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Theorem efgsp1 17077
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 ++  <" 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 17070 . . . . . . 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 463 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  F  e.  (Word  W  \  { (/) } ) )
109eldifad 3425 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  F  e. Word  W )
111, 2, 3, 4, 5, 6efgsf 17069 . . . . . . . . . . . 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 5718 . . . . . . . . . . . . 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 5706 . . . . . . . . . . . 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 6007 . . . . . . . . . 10  |-  ( F  e.  dom  S  -> 
( S `  F
)  e.  W )
1615adantr 463 . . . . . . . . 9  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( S `  F )  e.  W
)
171, 2, 3, 4efgtf 17062 . . . . . . . . 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 17 . . . . . . . 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 461 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( T `  ( S `  F
) ) : ( ( 0 ... ( # `
 ( S `  F ) ) )  X.  ( I  X.  2o ) ) --> W )
20 frn 5719 . . . . . . 7  |-  ( ( T `  ( S `
 F ) ) : ( ( 0 ... ( # `  ( S `  F )
) )  X.  (
I  X.  2o ) ) --> W  ->  ran  ( T `  ( S `
 F ) ) 
C_  W )
2119, 20syl 17 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ran  ( T `
 ( S `  F ) )  C_  W )
22 simpr 459 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  A  e.  ran  ( T `  ( S `  F )
) )
2321, 22sseldd 3442 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  A  e.  W )
2423s1cld 12667 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  <" A ">  e. Word  W )
25 ccatcl 12645 . . . 4  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W )  ->  ( F ++  <" A "> )  e. Word  W )
2610, 24, 25syl2anc 659 . . 3  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( F ++  <" A "> )  e. Word  W )
27 ccatlen 12646 . . . . . . 7  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W )  ->  ( # `
 ( F ++  <" A "> )
)  =  ( (
# `  F )  +  ( # `  <" A "> )
) )
2810, 24, 27syl2anc 659 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  ( F ++  <" A "> ) )  =  ( ( # `  F
)  +  ( # `  <" A "> ) ) )
29 s1len 12669 . . . . . . 7  |-  ( # `  <" A "> )  =  1
3029oveq2i 6288 . . . . . 6  |-  ( (
# `  F )  +  ( # `  <" A "> )
)  =  ( (
# `  F )  +  1 )
3128, 30syl6eq 2459 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  ( F ++  <" A "> ) )  =  ( ( # `  F
)  +  1 ) )
32 lencl 12612 . . . . . 6  |-  ( F  e. Word  W  ->  ( # `
 F )  e. 
NN0 )
33 nn0p1nn 10875 . . . . . 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 2490 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  ( F ++  <" A "> ) )  e.  NN )
36 wrdfin 12611 . . . . 5  |-  ( ( F ++  <" A "> )  e. Word  W  -> 
( F ++  <" A "> )  e.  Fin )
37 hashnncl 12482 . . . . 5  |-  ( ( F ++  <" A "> )  e.  Fin  ->  ( ( # `  ( F ++  <" A "> ) )  e.  NN  <->  ( F ++  <" A "> )  =/=  (/) ) )
3826, 36, 373syl 20 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( # `
 ( F ++  <" A "> )
)  e.  NN  <->  ( F ++  <" A "> )  =/=  (/) ) )
3935, 38mpbid 210 . . 3  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( F ++  <" A "> )  =/=  (/) )
40 eldifsn 4096 . . 3  |-  ( ( F ++  <" A "> )  e.  (Word  W  \  { (/) } )  <-> 
( ( F ++  <" A "> )  e. Word  W  /\  ( F ++ 
<" A "> )  =/=  (/) ) )
4126, 39, 40sylanbrc 662 . 2  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( F ++  <" A "> )  e.  (Word  W  \  { (/) } ) )
42 eldifsni 4097 . . . . . . 7  |-  ( F  e.  (Word  W  \  { (/) } )  ->  F  =/=  (/) )
439, 42syl 17 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  F  =/=  (/) )
44 wrdfin 12611 . . . . . . 7  |-  ( F  e. Word  W  ->  F  e.  Fin )
45 hashnncl 12482 . . . . . . 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 11892 . . . . 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 12647 . . . 4  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  0  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( F ++ 
<" A "> ) `  0 )  =  ( F ` 
0 ) )
