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Theorem efgsp1 17387
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 17380 . . . . . . 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 1023 . . . . . 6  |-  ( F  e.  dom  S  ->  F  e.  (Word  W  \  { (/) } ) )
98adantr 467 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  F  e.  (Word  W  \  { (/) } ) )
109eldifad 3416 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  F  e. Word  W )
111, 2, 3, 4, 5, 6efgsf 17379 . . . . . . . . . . . 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 5734 . . . . . . . . . . . . 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 5721 . . . . . . . . . . . 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 213 . . . . . . . . . . 11  |-  S : dom  S --> W
1514ffvelrni 6021 . . . . . . . . . 10  |-  ( F  e.  dom  S  -> 
( S `  F
)  e.  W )
1615adantr 467 . . . . . . . . 9  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( S `  F )  e.  W
)
171, 2, 3, 4efgtf 17372 . . . . . . . . 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 465 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( T `  ( S `  F
) ) : ( ( 0 ... ( # `
 ( S `  F ) ) )  X.  ( I  X.  2o ) ) --> W )
20 frn 5735 . . . . . . 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 463 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  A  e.  ran  ( T `  ( S `  F )
) )
2321, 22sseldd 3433 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  A  e.  W )
2423s1cld 12742 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  <" A ">  e. Word  W )
25 ccatcl 12720 . . . 4  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W )  ->  ( F ++  <" A "> )  e. Word  W )
2610, 24, 25syl2anc 667 . . 3  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( F ++  <" A "> )  e. Word  W )
27 ccatlen 12721 . . . . . . 7  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W )  ->  ( # `
 ( F ++  <" A "> )
)  =  ( (
# `  F )  +  ( # `  <" A "> )
) )
2810, 24, 27syl2anc 667 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  ( F ++  <" A "> ) )  =  ( ( # `  F
)  +  ( # `  <" A "> ) ) )
29 s1len 12744 . . . . . . 7  |-  ( # `  <" A "> )  =  1
3029oveq2i 6301 . . . . . 6  |-  ( (
# `  F )  +  ( # `  <" A "> )
)  =  ( (
# `  F )  +  1 )
3128, 30syl6eq 2501 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  ( F ++  <" A "> ) )  =  ( ( # `  F
)  +  1 ) )
32 lencl 12687 . . . . . 6  |-  ( F  e. Word  W  ->  ( # `
 F )  e. 
NN0 )
33 nn0p1nn 10909 . . . . . 6  |-  ( (
# `  F )  e.  NN0  ->  ( ( # `
 F )  +  1 )  e.  NN )
3410, 32, 333syl 18 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( # `
 F )  +  1 )  e.  NN )
3531, 34eqeltrd 2529 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  ( F ++  <" A "> ) )  e.  NN )
36 wrdfin 12686 . . . . 5  |-  ( ( F ++  <" A "> )  e. Word  W  -> 
( F ++  <" A "> )  e.  Fin )
37 hashnncl 12547 . . . . 5  |-  ( ( F ++  <" A "> )  e.  Fin  ->  ( ( # `  ( F ++  <" A "> ) )  e.  NN  <->  ( F ++  <" A "> )  =/=  (/) ) )
3826, 36, 373syl 18 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( # `
 ( F ++  <" A "> )
)  e.  NN  <->  ( F ++  <" A "> )  =/=  (/) ) )
3935, 38mpbid 214 . . 3  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( F ++  <" A "> )  =/=  (/) )
40 eldifsn 4097 . . 3  |-  ( ( F ++  <" A "> )  e.  (Word  W  \  { (/) } )  <-> 
( ( F ++  <" A "> )  e. Word  W  /\  ( F ++ 
<" A "> )  =/=  (/) ) )
4126, 39, 40sylanbrc 670 . 2  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( F ++  <" A "> )  e.  (Word  W  \  { (/) } ) )
42 eldifsni 4098 . . . . . . 7  |-  ( F  e.  (Word  W  \  { (/) } )  ->  F  =/=  (/) )
439, 42syl 17 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  F  =/=  (/) )
44 wrdfin 12686 . . . . . . 7  |-  ( F  e. Word  W  ->  F  e.  Fin )
45 hashnncl 12547 . . . . . . 7  |-  ( F  e.  Fin  ->  (
