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Theorem efgredlemc 17473
Description: The reduced word that forms the base of the sequence in efgsval 17459 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 ) )
Assertion
Ref Expression
efgredlemc  |-  ( ph  ->  ( P  e.  (
ZZ>= `  Q )  -> 
( 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    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 efgredlemc
Dummy variables  c 
i are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uzp1 11216 . 2  |-  ( P  e.  ( ZZ>= `  Q
)  ->  ( P  =  Q  \/  P  e.  ( ZZ>= `  ( Q  +  1 ) ) ) )
2 efgredlemb.8 . . . . . 6  |-  ( ph  ->  -.  ( A `  K )  =  ( B `  L ) )
3 efgval.w . . . . . . . . . . 11  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
4 fviss 5938 . . . . . . . . . . 11  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
53, 4eqsstri 3448 . . . . . . . . . 10  |-  W  C_ Word  ( I  X.  2o )
6 efgredlem.2 . . . . . . . . . . . . 13  |-  ( ph  ->  A  e.  dom  S
)
7 efgval.r . . . . . . . . . . . . . . 15  |-  .~  =  ( ~FG  `  I )
8 efgval2.m . . . . . . . . . . . . . . 15  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
9 efgval2.t . . . . . . . . . . . . . . 15  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
10 efgred.d . . . . . . . . . . . . . . 15  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
11 efgred.s . . . . . . . . . . . . . . 15  |-  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 ) ) )
123, 7, 8, 9, 10, 11efgsdm 17458 . . . . . . . . . . . . . 14  |-  ( 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 ) ) ) ) )
1312simp1bi 1045 . . . . . . . . . . . . 13  |-  ( A  e.  dom  S  ->  A  e.  (Word  W  \  { (/) } ) )
146, 13syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  (Word  W  \  { (/) } ) )
15 eldifi 3544 . . . . . . . . . . . 12  |-  ( A  e.  (Word  W  \  { (/) } )  ->  A  e. Word  W )
16 wrdf 12723 . . . . . . . . . . . 12  |-  ( A  e. Word  W  ->  A : ( 0..^ (
# `  A )
) --> W )
1714, 15, 163syl 18 . . . . . . . . . . 11  |-  ( ph  ->  A : ( 0..^ ( # `  A
) ) --> W )
18 fzossfz 11965 . . . . . . . . . . . . 13  |-  ( 0..^ ( ( # `  A
)  -  1 ) )  C_  ( 0 ... ( ( # `  A )  -  1 ) )
19 efgredlemb.k . . . . . . . . . . . . . 14  |-  K  =  ( ( ( # `  A )  -  1 )  -  1 )
20 efgredlem.1 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  ( # `  ( S `  A )
)  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) ) )
21 efgredlem.3 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  B  e.  dom  S
)
22 efgredlem.4 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( S `  A
)  =  ( S `
 B ) )
23 efgredlem.5 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  -.  ( A ` 
0 )  =  ( B `  0 ) )
243, 7, 8, 9, 10, 11, 20, 6, 21, 22, 23efgredlema 17468 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( ( # `  A )  -  1 )  e.  NN  /\  ( ( # `  B
)  -  1 )  e.  NN ) )
2524simpld 466 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( # `  A
)  -  1 )  e.  NN )
26 fzo0end 12032 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  A
)  -  1 )  e.  NN  ->  (
( ( # `  A
)  -  1 )  -  1 )  e.  ( 0..^ ( (
# `  A )  -  1 ) ) )
2725, 26syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( # `  A )  -  1 )  -  1 )  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) )
2819, 27syl5eqel 2553 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) )
2918, 28sseldi 3416 . . . . . . . . . . . 12  |-  ( ph  ->  K  e.  ( 0 ... ( ( # `  A )  -  1 ) ) )
30 lencl 12737 . . . . . . . . . . . . . . 15  |-  ( A  e. Word  W  ->  ( # `
 A )  e. 
NN0 )
3114, 15, 303syl 18 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( # `  A
)  e.  NN0 )
3231nn0zd 11061 . . . . . . . . . . . . 13  |-  ( ph  ->  ( # `  A
)  e.  ZZ )
33 fzoval 11948 . . . . . . . . . . . . 13  |-  ( (
# `  A )  e.  ZZ  ->  ( 0..^ ( # `  A
) )  =  ( 0 ... ( (
# `  A )  -  1 ) ) )
3432, 33syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( 0..^ ( # `  A ) )  =  ( 0 ... (
( # `  A )  -  1 ) ) )
3529, 34eleqtrrd 2552 . . . . . . . . . . 11  |-  ( ph  ->  K  e.  ( 0..^ ( # `  A
) ) )
3617, 35ffvelrnd 6038 . . . . . . . . . 10  |-  ( ph  ->  ( A `  K
)  e.  W )
375, 36sseldi 3416 . . . . . . . . 9  |-  ( ph  ->  ( A `  K
)  e. Word  ( I  X.  2o ) )
38 efgredlemb.p . . . . . . . . . 10  |-  ( ph  ->  P  e.  ( 0 ... ( # `  ( A `  K )
) ) )
39 elfzuz 11822 . . . . . . . . . 10  |-  ( P  e.  ( 0 ... ( # `  ( A `  K )
) )  ->  P  e.  ( ZZ>= `  0 )
)
40 eluzfz1 11832 . . . . . . . . . 10  |-  ( P  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... P
) )
4138, 39, 403syl 18 . . . . . . . . 9  |-  ( ph  ->  0  e.  ( 0 ... P ) )
42 lencl 12737 . . . . . . . . . . . 12  |-  ( ( A `  K )  e. Word  ( I  X.  2o )  ->  ( # `  ( A `  K
) )  e.  NN0 )
4337, 42syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  ( A `  K )
)  e.  NN0 )
44 nn0uz 11217 . . . . . . . . . . 11  |-  NN0  =  ( ZZ>= `  0 )
4543, 44syl6eleq 2559 . . . . . . . . . 10  |-  ( ph  ->  ( # `  ( A `  K )
)  e.  ( ZZ>= ` 
0 ) )
46 eluzfz2 11833 . . . . . . . . . 10  |-  ( (
# `  ( A `  K ) )  e.  ( ZZ>= `  0 )  ->  ( # `  ( A `  K )
)  e.  ( 0 ... ( # `  ( A `  K )
) ) )
4745, 46syl 17 . . . . . . . . 9  |-  ( ph  ->  ( # `  ( A `  K )
)  e.  ( 0 ... ( # `  ( A `  K )
) ) )
48 ccatswrd 12866 . . . . . . . . 9  |-  ( ( ( A `  K
)  e. Word  ( I  X.  2o )  /\  (
0  e.  ( 0 ... P )  /\  P  e.  ( 0 ... ( # `  ( A `  K )
) )  /\  ( # `
 ( A `  K ) )  e.  ( 0 ... ( # `
 ( A `  K ) ) ) ) )  ->  (
( ( A `  K ) substr  <. 0 ,  P >. ) ++  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K
) ) >. )
)  =  ( ( A `  K ) substr  <. 0 ,  ( # `  ( A `  K
) ) >. )
)
4937, 41, 38, 47, 48syl13anc 1294 . . . . . . . 8  |-  ( ph  ->  ( ( ( A `
 K ) substr  <. 0 ,  P >. ) ++  ( ( A `  K ) substr  <. P , 
( # `  ( A `
 K ) )