5110, 24, 49, 50syl3anc 1230 . . 3  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F ++  <" A "> ) `  0 )  =  ( F ` 
0 ) )
527simp2bi 1013 . . . 4  |-  ( F  e.  dom  S  -> 
( F `  0
)  e.  D )
5352adantr 463 . . 3  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( F `  0 )  e.  D )
5451, 53eqeltrd 2490 . 2  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F ++  <" 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 463 . . . . 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 11885 . . . . . . . . . . 11  |-  ( 1..^ ( # `  F
) )  C_  (
0..^ ( # `  F
) )
5857sseli 3437 . . . . . . . . . 10  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  i  e.  ( 0..^ ( # `  F
) ) )
59 ccatval1 12647 . . . . . . . . . 10  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  i  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( F ++ 
<" A "> ) `  i )  =  ( F `  i ) )
6058, 59syl3an3 1265 . . . . . . . . 9  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  i  e.  ( 1..^ ( # `  F ) ) )  ->  ( ( F ++ 
<" A "> ) `  i )  =  ( F `  i ) )
61 elfzoel2 11856 . . . . . . . . . . . . . . . 16  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( # `  F
)  e.  ZZ )
62 peano2zm 10947 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  e.  ZZ  ->  ( ( # `
 F )  - 
1 )  e.  ZZ )
6361, 62syl 17 . . . . . . . . . . . . . . 15  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( ( # `
 F )  - 
1 )  e.  ZZ )
6461zred 11007 . . . . . . . . . . . . . . . 16  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( # `  F
)  e.  RR )
6564lem1d 10518 . . . . . . . . . . . . . . 15  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( ( # `
 F )  - 
1 )  <_  ( # `
 F ) )
66 eluz2 11132 . . . . . . . . . . . . . . 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 11883 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  e.  ( ZZ>= `  ( ( # `
 F )  - 
1 ) )  -> 
( 0..^ ( (
# `  F )  -  1 ) ) 
C_  ( 0..^ (
# `  F )
) )
6967, 68syl 17 . . . . . . . . . . . . 13  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( 0..^ ( ( # `  F
)  -  1 ) )  C_  ( 0..^ ( # `  F
) ) )
70 elfzoelz 11857 . . . . . . . . . . . . . . 15  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  i  e.  ZZ )
71 elfzom1b 11946 . . . . . . . . . . . . . . 15  |-  ( ( i  e.  ZZ  /\  ( # `  F )  e.  ZZ )  -> 
( i  e.  ( 1..^ ( # `  F
) )  <->  ( i  -  1 )  e.  ( 0..^ ( (
# `  F )  -  1 ) ) ) )
7270, 61, 71syl2anc 659 . . . . . . . . . . . . . 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 3442 . . . . . . . . . . . 12  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( i  -  1 )  e.  ( 0..^ ( # `  F ) ) )
75 ccatval1 12647 . . . . . . . . . . . 12  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  ( i  -  1 )  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( F ++ 
<" A "> ) `  ( i  -  1 ) )  =  ( F `  ( i  -  1 ) ) )
7674, 75syl3an3 1265 . . . . . . . . . . 11  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  i  e.  ( 1..^ ( # `  F ) ) )  ->  ( ( F ++ 
<" A "> ) `  ( i  -  1 ) )  =  ( F `  ( i  -  1 ) ) )
7776fveq2d 5852 . . . . . . . . . 10  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  i  e.  ( 1..^ ( # `  F ) ) )  ->  ( T `  ( ( F ++  <" A "> ) `  ( i  -  1 ) ) )  =  ( T `  ( F `  ( i  -  1 ) ) ) )
7877rneqd 5050 . . . . . . . . 9  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  i  e.  ( 1..^ ( # `  F ) ) )  ->  ran  ( T `  ( ( F ++  <" A "> ) `  ( i  -  1 ) ) )  =  ran  ( T `  ( F `  ( i  -  1 ) ) ) )
7960, 78eleq12d 2484 . . . . . . . 8  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  i  e.  ( 1..^ ( # `  F ) ) )  ->  ( ( ( F ++  <" A "> ) `  i )  e.  ran  ( T `
 ( ( F ++ 
<" 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 ++ 
<" A "> ) `  i )  e.  ran  ( T `  ( ( F ++  <" A "> ) `  ( i  -  1 ) ) )  <->  ( F `  i )  e.  ran  ( T `  ( F `
 ( i  - 
1 ) ) ) ) )
8180ralbidva 2839 . . . . . 6  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W )  ->  ( A. i  e.  (
1..^ ( # `  F
) ) ( ( F ++  <" A "> ) `  i )  e.  ran  ( T `
 ( ( F ++ 
<" A "> ) `  ( i  -  1 ) ) )  <->  A. i  e.  ( 1..^ ( # `  F
) ) ( F `
 i )  e. 
