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Theorem efgsp1 15062
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 15055 . . . . . . 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 970 . . . . . 6  |-  ( F  e.  dom  S  ->  F  e.  (Word  W  \  { (/) } ) )
98adantr 451 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  F  e.  (Word  W  \  { (/) } ) )
10 eldifi 3311 . . . . 5  |-  ( F  e.  (Word  W  \  { (/) } )  ->  F  e. Word  W )
119, 10syl 15 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  F  e. Word  W )
121, 2, 3, 4, 5, 6efgsf 15054 . . . . . . . . . . . 12  |-  S : { t  e.  (Word 
W  \  { (/) } )  |  ( ( t `
 0 )  e.  D  /\  A. k  e.  ( 1..^ ( # `  t ) ) ( t `  k )  e.  ran  ( T `
 ( t `  ( k  -  1 ) ) ) ) } --> W
1312fdmi 5410 . . . . . . . . . . . . 13  |-  dom  S  =  { t  e.  (Word 
W  \  { (/) } )  |  ( ( t `
 0 )  e.  D  /\  A. k  e.  ( 1..^ ( # `  t ) ) ( t `  k )  e.  ran  ( T `
 ( t `  ( k  -  1 ) ) ) ) }
1413feq2i 5400 . . . . . . . . . . . 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 )
1512, 14mpbir 200 . . . . . . . . . . 11  |-  S : dom  S --> W
1615ffvelrni 5680 . . . . . . . . . 10  |-  ( F  e.  dom  S  -> 
( S `  F
)  e.  W )
1716adantr 451 . . . . . . . . 9  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( S `  F )  e.  W
)
181, 2, 3, 4efgtf 15047 . . . . . . . . 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 ) )
1917, 18syl 15 . . . . . . . 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 ) )
2019simprd 449 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( T `  ( S `  F
) ) : ( ( 0 ... ( # `
 ( S `  F ) ) )  X.  ( I  X.  2o ) ) --> W )
21 frn 5411 . . . . . . 7  |-  ( ( T `  ( S `
 F ) ) : ( ( 0 ... ( # `  ( S `  F )
) )  X.  (
I  X.  2o ) ) --> W  ->  ran  ( T `  ( S `
 F ) ) 
C_  W )
2220, 21syl 15 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ran  ( T `
 ( S `  F ) )  C_  W )
23 simpr 447 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  A  e.  ran  ( T `  ( S `  F )
) )
2422, 23sseldd 3194 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  A  e.  W )
2524s1cld 11458 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  <" A ">  e. Word  W )
26 ccatcl 11445 . . . 4  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W )  ->  ( F concat  <" A "> )  e. Word  W )
2711, 25, 26syl2anc 642 . . 3  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( F concat  <" A "> )  e. Word  W )
28 ccatlen 11446 . . . . . . 7  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W )  ->  ( # `
 ( F concat  <" A "> ) )  =  ( ( # `  F
)  +  ( # `  <" A "> ) ) )
2911, 25, 28syl2anc 642 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  ( F concat  <" A "> ) )  =  ( ( # `  F
)  +  ( # `  <" A "> ) ) )
30 s1len 11460 . . . . . . 7  |-  ( # `  <" A "> )  =  1
3130oveq2i 5885 . . . . . 6  |-  ( (
# `  F )  +  ( # `  <" A "> )
)  =  ( (
# `  F )  +  1 )
3229, 31syl6eq 2344 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  ( F concat  <" A "> ) )  =  ( ( # `  F
)  +  1 ) )
33 lencl 11437 . . . . . . 7  |-  ( F  e. Word  W  ->  ( # `
 F )  e. 
