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Theorem efgredlemd 16223
Description: The reduced word that forms the base of the sequence in efgsval 16210 is uniquely determined, given the ending representation. (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 ) ) )
efgredlem.1  |-  ( ph  ->  A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  ( # `  ( S `  A )
)  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) ) )
efgredlem.2  |-  ( ph  ->  A  e.  dom  S
)
efgredlem.3  |-  ( ph  ->  B  e.  dom  S
)
efgredlem.4  |-  ( ph  ->  ( S `  A
)  =  ( S `
 B ) )
efgredlem.5  |-  ( ph  ->  -.  ( A ` 
0 )  =  ( B `  0 ) )
efgredlemb.k  |-  K  =  ( ( ( # `  A )  -  1 )  -  1 )
efgredlemb.l  |-  L  =  ( ( ( # `  B )  -  1 )  -  1 )
efgredlemb.p  |-  ( ph  ->  P  e.  ( 0 ... ( # `  ( A `  K )
) ) )
efgredlemb.q  |-  ( ph  ->  Q  e.  ( 0 ... ( # `  ( B `  L )
) ) )
efgredlemb.u  |-  ( ph  ->  U  e.  ( I  X.  2o ) )
efgredlemb.v  |-  ( ph  ->  V  e.  ( I  X.  2o ) )
efgredlemb.6  |-  ( ph  ->  ( S `  A
)  =  ( P ( T `  ( A `  K )
) U ) )
efgredlemb.7  |-  ( ph  ->  ( S `  B
)  =  ( Q ( T `  ( B `  L )
) V ) )
efgredlemb.8  |-  ( ph  ->  -.  ( A `  K )  =  ( B `  L ) )
efgredlemd.9  |-  ( ph  ->  P  e.  ( ZZ>= `  ( Q  +  2
) ) )
efgredlemd.c  |-  ( ph  ->  C  e.  dom  S
)
efgredlemd.sc  |-  ( ph  ->  ( S `  C
)  =  ( ( ( B `  L
) substr  <. 0 ,  Q >. ) concat  ( ( A `
 K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) ) )
Assertion
Ref Expression
efgredlemd  |-  ( ph  ->  ( A `  0
)  =  ( B `
 0 ) )
Distinct variable groups:    a, b, A    y, a, z, b    L, a, b    K, a, b    t, n, v, w, y, z, P   
m, a, n, t, v, w, x, M, b    U, n, v, w, y, z    k, a, T, b, m, t, x    n, V, v, w, y, z    Q, n, t, v, w, y, z    W, a, b    k, n, v, w, y, z, W, m, t, x    .~ , a, b, m, t, x, y, z    B, a, b    C, a, b, k, m, n, t, v, w, x, y, z    S, a, b    I,
a, b, m, n, t, v, w, x, y, z    D, a, b, m, t
Allowed substitution hints:    ph( x, y, z, w, v, t, k, m, n, a, b)    A( x, y, z, w, v, t, k, m, n)    B( x, y, z, w, v, t, k, m, n)    D( x, y, z, w, v, k, n)    P( x, k, m, a, b)    Q( x, k, m, a, b)    .~ ( w, v, k, n)    S( x, y, z, w, v, t, k, m, n)    T( y, z, w, v, n)    U( x, t, k, m, a, b)    I( k)    K( x, y, z, w, v, t, k, m, n)    L( x, y, z, w, v, t, k, m, n)    M( y, z, k)    V( x, t, k, m, a, b)

Proof of Theorem efgredlemd
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 efgredlemd.c . . . . . . 7  |-  ( ph  ->  C  e.  dom  S
)
2 efgval.w . . . . . . . . 9  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
3 efgval.r . . . . . . . . 9  |-  .~  =  ( ~FG  `  I )
4 efgval2.m . . . . . . . . 9  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
5 efgval2.t . . . . . . . . 9  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
6 efgred.d . . . . . . . . 9  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
7 efgred.s . . . . . . . . 9  |-  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 ) ) )
82, 3, 4, 5, 6, 7efgsdm 16209 . . . . . . . 8  |-  ( C  e.  dom  S  <->  ( C  e.  (Word  W  \  { (/)
} )  /\  ( C `  0 )  e.  D  /\  A. i  e.  ( 1..^ ( # `  C ) ) ( C `  i )  e.  ran  ( T `
 ( C `  ( i  -  1 ) ) ) ) )
98simp1bi 998 . . . . . . 7  |-  ( C  e.  dom  S  ->  C  e.  (Word  W  \  { (/) } ) )
101, 9syl 16 . . . . . 6  |-  ( ph  ->  C  e.  (Word  W  \  { (/) } ) )
1110eldifad 3330 . . . . 5  |-  ( ph  ->  C  e. Word  W )
12 efgredlem.2 . . . . . . . . . 10  |-  ( ph  ->  A  e.  dom  S
)
132, 3, 4, 5, 6, 7efgsdm 16209 . . . . . . . . . . 11  |-  ( A  e.  dom  S  <->  ( A  e.  (Word  W  \  { (/)
} )  /\  ( A `  0 )  e.  D  /\  A. i  e.  ( 1..^ ( # `  A ) ) ( A `  i )  e.  ran  ( T `
 ( A `  ( i  -  1 ) ) ) ) )
1413simp1bi 998 . . . . . . . . . 10  |-  ( A  e.  dom  S  ->  A  e.  (Word  W  \  { (/) } ) )
1512, 14syl 16 . . . . . . . . 9  |-  ( ph  ->  A  e.  (Word  W  \  { (/) } ) )
1615eldifad 3330 . . . . . . . 8  |-  ( ph  ->  A  e. Word  W )
17 wrdf 12226 . . . . . . . 8  |-  ( A  e. Word  W  ->  A : ( 0..^ (
# `  A )
) --> W )
1816, 17syl 16 . . . . . . 7  |-  ( ph  ->  A : ( 0..^ ( # `  A
) ) --> W )
19 fzossfz 11556 . . . . . . . . 9  |-  ( 0..^ ( ( # `  A
)  -  1 ) )  C_  ( 0 ... ( ( # `  A )  -  1 ) )
20 lencl 12235 . . . . . . . . . . . 12  |-  ( A  e. Word  W  ->  ( # `
 A )  e. 