( # `  F )  e.  NN  <->  F  =/=  (/) ) )
4610, 44, 453syl 18 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( # `
 F )  e.  NN  <->  F  =/=  (/) ) )
4743, 46mpbird 236 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  F
)  e.  NN )
48 lbfzo0 11955 . . . . 5  |-  ( 0  e.  ( 0..^ (
# `  F )
)  <->  ( # `  F
)  e.  NN )
4947, 48sylibr 216 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  0  e.  ( 0..^ ( # `  F
) ) )
50 ccatval1 12722 . . . 4  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  0  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( F ++ 
<" A "> ) `  0 )  =  ( F ` 
0 ) )
5110, 24, 49, 50syl3anc 1268 . . 3  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F ++  <" A "> ) `  0 )  =  ( F ` 
0 ) )
527simp2bi 1024 . . . 4  |-  ( F  e.  dom  S  -> 
( F `  0
)  e.  D )
5352adantr 467 . . 3  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( F `  0 )  e.  D )
5451, 53eqeltrd 2529 . 2  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F ++  <" A "> ) `  0 )  e.  D )
557simp3bi 1025 . . . . . 6  |-  ( F  e.  dom  S  ->  A. i  e.  (
1..^ ( # `  F
) ) ( F `
 i )  e. 
ran  ( T `  ( F `  ( i  -  1 ) ) ) )
5655adantr 467 . . . . 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 11948 . . . . . . . . . . 11  |-  ( 1..^ ( # `  F
) )  C_  (
0..^ ( # `  F
) )
5857sseli 3428 . . . . . . . . . 10  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  i  e.  ( 0..^ ( # `  F
) ) )
59 ccatval1 12722 . . . . . . . . . 10  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  i  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( F ++ 
<" A "> ) `  i )  =  ( F `  i ) )
6058, 59syl3an3 1303 . . . . . . . . 9  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  i  e.  ( 1..^ ( # `  F ) ) )  ->  ( ( F ++ 
<" A "> ) `  i )  =  ( F `  i ) )
61 elfzoel2 11919 . . . . . . . . . . . . . . . 16  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( # `  F
)  e.  ZZ )
62 peano2zm 10980 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  e.  ZZ  ->  ( ( # `
 F )  - 
1 )  e.  ZZ )
6361, 62syl 17 . . . . . . . . . . . . . . 15  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( ( # `
 F )  - 
1 )  e.  ZZ )
6461zred 11040 . . . . . . . . . . . . . . . 16  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( # `  F
)  e.  RR )
6564lem1d 10540 . . . . . . . . . . . . . . 15  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( ( # `
 F )  - 
1 )  <_  ( # `
 F ) )
66 eluz2 11165 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  e.  ( ZZ>= `  ( ( # `
 F )  - 
1 ) )  <->  ( (
( # `  F )  -  1 )  e.  ZZ  /\  ( # `  F )  e.  ZZ  /\  ( ( # `  F
)  -  1 )  <_  ( # `  F
) ) )
6763, 61, 65, 66syl3anbrc 1192 . . . . . . . . . . . . . 14  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( # `  F
)  e.  ( ZZ>= `  ( ( # `  F
)  -  1 ) ) )
68 fzoss2 11946 . . . . . . . . . . . . . 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 11920 . . . . . . . . . . . . . . 15  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  i  e.  ZZ )
71 elfzom1b 12010 . . . . . . . . . . . . . . 15  |-  ( ( i  e.  ZZ  /\  ( # `  F )  e.  ZZ )  -> 
( i  e.  ( 1..^ ( # `  F
) )  <->  ( i  -  1 )  e.  ( 0..^ ( (
# `  F )  -  1 ) ) ) )
7270, 61, 71syl2anc 667 . . . . . . . . . . . . . 14  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( i  e.  ( 1..^ ( # `  F ) )  <->  ( i  -  1 )  e.  ( 0..^ ( (
# `  F )  -  1 ) ) ) )
7372ibi 245 . . . . . . . . . . . . 13  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( i  -  1 )  e.  ( 0..^ ( (
# `  F )  -  1 ) ) )
7469, 73sseldd 3433 . . . . . . . . . . . 12  |-  ( i  e.  ( 1..^ (
# `  F )
)  ->  ( i  -  1 )  e.  ( 0..^ ( # `  F ) ) )
75 ccatval1 12722 . . . . . . . . . . . 12  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  ( i  -  1 )  e.  ( 0..^ ( # `  F ) ) )  ->  ( ( F ++ 