ran  ( T `  ( F `  ( i  -  1 ) ) ) ) )
8210, 24, 81syl2anc 659 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( A. i  e.  ( 1..^ ( # `  F
) ) ( ( F ++  <" A "> ) `  i )  e.  ran  ( T `
 ( ( F ++ 
<" 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 ++  <" A "> ) `  i
)  e.  ran  ( T `  ( ( F ++  <" A "> ) `  ( i  -  1 ) ) ) )
8410, 32syl 17 . . . . . . . . . 10  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  F
)  e.  NN0 )
8584nn0cnd 10894 . . . . . . . . 9  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  F
)  e.  CC )
8685addid2d 9814 . . . . . . . 8  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( 0  +  ( # `  F
) )  =  (
# `  F )
)
8786fveq2d 5852 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F ++  <" A "> ) `  ( 0  +  ( # `  F
) ) )  =  ( ( F ++  <" A "> ) `  ( # `  F
) ) )
88 1nn 10586 . . . . . . . . . . 11  |-  1  e.  NN
8929, 88eqeltri 2486 . . . . . . . . . 10  |-  ( # `  <" A "> )  e.  NN
9089a1i 11 . . . . . . . . 9  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  <" A "> )  e.  NN )
91 lbfzo0 11892 . . . . . . . . 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 12649 . . . . . . . 8  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  0  e.  ( 0..^ ( # `  <" A "> ) ) )  -> 
( ( F ++  <" A "> ) `  ( 0  +  (
# `  F )
) )  =  (
<" A "> `  0 ) )
9410, 24, 92, 93syl3anc 1230 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F ++  <" A "> ) `  ( 0  +  ( # `  F
) ) )  =  ( <" A "> `  0 )
)
9587, 94eqtr3d 2445 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F ++  <" A "> ) `  ( # `  F ) )  =  ( <" A "> `  0 )
)
96 s1fv 12671 . . . . . . . 8  |-  ( A  e.  ran  ( T `
 ( S `  F ) )  -> 
( <" A "> `  0 )  =  A )
9796adantl 464 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( <" A "> `  0
)  =  A )
98 fzo0end 11939 . . . . . . . . . . . 12  |-  ( (
# `  F )  e.  NN  ->  ( ( # `
 F )  - 
1 )  e.  ( 0..^ ( # `  F
) ) )
9947, 98syl 17 . . . . . . . . . . 11  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( # `
 F )  - 
1 )  e.  ( 0..^ ( # `  F
) ) )
100 ccatval1 12647 . . . . . . . . . . 11  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  ( (
# `  F )  -  1 )  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( F ++ 
<" A "> ) `  ( ( # `
 F )  - 
1 ) )  =  ( F `  (
( # `  F )  -  1 ) ) )
10110, 24, 99, 100syl3anc 1230 . . . . . . . . . 10  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F ++  <" A "> ) `  ( (
# `  F )  -  1 ) )  =  ( F `  ( ( # `  F
)  -  1 ) ) )
1021, 2, 3, 4, 5, 6efgsval 17071 . . . . . . . . . . 11  |-  ( F  e.  dom  S  -> 
( S `  F
)  =  ( F `
 ( ( # `  F )  -  1 ) ) )
103102adantr 463 . . . . . . . . . 10  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( S `  F )  =  ( F `  ( (
# `  F )  -  1 ) ) )
104101, 103eqtr4d 2446 . . . . . . . . 9  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F ++  <" A "> ) `  ( (
# `  F )  -  1 ) )  =  ( S `  F ) )
105104fveq2d 5852 . . . . . . . 8  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( T `  ( ( F ++  <" A "> ) `  ( ( # `  F
)  -  1 ) ) )  =  ( T `  ( S `
 F ) ) )
106105rneqd 5050 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ran  ( T `
 ( ( F ++ 
<" A "> ) `  ( ( # `
 F )  - 
1 ) ) )  =  ran  ( T `
 ( S `  F ) ) )
10722, 97, 1063eltr4d 2505 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( <" A "> `  0
)  e.  ran  ( T `  ( ( F ++  <" A "> ) `  ( (
# `  F )  -  1 ) ) ) )
10895, 107eqeltrd 2490 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F ++  <" A "> ) `  ( # `  F ) )  e. 