>. ) )  =  ( ( A `  K
) substr  <. 0 ,  (
# `  ( A `  K ) ) >.
) )
50 swrdid 12838 . . . . . . . . 9  |-  ( ( A `  K )  e. Word  ( I  X.  2o )  ->  ( ( A `  K ) substr  <. 0 ,  ( # `  ( A `  K
) ) >. )  =  ( A `  K ) )
5137, 50syl 17 . . . . . . . 8  |-  ( ph  ->  ( ( A `  K ) substr  <. 0 ,  ( # `  ( A `  K )
) >. )  =  ( A `  K ) )
5249, 51eqtrd 2505 . . . . . . 7  |-  ( ph  ->  ( ( ( A `
 K ) substr  <. 0 ,  P >. ) ++  ( ( A `  K ) substr  <. P , 
( # `  ( A `
 K ) )
>. ) )  =  ( A `  K ) )
533, 7, 8, 9, 10, 11efgsdm 17458 . . . . . . . . . . . . . 14  |-  ( 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 ) ) ) ) )
5453simp1bi 1045 . . . . . . . . . . . . 13  |-  ( B  e.  dom  S  ->  B  e.  (Word  W  \  { (/) } ) )
5521, 54syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  (Word  W  \  { (/) } ) )
56 eldifi 3544 . . . . . . . . . . . 12  |-  ( B  e.  (Word  W  \  { (/) } )  ->  B  e. Word  W )
57 wrdf 12723 . . . . . . . . . . . 12  |-  ( B  e. Word  W  ->  B : ( 0..^ (
# `  B )
) --> W )
5855, 56, 573syl 18 . . . . . . . . . . 11  |-  ( ph  ->  B : ( 0..^ ( # `  B
) ) --> W )
59 fzossfz 11965 . . . . . . . . . . . . 13  |-  ( 0..^ ( ( # `  B
)  -  1 ) )  C_  ( 0 ... ( ( # `  B )  -  1 ) )
60 efgredlemb.l . . . . . . . . . . . . . 14  |-  L  =  ( ( ( # `  B )  -  1 )  -  1 )
6124simprd 470 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( # `  B
)  -  1 )  e.  NN )
62 fzo0end 12032 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  B
)  -  1 )  e.  NN  ->  (
( ( # `  B
)  -  1 )  -  1 )  e.  ( 0..^ ( (
# `  B )  -  1 ) ) )
6361, 62syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( # `  B )  -  1 )  -  1 )  e.  ( 0..^ ( ( # `  B
)  -  1 ) ) )
6460, 63syl5eqel 2553 . . . . . . . . . . . . 13  |-  ( ph  ->  L  e.  ( 0..^ ( ( # `  B
)  -  1 ) ) )
6559, 64sseldi 3416 . . . . . . . . . . . 12  |-  ( ph  ->  L  e.  ( 0 ... ( ( # `  B )  -  1 ) ) )
66 lencl 12737 . . . . . . . . . . . . . . 15  |-  ( B  e. Word  W  ->  ( # `
 B )  e. 
NN0 )
6755, 56, 663syl 18 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( # `  B
)  e.  NN0 )
6867nn0zd 11061 . . . . . . . . . . . . 13  |-  ( ph  ->  ( # `  B
)  e.  ZZ )
69 fzoval 11948 . . . . . . . . . . . . 13  |-  ( (
# `  B )  e.  ZZ  ->  ( 0..^ ( # `  B
) )  =  ( 0 ... ( (
# `  B )  -  1 ) ) )
7068, 69syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( 0..^ ( # `  B ) )  =  ( 0 ... (
( # `  B )  -  1 ) ) )
7165, 70eleqtrrd 2552 . . . . . . . . . . 11  |-  ( ph  ->  L  e.  ( 0..^ ( # `  B
) ) )
7258, 71ffvelrnd 6038 . . . . . . . . . 10  |-  ( ph  ->  ( B `  L
)  e.  W )
735, 72sseldi 3416 . . . . . . . . 9  |-  ( ph  ->  ( B `  L
)  e. Word  ( I  X.  2o ) )
74 efgredlemb.q . . . . . . . . . 10  |-  ( ph  ->  Q  e.  ( 0 ... ( # `  ( B `  L )
) ) )
75 elfzuz 11822 . . . . . . . . . 10  |-  ( Q  e.  ( 0 ... ( # `  ( B `  L )
) )  ->  Q  e.  ( ZZ>= `  0 )
)
76 eluzfz1 11832 . . . . . . . . . 10  |-  ( Q  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... Q
) )
7774, 75, 763syl 18 . . . . . . . . 9  |-  ( ph  ->  0  e.  ( 0 ... Q ) )
78 lencl 12737 . . . . . . . . . . . 12  |-  ( ( B `  L )  e. Word  ( I  X.  2o )  ->  ( # `  ( B `  L
) )  e.  NN0 )
7973, 78syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  ( B `  L )
)  e.  NN0 )
8079, 44syl6eleq 2559 . . . . . . . . . 10  |-  ( ph  ->  ( # `  ( B `  L )
)  e.  ( ZZ>= ` 
0 ) )
81 eluzfz2 11833 . . . . . . . . . 10  |-  ( (
# `  ( B `  L ) )  e.  ( ZZ>= `  0 )  ->  ( # `  ( B `  L )
)  e.  ( 0 ... ( # `  ( B `  L )
) ) )
8280, 81syl 17 . . . . . . . . 9  |-  ( ph  ->  ( # `  ( B `  L )
)  e.  ( 0 ... ( # `  ( B `  L )
) ) )
83 ccatswrd 12866 . . . . . . . . 9  |-  ( ( ( B `  L
)  e. Word  ( I  X.  2o )  /\  (
0  e.  ( 0 ... Q )  /\  Q  e.  ( 0 ... ( # `  ( B `  L )
) )  /\  ( # `
 ( B `  L ) )  e.  ( 0 ... ( # `
 ( B `  L ) ) ) ) )  ->  (
( ( B `  L ) substr  <. 0 ,  Q >. ) ++  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L
) ) >. )
)  =  ( ( B `  L ) substr  <. 0 ,  ( # `  ( B `  L
) ) >. )
)
8473, 77, 74, 82, 83syl13anc 1294 . . . . . . . 8  |-  ( ph  ->  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( B `  L ) substr  <. Q , 
( # `  ( B `
 L ) )