NN0 )
2116, 20syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  A
)  e.  NN0 )
2221nn0zd 10735 . . . . . . . . . 10  |-  ( ph  ->  ( # `  A
)  e.  ZZ )
23 fzoval 11540 . . . . . . . . . 10  |-  ( (
# `  A )  e.  ZZ  ->  ( 0..^ ( # `  A
) )  =  ( 0 ... ( (
# `  A )  -  1 ) ) )
2422, 23syl 16 . . . . . . . . 9  |-  ( ph  ->  ( 0..^ ( # `  A ) )  =  ( 0 ... (
( # `  A )  -  1 ) ) )
2519, 24syl5sseqr 3395 . . . . . . . 8  |-  ( ph  ->  ( 0..^ ( (
# `  A )  -  1 ) ) 
C_  ( 0..^ (
# `  A )
) )
26 efgredlemb.k . . . . . . . . 9  |-  K  =  ( ( ( # `  A )  -  1 )  -  1 )
27 efgredlem.1 . . . . . . . . . . . 12  |-  ( ph  ->  A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  ( # `  ( S `  A )
)  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) ) )
28 efgredlem.3 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  dom  S
)
29 efgredlem.4 . . . . . . . . . . . 12  |-  ( ph  ->  ( S `  A
)  =  ( S `
 B ) )
30 efgredlem.5 . . . . . . . . . . . 12  |-  ( ph  ->  -.  ( A ` 
0 )  =  ( B `  0 ) )
312, 3, 4, 5, 6, 7, 27, 12, 28, 29, 30efgredlema 16219 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( # `  A )  -  1 )  e.  NN  /\  ( ( # `  B
)  -  1 )  e.  NN ) )
3231simpld 456 . . . . . . . . . 10  |-  ( ph  ->  ( ( # `  A
)  -  1 )  e.  NN )
33 fzo0end 11605 . . . . . . . . . 10  |-  ( ( ( # `  A
)  -  1 )  e.  NN  ->  (
( ( # `  A
)  -  1 )  -  1 )  e.  ( 0..^ ( (
# `  A )  -  1 ) ) )
3432, 33syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( ( # `  A )  -  1 )  -  1 )  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) )
3526, 34syl5eqel 2519 . . . . . . . 8  |-  ( ph  ->  K  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) )
3625, 35sseldd 3347 . . . . . . 7  |-  ( ph  ->  K  e.  ( 0..^ ( # `  A
) ) )
3718, 36ffvelrnd 5834 . . . . . 6  |-  ( ph  ->  ( A `  K
)  e.  W )
3837s1cld 12280 . . . . 5  |-  ( ph  ->  <" ( A `
 K ) ">  e. Word  W )
39 eldifsn 3990 . . . . . . . 8  |-  ( C  e.  (Word  W  \  { (/) } )  <->  ( C  e. Word  W  /\  C  =/=  (/) ) )
40 lennncl 12236 . . . . . . . 8  |-  ( ( C  e. Word  W  /\  C  =/=  (/) )  ->  ( # `
 C )  e.  NN )
4139, 40sylbi 195 . . . . . . 7  |-  ( C  e.  (Word  W  \  { (/) } )  -> 
( # `  C )  e.  NN )
4210, 41syl 16 . . . . . 6  |-  ( ph  ->  ( # `  C
)  e.  NN )
43 lbfzo0 11572 . . . . . 6  |-  ( 0  e.  ( 0..^ (
# `  C )
)  <->  ( # `  C
)  e.  NN )
4442, 43sylibr 212 . . . . 5  |-  ( ph  ->  0  e.  ( 0..^ ( # `  C
) ) )
45 ccatval1 12262 . . . . 5  |-  ( ( C  e. Word  W  /\  <" ( A `  K ) ">  e. Word  W  /\  0  e.  ( 0..^ ( # `  C ) ) )  ->  ( ( C concat  <" ( A `  K ) "> ) `  0 )  =  ( C ` 
0 ) )
4611, 38, 44, 45syl3anc 1213 . . . 4  |-  ( ph  ->  ( ( C concat  <" ( A `  K ) "> ) `  0
)  =  ( C `
 0 ) )
472, 3, 4, 5, 6, 7efgsdm 16209 . . . . . . . . . . 11  |-  ( B  e.  dom  S  <->  ( B  e.  (Word  W  \  { (/)
} )  /\  ( B `  0 )  e.  D  /\  A. i  e.  ( 1..^ ( # `  B ) ) ( B `  i )  e.  ran  ( T `
 ( B `  ( i  -  1 ) ) ) ) )
4847simp1bi 998 . . . . . . . . . 10  |-  ( B  e.  dom  S  ->  B  e.  (Word  W  \  { (/) } ) )
4928, 48syl 16 . . . . . . . . 9  |-  ( ph  ->  B  e.  (Word  W  \  { (/) } ) )
5049eldifad 3330 . . . . . . . 8  |-  ( ph  ->  B  e. Word  W )
51 wrdf 12226 . . . . . . . 8  |-  ( B  e. Word  W  ->  B : ( 0..^ (
# `  B )
) --> W )
5250, 51syl 16 . . . . . . 7  |-  ( ph  ->  B : ( 0..^ ( # `  B
) ) --> W )
53 fzossfz 11556 . . . . . . . . 9  |-  ( 0..^ ( ( # `  B
)  -  1 ) )  C_  ( 0 ... ( ( # `  B )  -  1 ) )
54 lencl 12235 . . . . . . . . . . . 12  |-  ( B  e. Word  W  ->  ( # `
 B )  e. 
NN0 )
5550, 54syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  B
)  e.  NN0 )
5655nn0zd 10735 . . . . . . . . . 10  |-  ( ph  ->  ( # `  B
)  e.  ZZ )
57 fzoval 11540 . . . . . . . . . 10  |-  ( (
# `  B )  e.  ZZ  ->  ( 0..^ ( # `  B
) )  =  ( 0 ... ( (
# `  B )  -  1 ) ) )
5856, 57syl 16 . . . . . . . . 9  |-  ( ph  ->  ( 0..^ ( # `  B ) )  =  ( 0 ... (
( # `  B )  -  1 ) ) )
5953, 58syl5sseqr 3395 . . . . . . . 8  |-  ( ph  ->  ( 0..^ ( (
# `  B )  -  1 ) ) 
C_  ( 0..^ (
# `  B )
) )
60 efgredlemb.l . . . . . . . . 9  |-  L  =  ( ( ( # `  B )  -  1 )  -  1 )
6131simprd 460 . . . . . . . . . 10  |-  ( ph  ->  ( ( # `  B
)  -  1 )  e.  NN )
62 fzo0end 11605 . . . . . . . . . 10  |-  ( ( ( # `  B
)  -  1 )  e.  NN  ->  (
( ( # `  B
)  -  1 )  -  1 )  e.  ( 0..^ ( (
# `  B )  -  1 ) ) )
6361, 62syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( ( # `  B )  -  1 )  -  1 )  e.  ( 0..^ ( ( # `  B
)  -  1 ) ) )
6460, 63syl5eqel 2519 . . . . . . . 8  |-  ( ph  ->  L  e.  ( 0..^ ( ( # `  B
)  -  1 ) ) )
6559, 64sseldd 3347 . . . . . . 7  |-  ( ph  ->  L  e.  ( 0..^ ( # `  B
) ) )
6652, 65ffvelrnd 5834 . . . . . 6  |-  ( ph  ->  ( B `  L
)  e.  W )
6766s1cld 12280 . . . . 5  |-  ( ph  ->  <" ( B `
 L ) ">  e. Word  W )
68 ccatval1 12262 . . . . 5  |-  ( ( C  e. Word  W  /\  <" ( B `  L ) ">  e. Word  W  /\  0  e.  ( 0..^ ( # `  C ) ) )  ->  ( ( C concat  <" ( B `  L ) "> ) `  0 )  =  ( C ` 
0 ) )
6911, 67, 44, 68syl3anc 1213 . . . 4  |-  ( ph  ->  ( ( C concat  <" ( B `  L ) "> ) `  0
)  =  ( C `
 0 ) )
7046, 69eqtr4d 2470 . . 3  |-  ( ph  ->  ( ( C concat  <" ( A `  K ) "> ) `  0
)  =  ( ( C concat  <" ( B `
 L ) "> ) `  0
) )
71 fviss 5739 . . . . . . . . . 10  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
722, 71eqsstri 3376 . . . . . . . . 9  |-  W  C_ Word  ( I  X.  2o )
7372, 37sseldi 3344 . . . . . . . 8  |-  ( ph  ->  ( A `  K
)  e. Word  ( I  X.  2o ) )
74 lencl 12235 . . . . . . . 8  |-  ( ( A `  K )  e. Word  ( I  X.  2o )  ->  ( # `  ( A `  K
) )  e.  NN0 )
7573, 74syl 16 . . . . . . 7  |-  ( ph  ->  ( # `  ( A `  K )
)  e.  NN0 )
7675nn0red 10627 . . . . . 6  |-  ( ph  ->  ( # `  ( A `  K )
)  e.  RR )
77 2rp 10986 . . . . . 6  |-  2  e.  RR+
78 ltaddrp 11013 . . . . . 6  |-  ( ( ( # `  ( A `  K )
)  e.  RR  /\  2  e.  RR+ )  -> 
( # `  ( A `
 K ) )  <  ( ( # `  ( A `  K
) )  +  2 ) )
7976, 77, 78sylancl 657 . . . . 5  |-  ( ph  ->  ( # `  ( A `  K )
)  <  ( ( # `
 ( A `  K ) )  +  2 ) )
8021nn0red 10627 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  A
)  e.  RR )
8180lem1d 10256 . . . . . . . . . 10  |-  ( ph  ->  ( ( # `  A
)  -  1 )  <_  ( # `  A
) )
82 fznn 11512 . . . . . . . . . . 11  |-  ( (
# `  A )  e.  ZZ  ->  ( (
( # `  A )  -  1 )  e.  ( 1 ... ( # `
 A ) )  <-> 
( ( ( # `  A )  -  1 )  e.  NN  /\  ( ( # `  A
)  -  1 )  <_  ( # `  A
) ) ) )
8322, 82syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( # `  A )  -  1 )  e.  ( 1 ... ( # `  A
) )  <->  ( (
( # `  A )  -  1 )  e.  NN  /\  ( (
# `  A )  -  1 )  <_ 
( # `  A ) ) ) )
8432, 81, 83mpbir2and 908 . . . . . . . . 9  |-  ( ph  ->  ( ( # `  A
)  -  1 )  e.  ( 1 ... ( # `  A
) ) )
852, 3, 4, 5, 6, 7efgsres 16217 . . . . . . . . 9  |-  ( ( A  e.  dom  S  /\  ( ( # `  A
)  -  1 )  e.  ( 1 ... ( # `  A
) ) )  -> 
( A  |`  (
0..^ ( ( # `  A )  -  1 ) ) )  e. 