ran  ( T `  ( ( F ++  <" A "> ) `  ( ( # `  F
)  -  1 ) ) ) )
109 fvex 5858 . . . . . 6  |-  ( # `  F )  e.  _V
110 fveq2 5848 . . . . . . 7  |-  ( i  =  ( # `  F
)  ->  ( ( F ++  <" A "> ) `  i )  =  ( ( F ++ 
<" A "> ) `  ( # `  F
) ) )
111 oveq1 6284 . . . . . . . . . 10  |-  ( i  =  ( # `  F
)  ->  ( i  -  1 )  =  ( ( # `  F
)  -  1 ) )
112111fveq2d 5852 . . . . . . . . 9  |-  ( i  =  ( # `  F
)  ->  ( ( F ++  <" A "> ) `  ( i  -  1 ) )  =  ( ( F ++ 
<" A "> ) `  ( ( # `
 F )  - 
1 ) ) )
113112fveq2d 5852 . . . . . . . 8  |-  ( i  =  ( # `  F
)  ->  ( T `  ( ( F ++  <" A "> ) `  ( i  -  1 ) ) )  =  ( T `  (
( F ++  <" A "> ) `  (
( # `  F )  -  1 ) ) ) )
114113rneqd 5050 . . . . . . 7  |-  ( i  =  ( # `  F
)  ->  ran  ( T `
 ( ( F ++ 
<" A "> ) `  ( i  -  1 ) ) )  =  ran  ( T `  ( ( F ++  <" A "> ) `  ( (
# `  F )  -  1 ) ) ) )
115110, 114eleq12d 2484 . . . . . 6  |-  ( i  =  ( # `  F
)  ->  ( (
( F ++  <" A "> ) `  i
)  e.  ran  ( T `  ( ( F ++  <" A "> ) `  ( i  -  1 ) ) )  <->  ( ( F ++ 
<" A "> ) `  ( # `  F
) )  e.  ran  ( T `  ( ( F ++  <" A "> ) `  ( (
# `  F )  -  1 ) ) ) ) )
116109, 115ralsn 4010 . . . . 5  |-  ( A. i  e.  { ( # `
 F ) }  ( ( F ++  <" A "> ) `  i )  e.  ran  ( T `  ( ( F ++  <" A "> ) `  ( i  -  1 ) ) )  <->  ( ( F ++ 
<" A "> ) `  ( # `  F
) )  e.  ran  ( T `  ( ( F ++  <" A "> ) `  ( (
# `  F )  -  1 ) ) ) )
117108, 116sylibr 212 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  A. i  e.  { ( # `  F
) }  ( ( F ++  <" A "> ) `  i )  e.  ran  ( T `
 ( ( F ++ 
<" A "> ) `  ( i  -  1 ) ) ) )
118 ralunb 3623 . . . 4  |-  ( A. i  e.  ( (
1..^ ( # `  F
) )  u.  {
( # `  F ) } ) ( ( F ++  <" A "> ) `  i )  e.  ran  ( T `
 ( ( F ++ 
<" A "> ) `  ( i  -  1 ) ) )  <->  ( A. i  e.  ( 1..^ ( # `  F ) ) ( ( F ++  <" A "> ) `  i
)  e.  ran  ( T `  ( ( F ++  <" A "> ) `  ( i  -  1 ) ) )  /\  A. i  e.  { ( # `  F
) }  ( ( F ++  <" A "> ) `  i )  e.  ran  ( T `
 ( ( F ++ 
<" A "> ) `  ( i  -  1 ) ) ) ) )
11983, 117, 118sylanbrc 662 . . 3  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  A. i  e.  ( ( 1..^ (
# `  F )
)  u.  { (
# `  F ) } ) ( ( F ++  <" A "> ) `  i )  e.  ran  ( T `
 ( ( F ++ 
<" A "> ) `  ( i  -  1 ) ) ) )
12031oveq2d 6293 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( 1..^ ( # `  ( F ++  <" A "> ) ) )  =  ( 1..^ ( (
# `  F )  +  1 ) ) )
121 nnuz 11161 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
12247, 121syl6eleq 2500 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  F
)  e.  ( ZZ>= ` 
1 ) )
123 fzosplitsn 11953 . . . . . 6  |-  ( (