NN0 )
3411, 33syl 15 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  F
)  e.  NN0 )
35 nn0p1nn 10019 . . . . . 6  |-  ( (
# `  F )  e.  NN0  ->  ( ( # `
 F )  +  1 )  e.  NN )
3634, 35syl 15 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( # `
 F )  +  1 )  e.  NN )
3732, 36eqeltrd 2370 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  ( F concat  <" A "> ) )  e.  NN )
38 wrdfin 11436 . . . . . 6  |-  ( ( F concat  <" A "> )  e. Word  W  -> 
( F concat  <" A "> )  e.  Fin )
3927, 38syl 15 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( F concat  <" A "> )  e.  Fin )
40 hashnncl 11370 . . . . 5  |-  ( ( F concat  <" A "> )  e.  Fin  ->  ( ( # `  ( F concat  <" A "> ) )  e.  NN  <->  ( F concat  <" A "> )  =/=  (/) ) )
4139, 40syl 15 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( # `
 ( F concat  <" A "> ) )  e.  NN  <->  ( F concat  <" A "> )  =/=  (/) ) )
4237, 41mpbid 201 . . 3  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( F concat  <" A "> )  =/=  (/) )
43 eldifsn 3762 . . 3  |-  ( ( F concat  <" A "> )  e.  (Word  W  \  { (/) } )  <-> 
( ( F concat  <" A "> )  e. Word  W  /\  ( F concat  <" A "> )  =/=  (/) ) )
4427, 42, 43sylanbrc 645 . 2  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( F concat  <" A "> )  e.  (Word  W  \  { (/) } ) )
45 eldifsni 3763 . . . . . . 7  |-  ( F  e.  (Word  W  \  { (/) } )  ->  F  =/=  (/) )
469, 45syl 15 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  F  =/=  (/) )
47 wrdfin 11436 . . . . . . . 8  |-  ( F  e. Word  W  ->  F  e.  Fin )
4811, 47syl 15 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  F  e.  Fin )
49 hashnncl 11370 . . . . . . 7  |-  ( F  e.  Fin  ->  (
( # `  F )  e.  NN  <->  F  =/=  (/) ) )
5048, 49syl 15 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( # `
 F )  e.  NN  <->  F  =/=  (/) ) )
5146, 50mpbird 223 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  F
)  e.  NN )
52 lbfzo0 10919 . . . . 5  |-  ( 0  e.  ( 0..^ (
# `  F )
)  <->  ( # `  F
)  e.  NN )
5351, 52sylibr 203 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  0  e.  ( 0..^ ( # `  F
) ) )
54 ccatval1 11447 . . . 4  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  0  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( F concat  <" A "> ) `  0 )  =  ( F ` 
0 ) )
5511, 25, 53, 54syl3anc 1182 . . 3  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  0 )  =  ( F ` 
0 ) )
567simp2bi 971 . . . 4  |-  ( F  e.  dom  S  -> 
( F `  0
)  e.  D )
5756adantr 451 . . 3  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( F `  0 )  e.  D )
5855, 57eqeltrd 2370 . 2  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  0 )  e.  D )
597simp3bi 972 . . . . . 6  |-  ( F  e.  dom  S  ->  A. i  e.  (
1..^ ( # `  F
) ) ( F `
 i )  e. 
ran  ( T `  ( F `  ( i  -  1 ) ) ) )
6059adantr 451 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  A. i  e.  ( 1..^ ( # `  F ) ) ( F `  i )  e.  ran  ( T `
 ( F `  ( i  -  1 ) ) ) )
61 1nn0 9997 . . . . . . . . . . . . 13  |-  1  e.  NN0
62 nn0uz 10278 . . . . . . . . . . . . 13  |-  NN0  =  ( ZZ>= `  0 )
6361, 62eleqtri 2368 . . . . . . . . . . . 12  |-  1  e.  ( ZZ>= `  0 )
64 fzoss1 10912 . . . . . . . . . . . 12  |-  ( 1  e.  ( ZZ>= `  0
)  ->  ( 1..^ ( # `  F
) )  C_  (
0..^ ( # `  F
) ) )
6563, 64ax-mp 8 . . . . . . . . . . 11  |-  ( 1..^ ( # `  F
) )  C_  (
0..^ ( # `  F
) )
6665sseli 3189 . . . . . . . . . 10  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  i  e.  ( 0..^ ( # `  F
) ) )