dom  S )
8612, 84, 85syl2anc 656 . . . . . . . 8  |-  ( ph  ->  ( A  |`  (
0..^ ( ( # `  A )  -  1 ) ) )  e. 
dom  S )
872, 3, 4, 5, 6, 7efgsval 16210 . . . . . . . 8  |-  ( ( A  |`  ( 0..^ ( ( # `  A
)  -  1 ) ) )  e.  dom  S  ->  ( S `  ( A  |`  ( 0..^ ( ( # `  A
)  -  1 ) ) ) )  =  ( ( A  |`  ( 0..^ ( ( # `  A )  -  1 ) ) ) `  ( ( # `  ( A  |`  ( 0..^ ( ( # `  A
)  -  1 ) ) ) )  - 
1 ) ) )
8886, 87syl 16 . . . . . . 7  |-  ( ph  ->  ( S `  ( A  |`  ( 0..^ ( ( # `  A
)  -  1 ) ) ) )  =  ( ( A  |`  ( 0..^ ( ( # `  A )  -  1 ) ) ) `  ( ( # `  ( A  |`  ( 0..^ ( ( # `  A
)  -  1 ) ) ) )  - 
1 ) ) )
89 1nn0 10585 . . . . . . . . . . . . . . . 16  |-  1  e.  NN0
90 nn0uz 10885 . . . . . . . . . . . . . . . 16  |-  NN0  =  ( ZZ>= `  0 )
9189, 90eleqtri 2507 . . . . . . . . . . . . . . 15  |-  1  e.  ( ZZ>= `  0 )
92 fzss1 11486 . . . . . . . . . . . . . . 15  |-  ( 1  e.  ( ZZ>= `  0
)  ->  ( 1 ... ( # `  A
) )  C_  (
0 ... ( # `  A
) ) )
9391, 92ax-mp 5 . . . . . . . . . . . . . 14  |-  ( 1 ... ( # `  A
) )  C_  (
0 ... ( # `  A
) )
9493, 84sseldi 3344 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( # `  A
)  -  1 )  e.  ( 0 ... ( # `  A
) ) )
95 swrd0val 12303 . . . . . . . . . . . . 13  |-  ( ( A  e. Word  W  /\  ( ( # `  A
)  -  1 )  e.  ( 0 ... ( # `  A
) ) )  -> 
( A substr  <. 0 ,  ( ( # `  A
)  -  1 )
>. )  =  ( A  |`  ( 0..^ ( ( # `  A
)  -  1 ) ) ) )
9616, 94, 95syl2anc 656 . . . . . . . . . . . 12  |-  ( ph  ->  ( A substr  <. 0 ,  ( ( # `  A )  -  1 ) >. )  =  ( A  |`  ( 0..^ ( ( # `  A
)  -  1 ) ) ) )
9796fveq2d 5685 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  ( A substr  <. 0 ,  ( ( # `  A
)  -  1 )
>. ) )  =  (
# `  ( A  |`  ( 0..^ ( (
# `  A )  -  1 ) ) ) ) )
98 swrd0len 12304 . . . . . . . . . . . 12  |-  ( ( A  e. Word  W  /\  ( ( # `  A
)  -  1 )  e.  ( 0 ... ( # `  A
) ) )  -> 
( # `  ( A substr  <. 0 ,  ( (
# `  A )  -  1 ) >.
) )  =  ( ( # `  A
)  -  1 ) )
9916, 94, 98syl2anc 656 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  ( A substr  <. 0 ,  ( ( # `  A
)  -  1 )
>. ) )  =  ( ( # `  A
)  -  1 ) )
10097, 99eqtr3d 2469 . . . . . . . . . 10  |-  ( ph  ->  ( # `  ( A  |`  ( 0..^ ( ( # `  A
)  -  1 ) ) ) )  =  ( ( # `  A
)  -  1 ) )
101100oveq1d 6097 . . . . . . . . 9  |-  ( ph  ->  ( ( # `  ( A  |`  ( 0..^ ( ( # `  A
)  -  1 ) ) ) )  - 
1 )  =  ( ( ( # `  A
)  -  1 )  -  1 ) )
102101, 26syl6eqr 2485 . . . . . . . 8  |-  ( ph  ->  ( ( # `  ( A  |`  ( 0..^ ( ( # `  A
)  -  1 ) ) ) )  - 
1 )  =  K )
103102fveq2d 5685 . . . . . . 7  |-  ( ph  ->  ( ( A  |`  ( 0..^ ( ( # `  A )  -  1 ) ) ) `  ( ( # `  ( A  |`  ( 0..^ ( ( # `  A
)  -  1 ) ) ) )  - 
1 ) )  =  ( ( A  |`  ( 0..^ ( ( # `  A )  -  1 ) ) ) `  K ) )
104 fvres 5694 . . . . . . . 8  |-  ( K  e.  ( 0..^ ( ( # `  A
)  -  1 ) )  ->  ( ( A  |`  ( 0..^ ( ( # `  A
)  -  1 ) ) ) `  K
)  =  ( A `
 K ) )
10535, 104syl 16 . . . . . . 7  |-  ( ph  ->  ( ( A  |`  ( 0..^ ( ( # `  A )  -  1 ) ) ) `  K )  =  ( A `  K ) )
10688, 103, 1053eqtrd 2471 . . . . . 6  |-  ( ph  ->  ( S `  ( A  |`  ( 0..^ ( ( # `  A
)  -  1 ) ) ) )  =  ( A `  K
) )
107106fveq2d 5685 . . . . 5  |-  ( ph  ->  ( # `  ( S `  ( A  |`  ( 0..^ ( (
# `  A )  -  1 ) ) ) ) )  =  ( # `  ( A `  K )
) )
1082, 3, 4, 5, 6, 7efgsdmi 16211 . . . . . . . 8  |-  ( ( A  e.  dom  S  /\  ( ( # `  A
)  -  1 )  e.  NN )  -> 
( S `  A
)  e.  ran  ( T `  ( A `  ( ( ( # `  A )  -  1 )  -  1 ) ) ) )
10912, 32, 108syl2anc 656 . . . . . . 7  |-  ( ph  ->  ( S `  A
)  e.  ran  ( T `  ( A `  ( ( ( # `  A )  -  1 )  -  1 ) ) ) )
11026fveq2i 5684 . . . . . . . . 9  |-  ( A `
 K )  =  ( A `  (
( ( # `  A
)  -  1 )  -  1 ) )
111110fveq2i 5684 . . . . . . . 8  |-  ( T `
 ( A `  K ) )  =  ( T `  ( A `  ( (
( # `  A )  -  1 )  - 
1 ) ) )
112111rneqi 5055 . . . . . . 7  |-  ran  ( T `  ( A `  K ) )  =  ran  ( T `  ( A `  ( ( ( # `  A
)  -  1 )  -  1 ) ) )
113109, 112syl6eleqr 2526 . . . . . 6  |-  ( ph  ->  ( S `  A
)  e.  ran  ( T `  ( A `  K ) ) )
1142, 3, 4, 5efgtlen 16205 . . . . . 6  |-  ( ( ( A `  K
)  e.  W  /\  ( S `  A )  e.  ran  ( T `
 ( A `  K ) ) )  ->  ( # `  ( S `  A )
)  =  ( (
# `  ( A `  K ) )  +  2 ) )
11537, 113, 114syl2anc 656 . . . . 5  |-  ( ph  ->  ( # `  ( S `  A )