>. ) )  =  ( ( B `  L
) substr  <. 0 ,  (
# `  ( B `  L ) ) >.
) )
85 swrdid 12838 . . . . . . . . 9  |-  ( ( B `  L )  e. Word  ( I  X.  2o )  ->  ( ( B `  L ) substr  <. 0 ,  ( # `  ( B `  L
) ) >. )  =  ( B `  L ) )
8673, 85syl 17 . . . . . . . 8  |-  ( ph  ->  ( ( B `  L ) substr  <. 0 ,  ( # `  ( B `  L )
) >. )  =  ( B `  L ) )
8784, 86eqtrd 2505 . . . . . . 7  |-  ( ph  ->  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( B `  L ) substr  <. Q , 
( # `  ( B `
 L ) )
>. ) )  =  ( B `  L ) )
8852, 87eqeq12d 2486 . . . . . 6  |-  ( ph  ->  ( ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++  ( ( A `  K ) substr  <. P , 
( # `  ( A `
 K ) )
>. ) )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L
) ) >. )
)  <->  ( A `  K )  =  ( B `  L ) ) )
892, 88mtbird 308 . . . . 5  |-  ( ph  ->  -.  ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++  ( ( A `  K ) substr  <. P , 
( # `  ( A `
 K ) )
>. ) )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L
) ) >. )
) )
90 efgredlemb.6 . . . . . . . . . . . . 13  |-  ( ph  ->  ( S `  A
)  =  ( P ( T `  ( A `  K )
) U ) )
91 efgredlemb.u . . . . . . . . . . . . . 14  |-  ( ph  ->  U  e.  ( I  X.  2o ) )
923, 7, 8, 9efgtval 17451 . . . . . . . . . . . . . 14  |-  ( ( ( A `  K
)  e.  W  /\  P  e.  ( 0 ... ( # `  ( A `  K )
) )  /\  U  e.  ( I  X.  2o ) )  ->  ( P ( T `  ( A `  K ) ) U )  =  ( ( A `  K ) splice  <. P ,  P ,  <" U
( M `  U
) "> >. )
)
9336, 38, 91, 92syl3anc 1292 . . . . . . . . . . . . 13  |-  ( ph  ->  ( P ( T `
 ( A `  K ) ) U )  =  ( ( A `  K ) splice  <. P ,  P ,  <" U ( M `
 U ) "> >. ) )
948efgmf 17441 . . . . . . . . . . . . . . . . 17  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
9594ffvelrni 6036 . . . . . . . . . . . . . . . 16  |-  ( U  e.  ( I  X.  2o )  ->  ( M `
 U )  e.  ( I  X.  2o ) )
9691, 95syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( M `  U
)  e.  ( I  X.  2o ) )
9791, 96s2cld 13025 . . . . . . . . . . . . . 14  |-  ( ph  ->  <" U ( M `  U ) ">  e. Word  (
I  X.  2o ) )
98 splval 12912 . . . . . . . . . . . . . 14  |-  ( ( ( A `  K
)  e.  W  /\  ( P  e.  (
0 ... ( # `  ( A `  K )
) )  /\  P  e.  ( 0 ... ( # `
 ( A `  K ) ) )  /\  <" U ( M `  U ) ">  e. Word  (
I  X.  2o ) ) )  ->  (
( A `  K
) splice  <. P ,  P ,  <" U ( M `  U ) "> >. )  =  ( ( ( ( A `  K
) substr  <. 0 ,  P >. ) ++  <" U ( M `  U ) "> ) ++  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) ) )
9936, 38, 38, 97, 98syl13anc 1294 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( A `  K ) splice  <. P ,  P ,  <" U
( M `  U
) "> >. )  =  ( ( ( ( A `  K
) substr  <. 0 ,  P >. ) ++  <" U ( M `  U ) "> ) ++  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) ) )
10090, 93, 993eqtrd 2509 . . . . . . . . . . . 12  |-  ( ph  ->  ( S `  A
)  =  ( ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++  <" U
( M `  U
) "> ) ++  ( ( A `  K ) substr  <. P , 
( # `  ( A `
 K ) )
>. ) ) )
101 efgredlemb.7 . . . . . . . . . . . . 13  |-  ( ph  ->  ( S `  B
)  =  ( Q ( T `  ( B `  L )
) V ) )
102 efgredlemb.v . . . . . . . . . . . . . 14  |-  ( ph  ->  V  e.  ( I  X.  2o ) )
1033, 7, 8, 9efgtval 17451 . . . . . . . . . . . . . 14  |-  ( ( ( B `  L
)  e.  W  /\  Q  e.  ( 0 ... ( # `  ( B `  L )
) )  /\  V  e.  ( I  X.  2o ) )  ->  ( Q ( T `  ( B `  L ) ) V )  =  ( ( B `  L ) splice  <. Q ,  Q ,  <" V
( M `  V
) "> >. )
)
10472, 74, 102, 103syl3anc 1292 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Q ( T `
 ( B `  L ) ) V )  =  ( ( B `  L ) splice  <. Q ,  Q ,  <" V ( M `
 V ) "> >. ) )
10594ffvelrni 6036 . . . . . . . . . . . . . . . 16  |-  ( V  e.  ( I  X.  2o )  ->  ( M `
 V )  e.  ( I  X.  2o ) )
106102, 105syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( M `  V
)  e.  ( I  X.  2o ) )
107102, 106s2cld 13025 . . . . . . . . . . . . . 14  |-  ( ph  ->  <" V ( M `  V ) ">  e. Word  (
I  X.  2o ) )
108 splval 12912 . . . . . . . . . . . . . 14  |-  ( ( ( B `  L
)  e.  W  /\  ( Q  e.  (
0 ... ( # `  ( B `  L )
) )  /\  Q  e.  ( 0 ... ( # `
 ( B `  L ) ) )  /\  <" V ( M `  V ) ">  e. Word  (
I  X.  2o ) ) )  ->  (
( B `  L
) splice  <. Q ,  Q ,  <" V ( M `  V ) "> >. )  =  ( ( ( ( B `  L
) substr  <. 0 ,  Q >. ) ++  <" V ( M `  V ) "> ) ++  ( ( B `  L
) substr  <. Q ,  (
# `  ( B `  L ) ) >.
) ) )
10972, 74, 74, 107, 108syl13anc 1294 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( B `  L ) splice  <. Q ,  Q ,  <" V
