( F concat  <" A "> )  e. Word  W )
2610, 24, 25syl2anc 643 . . 3  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( F concat  <" A "> )  e. Word  W )
27 ccatlen 11699 . . . . . . 7  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W )  ->  ( # `
 ( F concat  <" A "> ) )  =  ( ( # `  F
)  +  ( # `  <" A "> ) ) )
2810, 24, 27syl2anc 643 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  ( F concat  <" A "> ) )  =  ( ( # `  F
)  +  ( # `  <" A "> ) ) )
29 s1len 11713 . . . . . . 7  |-  ( # `  <" A "> )  =  1
3029oveq2i 6051 . . . . . 6  |-  ( (
# `  F )  +  ( # `  <" A "> )
)  =  ( (
# `  F )  +  1 )
3128, 30syl6eq 2452 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  ( F concat  <" A "> ) )  =  ( ( # `  F
)  +  1 ) )
32 lencl 11690 . . . . . 6  |-  ( F  e. Word  W  ->  ( # `
 F )  e. 
NN0 )
33 nn0p1nn 10215 . . . . . 6  |-  ( (
# `  F )  e.  NN0  ->  ( ( # `
 F )  +  1 )  e.  NN )
3410, 32, 333syl 19 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( # `
 F )  +  1 )  e.  NN )
3531, 34eqeltrd 2478 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  ( F concat  <" A "> ) )  e.  NN )
36 wrdfin 11689 . . . . 5  |-  ( ( F concat  <" A "> )  e. Word  W  -> 
( F concat  <" A "> )  e.  Fin )
37 hashnncl 11600 . . . . 5  |-  ( ( F concat  <" A "> )  e.  Fin  ->  ( ( # `  ( F concat  <" A "> ) )  e.  NN  <->  ( F concat  <" A "> )  =/=  (/) ) )
3826, 36, 373syl 19 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( # `
 ( F concat  <" A "> ) )  e.  NN  <->  ( F concat  <" A "> )  =/=  (/) ) )
3935, 38mpbid 202 . . 3  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( F concat  <" A "> )  =/=  (/) )
40 eldifsn 3887 . . 3  |-  ( ( F concat  <" A "> )  e.  (Word  W  \  { (/) } )  <-> 
( ( F concat  <" A "> )  e. Word  W  /\  ( F concat  <" A "> )  =/=  (/) ) )
4126, 39, 40sylanbrc 646 . 2  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( F concat  <" A "> )  e.  (Word  W  \  { (/) } ) )
42 eldifsni 3888 . . . . . . 7  |-  ( F  e.  (Word  W  \  { (/) } )  ->  F  =/=  (/) )
439, 42syl 16 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  F  =/=  (/) )
44 wrdfin 11689 . . . . . . 7  |-  ( F  e. Word  W  ->  F  e.  Fin )
45 hashnncl 11600 . . . . . . 7  |-  ( F  e.  Fin  ->  (
( # `  F )  e.  NN  <->  F  =/=  (/) ) )
4610, 44, 453syl 19 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( # `
 F )  e.  NN  <->  F  =/=  (/) ) )
4743, 46mpbird 224 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  F
)  e.  NN )
48 lbfzo0 11125 . . . . 5  |-  ( 0  e.  ( 0..^ (
# `  F )
)  <->  ( # `  F
)  e.  NN )
4947, 48sylibr 204 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  0  e.  ( 0..^ ( # `  F
) ) )
50 ccatval1 11700 . . . 4  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  0  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( F concat  <" A "> ) `  0 )  =  ( F ` 
0 ) )
5110, 24, 49, 50syl3anc 1184 . . 3  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  0 )  =  ( F ` 
0 ) )
527simp2bi 973 . . . 4  |-  ( F  e.  dom  S  -> 
( F `  0
)  e.  D )
5352adantr 452 . . 3  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( F `  0 )  e.  D )
5451, 53eqeltrd 2478 . 2  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  0 )  e.  D )
557simp3bi 974 . . . . . 6  |-  ( F  e.  dom  S  ->  A. i  e.  (
1..^ ( # `  F
) ) ( F `
 i )  e. 