)  =  ( (
# `  ( A `  K ) )  +  2 ) )
11679, 107, 1153brtr4d 4312 . . . 4  |-  ( ph  ->  ( # `  ( S `  ( A  |`  ( 0..^ ( (
# `  A )  -  1 ) ) ) ) )  < 
( # `  ( S `
 A ) ) )
117 efgredlemb.p . . . . . . . . 9  |-  ( ph  ->  P  e.  ( 0 ... ( # `  ( A `  K )
) ) )
118 efgredlemb.q . . . . . . . . 9  |-  ( ph  ->  Q  e.  ( 0 ... ( # `  ( B `  L )
) ) )
119 efgredlemb.u . . . . . . . . 9  |-  ( ph  ->  U  e.  ( I  X.  2o ) )
120 efgredlemb.v . . . . . . . . 9  |-  ( ph  ->  V  e.  ( I  X.  2o ) )
121 efgredlemb.6 . . . . . . . . 9  |-  ( ph  ->  ( S `  A
)  =  ( P ( T `  ( A `  K )
) U ) )
122 efgredlemb.7 . . . . . . . . 9  |-  ( ph  ->  ( S `  B
)  =  ( Q ( T `  ( B `  L )
) V ) )
123 efgredlemb.8 . . . . . . . . 9  |-  ( ph  ->  -.  ( A `  K )  =  ( B `  L ) )
124 efgredlemd.9 . . . . . . . . 9  |-  ( ph  ->  P  e.  ( ZZ>= `  ( Q  +  2
) ) )
125 efgredlemd.sc . . . . . . . . 9  |-  ( ph  ->  ( S `  C
)  =  ( ( ( B `  L
) substr  <. 0 ,  Q >. ) concat  ( ( A `
 K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) ) )
1262, 3, 4, 5, 6, 7, 27, 12, 28, 29, 30, 26, 60, 117, 118, 119, 120, 121, 122, 123, 124, 1, 125efgredleme 16222 . . . . . . . 8  |-  ( ph  ->  ( ( A `  K )  e.  ran  ( T `  ( S `
 C ) )  /\  ( B `  L )  e.  ran  ( T `  ( S `
 C ) ) ) )
127126simpld 456 . . . . . . 7  |-  ( ph  ->  ( A `  K
)  e.  ran  ( T `  ( S `  C ) ) )
1282, 3, 4, 5, 6, 7efgsp1 16216 . . . . . . 7  |-  ( ( C  e.  dom  S  /\  ( A `  K
)  e.  ran  ( T `  ( S `  C ) ) )  ->  ( C concat  <" ( A `  K ) "> )  e.  dom  S )
1291, 127, 128syl2anc 656 . . . . . 6  |-  ( ph  ->  ( C concat  <" ( A `  K ) "> )  e.  dom  S )
1302, 3, 4, 5, 6, 7efgsval2 16212 . . . . . 6  |-  ( ( C  e. Word  W  /\  ( A `  K )  e.  W  /\  ( C concat  <" ( A `
 K ) "> )  e.  dom  S )  ->  ( S `  ( C concat  <" ( A `  K ) "> ) )  =  ( A `  K
) )
13111, 37, 129, 130syl3anc 1213 . . . . 5  |-  ( ph  ->  ( S `  ( C concat  <" ( A `
 K ) "> ) )  =  ( A `  K
) )
132106, 131eqtr4d 2470 . . . 4  |-  ( ph  ->  ( S `  ( A  |`  ( 0..^ ( ( # `  A
)  -  1 ) ) ) )  =  ( S `  ( C concat  <" ( A `
 K ) "> ) ) )
133 fveq2 5681 . . . . . . . . 9  |-  ( a  =  ( A  |`  ( 0..^ ( ( # `  A )  -  1 ) ) )  -> 
( S `  a
)  =  ( S `
 ( A  |`  ( 0..^ ( ( # `  A )  -  1 ) ) ) ) )
134133fveq2d 5685 . . . . . . . 8  |-  ( a  =  ( A  |`  ( 0..^ ( ( # `  A )  -  1 ) ) )  -> 
( # `  ( S `
 a ) )  =  ( # `  ( S `  ( A  |`  ( 0..^ ( (
# `  A )  -  1 ) ) ) ) ) )
135134breq1d 4292 . . . . . . 7  |-  ( a  =  ( A  |`  ( 0..^ ( ( # `  A )  -  1 ) ) )  -> 
( ( # `  ( S `  a )
)  <  ( # `  ( S `  A )
)  <->  ( # `  ( S `  ( A  |`  ( 0..^ ( (
# `  A )  -  1 ) ) ) ) )  < 
( # `  ( S `
 A ) ) ) )
136133eqeq1d 2443 . . . . . . . 8  |-  ( a  =  ( A  |`  ( 0..^ ( ( # `  A )  -  1 ) ) )  -> 
( ( S `  a )  =  ( S `  b )  <-> 
( S `  ( A  |`  ( 0..^ ( ( # `  A
)  -  1 ) ) ) )  =  ( S `  b
) ) )
137 fveq1 5680 . . . . . . . . 9  |-  ( a  =  ( A  |`  ( 0..^ ( ( # `  A )  -  1 ) ) )  -> 
( a `  0
)  =  ( ( A  |`  ( 0..^ ( ( # `  A
)  -  1 ) ) ) `  0
) )
138137eqeq1d 2443 . . . . . . . 8  |-  ( a  =  ( A  |`  ( 0..^ ( ( # `  A )  -  1 ) ) )  -> 
( ( a ` 
0 )  =  ( b `  0 )  <-> 
( ( A  |`  ( 0..^ ( ( # `  A )  -  1 ) ) ) ` 
0 )  =  ( b `  0 ) ) )
139136, 138imbi12d 320 . . . . . . 7  |-  ( a  =  ( A  |`  ( 0..^ ( ( # `  A )  -  1 ) ) )  -> 
( ( ( S `
 a )  =  ( S `  b
)  ->  ( a `  0 )  =  ( b `  0
) )  <->  ( ( S `  ( A  |`  ( 0..^ ( (
# `  A )  -  1 ) ) ) )  =  ( S `  b )  ->  ( ( A  |`  ( 0..^ ( (
# `  A )  -  1 ) ) ) `  0 )  =  ( b ` 
0 ) ) ) )
140135, 139imbi12d 320 . . . . . 6  |-  ( a  =  ( A  |`  ( 0..^ ( ( # `  A )  -  1 ) ) )  -> 
( ( ( # `  ( S `  a
) )  <  ( # `
 ( S `  A ) )  -> 
( ( S `  a )  =  ( S `  b )  ->  ( a ` 
0 )  =  ( b `  0 ) ) )  <->  ( ( # `
 ( S `  ( A  |`  ( 0..^ ( ( # `  A
)  -  1 ) ) ) ) )  <  ( # `  ( S `  A )
)  ->  ( ( S `  ( A  |`  ( 0..^ ( (
# `  A )  -  1 ) ) ) )  =  ( S `  b )  ->  ( ( A  |`  ( 0..^ ( (
# `  A )  -  1 ) ) ) `  0 )  =  ( b ` 
0 ) ) ) ) )
141 fveq2 5681 . . . . . . . . 9  |-  ( b  =  ( C concat  <" ( A `  K ) "> )  ->  ( S `  b )  =  ( S `  ( C concat  <" ( A `  K ) "> ) ) )
142141eqeq2d 2446 . . . . . . . 8  |-  ( b  =  ( C concat  <" ( A `  K ) "> )  ->  (