( M `  V
) "> >. )  =  ( ( ( ( B `  L
) substr  <. 0 ,  Q >. ) ++  <" V ( M `  V ) "> ) ++  ( ( B `  L
) substr  <. Q ,  (
# `  ( B `  L ) ) >.
) ) )
110101, 104, 1093eqtrd 2509 . . . . . . . . . . . 12  |-  ( ph  ->  ( S `  B
)  =  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V
( M `  V
) "> ) ++  ( ( B `  L ) substr  <. Q , 
( # `  ( B `
 L ) )
>. ) ) )
11122, 100, 1103eqtr3d 2513 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++ 
<" U ( M `
 U ) "> ) ++  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K
) ) >. )
)  =  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V
( M `  V
) "> ) ++  ( ( B `  L ) substr  <. Q , 
( # `  ( B `
 L ) )
>. ) ) )
112111adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  P  =  Q )  ->  (
( ( ( A `
 K ) substr  <. 0 ,  P >. ) ++ 
<" U ( M `
 U ) "> ) ++  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K
) ) >. )
)  =  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V
( M `  V
) "> ) ++  ( ( B `  L ) substr  <. Q , 
( # `  ( B `
 L ) )
>. ) ) )
113 swrdcl 12829 . . . . . . . . . . . . . 14  |-  ( ( A `  K )  e. Word  ( I  X.  2o )  ->  ( ( A `  K ) substr  <. 0 ,  P >. )  e. Word  ( I  X.  2o ) )
11437, 113syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( A `  K ) substr  <. 0 ,  P >. )  e. Word  (
I  X.  2o ) )
115114adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  P  =  Q )  ->  (
( A `  K
) substr  <. 0 ,  P >. )  e. Word  ( I  X.  2o ) )
11697adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  P  =  Q )  ->  <" U
( M `  U
) ">  e. Word  ( I  X.  2o ) )
117 ccatcl 12771 . . . . . . . . . . . 12  |-  ( ( ( ( A `  K ) substr  <. 0 ,  P >. )  e. Word  (
I  X.  2o )  /\  <" U ( M `  U ) ">  e. Word  (
I  X.  2o ) )  ->  ( (
( A `  K
) substr  <. 0 ,  P >. ) ++  <" U ( M `  U ) "> )  e. Word 
( I  X.  2o ) )
118115, 116, 117syl2anc 673 . . . . . . . . . . 11  |-  ( (
ph  /\  P  =  Q )  ->  (
( ( A `  K ) substr  <. 0 ,  P >. ) ++  <" U
( M `  U
) "> )  e. Word  ( I  X.  2o ) )
119 swrdcl 12829 . . . . . . . . . . . . 13  |-  ( ( A `  K )  e. Word  ( I  X.  2o )  ->  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K
) ) >. )  e. Word  ( I  X.  2o ) )
12037, 119syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( A `  K ) substr  <. P , 
( # `  ( A `
 K ) )
>. )  e. Word  ( I  X.  2o ) )
121120adantr 472 . . . . . . . . . . 11  |-  ( (
ph  /\  P  =  Q )  ->  (
( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
)  e. Word  ( I  X.  2o ) )
122 swrdcl 12829 . . . . . . . . . . . . . 14  |-  ( ( B `  L )  e. Word  ( I  X.  2o )  ->  ( ( B `  L ) substr  <. 0 ,  Q >. )  e. Word  ( I  X.  2o ) )
12373, 122syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( B `  L ) substr  <. 0 ,  Q >. )  e. Word  (
I  X.  2o ) )
124123adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  P  =  Q )  ->  (
( B `  L
) substr  <. 0 ,  Q >. )  e. Word  ( I  X.  2o ) )
125107adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  P  =  Q )  ->  <" V
( M `  V
) ">  e. Word  ( I  X.  2o ) )
126 ccatcl 12771 . . . . . . . . . . . 12  |-  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. )  e. Word  (
I  X.  2o )  /\  <" V ( M `  V ) ">  e. Word  (
I  X.  2o ) )  ->  ( (
( B `  L
) substr  <. 0 ,  Q >. ) ++  <" V ( M `  V ) "> )  e. Word 
( I  X.  2o ) )
127124, 125, 126syl2anc 673 . . . . . . . . . . 11  |-  ( (
ph  /\  P  =  Q )  ->  (
( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V
( M `  V
) "> )  e. Word  ( I  X.  2o ) )
128 swrdcl 12829 . . . . . . . . . . . . 13  |-  ( ( B `  L )  e. Word  ( I  X.  2o )  ->  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L
) ) >. )  e. Word  ( I  X.  2o ) )
12973, 128syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( B `  L ) substr  <. Q , 
( # `  ( B `
 L ) )
>. )  e. Word  ( I  X.  2o ) )
130129adantr 472 . . . . . . . . . . 11  |-  ( (
ph  /\  P  =  Q )  ->  (
( B `  L
) substr  <. Q ,  (
# `  ( B `  L ) ) >.
)  e. Word  ( I  X.  2o ) )
131 swrd0len 12832 . . . . . . . . . . . . . . . 16  |-  ( ( ( A `  K
)  e. Word  ( I  X.  2o )  /\  P  e.  ( 0 ... ( # `
 ( A `  K ) ) ) )  ->  ( # `  (
( A `  K
) substr  <. 0 ,  P >. ) )  =  P )
13237, 38, 131syl2anc 673 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( # `  (
( A `  K
) substr  <. 0 ,  P >. ) )  =  P )
133 swrd0len 12832 . . . . . . . . . . . . . . . 16  |-  ( ( ( B `  L
)  e. Word  ( I  X.  2o )  /\  Q  e.  ( 0 ... ( # `
 ( B `  L ) ) ) )  ->  ( # `  (
( B `  L
) substr  <. 0 ,  Q >. ) )  =  Q )
13473, 74, 133syl2anc 673 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( # `  (
( B `  L
) substr  <. 0 ,  Q >. ) )  =  Q )
135132, 134eqeq12d 2486 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( # `  (
( A `  K
) substr  <. 0 ,  P >. ) )  =  (
# `  ( ( B `  L ) substr  <.
0 ,  Q >. ) )  <->  P  =  Q
) )
136135biimpar 493 . . . . . . . . . . . . 13  |-  ( (
ph  /\  P  =  Q )  ->  ( # `
 ( ( A `
 K ) substr  <. 0 ,  P >. ) )  =  ( # `  ( ( B `  L ) substr  <. 0 ,  Q >. ) ) )
137 s2len 13043 . . . . . . . . . . . . . . 15  |-  ( # `  <" U ( M `  U ) "> )  =  2
138 s2len 13043 . . . . . . . . . . . . . . 15  |-  ( # `  <" V ( M `  V ) "> )  =  2
139137, 138eqtr4i 2496 . . . . . . . . . . . . . 14  |-  ( # `  <" U ( M `  U ) "> )  =  ( # `  <" V ( M `  V ) "> )
140139a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  P  =  Q )  ->  ( # `
 <" U ( M `  U ) "> )  =  ( # `  <" V ( M `  V ) "> ) )
141136, 140oveq12d 6326 . . . . . . . . . . . 12  |-  ( (
ph  /\  P  =  Q )  ->  (
( # `  ( ( A `  K ) substr  <. 0 ,  P >. ) )  +  ( # `  <" U ( M `  U ) "> ) )  =  ( ( # `  ( ( B `  L ) substr  <. 0 ,  Q >. ) )  +  ( # `  <" V ( M `  V ) "> ) ) )
142 ccatlen 12772 . . . . . . . . . . . . 13  |-  ( ( ( ( A `  K ) substr  <. 0 ,  P >. )  e. Word  (
I  X.  2o )  /\  <" U ( M `  U ) ">  e. Word  (
I  X.  2o ) )  ->  ( # `  (
( ( A `  K ) substr  <. 0 ,  P >. ) ++  <" U
( M `  U
) "> )