<" A "> ) `  ( i  -  1 ) )  =  ( F `  ( i  -  1 ) ) )
7674, 75syl3an3 1303 . . . . . . . . . . 11  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  i  e.  ( 1..^ ( # `  F ) ) )  ->  ( ( F ++ 
<" A "> ) `  ( i  -  1 ) )  =  ( F `  ( i  -  1 ) ) )
7776fveq2d 5869 . . . . . . . . . 10  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  i  e.  ( 1..^ ( # `  F ) ) )  ->  ( T `  ( ( F ++  <" A "> ) `  ( i  -  1 ) ) )  =  ( T `  ( F `  ( i  -  1 ) ) ) )
7877rneqd 5062 . . . . . . . . 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 2523 . . . . . . . 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 1208 . . . . . . 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 2824 . . . . . 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 667 . . . . 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 236 . . . 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 10927 . . . . . . . . 9  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  F
)  e.  CC )
8685addid2d 9834 . . . . . . . 8  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( 0  +  ( # `  F
) )  =  (
# `  F )
)
8786fveq2d 5869 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F ++  <" A "> ) `  ( 0  +  ( # `  F
) ) )  =  ( ( F ++  <" A "> ) `  ( # `  F
) ) )
88 1nn 10620 . . . . . . . . . . 11  |-  1  e.  NN
8929, 88eqeltri 2525 . . . . . . . . . 10  |-  ( # `  <" A "> )  e.  NN
9089a1i 11 . . . . . . . . 9  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  <" A "> )  e.  NN )
91 lbfzo0 11955 . . . . . . . . 9  |-  ( 0  e.  ( 0..^ (
# `  <" A "> ) )  <->  ( # `  <" A "> )  e.  NN )
9290, 91sylibr 216 . . . . . . . 8  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  0  e.  ( 0..^ ( # `  <" A "> )
) )
93 ccatval3 12724 . . . . . . . 8  |-  ( ( F  e. Word  W  /\  <" A ">  e. Word  W  /\  0  e.  ( 0..^ ( # `  <" A "> ) ) )  -> 
( ( F ++  <" A "> ) `  ( 0  +  (
# `  F )
) )  =  (
<" A "> `  0 ) )
9410, 24, 92, 93syl3anc 1268 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F ++  <" A "> ) `  ( 0  +  ( # `  F
) ) )  =  ( <" A "> `  0 )
)
9587, 94eqtr3d 2487 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F ++  <" A "> ) `  ( # `  F ) )  =  ( <" A "> `  0 )
)
96 s1fv 12748 . . . . . . . 8  |-  ( A  e.  ran  ( T `
 ( S `  F ) )  -> 
( <" A "> `  0 )  =  A )
9796adantl 468 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( <" A "> `  0
)  =  A )
98 fzo0end 12003 . . . . . . . . . . . 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 12722 . . . . . . . . . . 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 1268 . . . . . . . . . 10  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F ++  <" A "> ) `  ( (
# `  F )  -  1 ) )  =  ( F `  ( ( # `  F
)  -  1 ) ) )
1021, 2, 3, 4, 5, 6efgsval 17381 . . . . . . . . . . 11  |-  ( F  e.  dom  S  -> 
( S `  F
)  =  ( F `
 ( ( # `  F )  -  1 ) ) )
103102adantr 467 . . . . . . . . . 10  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( S `  F )  =  ( F `  ( (
# `  F )  -  1 ) ) )
104101, 103eqtr4d 2488 . . . . . . . . 9  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F ++  <" A "> ) `  ( (
# `  F )  -  1 ) )  =  ( S `  F ) )
105104fveq2d 5869 . . . . . . . 8  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( T `  ( ( F ++  <" A "> ) `  ( ( # `  F
)  -  1 ) ) )  =  ( T `  ( S `
 F ) ) )
106105rneqd 5062 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ran  ( T `
 ( ( F ++ 
<" A "> ) `  ( ( # `
 F )  - 
1 ) ) )  =  ran  ( T `
 ( S `  F ) ) )
10722, 97, 1063eltr4d 2544 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( <" A "> `  0
)  e.  ran  ( T `  ( ( F ++  <" A "> ) `  ( (
# `  F )  -  1 ) ) ) )
10895, 107eqeltrd 2529 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( ( F ++  <" A "> ) `  ( # `  F ) )  e. 