( S `  ( A  |`  ( 0..^ ( ( # `  A
)  -  1 ) ) ) )  =  ( S `  b
)  <->  ( S `  ( A  |`  ( 0..^ ( ( # `  A
)  -  1 ) ) ) )  =  ( S `  ( C concat  <" ( A `
 K ) "> ) ) ) )
143 fveq1 5680 . . . . . . . . 9  |-  ( b  =  ( C concat  <" ( A `  K ) "> )  ->  (
b `  0 )  =  ( ( C concat  <" ( A `  K ) "> ) `  0 )
)
144143eqeq2d 2446 . . . . . . . 8  |-  ( b  =  ( C concat  <" ( A `  K ) "> )  ->  (
( ( A  |`  ( 0..^ ( ( # `  A )  -  1 ) ) ) ` 
0 )  =  ( b `  0 )  <-> 
( ( A  |`  ( 0..^ ( ( # `  A )  -  1 ) ) ) ` 
0 )  =  ( ( C concat  <" ( A `  K ) "> ) `  0
) ) )
145142, 144imbi12d 320 . . . . . . 7  |-  ( b  =  ( C concat  <" ( A `  K ) "> )  ->  (
( ( S `  ( A  |`  ( 0..^ ( ( # `  A
)  -  1 ) ) ) )  =  ( S `  b
)  ->  ( ( A  |`  ( 0..^ ( ( # `  A
)  -  1 ) ) ) `  0
)  =  ( b `
 0 ) )  <-> 
( ( S `  ( A  |`  ( 0..^ ( ( # `  A
)  -  1 ) ) ) )  =  ( S `  ( C concat  <" ( A `
 K ) "> ) )  -> 
( ( A  |`  ( 0..^ ( ( # `  A )  -  1 ) ) ) ` 
0 )  =  ( ( C concat  <" ( A `  K ) "> ) `  0
) ) ) )
146145imbi2d 316 . . . . . 6  |-  ( b  =  ( C concat  <" ( A `  K ) "> )  ->  (
( ( # `  ( S `  ( A  |`  ( 0..^ ( (
# `  A )  -  1 ) ) ) ) )  < 
( # `  ( S `
 A ) )  ->  ( ( S `
 ( A  |`  ( 0..^ ( ( # `  A )  -  1 ) ) ) )  =  ( S `  b )  ->  (
( A  |`  (
0..^ ( ( # `  A )  -  1 ) ) ) ` 
0 )  =  ( b `  0 ) ) )  <->  ( ( # `
 ( S `  ( A  |`  ( 0..^ ( ( # `  A
)  -  1 ) ) ) ) )  <  ( # `  ( S `  A )
)  ->  ( ( S `  ( A  |`  ( 0..^ ( (
# `  A )  -  1 ) ) ) )  =  ( S `  ( C concat  <" ( A `  K ) "> ) )  ->  (
( A  |`  (
0..^ ( ( # `  A )  -  1 ) ) ) ` 
0 )  =  ( ( C concat  <" ( A `  K ) "> ) `  0
) ) ) ) )
147140, 146rspc2va 3071 . . . . 5  |-  ( ( ( ( A  |`  ( 0..^ ( ( # `  A )  -  1 ) ) )  e. 
dom  S  /\  ( C concat  <" ( A `
 K ) "> )  e.  dom  S )  /\  A. a  e.  dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  < 
( # `  ( S `
 A ) )  ->  ( ( S `
 a )  =  ( S `  b
)  ->  ( a `  0 )  =  ( b `  0
) ) ) )  ->  ( ( # `  ( S `  ( A  |`  ( 0..^ ( ( # `  A
)  -  1 ) ) ) ) )  <  ( # `  ( S `  A )
)  ->  ( ( S `  ( A  |`  ( 0..^ ( (
# `  A )  -  1 ) ) ) )  =  ( S `  ( C concat  <" ( A `  K ) "> ) )  ->  (
( A  |`  (
0..^ ( ( # `  A )  -  1 ) ) ) ` 
0 )  =  ( ( C concat  <" ( A `  K ) "> ) `  0
) ) ) )
14886, 129, 27, 147syl21anc 1212 . . . 4  |-  ( ph  ->  ( ( # `  ( S `  ( A  |`  ( 0..^ ( (
# `  A )  -  1 ) ) ) ) )  < 
( # `  ( S `
 A ) )  ->  ( ( S `
 ( A  |`  ( 0..^ ( ( # `  A )  -  1 ) ) ) )  =  ( S `  ( C concat  <" ( A `  K ) "> ) )  -> 
( ( A  |`  ( 0..^ ( ( # `  A )  -  1 ) ) ) ` 
0 )  =  ( ( C concat  <" ( A `  K ) "> ) `  0
) ) ) )
149116, 132, 148mp2d 45 . . 3  |-  ( ph  ->  ( ( A  |`  ( 0..^ ( ( # `  A )  -  1 ) ) ) ` 
0 )  =  ( ( C concat  <" ( A `  K ) "> ) `  0
) )
15072, 66sseldi 3344 . . . . . . . 8  |-  ( ph  ->  ( B `  L
)  e. Word  ( I  X.  2o ) )
151 lencl 12235 . . . . . . . 8  |-  ( ( B `  L )  e. Word  ( I  X.  2o )  ->  ( # `  ( B `  L
) )  e.  NN0 )
152150, 151syl 16 . . . . . . 7  |-  ( ph  ->  ( # `  ( B `  L )
)  e.  NN0 )
153152nn0red 10627 . . . . . 6  |-  ( ph  ->  ( # `  ( B `  L )
)  e.  RR )
154 ltaddrp 11013 . . . . . 6  |-  ( ( ( # `  ( B `  L )
)  e.  RR  /\  2  e.  RR+ )  -> 
( # `  ( B `
 L ) )  <  ( ( # `  ( B `  L
) )  +  2 ) )
155153, 77, 154sylancl 657 . . . . 5  |-  ( ph  ->  ( # `  ( B `  L )
)  <  ( ( # `
 ( B `  L ) )  +  2 ) )
15655nn0red 10627 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  B
)  e.  RR )
157156lem1d 10256 . . . . . . . . . 10  |-  ( ph  ->  ( ( # `  B
)  -  1 )  <_  ( # `  B
) )
158 fznn 11512 . . . . . . . . . . 11  |-  ( (
# `  B )  e.  ZZ  ->  ( (
( # `  B )  -  1 )  e.  ( 1 ... ( # `
 B ) )  <-> 
( ( ( # `  B )  -  1 )  e.  NN  /\  ( ( # `  B
)  -  1 )  <_  ( # `  B
) ) ) )
15956, 158syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( # `  B )  -  1 )  e.  ( 1 ... ( # `  B
) )  <->  ( (
( # `  B )  -  1 )  e.  NN  /\  ( (
# `  B )  -  1 )  <_ 
( # `  B ) ) ) )
16061, 157, 159mpbir2and 908 . . . . . . . . 9  |-  ( ph  ->  ( ( # `  B
)  -  1 )  e.  ( 1 ... ( # `  B
) ) )
1612, 3, 4, 5, 6, 7efgsres 16217 . . . . . . . . 9  |-  ( ( B  e.  dom  S  /\  ( ( # `  B
)  -  1 )  e.  ( 1 ... ( # `  B
) ) )  -> 
( B  |`  (
0..^ ( ( # `  B )  -  1 ) ) )  e. 