ran  ( T `  ( F `  ( i  -  1 ) ) ) )
5655adantr 452 . . . . 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 10193 . . . . . . . . . . . . 13  |-  1  e.  NN0
58 nn0uz 10476 . . . . . . . . . . . . 13  |-  NN0  =  ( ZZ>= `  0 )
5957, 58eleqtri 2476 . . . . . . . . . . . 12  |-  1  e.  ( ZZ>= `  0 )
60 fzoss1 11117 . . . . . . . . . . . 12  |-  ( 1  e.  ( ZZ>= `  0
)  ->  ( 1..^ ( # `  F
) )  C_  (
0..^ ( # `  F
) ) )
6159, 60ax-mp 8 . . . . . . . . . . 11  |-  ( 1..^ ( # `  F
) )  C_  (
0..^ ( # `  F
) )
6261sseli 3304 . . . . . . . . . 10  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  i  e.  ( 0..^ ( # `  F
) ) )
63 ccatval1 11700 . . . . . . . . . 10  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  i  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( F concat  <" A "> ) `  i )  =  ( F `  i ) )
6462, 63syl3an3 1219 . . . . . . . . 9  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  i  e.  ( 1..^ ( # `  F ) ) )  ->  ( ( F concat  <" A "> ) `  i )  =  ( F `  i ) )
65 elfzoel2 11094 . . . . . . . . . . . . . . . 16  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( # `  F
)  e.  ZZ )
66 peano2zm 10276 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  e.  ZZ  ->  ( ( # `
 F )  - 
1 )  e.  ZZ )
6765, 66syl 16 . . . . . . . . . . . . . . 15  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( ( # `
 F )  - 
1 )  e.  ZZ )
6865zred 10331 . . . . . . . . . . . . . . . 16  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( # `  F
)  e.  RR )
6968lem1d 9900 . . . . . . . . . . . . . . 15  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( ( # `
 F )  - 
1 )  <_  ( # `
 F ) )
70 eluz2 10450 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  e.  ( ZZ>= `  ( ( # `
 F )  - 
1 ) )  <->  ( (
( # `  F )  -  1 )  e.  ZZ  /\  ( # `  F )  e.  ZZ  /\  ( ( # `  F
)  -  1 )  <_  ( # `  F
) ) )
7167, 65, 69, 70syl3anbrc 1138 . . . . . . . . . . . . . 14  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( # `  F
)  e.  ( ZZ>= `  ( ( # `  F
)  -  1 ) ) )
72 fzoss2 11118 . . . . . . . . . . . . . 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 11095 . . . . . . . . . . . . . . 15  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  i  e.  ZZ )
75 elfzom1b 11146 . . . . . . . . . . . . . . 15  |-  ( ( i  e.  ZZ  /\  ( # `  F )  e.  ZZ )  -> 
( i  e.  ( 1..^ ( # `  F
) )  <->  ( i  -  1 )  e.  ( 0..^ ( (
# `  F )  -  1 ) ) ) )
7674, 65, 75syl2anc 643 . . . . . . . . . . . . . 14  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( i  e.  ( 1..^ ( # `  F ) )  <->  ( i  -  1 )  e.  ( 0..^ ( (
# `  F )  -  1 ) ) ) )
7776ibi 233 . . . . . . . . . . . . 13  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( i  -  1 )  e.  ( 0..^ ( (
# `  F )  -  1 ) ) )
7873, 77sseldd 3309 . . . . . . . . . . . 12  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( i  -  1 )  e.  ( 0..^ ( # `  F ) ) )
79 ccatval1 11700 . . . . . . . . . . . 12  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  ( i  -  1 )  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( F concat  <" A "> ) `  ( i  -  1 ) )  =  ( F `  ( i  -  1 ) ) )
8078, 79syl3an3 1219 . . . . . . . . . . 11  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  i  e.  ( 1..^ ( # `  F ) ) )  ->  ( ( F concat  <" A "> ) `  ( i  -  1 ) )  =  ( F `  ( i  -  1 ) ) )
8180fveq2d 5691 . . . . . . . . . 10  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  i  e.  ( 1..^ ( # `  F ) ) )  ->  ( T `  ( ( F concat  <" A "> ) `  (
i  -  1 ) ) )  =  ( T `  ( F `
 ( i  - 
1 ) ) ) )
8281rneqd 5056 . . . . . . . . 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 2472 . . . . . . . 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 1153 . . . . . . 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 2682 . . . . . 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 643 . . . . 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 224 . . . 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 10232 . . . . . . . . 9  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  F
)  e.  CC )
9089addid2d 9223 . . . . . . . 8  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( 0  +  ( # `  F
) )  =  (
# `  F )
)
9190fveq2d 5691 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  ( 0  +  ( # `  F
) ) )  =  ( ( F concat  <" A "> ) `  ( # `
 F ) ) )
92 1nn 9967 . . . . . . . . . . 11  |-  1  e.  NN
9329, 92eqeltri 2474 . . . . . . . . . 10  |-  ( # `  <" A "> )  e.  NN
9493a1i 11 . . . . . . . . 9  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  <" A "> )  e.  NN )
95 lbfzo0 11125 . . . . . . . . 9  |-  ( 0  e.  ( 0..^ (
# `  <" A "> ) )  <->  ( # `  <" A "> )  e.  NN )
9694, 95sylibr 204 . . . . . . . 8  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  0  e.  ( 0..^ ( # `  <" A "> )