)  =  ( (
# `  ( ( A `  K ) substr  <.
0 ,  P >. ) )  +  ( # `  <" U ( M `  U ) "> ) ) )
143115, 116, 142syl2anc 673 . . . . . . . . . . . 12  |-  ( (
ph  /\  P  =  Q )  ->  ( # `
 ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++ 
<" U ( M `
 U ) "> ) )  =  ( ( # `  (
( A `  K
) substr  <. 0 ,  P >. ) )  +  (
# `  <" U
( M `  U
) "> )
) )
144 ccatlen 12772 . . . . . . . . . . . . 13  |-  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. )  e. Word  (
I  X.  2o )  /\  <" V ( M `  V ) ">  e. Word  (
I  X.  2o ) )  ->  ( # `  (
( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V
( M `  V
) "> )
)  =  ( (
# `  ( ( B `  L ) substr  <.
0 ,  Q >. ) )  +  ( # `  <" V ( M `  V ) "> ) ) )
145124, 125, 144syl2anc 673 . . . . . . . . . . . 12  |-  ( (
ph  /\  P  =  Q )  ->  ( # `
 ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++ 
<" V ( M `
 V ) "> ) )  =  ( ( # `  (
( B `  L
) substr  <. 0 ,  Q >. ) )  +  (
# `  <" V
( M `  V
) "> )
) )
146141, 143, 1453eqtr4d 2515 . . . . . . . . . . 11  |-  ( (
ph  /\  P  =  Q )  ->  ( # `
 ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++ 
<" U ( M `
 U ) "> ) )  =  ( # `  (
( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V
( M `  V
) "> )
) )
147 ccatopth 12880 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++ 
<" U ( M `
 U ) "> )  e. Word  (
I  X.  2o )  /\  ( ( A `
 K ) substr  <. P ,  ( # `  ( A `  K )
) >. )  e. Word  (
I  X.  2o ) )  /\  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V
( M `  V
) "> )  e. Word  ( I  X.  2o )  /\  ( ( B `
 L ) substr  <. Q ,  ( # `  ( B `  L )
) >. )  e. Word  (
I  X.  2o ) )  /\  ( # `  ( ( ( A `
 K ) substr  <. 0 ,  P >. ) ++ 
<" U ( M `
 U ) "> ) )  =  ( # `  (
( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V
( M `  V
) "> )
) )  ->  (
( ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++ 
<" U ( M `
 U ) "> ) ++  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K
) ) >. )
)  =  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V
( M `  V
) "> ) ++  ( ( B `  L ) substr  <. Q , 
( # `  ( B `
 L ) )
>. ) )  <->  ( (
( ( A `  K ) substr  <. 0 ,  P >. ) ++  <" U
( M `  U
) "> )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++ 
<" V ( M `
 V ) "> )  /\  (
( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
)  =  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L
) ) >. )
) ) )
148118, 121, 127, 130, 146, 147syl221anc 1303 . . . . . . . . . 10  |-  ( (
ph  /\  P  =  Q )  ->  (
( ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++ 
<" U ( M `
 U ) "> ) ++  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K
) ) >. )
)  =  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V
( M `  V
) "> ) ++  ( ( B `  L ) substr  <. Q , 
( # `  ( B `
 L ) )
>. ) )  <->  ( (
( ( A `  K ) substr  <. 0 ,  P >. ) ++  <" U
( M `  U
) "> )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++ 
<" V ( M `
 V ) "> )  /\  (
( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
)  =  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L
) ) >. )
) ) )
149112, 148mpbid 215 . . . . . . . . 9  |-  ( (
ph  /\  P  =  Q )  ->  (
( ( ( A `
 K ) substr  <. 0 ,  P >. ) ++ 
<" U ( M `
 U ) "> )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V
( M `  V
) "> )  /\  ( ( A `  K ) substr  <. P , 
( # `  ( A `
 K ) )
>. )  =  (
( B `  L
) substr  <. Q ,  (
# `  ( B `  L ) ) >.
) ) )
150149simpld 466 . . . . . . . 8  |-  ( (
ph  /\  P  =  Q )  ->  (
( ( A `  K ) substr  <. 0 ,  P >. ) ++  <" U
( M `  U
) "> )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++ 
<" V ( M `
 V ) "> ) )
151 ccatopth 12880 . . . . . . . . 9  |-  ( ( ( ( ( A `
 K ) substr  <. 0 ,  P >. )  e. Word  ( I  X.  2o )  /\  <" U
( M `  U
) ">  e. Word  ( I  X.  2o ) )  /\  ( ( ( B `  L
) substr  <. 0 ,  Q >. )  e. Word  ( I  X.  2o )  /\  <" V ( M `
 V ) ">  e. Word  ( I  X.  2o ) )  /\  ( # `  ( ( A `  K ) substr  <. 0 ,  P >. ) )  =  ( # `  ( ( B `  L ) substr  <. 0 ,  Q >. ) ) )  ->  ( ( ( ( A `  K
) substr  <. 0 ,  P >. ) ++  <" U ( M `  U ) "> )  =  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++ 
<" V ( M `
 V ) "> )  <->  ( (
( A `  K
) substr  <. 0 ,  P >. )  =  ( ( B `  L ) substr  <. 0 ,  Q >. )  /\  <" U ( M `  U ) ">  =  <" V ( M `  V ) "> ) ) )
152115, 116, 124, 125, 136, 151syl221anc 1303 . . . . . . . 8  |-  ( (
ph  /\  P  =  Q )  ->  (
( ( ( A `
 K ) substr  <. 0 ,  P >. ) ++ 
<" U ( M `
 U ) "> )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V
( M `  V
) "> )  <->  ( ( ( A `  K ) substr  <. 0 ,  P >. )  =  ( ( B `  L
) substr  <. 0 ,  Q >. )  /\  <" U
( M `  U
) ">  =  <" V ( M `
 V ) "> ) ) )
153150, 152mpbid 215 . . . . . . 7  |-  ( (
ph  /\  P  =  Q )  ->  (
( ( A `  K ) substr  <. 0 ,  P >. )  =  ( ( B `  L
) substr  <. 0 ,  Q >. )  /\  <" U
( M `  U
) ">  =  <" V ( M `
 V ) "> ) )
154153simpld 466 . . . . . 6  |-  ( (
ph  /\  P  =  Q )  ->  (
( A `  K
) substr  <. 0 ,  P >. )  =  ( ( B `  L ) substr  <. 0 ,  Q >. ) )
155149simprd 470 . . . . . 6  |-  ( (
ph  /\  P  =  Q )  ->  (
( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
)  =  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L
) ) >. )
)
156154, 155oveq12d 6326 . . . . 5  |-  ( (
ph  /\  P  =  Q )  ->  (
( ( A `  K ) substr  <. 0 ,  P >. ) ++  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K
) ) >. )
)  =  ( ( ( B `  L
) substr  <. 0 ,  Q >. ) ++  ( ( B `
 L ) substr  <. Q ,  ( # `  ( B `  L )
) >. ) ) )
15789, 156mtand 671 . . . 4  |-  ( ph  ->  -.  P  =  Q )
158157pm2.21d 109 . . 3  |-  ( ph  ->  ( P  =  Q  ->  ( A ` 
0 )  =  ( B `  0 ) ) )
159 uzp1 11216 . . . 4  |-  ( P  e.  ( ZZ>= `  ( Q  +  1 ) )  ->  ( P  =  ( Q  + 
1 )  \/  P  e.  ( ZZ>= `  ( ( Q  +  1 )  +  1 ) ) ) )
16091s1cld 12795 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  <" U ">  e. Word  ( I  X.  2o ) )
161 ccatcl 12771 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A `  K ) substr  <. 0 ,  P >. )  e. Word  (
I  X.  2o )  /\  <" U ">  e. Word  ( I  X.  2o ) )  ->  (
( ( A `  K ) substr  <. 0 ,  P >. ) ++  <" U "> )  e. Word  (
I  X.  2o ) )
162114, 160, 161syl2anc 673 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( ( A `
 K ) substr  <. 0 ,  P >. ) ++ 
<" U "> )  e. Word  ( I  X.  2o ) )
16396s1cld 12795 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  <" ( M `
 U ) ">  e. Word  ( I  X.  2o ) )
164 ccatass 12783 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( A `
 K ) substr  <. 0 ,  P >. ) ++ 
<" U "> )  e. Word  ( I  X.  2o )  /\  <" ( M `  U
) ">  e. Word  ( I  X.  2o )  /\  ( ( A `
 K ) substr  <. P ,  ( # `  ( A `  K )
) >. )  e. Word  (
I  X.  2o ) )  ->  ( (
( ( ( A `
 K ) substr  <. 0 ,  P >. ) ++ 
<" U "> ) ++  <" ( M `
 U ) "> ) ++  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K
) ) >. )