dom  S )
16228, 160, 161syl2anc 656 . . . . . . . 8  |-  ( ph  ->  ( B  |`  (
0..^ ( ( # `  B )  -  1 ) ) )  e. 
dom  S )
1632, 3, 4, 5, 6, 7efgsval 16210 . . . . . . . 8  |-  ( ( B  |`  ( 0..^ ( ( # `  B
)  -  1 ) ) )  e.  dom  S  ->  ( S `  ( B  |`  ( 0..^ ( ( # `  B
)  -  1 ) ) ) )  =  ( ( B  |`  ( 0..^ ( ( # `  B )  -  1 ) ) ) `  ( ( # `  ( B  |`  ( 0..^ ( ( # `  B
)  -  1 ) ) ) )  - 
1 ) ) )
164162, 163syl 16 . . . . . . 7  |-  ( ph  ->  ( S `  ( B  |`  ( 0..^ ( ( # `  B
)  -  1 ) ) ) )  =  ( ( B  |`  ( 0..^ ( ( # `  B )  -  1 ) ) ) `  ( ( # `  ( B  |`  ( 0..^ ( ( # `  B
)  -  1 ) ) ) )  - 
1 ) ) )
165 fzss1 11486 . . . . . . . . . . . . . . 15  |-  ( 1  e.  ( ZZ>= `  0
)  ->  ( 1 ... ( # `  B
) )  C_  (
0 ... ( # `  B
) ) )
16691, 165ax-mp 5 . . . . . . . . . . . . . 14  |-  ( 1 ... ( # `  B
) )  C_  (
0 ... ( # `  B
) )
167166, 160sseldi 3344 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( # `  B
)  -  1 )  e.  ( 0 ... ( # `  B
) ) )
168 swrd0val 12303 . . . . . . . . . . . . 13  |-  ( ( B  e. Word  W  /\  ( ( # `  B
)  -  1 )  e.  ( 0 ... ( # `  B
) ) )  -> 
( B substr  <. 0 ,  ( ( # `  B
)  -  1 )
>. )  =  ( B  |`  ( 0..^ ( ( # `  B
)  -  1 ) ) ) )
16950, 167, 168syl2anc 656 . . . . . . . . . . . 12  |-  ( ph  ->  ( B substr  <. 0 ,  ( ( # `  B )  -  1 ) >. )  =  ( B  |`  ( 0..^ ( ( # `  B
)  -  1 ) ) ) )
170169fveq2d 5685 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  ( B substr  <. 0 ,  ( ( # `  B
)  -  1 )
>. ) )  =  (
# `  ( B  |`  ( 0..^ ( (
# `  B )  -  1 ) ) ) ) )
171 swrd0len 12304 . . . . . . . . . . . 12  |-  ( ( B  e. Word  W  /\  ( ( # `  B
)  -  1 )  e.  ( 0 ... ( # `  B
) ) )  -> 
( # `  ( B substr  <. 0 ,  ( (
# `  B )  -  1 ) >.
) )  =  ( ( # `  B
)  -  1 ) )
17250, 167, 171syl2anc 656 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  ( B substr  <. 0 ,  ( ( # `  B
)  -  1 )
>. ) )  =  ( ( # `  B
)  -  1 ) )
173170, 172eqtr3d 2469 . . . . . . . . . 10  |-  ( ph  ->  ( # `  ( B  |`  ( 0..^ ( ( # `  B
)  -  1 ) ) ) )  =  ( ( # `  B
)  -  1 ) )
174173oveq1d 6097 . . . . . . . . 9  |-  ( ph  ->  ( ( # `  ( B  |`  ( 0..^ ( ( # `  B
)  -  1 ) ) ) )  - 
1 )  =  ( ( ( # `  B
)  -  1 )  -  1 ) )
175174, 60syl6eqr 2485 . . . . . . . 8  |-  ( ph  ->  ( ( # `  ( B  |`  ( 0..^ ( ( # `  B
)  -  1 ) ) ) )  - 
1 )  =  L )
176175fveq2d 5685 . . . . . . 7  |-  ( ph  ->  ( ( B  |`  ( 0..^ ( ( # `  B )  -  1 ) ) ) `  ( ( # `  ( B  |`  ( 0..^ ( ( # `  B
)  -  1 ) ) ) )  - 
1 ) )  =  ( ( B  |`  ( 0..^ ( ( # `  B )  -  1 ) ) ) `  L ) )
177 fvres 5694 . . . . . . . 8  |-  ( L  e.  ( 0..^ ( ( # `  B
)  -  1 ) )  ->  ( ( B  |`  ( 0..^ ( ( # `  B
)  -  1 ) ) ) `  L
)  =  ( B `
 L ) )
17864, 177syl 16 . . . . . . 7  |-  ( ph  ->  ( ( B  |`  ( 0..^ ( ( # `  B )  -  1 ) ) ) `  L )  =  ( B `  L ) )
179164, 176, 1783eqtrd 2471 . . . . . 6  |-  ( ph  ->  ( S `  ( B  |`  ( 0..^ ( ( # `  B
)  -  1 ) ) ) )  =  ( B `  L
) )
180179fveq2d 5685 . . . . 5  |-  ( ph  ->  ( # `  ( S `  ( B  |`  ( 0..^ ( (
# `  B )  -  1 ) ) ) ) )  =  ( # `  ( B `  L )
) )
1812, 3, 4, 5, 6, 7efgsdmi 16211 . . . . . . . . 9  |-  ( ( B  e.  dom  S  /\  ( ( # `  B
)  -  1 )  e.  NN )  -> 
( S `  B
)  e.  ran  ( T `  ( B `  ( ( ( # `  B )  -  1 )  -  1 ) ) ) )
18228, 61, 181syl2anc 656 . . . . . . . 8  |-  ( ph  ->  ( S `  B
)  e.  ran  ( T `  ( B `  ( ( ( # `  B )  -  1 )  -  1 ) ) ) )
18329, 182eqeltrd 2509 . . . . . . 7  |-  ( ph  ->  ( S `  A
)  e.  ran  ( T `  ( B `  ( ( ( # `  B )  -  1 )  -  1 ) ) ) )
18460fveq2i 5684 . . . . . . . . 9  |-  ( B `
 L )  =  ( B `  (
( ( # `  B
)  -  1 )  -  1 ) )
185184fveq2i 5684 . . . . . . . 8  |-  ( T `
 ( B `  L ) )  =  ( T `  ( B `  ( (
( # `  B )  -  1 )  - 
1 ) ) )
186185rneqi 5055 . . . . . . 7  |-  ran  ( T `  ( B `  L ) )  =  ran  ( T `  ( B `  ( ( ( # `  B
)  -  1 )  -  1 ) ) )
187183, 186syl6eleqr 2526 . . . . . 6  |-  ( ph  ->  ( S `  A
)  e.  ran  ( T `  ( B `  L ) ) )
1882, 3, 4, 5efgtlen 16205 . . . . . 6  |-  ( ( ( B `  L
)  e.  W  /\  ( S `  A )  e.  ran  ( T `
 ( B `  L ) ) )  ->  ( # `  ( S `  A )
)  =  ( (
# `  ( B `  L ) )  +  2 ) )
18966, 187, 188syl2anc 656 . . . . 5  |-  ( ph  ->  ( # `  ( S `  A )
)  =  ( (
# `  ( B `  L ) )  +  2 ) )
190155, 180, 1893brtr4d 4312 . . . 4  |-  ( ph  ->  ( # `  ( S `  ( B  |`  ( 0..^ ( (
# `  B )  -  1 ) ) ) ) )  < 
( # `  ( S `
 A ) ) )
191126simprd 460 . . . . . . 7  |-  ( ph  ->  ( B `  L
)  e.  ran  ( T `  ( S `  C ) ) )
1922, 3, 4, 5, 6, 7efgsp1 16216 . . . . . . 7  |-  ( ( C  e.  dom  S  /\  ( B `  L