# `  F )  e.  ( ZZ>= `  1 )  ->  ( 1..^ ( (
# `  F )  +  1 ) )  =  ( ( 1..^ ( # `  F
) )  u.  {
( # `  F ) } ) )
124122, 123syl 17 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( 1..^ ( ( # `  F
)  +  1 ) )  =  ( ( 1..^ ( # `  F
) )  u.  {
( # `  F ) } ) )
125120, 124eqtrd 2443 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( 1..^ ( # `  ( F ++  <" A "> ) ) )  =  ( ( 1..^ (
# `  F )
)  u.  { (
# `  F ) } ) )
126125raleqdv 3009 . . 3  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( A. i  e.  ( 1..^ ( # `  ( F ++  <" A "> ) ) ) ( ( F ++  <" A "> ) `  i
)  e.  ran  ( T `  ( ( F ++  <" A "> ) `  ( i  -  1 ) ) )  <->  A. i  e.  ( ( 1..^ ( # `  F ) )  u. 
{ ( # `  F
) } ) ( ( F ++  <" A "> ) `  i
)  e.  ran  ( T `  ( ( F ++  <" A "> ) `  ( i  -  1 ) ) ) ) )
127119, 126mpbird 232 . 2  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  A. i  e.  ( 1..^ ( # `  ( F ++  <" A "> ) ) ) ( ( F ++  <" A "> ) `  i )  e.  ran  ( T `  ( ( F ++  <" A "> ) `  ( i  -  1 ) ) ) )
1281, 2, 3, 4, 5, 6efgsdm 17070 . 2  |-  ( ( F ++  <" A "> )  e.  dom  S  <-> 
( ( F ++  <" A "> )  e.  (Word  W  \  { (/)
} )  /\  (
( F ++  <" A "> ) `  0
)  e.  D  /\  A. i  e.  ( 1..^ ( # `  ( F ++  <" A "> ) ) ) ( ( F ++  <" A "> ) `  i
)  e.  ran  ( T `  ( ( F ++  <" A "> ) `  ( i  -  1 ) ) ) ) )
12941, 54, 127, 128syl3anbrc 1181 1  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( F ++  <" A "> )  e.  dom  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2753   {crab 2757    \ cdif 3410    u. cun 3411    C_ wss 3413   (/)c0 3737   {csn 3971   <.cop 3977   <.cotp 3979   U_ciun 4270   class class class wbr 4394    |-> cmpt 4452    _I cid 4732    X. cxp 4820   dom cdm 4822   ran crn 4823   -->wf 5564   ` cfv 5568  (class class class)co 6277    |-> cmpt2 6279   1oc1o 7159   2oc2o 7160   Fincfn 7553   0cc0 9521   1c1 9522    + caddc 9524    <_ cle 9658    - cmin 9840   NNcn 10575   NN0cn0 10835   ZZcz 10904   ZZ>=cuz 11126   ...cfz 11724  ..^cfzo 11852   #chash 12450  Word cword 12581   ++ cconcat 12583   <"cs1 12584   splice csplice 12586   <"cs2 12860   ~FG cefg 17046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-ot 3980  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-2o 7167  df-oadd 7170  df-er 7347  df-map 7458  df-pm 7459  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-n0 10836  df-z 10905  df-uz 11127  df-fz 11725  df-fzo 11853  df-hash 12451  df-word 12589  df-concat 12591  df-s1 12592  df-substr 12593  df-splice 12594  df-s2 12867
This theorem is referenced by:  efgsfo  17079  efgredlemd  17084  efgrelexlemb  17090
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