67 ccatval1 11447 . . . . . . . . . 10  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  i  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( F concat  <" A "> ) `  i )  =  ( F `  i ) )
6866, 67syl3an3 1217 . . . . . . . . 9  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  i  e.  ( 1..^ ( # `  F ) ) )  ->  ( ( F concat  <" A "> ) `  i )  =  ( F `  i ) )
69 elfzoel2 10890 . . . . . . . . . . . . . . . 16  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( # `  F
)  e.  ZZ )
70 peano2zm 10078 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  e.  ZZ  ->  ( ( # `
 F )  - 
1 )  e.  ZZ )
7169, 70syl 15 . . . . . . . . . . . . . . 15  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( ( # `
 F )  - 
1 )  e.  ZZ )
7269zred 10133 . . . . . . . . . . . . . . . 16  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( # `  F
)  e.  RR )
7372lem1d 9706 . . . . . . . . . . . . . . 15  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( ( # `
 F )  - 
1 )  <_  ( # `
 F ) )
74 eluz2 10252 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  e.  ( ZZ>= `  ( ( # `
 F )  - 
1 ) )  <->  ( (
( # `  F )  -  1 )  e.  ZZ  /\  ( # `  F )  e.  ZZ  /\  ( ( # `  F
)  -  1 )  <_  ( # `  F
) ) )
7571, 69, 73, 74syl3anbrc 1136 . . . . . . . . . . . . . 14  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( # `  F
)  e.  ( ZZ>= `  ( ( # `  F
)  -  1 ) ) )
76 fzoss2 10913 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  e.  ( ZZ>= `  ( ( # `
 F )  - 
1 ) )  -> 
( 0..^ ( (
# `  F )  -  1 ) ) 
C_  ( 0..^ (
# `  F )
) )
7775, 76syl 15 . . . . . . . . . . . . 13  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( 0..^ ( ( # `  F
)  -  1 ) )  C_  ( 0..^ ( # `  F
) ) )
78 elfzoelz 10891 . . . . . . . . . . . . . . 15  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  i  e.  ZZ )
79 elfzom1b 10934 . . . . . . . . . . . . . . 15  |-  ( ( i  e.  ZZ  /\  ( # `  F )  e.  ZZ )  -> 
( i  e.  ( 1..^ ( # `  F
) )  <->  ( i  -  1 )  e.  ( 0..^ ( (
# `  F )  -  1 ) ) ) )
8078, 69, 79syl2anc 642 . . . . . . . . . . . . . 14  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( i  e.  ( 1..^ ( # `  F ) )  <->  ( i  -  1 )  e.  ( 0..^ ( (
# `  F )  -  1 ) ) ) )
8180ibi 232 . . . . . . . . . . . . 13  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( i  -  1 )  e.  ( 0..^ ( (
# `  F )  -  1 ) ) )
8277, 81sseldd 3194 . . . . . . . . . . . 12  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( i  -  1 )  e.  ( 0..^ ( # `  F ) ) )
83 ccatval1 11447 . . . . . . . . . . . 12  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  ( i  -  1 )  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( F concat  <" A "> ) `  ( i  -  1 ) )  =  ( F `  ( i  -  1 ) ) )
8482, 83syl3an3 1217 . . . . . . . . . . 11  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  i  e.  ( 1..^ ( # `  F ) ) )  ->  ( ( F concat  <" A "> ) `  ( i  -  1 ) )  =  ( F `  ( i  -  1 ) ) )
8584fveq2d 5545 . . . . . . . . . 10  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  i  e.  ( 1..^ ( # `  F ) ) )  ->  ( T `  ( ( F concat  <" A "> ) `  (
i  -  1 ) ) )  =  ( T `  ( F `
 ( i  - 
1 ) ) ) )
8685rneqd 4922 . . . . . . . . 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 ) ) ) )
8768, 86eleq12d 2364 . . . . . . . 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 ) ) ) ) )
88873expa 1151 . . . . . . 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 ) ) ) ) )
8988ralbidva 2572 . . . . . 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 ) ) ) ) )
9011, 25, 89syl2anc 642 . . . . 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 ) ) ) ) )