)  e.  ran  ( T `  ( S `  C ) ) )  ->  ( C concat  <" ( B `  L ) "> )  e.  dom  S )
1931, 191, 192syl2anc 656 . . . . . 6  |-  ( ph  ->  ( C concat  <" ( B `  L ) "> )  e.  dom  S )
1942, 3, 4, 5, 6, 7efgsval2 16212 . . . . . 6  |-  ( ( C  e. Word  W  /\  ( B `  L )  e.  W  /\  ( C concat  <" ( B `
 L ) "> )  e.  dom  S )  ->  ( S `  ( C concat  <" ( B `  L ) "> ) )  =  ( B `  L
) )
19511, 66, 193, 194syl3anc 1213 . . . . 5  |-  ( ph  ->  ( S `  ( C concat  <" ( B `
 L ) "> ) )  =  ( B `  L
) )
196179, 195eqtr4d 2470 . . . 4  |-  ( ph  ->  ( S `  ( B  |`  ( 0..^ ( ( # `  B
)  -  1 ) ) ) )  =  ( S `  ( C concat  <" ( B `
 L ) "> ) ) )
197 fveq2 5681 . . . . . . . . 9  |-  ( a  =  ( B  |`  ( 0..^ ( ( # `  B )  -  1 ) ) )  -> 
( S `  a
)  =  ( S `
 ( B  |`  ( 0..^ ( ( # `  B )  -  1 ) ) ) ) )
198197fveq2d 5685 . . . . . . . 8  |-  ( a  =  ( B  |`  ( 0..^ ( ( # `  B )  -  1 ) ) )  -> 
( # `  ( S `
 a ) )  =  ( # `  ( S `  ( B  |`  ( 0..^ ( (
# `  B )  -  1 ) ) ) ) ) )
199198breq1d 4292 . . . . . . 7  |-  ( a  =  ( B  |`  ( 0..^ ( ( # `  B )  -  1 ) ) )  -> 
( ( # `  ( S `  a )
)  <  ( # `  ( S `  A )
)  <->  ( # `  ( S `  ( B  |`  ( 0..^ ( (
# `  B )  -  1 ) ) ) ) )  < 
( # `  ( S `
 A ) ) ) )
200197eqeq1d 2443 . . . . . . . 8  |-  ( a  =  ( B  |`  ( 0..^ ( ( # `  B )  -  1 ) ) )  -> 
( ( S `  a )  =  ( S `  b )  <-> 
( S `  ( B  |`  ( 0..^ ( ( # `  B
)  -  1 ) ) ) )  =  ( S `  b
) ) )
201 fveq1 5680 . . . . . . . . 9  |-  ( a  =  ( B  |`  ( 0..^ ( ( # `  B )  -  1 ) ) )  -> 
( a `  0
)  =  ( ( B  |`  ( 0..^ ( ( # `  B
)  -  1 ) ) ) `  0
) )
202201eqeq1d 2443 . . . . . . . 8  |-  ( a  =  ( B  |`  ( 0..^ ( ( # `  B )  -  1 ) ) )  -> 
( ( a ` 
0 )  =  ( b `  0 )  <-> 
( ( B  |`  ( 0..^ ( ( # `  B )  -  1 ) ) ) ` 
0 )  =  ( b `  0 ) ) )
203200, 202imbi12d 320 . . . . . . 7  |-  ( a  =  ( B  |`  ( 0..^ ( ( # `  B )  -  1 ) ) )  -> 
( ( ( S `
 a )  =  ( S `  b
)  ->  ( a `  0 )  =  ( b `  0
) )  <->  ( ( S `  ( B  |`  ( 0..^ ( (
# `  B )  -  1 ) ) ) )  =  ( S `  b )  ->  ( ( B  |`  ( 0..^ ( (
# `  B )  -  1 ) ) ) `  0 )  =  ( b ` 
0 ) ) ) )
204199, 203imbi12d 320 . . . . . 6  |-  ( a  =  ( B  |`  ( 0..^ ( ( # `  B )  -  1 ) ) )  -> 
( ( ( # `  ( S `  a
) )  <  ( # `
 ( S `  A ) )  -> 
( ( S `  a )  =  ( S `  b )  ->  ( a ` 
0 )  =  ( b `  0 ) ) )  <->  ( ( # `
 ( S `  ( B  |`  ( 0..^ ( ( # `  B
)  -  1 ) ) ) ) )  <  ( # `  ( S `  A )
)  ->  ( ( S `  ( B  |`  ( 0..^ ( (
# `  B )  -  1 ) ) ) )  =  ( S `  b )  ->  ( ( B  |`  ( 0..^ ( (
# `  B )  -  1 ) ) ) `  0 )  =  ( b ` 
0 ) ) ) ) )
205 fveq2 5681 . . . . . . . . 9  |-  ( b  =  ( C concat  <" ( B `  L ) "> )  ->  ( S `  b )  =  ( S `  ( C concat  <" ( B `  L ) "> ) ) )
206205eqeq2d 2446 . . . . . . . 8  |-  ( b  =  ( C concat  <" ( B `  L ) "> )  ->  (
( S `  ( B  |`  ( 0..^ ( ( # `  B
)  -  1 ) ) ) )  =  ( S `  b
)  <->  ( S `  ( B  |`  ( 0..^ ( ( # `  B
)  -  1 ) ) ) )  =  ( S `  ( C concat  <" ( B `
 L ) "> ) ) ) )
207 fveq1 5680 . . . . . . . . 9  |-  ( b  =  ( C concat  <" ( B `  L ) "> )  ->  (
b `  0 )  =  ( ( C concat  <" ( B `  L ) "> ) `  0 )
)
208207eqeq2d 2446 . . . . . . . 8  |-  ( b  =  ( C concat  <" ( B `  L ) "> )  ->  (
( ( B  |`  ( 0..^ ( ( # `  B )  -  1 ) ) ) ` 
0 )  =  ( b `  0 )  <-> 
( ( B  |`  ( 0..^ ( ( # `  B )  -  1 ) ) ) ` 
0 )  =  ( ( C concat  <" ( B `  L ) "> ) `  0
) ) )
209206, 208imbi12d 320 . . . . . . 7  |-  ( b  =  ( C concat  <" ( B `  L ) "> )  ->  (
( ( S `  ( B  |`  ( 0..^ ( ( # `  B
)  -  1 ) ) ) )  =  ( S `  b
)  ->  ( ( B  |`  ( 0..^ ( ( # `  B
)  -  1 ) ) ) `  0
)  =  ( b `
 0 ) )  <-> 
( ( S `  ( B  |`  ( 0..^ ( ( # `  B
)  -  1 ) ) ) )  =  ( S `  ( C concat  <" ( B `
 L ) "> ) )  -> 
( ( B  |`  ( 0..^ ( ( # `  B )  -  1 ) ) ) ` 
0 )  =  ( ( C concat  <" ( B `  L ) "> ) `  0
) ) ) )
210209imbi2d 316 . . . . . 6  |-  ( b  =  ( C concat  <" ( B `  L ) "> )  ->  (
( ( # `  ( S `  ( B  |`  ( 0..^ ( (
# `  B )  -  1 ) ) ) ) )  < 
( # `  ( S `
 A ) )  ->  ( ( S `
 ( B  |`  ( 0..^ ( ( # `  B )  -  1 ) ) ) )  =  ( S `  b )  ->  (
( B  |`  (
0..^ ( ( # `  B )  -  1 ) ) ) ` 
0 )  =  ( b `  0 ) ) )  <->  ( ( # `
 ( S `  ( B  |`  ( 0..^ ( ( # `  B
)  -  1 ) ) ) ) )  <  ( # `  ( S `  A )
)  ->  ( ( S `  ( B  |`  ( 0..^ ( (
# `  B )  -  1 ) ) ) )  =  ( S `  ( C concat  <" ( B `  L ) "> ) )  ->  (
( B  |`  (
0..^ ( ( # `  B )  -  1 ) ) ) ` 
0 )  =  ( ( C concat  <" ( B `  L ) "> ) `  0
) ) ) ) )
211204, 210rspc2va 3071 . . . . 5  |-  ( ( ( ( B  |`  ( 0..^ ( ( # `  B )  -  1 ) ) )  e. 