ran  ( T `  ( ( F ++  <" A "> ) `  ( ( # `  F
)  -  1 ) ) ) )
109 fvex 5875 . . . . . 6  |-  ( # `  F )  e.  _V
110 fveq2 5865 . . . . . . 7  |-  ( i  =  ( # `  F
)  ->  ( ( F ++  <" A "> ) `  i )  =  ( ( F ++ 
<" A "> ) `  ( # `  F
) ) )
111 oveq1 6297 . . . . . . . . . 10  |-  ( i  =  ( # `  F
)  ->  ( i  -  1 )  =  ( ( # `  F
)  -  1 ) )
112111fveq2d 5869 . . . . . . . . 9  |-  ( i  =  ( # `  F
)  ->  ( ( F ++  <" A "> ) `  ( i  -  1 ) )  =  ( ( F ++ 
<" A "> ) `  ( ( # `
 F )  - 
1 ) ) )
113112fveq2d 5869 . . . . . . . 8  |-  ( i  =  ( # `  F
)  ->  ( T `  ( ( F ++  <" A "> ) `  ( i  -  1 ) ) )  =  ( T `  (
( F ++  <" A "> ) `  (
( # `  F )  -  1 ) ) ) )
114113rneqd 5062 . . . . . . 7  |-  ( i  =  ( # `  F
)  ->  ran  ( T `
 ( ( F ++ 
<" A "> ) `  ( i  -  1 ) ) )  =  ran  ( T `  ( ( F ++  <" A "> ) `  ( (
# `  F )  -  1 ) ) ) )
115110, 114eleq12d 2523 . . . . . 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 216 . . . 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 3615 . . . 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 670 . . 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 6306 . . . . 5  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( 1..^ ( # `  ( F ++  <" A "> ) ) )  =  ( 1..^ ( (
# `  F )  +  1 ) ) )
121 nnuz 11194 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
12247, 121syl6eleq 2539 . . . . . 6  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( # `  F
)  e.  ( ZZ>= ` 
1 ) )
123 fzosplitsn 12017 . . . . . 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 2485 . . . 4  |-  ( ( F  e.  dom  S  /\  A  e.  ran  ( T `  ( S `
 F ) ) )  ->  ( 1..^ ( # `  ( F ++  <" A "> ) ) )  =  ( ( 1..^ (
# `  F )
)  u.  { (
# `  F ) } ) )
126125raleqdv 2993 . . 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 236 . 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 17380 . 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 1192 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 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   {crab 2741    \ cdif 3401    u. cun 3402    C_ wss 3404   (/)c0 3731   {csn 3968   <.cop 3974   <.cotp 3976   U_ciun 4278   class class class wbr 4402    |-> cmpt 4461    _I cid 4744    X. cxp 4832   dom cdm 4834   ran crn 4835   -->wf 5578   ` cfv 5582  (class class class)co 6290    |-> cmpt2 6292   1oc1o 7175   2oc2o 7176   Fincfn 7569   0cc0 9539   1c1 9540    + caddc 9542    <_ cle 9676    - cmin 9860   NNcn 10609   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   ...cfz 11784  ..^cfzo 11915   #chash 12515  Word cword 12656   ++ cconcat 12658   <"cs1 12659   splice csplice 12661   <"cs2 12937   ~FG cefg 17356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-ot 3977  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-hash 12516  df-word 12664  df-concat 12666  df-s1 12667  df-substr 12668  df-splice 12669  df-s2 12944
This theorem is referenced by:  efgsfo  17389  efgredlemd  17394  efgrelexlemb  17400
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