)  =  ( ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++  <" U "> ) ++  ( <" ( M `  U ) "> ++  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) ) ) )
165162, 163, 120, 164syl3anc 1292 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( ( ( ( A `  K
) substr  <. 0 ,  P >. ) ++  <" U "> ) ++  <" ( M `  U ) "> ) ++  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K
) ) >. )
)  =  ( ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++  <" U "> ) ++  ( <" ( M `  U ) "> ++  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) ) ) )
166 ccatass 12783 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( A `  K ) substr  <. 0 ,  P >. )  e. Word  (
I  X.  2o )  /\  <" U ">  e. Word  ( I  X.  2o )  /\  <" ( M `  U ) ">  e. Word  ( I  X.  2o ) )  -> 
( ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++ 
<" U "> ) ++  <" ( M `
 U ) "> )  =  ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++  ( <" U "> ++  <" ( M `  U ) "> ) ) )
167114, 160, 163, 166syl3anc 1292 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++ 
<" U "> ) ++  <" ( M `
 U ) "> )  =  ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++  ( <" U "> ++  <" ( M `  U ) "> ) ) )
168 df-s2 13003 . . . . . . . . . . . . . . . . . . 19  |-  <" U
( M `  U
) ">  =  ( <" U "> ++  <" ( M `
 U ) "> )
169168oveq2i 6319 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A `  K
) substr  <. 0 ,  P >. ) ++  <" U ( M `  U ) "> )  =  ( ( ( A `
 K ) substr  <. 0 ,  P >. ) ++  ( <" U "> ++  <" ( M `
 U ) "> ) )
170167, 169syl6eqr 2523 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++ 
<" U "> ) ++  <" ( M `
 U ) "> )  =  ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++  <" U
( M `  U
) "> )
)
171170oveq1d 6323 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( ( ( ( A `  K
) substr  <. 0 ,  P >. ) ++  <" U "> ) ++  <" ( M `  U ) "> ) ++  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K
) ) >. )
)  =  ( ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++  <" U
( M `  U
) "> ) ++  ( ( A `  K ) substr  <. P , 
( # `  ( A `
 K ) )
>. ) ) )
172102s1cld 12795 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  <" V ">  e. Word  ( I  X.  2o ) )
173106s1cld 12795 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  <" ( M `
 V ) ">  e. Word  ( I  X.  2o ) )
174 ccatass 12783 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. )  e. Word  (
I  X.  2o )  /\  <" V ">  e. Word  ( I  X.  2o )  /\  <" ( M `  V ) ">  e. Word  ( I  X.  2o ) )  -> 
( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++ 
<" V "> ) ++  <" ( M `
 V ) "> )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++  ( <" V "> ++  <" ( M `  V ) "> ) ) )
175123, 172, 173, 174syl3anc 1292 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++ 
<" V "> ) ++  <" ( M `
 V ) "> )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++  ( <" V "> ++  <" ( M `  V ) "> ) ) )
176 df-s2 13003 . . . . . . . . . . . . . . . . . . 19  |-  <" V
( M `  V
) ">  =  ( <" V "> ++  <" ( M `
 V ) "> )
177176oveq2i 6319 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( B `  L
) substr  <. 0 ,  Q >. ) ++  <" V ( M `  V ) "> )  =  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( <" V "> ++  <" ( M `
 V ) "> ) )
178175, 177syl6eqr 2523 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++ 
<" V "> ) ++  <" ( M `
 V ) "> )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V
( M `  V
) "> )
)
179178oveq1d 6323 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( ( ( ( B `  L
) substr  <. 0 ,  Q >. ) ++  <" V "> ) ++  <" ( M `  V ) "> ) ++  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L
) ) >. )
)  =  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V
( M `  V
) "> ) ++  ( ( B `  L ) substr  <. Q , 
( # `  ( B `
 L ) )
>. ) ) )
180111, 171, 1793eqtr4d 2515 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( ( ( ( A `  K
) substr  <. 0 ,  P >. ) ++  <" U "> ) ++  <" ( M `  U ) "> ) ++  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K
) ) >. )
)  =  ( ( ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++ 
<" V "> ) ++  <" ( M `
 V ) "> ) ++  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L
) ) >. )
) )
181165, 180eqtr3d 2507 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++ 
<" U "> ) ++  ( <" ( M `  U ) "> ++  ( ( A `
 K ) substr  <. P ,  ( # `  ( A `  K )
) >. ) ) )  =  ( ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V "> ) ++  <" ( M `  V ) "> ) ++  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L
) ) >. )
) )
182181adantr 472 . . . . . . . . . . . . 13  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( ( A `
 K ) substr  <. 0 ,  P >. ) ++ 
<" U "> ) ++  ( <" ( M `  U ) "> ++  ( ( A `
 K ) substr  <. P ,  ( # `  ( A `  K )
) >. ) ) )  =  ( ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V "> ) ++  <" ( M `  V ) "> ) ++  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L
) ) >. )
) )
183162adantr 472 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( A `  K ) substr  <. 0 ,  P >. ) ++  <" U "> )  e. Word  (
I  X.  2o ) )
184163adantr 472 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  <" ( M `  U ) ">  e. Word  ( I  X.  2o ) )
185120adantr 472 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
)  e. Word  ( I  X.  2o ) )
186 ccatcl 12771 . . . . . . . . . . . . . . 15  |-  ( (
<" ( M `  U ) ">  e. Word  ( I  X.  2o )  /\  ( ( A `
 K ) substr  <. P ,  ( # `  ( A `  K )
) >. )  e. Word  (
I  X.  2o ) )  ->  ( <" ( M `  U
) "> ++  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K
) ) >. )
)  e. Word  ( I  X.  2o ) )
187184, 185, 186syl2anc 673 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( <" ( M `  U ) "> ++  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) )  e. Word  (
I  X.  2o ) )
188 ccatcl 12771 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. )  e. Word  (
I  X.  2o )  /\  <" V ">  e. Word  ( I  X.  2o ) )  ->  (
( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V "> )  e. Word  (
I  X.  2o ) )
189123, 172, 188syl2anc 673 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++ 
<" V "> )  e. Word  ( I  X.  2o ) )
190189adantr 472 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V "> )  e. Word  (
I  X.  2o ) )
191173adantr 472 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  <" ( M `  V ) ">  e. Word  ( I  X.  2o ) )
192 ccatcl 12771 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++ 
<" V "> )  e. Word  ( I  X.  2o )  /\  <" ( M `  V
) ">  e. Word  ( I  X.  2o ) )  ->  ( (
( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V "> ) ++  <" ( M `  V ) "> )  e. Word  (
I  X.  2o ) )
193190, 191, 192syl2anc 673 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++ 
<" V "> ) ++  <" ( M `
 V ) "> )  e. Word  (
I  X.  2o ) )