dom  S  /\  ( C concat  <" ( B `
 L ) "> )  e.  dom  S )  /\  A. a  e.  dom  S A. b  e.  dom  S ( (
# `  ( S `  a ) )  < 
( # `  ( S `
 A ) )  ->  ( ( S `
 a )  =  ( S `  b
)  ->  ( a `  0 )  =  ( b `  0
) ) ) )  ->  ( ( # `  ( S `  ( B  |`  ( 0..^ ( ( # `  B
)  -  1 ) ) ) ) )  <  ( # `  ( S `  A )
)  ->  ( ( S `  ( B  |`  ( 0..^ ( (
# `  B )  -  1 ) ) ) )  =  ( S `  ( C concat  <" ( B `  L ) "> ) )  ->  (
( B  |`  (
0..^ ( ( # `  B )  -  1 ) ) ) ` 
0 )  =  ( ( C concat  <" ( B `  L ) "> ) `  0
) ) ) )
212162, 193, 27, 211syl21anc 1212 . . . 4  |-  ( ph  ->  ( ( # `  ( S `  ( B  |`  ( 0..^ ( (
# `  B )  -  1 ) ) ) ) )  < 
( # `  ( S `
 A ) )  ->  ( ( S `
 ( B  |`  ( 0..^ ( ( # `  B )  -  1 ) ) ) )  =  ( S `  ( C concat  <" ( B `  L ) "> ) )  -> 
( ( B  |`  ( 0..^ ( ( # `  B )  -  1 ) ) ) ` 
0 )  =  ( ( C concat  <" ( B `  L ) "> ) `  0
) ) ) )
213190, 196, 212mp2d 45 . . 3  |-  ( ph  ->  ( ( B  |`  ( 0..^ ( ( # `  B )  -  1 ) ) ) ` 
0 )  =  ( ( C concat  <" ( B `  L ) "> ) `  0
) )
21470, 149, 2133eqtr4d 2477 . 2  |-  ( ph  ->  ( ( A  |`  ( 0..^ ( ( # `  A )  -  1 ) ) ) ` 
0 )  =  ( ( B  |`  (
0..^ ( ( # `  B )  -  1 ) ) ) ` 
0 ) )
215 lbfzo0 11572 . . . 4  |-  ( 0  e.  ( 0..^ ( ( # `  A
)  -  1 ) )  <->  ( ( # `  A )  -  1 )  e.  NN )
21632, 215sylibr 212 . . 3  |-  ( ph  ->  0  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) )
217 fvres 5694 . . 3  |-  ( 0  e.  ( 0..^ ( ( # `  A
)  -  1 ) )  ->  ( ( A  |`  ( 0..^ ( ( # `  A
)  -  1 ) ) ) `  0
)  =  ( A `
 0 ) )
218216, 217syl 16 . 2  |-  ( ph  ->  ( ( A  |`  ( 0..^ ( ( # `  A )  -  1 ) ) ) ` 
0 )  =  ( A `  0 ) )
219 lbfzo0 11572 . . . 4  |-  ( 0  e.  ( 0..^ ( ( # `  B
)  -  1 ) )  <->  ( ( # `  B )  -  1 )  e.  NN )
22061, 219sylibr 212 . . 3  |-  ( ph  ->  0  e.  ( 0..^ ( ( # `  B
)  -  1 ) ) )
221 fvres 5694 . . 3  |-  ( 0  e.  ( 0..^ ( ( # `  B
)  -  1 ) )  ->  ( ( B  |`  ( 0..^ ( ( # `  B
)  -  1 ) ) ) `  0
)  =  ( B `
 0 ) )
222220, 221syl 16 . 2  |-  ( ph  ->  ( ( B  |`  ( 0..^ ( ( # `  B )  -  1 ) ) ) ` 
0 )  =  ( B `  0 ) )
223214, 218, 2223eqtr3d 2475 1  |-  ( ph  ->  ( A `  0
)  =  ( B `
 0 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1757    =/= wne 2598   A.wral 2707   {crab 2711    \ cdif 3315    C_ wss 3318   (/)c0 3627   {csn 3867   <.cop 3873   <.cotp 3875   U_ciun 4161   class class class wbr 4282    e. cmpt 4340    _I cid 4620    X. cxp 4827   dom cdm 4829   ran crn 4830    |` cres 4831   -->wf 5404   ` cfv 5408  (class class class)co 6082    e. cmpt2 6084   1oc1o 6903   2oc2o 6904   RRcr 9271   0cc0 9272   1c1 9273    + caddc 9275    < clt 9408    <_ cle 9409    - cmin 9585   NNcn 10312   2c2 10361   NN0cn0 10569   ZZcz 10636   ZZ>=cuz 10851   RR+crp 10981   ...cfz 11426  ..^cfzo 11534   #chash 12089  Word cword 12207   concat cconcat 12209   <"cs1 12210   substr csubstr 12211   splice csplice 12212   <"cs2 12454   ~FG cefg 16185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2416  ax-rep 4393  ax-sep 4403  ax-nul 4411  ax-pow 4460  ax-pr 4521  ax-un 6363  ax-cnex 9328  ax-resscn 9329  ax-1cn 9330  ax-icn 9331  ax-addcl 9332  ax-addrcl 9333  ax-mulcl 9334  ax-mulrcl 9335  ax-mulcom 9336  ax-addass 9337  ax-mulass 9338  ax-distr 9339  ax-i2m1 9340  ax-1ne0 9341  ax-1rid 9342  ax-rnegex 9343  ax-rrecex 9344  ax-cnre 9345  ax-pre-lttri 9346  ax-pre-lttrn 9347  ax-pre-ltadd 9348  ax-pre-mulgt0 9349
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1702  df-eu 2260  df-mo 2261  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2966  df-sbc 3178  df-csb 3279  df-dif 3321  df-un 3323  df-in 3325  df-ss 3332  df-pss 3334  df-nul 3628  df-if 3782  df-pw 3852  df-sn 3868  df-pr 3870  df-tp 3872  df-op 3874  df-ot 3876  df-uni 4082  df-int 4119  df-iun 4163  df-br 4283  df-opab 4341  df-mpt 4342  df-tr 4376  df-eprel 4621  df-id 4625  df-po 4630  df-so 4631  df-fr 4668  df-we 4670  df-ord 4711  df-on 4712  df-lim 4713  df-suc 4714  df-xp 4835  df-rel 4836  df-cnv 4837  df-co 4838  df-dm 4839  df-rn 4840  df-res 4841  df-ima 4842  df-iota 5371  df-fun 5410  df-fn 5411  df-f 5412  df-f1 5413  df-fo 5414  df-f1o 5415  df-fv 5416  df-riota 6041  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-om 6468  df-1st 6568  df-2nd 6569  df-recs 6820  df-rdg 6854  df-1o 6910  df-2o 6911  df-oadd 6914  df-er 7091  df-map 7206  df-pm 7207  df-en 7301  df-dom 7302  df-sdom 7303  df-fin 7304  df-card 8099  df-pnf 9410  df-mnf 9411  df-xr 9412  df-ltxr 9413  df-le 9414  df-sub 9587  df-neg 9588  df-nn 10313  df-2 10370  df-n0 10570  df-z 10637  df-uz 10852  df-rp 10982  df-fz 11427  df-fzo 11535  df-hash 12090  df-word 12215  df-concat 12217  df-s1 12218  df-substr 12219  df-splice 12220  df-s2 12461
This theorem is referenced by:  efgredlemc  16224
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