9160, 90mpbird 223 . . . 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 ) ) ) )
9234nn0cnd 10036 . . . . . . . . 9  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  F
)  e.  CC )
9392addid2d 9029 . . . . . . . 8  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( 0  +  ( # `  F
) )  =  (
# `  F )
)
9493fveq2d 5545 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  ( 0  +  ( # `  F
) ) )  =  ( ( F concat  <" A "> ) `  ( # `
 F ) ) )
95 1nn 9773 . . . . . . . . . . 11  |-  1  e.  NN
9630, 95eqeltri 2366 . . . . . . . . . 10  |-  ( # `  <" A "> )  e.  NN
9796a1i 10 . . . . . . . . 9  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  <" A "> )  e.  NN )
98 lbfzo0 10919 . . . . . . . . 9  |-  ( 0  e.  ( 0..^ (
# `  <" A "> ) )  <->  ( # `  <" A "> )  e.  NN )
9997, 98sylibr 203 . . . . . . . 8  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  0  e.  ( 0..^ ( # `  <" A "> )
) )
100 ccatval3 11449 . . . . . . . 8  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  0  e.  ( 0..^ ( # `  <" A "> ) ) )  -> 
( ( F concat  <" A "> ) `  (
0  +  ( # `  F ) ) )  =  ( <" A "> `  0 )
)
10111, 25, 99, 100syl3anc 1182 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  ( 0  +  ( # `  F
) ) )  =  ( <" A "> `  0 )
)
10294, 101eqtr3d 2330 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  ( # `  F ) )  =  ( <" A "> `  0 )
)
103 s1fv 11462 . . . . . . . . 9  |-  ( A  e.  ran  ( T `
 ( S `  F ) )  -> 
( <" A "> `  0 )  =  A )
104103adantl 452 . . . . . . . 8  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( <" A "> `  0
)  =  A )
105 fzo0end 10931 . . . . . . . . . . . . 13  |-  ( (
# `  F )  e.  NN  ->  ( ( # `
 F )  - 
1 )  e.  ( 0..^ ( # `  F
) ) )
10651, 105syl 15 . . . . . . . . . . . 12  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( # `
 F )  - 
1 )  e.  ( 0..^ ( # `  F
) ) )
107 ccatval1 11447 . . . . . . . . . . . 12  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  ( (
# `  F )  -  1 )  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( F concat  <" A "> ) `  ( ( # `
 F )  - 
1 ) )  =  ( F `  (
( # `  F )  -  1 ) ) )
10811, 25, 106, 107syl3anc 1182 . . . . . . . . . . 11  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  ( (
# `  F )  -  1 ) )  =  ( F `  ( ( # `  F
)  -  1 ) ) )
1091, 2, 3, 4, 5, 6efgsval 15056 . . . . . . . . . . . 12  |-  ( F  e.  dom  S  -> 
( S `  F
)  =  ( F `
 ( ( # `  F )  -  1 ) ) )
110109adantr 451 . . . . . . . . . . 11  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( S `  F )  =  ( F `  ( (
# `  F )  -  1 ) ) )
111108, 110eqtr4d 2331 . . . . . . . . . 10  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  ( (
# `  F )  -  1 ) )  =  ( S `  F ) )
112111fveq2d 5545 . . . . . . . . 9  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( T `  ( ( F concat  <" A "> ) `  (
( # `  F )  -  1 ) ) )  =  ( T `
 ( S `  F ) ) )
113112rneqd 4922 . . . . . . . 8  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ran  ( T `
 ( ( F concat  <" A "> ) `  ( ( # `
 F )  - 
1 ) ) )  =  ran  ( T `
 ( S `  F ) ) )
114104, 113eleq12d 2364 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( <" A "> `  0 )  e.  ran  ( T `  ( ( F concat  <" A "> ) `  ( (
# `  F )  -  1 ) ) )  <->  A  e.  ran  ( T `  ( S `
 F ) ) ) )
11523, 114mpbird 223 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( <" A "> `  0
)  e.  ran  ( T `  ( ( F concat  <" A "> ) `  ( (
# `  F )  -  1 ) ) ) )
116102, 115eqeltrd 2370 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F concat  <" A "> ) `  ( # `  F ) )  e. 