194129adantr 472 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( B `  L
) substr  <. Q ,  (
# `  ( B `  L ) ) >.
)  e. Word  ( I  X.  2o ) )
195 ccatlen 12772 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. )  e. Word  (
I  X.  2o )  /\  <" V ">  e. Word  ( I  X.  2o ) )  ->  ( # `
 ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++ 
<" V "> ) )  =  ( ( # `  (
( B `  L
) substr  <. 0 ,  Q >. ) )  +  (
# `  <" V "> ) ) )
196123, 172, 195syl2anc 673 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( # `  (
( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V "> ) )  =  ( ( # `  (
( B `  L
) substr  <. 0 ,  Q >. ) )  +  (
# `  <" V "> ) ) )
197 s1len 12797 . . . . . . . . . . . . . . . . . . . . 21  |-  ( # `  <" V "> )  =  1
198197a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( # `  <" V "> )  =  1 )
199134, 198oveq12d 6326 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( ( # `  (
( B `  L
) substr  <. 0 ,  Q >. ) )  +  (
# `  <" V "> ) )  =  ( Q  +  1 ) )
200196, 199eqtrd 2505 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( # `  (
( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V "> ) )  =  ( Q  +  1 ) )
201132, 200eqeq12d 2486 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( # `  (
( A `  K
) substr  <. 0 ,  P >. ) )  =  (
# `  ( (
( B `  L
) substr  <. 0 ,  Q >. ) ++  <" V "> ) )  <->  P  =  ( Q  +  1
) ) )
202201biimpar 493 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( # `
 ( ( A `
 K ) substr  <. 0 ,  P >. ) )  =  ( # `  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++ 
<" V "> ) ) )
203 s1len 12797 . . . . . . . . . . . . . . . . . 18  |-  ( # `  <" U "> )  =  1
204 s1len 12797 . . . . . . . . . . . . . . . . . 18  |-  ( # `  <" ( M `
 V ) "> )  =  1
205203, 204eqtr4i 2496 . . . . . . . . . . . . . . . . 17  |-  ( # `  <" U "> )  =  ( # `
 <" ( M `
 V ) "> )
206205a1i 11 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( # `
 <" U "> )  =  ( # `
 <" ( M `
 V ) "> ) )
207202, 206oveq12d 6326 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( # `  ( ( A `  K ) substr  <. 0 ,  P >. ) )  +  ( # `  <" U "> ) )  =  ( ( # `  (
( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V "> ) )  +  ( # `  <" ( M `  V
) "> )
) )
208114adantr 472 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( A `  K
) substr  <. 0 ,  P >. )  e. Word  ( I  X.  2o ) )
209160adantr 472 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  <" U ">  e. Word  ( I  X.  2o ) )
210 ccatlen 12772 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A `  K ) substr  <. 0 ,  P >. )  e. Word  (
I  X.  2o )  /\  <" U ">  e. Word  ( I  X.  2o ) )  ->  ( # `
 ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++ 
<" U "> ) )  =  ( ( # `  (
( A `  K
) substr  <. 0 ,  P >. ) )  +  (
# `  <" U "> ) ) )
211208, 209, 210syl2anc 673 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( # `
 ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++ 
<" U "> ) )  =  ( ( # `  (
( A `  K
) substr  <. 0 ,  P >. ) )  +  (
# `  <" U "> ) ) )
212 ccatlen 12772 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++ 
<" V "> )  e. Word  ( I  X.  2o )  /\  <" ( M `  V
) ">  e. Word  ( I  X.  2o ) )  ->  ( # `  (
( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++ 
<" V "> ) ++  <" ( M `
 V ) "> ) )  =  ( ( # `  (
( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V "> ) )  +  ( # `  <" ( M `  V
) "> )
) )
213190, 191, 212syl2anc 673 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( # `
 ( ( ( ( B `  L
) substr  <. 0 ,  Q >. ) ++  <" V "> ) ++  <" ( M `  V ) "> ) )  =  ( ( # `  (
( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V "> ) )  +  ( # `  <" ( M `  V
) "> )
) )
214207, 211, 2133eqtr4d 2515 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( # `
 ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++ 
<" U "> ) )  =  (
# `  ( (
( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V "> ) ++  <" ( M `  V ) "> ) ) )
215 ccatopth 12880 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++ 
<" U "> )  e. Word  ( I  X.  2o )  /\  ( <" ( M `  U ) "> ++  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) )  e. Word  (
I  X.  2o ) )  /\  ( ( ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++ 
<" V "> ) ++  <" ( M `
 V ) "> )  e. Word  (
I  X.  2o )  /\  ( ( B `
 L ) substr  <. Q ,  ( # `  ( B `  L )
) >. )  e. Word  (
I  X.  2o ) )  /\  ( # `  ( ( ( A `
 K ) substr  <. 0 ,  P >. ) ++ 
<" U "> ) )  =  (
# `  ( (
( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V "> ) ++  <" ( M `  V ) "> ) ) )  ->  ( ( ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++  <" U "> ) ++  ( <" ( M `  U ) "> ++  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) ) )  =  ( ( ( ( ( B `  L
) substr  <. 0 ,  Q >. ) ++  <" V "> ) ++  <" ( M `  V ) "> ) ++  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L
) ) >. )
)  <->  ( ( ( ( A `  K
) substr  <. 0 ,  P >. ) ++  <" U "> )  =  (
( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++ 
<" V "> ) ++  <" ( M `
 V ) "> )  /\  ( <" ( M `  U ) "> ++  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) )  =  ( ( B `  L