) )
97 ccatval3 11702 . . . . . . . 8  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  0  e.  ( 0..^ ( # `  <" A "> ) ) )  -> 
( ( F concat  <" A "> ) `  (
0  +  ( # `  F ) ) )  =  ( <" A "> `  0 )
)
9810, 24, 96, 97syl3anc 1184 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  ( 0  +  ( # `  F
) ) )  =  ( <" A "> `  0 )
)
9991, 98eqtr3d 2438 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  ( # `  F ) )  =  ( <" A "> `  0 )
)
100 s1fv 11715 . . . . . . . 8  |-  ( A  e.  ran  ( T `
 ( S `  F ) )  -> 
( <" A "> `  0 )  =  A )
101100adantl 453 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( <" A "> `  0
)  =  A )
102 fzo0end 11143 . . . . . . . . . . . 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 11700 . . . . . . . . . . 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 1184 . . . . . . . . . 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 15318 . . . . . . . . . . 11  |-  ( F  e.  dom  S  -> 
( S `  F
)  =  ( F `
 ( ( # `  F )  -  1 ) ) )
107106adantr 452 . . . . . . . . . 10  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( S `  F )  =  ( F `  ( (
# `  F )  -  1 ) ) )
108105, 107eqtr4d 2439 . . . . . . . . 9  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  ( (
# `  F )  -  1 ) )  =  ( S `  F ) )
109108fveq2d 5691 . . . . . . . 8  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( T `  ( ( F concat  <" A "> ) `  (
( # `  F )  -  1 ) ) )  =  ( T `
 ( S `  F ) ) )
110109rneqd 5056 . . . . . . 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 2485 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( <" A "> `  0
)  e.  ran  ( T `  ( ( F concat  <" A "> ) `  ( (
# `  F )  -  1 ) ) ) )
11299, 111eqeltrd 2478 . . . . 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 5701 . . . . . 6  |-  ( # `  F )  e.  _V
114 fveq2 5687 . . . . . . 7  |-  ( i  =  ( # `  F
)  ->  ( ( F concat  <" A "> ) `  i )  =  ( ( F concat  <" A "> ) `  ( # `  F
) ) )
115 oveq1 6047 . . . . . . . . . 10  |-  ( i  =  ( # `  F
)  ->  ( i  -  1 )  =  ( ( # `  F
)  -  1 ) )
116115fveq2d 5691 . . . . . . . . 9  |-  ( i  =  ( # `  F
)  ->  ( ( F concat  <" A "> ) `  ( i  -  1 ) )  =  ( ( F concat  <" A "> ) `  ( ( # `
 F )  - 
1 ) ) )
117116fveq2d 5691 . . . . . . . 8  |-  ( i  =  ( # `  F
)  ->  ( T `  ( ( F concat  <" A "> ) `  (
i  -  1 ) ) )  =  ( T `  ( ( F concat  <" A "> ) `  ( (
# `  F )  -  1 ) ) ) )
118117rneqd 5056 . . . . . . 7  |-  ( i  =  ( # `  F
)  ->  ran  ( T `
 ( ( F concat  <" A "> ) `  ( i  -  1 ) ) )  =  ran  ( T `  ( ( F concat  <" A "> ) `  ( (
# `  F )  -  1 ) ) ) )
119114, 118eleq12d 2472 . . . . . 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 3809 . . . . 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 204 . . . 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 3488 . . . 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 646 . . 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 6056 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( 1..^ ( # `  ( F concat  <" A "> ) ) )  =  ( 1..^ ( (
# `  F )  +  1 ) ) )
125 nnuz 10477 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
12647, 125syl6eleq 2494 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  F
)  e.  ( ZZ>= ` 
1 ) )
127 fzosplitsn 11150 . . . . . 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 2436 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( 1..^ ( # `  ( F concat  <" A "> ) ) )  =  ( ( 1..^ (
# `  F )
)  u.  { (
# `  F ) } ) )
130129raleqdv 2870 . . 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 224 . 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 15317 . 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 1138 1  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( F concat  <" A "> )  e.  dom  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   {crab 2670    \ cdif 3277    u. cun 3278    C_ wss 3280   (/)c0 3588   {csn 3774   <.cop 3777   <.cotp 3778   U_ciun 4053   class class class wbr 4172    e. cmpt 4226    _I cid 4453    X. cxp 4835   dom cdm 4837   ran crn 4838   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1oc1o 6676   2oc2o 6677   Fincfn 7068   0cc0 8946   1c1 8947    + caddc 8949    <_ cle 9077    - cmin 9247   NNcn 9956   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...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:  efgsfo  15326  efgredlemd  15331  efgrelexlemb  15337
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-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-n0 10178  df-z 10239  df-uz 10445  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
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