ran  ( T `  ( ( F concat  <" A "> ) `  (
( # `  F )  -  1 ) ) ) )
117 fvex 5555 . . . . . 6  |-  ( # `  F )  e.  _V
118 fveq2 5541 . . . . . . 7  |-  ( i  =  ( # `  F
)  ->  ( ( F concat  <" A "> ) `  i )  =  ( ( F concat  <" A "> ) `  ( # `  F
) ) )
119 oveq1 5881 . . . . . . . . . 10  |-  ( i  =  ( # `  F
)  ->  ( i  -  1 )  =  ( ( # `  F
)  -  1 ) )
120119fveq2d 5545 . . . . . . . . 9  |-  ( i  =  ( # `  F
)  ->  ( ( F concat  <" A "> ) `  ( i  -  1 ) )  =  ( ( F concat  <" A "> ) `  ( ( # `
 F )  - 
1 ) ) )
121120fveq2d 5545 . . . . . . . 8  |-  ( i  =  ( # `  F
)  ->  ( T `  ( ( F concat  <" A "> ) `  (
i  -  1 ) ) )  =  ( T `  ( ( F concat  <" A "> ) `  ( (
# `  F )  -  1 ) ) ) )
122121rneqd 4922 . . . . . . 7  |-  ( i  =  ( # `  F
)  ->  ran  ( T `
 ( ( F concat  <" A "> ) `  ( i  -  1 ) ) )  =  ran  ( T `  ( ( F concat  <" A "> ) `  ( (
# `  F )  -  1 ) ) ) )
123118, 122eleq12d 2364 . . . . . 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 ) ) ) ) )
124117, 123ralsn 3687 . . . . 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 ) ) ) )
125116, 124sylibr 203 . . . 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 ) ) ) )
126 ralunb 3369 . . . 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 ) ) ) ) )
12791, 125, 126sylanbrc 645 . . 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 ) ) ) )
12832oveq2d 5890 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( 1..^ ( # `  ( F concat  <" A "> ) ) )  =  ( 1..^ ( (
# `  F )  +  1 ) ) )
129 nnuz 10279 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
13051, 129syl6eleq 2386 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  F
)  e.  ( ZZ>= ` 
1 ) )
131 fzosplitsn 10936 . . . . . 6  |-  ( (
# `  F )  e.  ( ZZ>= `  1 )  ->  ( 1..^ ( (
# `  F )  +  1 ) )  =  ( ( 1..^ ( # `  F
) )  u.  {
( # `  F ) } ) )
132130, 131syl 15 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( 1..^ ( ( # `  F
)  +  1 ) )  =  ( ( 1..^ ( # `  F
) )  u.  {
( # `  F ) } ) )
133128, 132eqtrd 2328 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( 1..^ ( # `  ( F concat  <" A "> ) ) )  =  ( ( 1..^ (
# `  F )
)  u.  { (
# `  F ) } ) )
134133raleqdv 2755 . . 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 ) ) ) ) )
135127, 134mpbird 223 . 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 ) ) ) )
1361, 2, 3, 4, 5, 6efgsdm 15055 . 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 ) ) ) ) )
13744, 58, 135, 136syl3anbrc 1136 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   {crab 2560    \ cdif 3162    u. cun 3163    C_ wss 3165   (/)c0 3468   {csn 3653   <.cop 3656   <.cotp 3657   U_ciun 3921   class class class wbr 4039    e. cmpt 4093    _I cid 4320    X. cxp 4703   dom cdm 4705   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1oc1o 6488   2oc2o 6489   Fincfn 6879   0cc0 8753   1c1 8754    + caddc 8756    <_ cle 8884    - cmin 9053   NNcn 9762   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798  ..^cfzo 10886   #chash 11353  Word cword 11419   concat cconcat 11420   <"cs1 11421   splice csplice 11423   <"cs2 11507   ~FG cefg 15031
This theorem is referenced by:  efgsfo  15064  efgredlemd  15069  efgrelexlemb  15075
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-ot 3663  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-hash 11354  df-word 11425  df-concat 11426  df-s1 11427  df-substr 11428  df-splice 11429  df-s2 11514
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