) substr  <. Q ,  (
# `  ( B `  L ) ) >.
) ) ) )
216183, 187, 193, 194, 214, 215syl221anc 1303 . . . . . . . . . . . . 13  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++ 
<" U "> ) ++  ( <" ( M `  U ) "> ++  ( ( A `
 K ) substr  <. P ,  ( # `  ( A `  K )
) >. ) ) )  =  ( ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V "> ) ++  <" ( M `  V ) "> ) ++  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L
) ) >. )
)  <->  ( ( ( ( A `  K
) substr  <. 0 ,  P >. ) ++  <" U "> )  =  (
( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++ 
<" V "> ) ++  <" ( M `
 V ) "> )  /\  ( <" ( M `  U ) "> ++  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) )  =  ( ( B `  L
) substr  <. Q ,  (
# `  ( B `  L ) ) >.
) ) ) )
217182, 216mpbid 215 . . . . . . . . . . . 12  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( ( A `
 K ) substr  <. 0 ,  P >. ) ++ 
<" U "> )  =  ( (
( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V "> ) ++  <" ( M `  V ) "> )  /\  ( <" ( M `  U ) "> ++  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) )  =  ( ( B `  L
) substr  <. Q ,  (
# `  ( B `  L ) ) >.
) ) )
218217simpld 466 . . . . . . . . . . 11  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( A `  K ) substr  <. 0 ,  P >. ) ++  <" U "> )  =  ( ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++ 
<" V "> ) ++  <" ( M `
 V ) "> ) )
219 ccatopth 12880 . . . . . . . . . . . 12  |-  ( ( ( ( ( A `
 K ) substr  <. 0 ,  P >. )  e. Word  ( I  X.  2o )  /\  <" U ">  e. Word  ( I  X.  2o ) )  /\  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++ 
<" V "> )  e. Word  ( I  X.  2o )  /\  <" ( M `  V
) ">  e. Word  ( I  X.  2o ) )  /\  ( # `  ( ( A `  K ) substr  <. 0 ,  P >. ) )  =  ( # `  (
( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V "> ) ) )  ->  ( ( ( ( A `  K
) substr  <. 0 ,  P >. ) ++  <" U "> )  =  (
( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++ 
<" V "> ) ++  <" ( M `
 V ) "> )  <->  ( (
( A `  K
) substr  <. 0 ,  P >. )  =  ( ( ( B `  L
) substr  <. 0 ,  Q >. ) ++  <" V "> )  /\  <" U ">  =  <" ( M `  V ) "> ) ) )
220208, 209, 190, 191, 202, 219syl221anc 1303 . . . . . . . . . . 11  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( ( A `
 K ) substr  <. 0 ,  P >. ) ++ 
<" U "> )  =  ( (
( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V "> ) ++  <" ( M `  V ) "> )  <->  ( (
( A `  K
) substr  <. 0 ,  P >. )  =  ( ( ( B `  L
) substr  <. 0 ,  Q >. ) ++  <" V "> )  /\  <" U ">  =  <" ( M `  V ) "> ) ) )
221218, 220mpbid 215 . . . . . . . . . 10  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( A `  K ) substr  <. 0 ,  P >. )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V "> )  /\  <" U ">  =  <" ( M `  V ) "> ) )
222221simpld 466 . . . . . . . . 9  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( A `  K
) substr  <. 0 ,  P >. )  =  ( ( ( B `  L
) substr  <. 0 ,  Q >. ) ++  <" V "> ) )
223222oveq1d 6323 . . . . . . . 8  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( A `  K ) substr  <. 0 ,  P >. ) ++  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K
) ) >. )
)  =  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V "> ) ++  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K
) ) >. )
) )
224123adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( B `  L
) substr  <. 0 ,  Q >. )  e. Word  ( I  X.  2o ) )
225172adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  <" V ">  e. Word  ( I  X.  2o ) )
226 ccatass 12783 . . . . . . . . 9  |-  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. )  e. Word  (
I  X.  2o )  /\  <" V ">  e. Word  ( I  X.  2o )  /\  (
( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
)  e. Word  ( I  X.  2o ) )  -> 
( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++ 
<" V "> ) ++  ( ( A `  K ) substr  <. P , 
( # `  ( A `
 K ) )
>. ) )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++  ( <" V "> ++  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) ) ) )
227224, 225, 185, 226syl3anc 1292 . . . . . . . 8  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++ 
<" V "> ) ++  ( ( A `  K ) substr  <. P , 
( # `  ( A `
 K ) )
>. ) )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++  ( <" V "> ++  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) ) ) )
228221simprd 470 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  <" U ">  =  <" ( M `  V ) "> )
229 s111 12806 . . . . . . . . . . . . . . . . 17  |-  ( ( U  e.  ( I  X.  2o )  /\  ( M `  V )  e.  ( I  X.  2o ) )  ->  ( <" U ">  =  <" ( M `
 V ) ">  <->  U  =  ( M `  V )
) )
23091, 106, 229syl2anc 673 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( <" U ">  =  <" ( M `  V ) ">  <->  U  =  ( M `  V )
) )
231230adantr 472 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( <" U ">  =  <" ( M `
 V ) ">  <->  U  =  ( M `  V )
) )
232228, 231mpbid 215 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  U  =  ( M `  V ) )
233232fveq2d 5883 . . . . . . . . . . . . 13  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( M `  U )  =  ( M `  ( M `  V ) ) )
2348efgmnvl 17442 . . . . . . . . . . . . . . 15  |-  ( V  e.  ( I  X.  2o )  ->  ( M `
 ( M `  V ) )  =  V )
235102, 234syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M `  ( M `  V )
)  =  V )
236235adantr 472 . . . . . . . . . . . . 13  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( M `  ( M `  V ) )  =  V )
237233, 236eqtrd 2505 . . . . . . . . . . . 12  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( M `  U )  =  V )
238237s1eqd 12793 . . . . . . . . . . 11  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  <" ( M `  U ) ">  =  <" V "> )
239238oveq1d 6323 . . . . . . . . . 10  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( <" ( M `  U ) "> ++  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) )  =  (
<" V "> ++  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) ) )
240217simprd 470 . . . . . . . . . 10  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( <" ( M `  U ) "> ++  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) )  =  ( ( B `  L
) substr  <. Q ,  (
# `  ( B `  L ) ) >.
) )
241239, 240eqtr3d 2507 . . . . . . . . 9  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( <" V "> ++  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) )  =  ( ( B `  L
) substr  <. Q ,  (
# `  ( B `  L ) ) >.
) )
242241oveq2d 6324 . . . . . . . 8  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( B `  L ) substr  <. 0 ,  Q >. ) ++  ( <" V "> ++  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) ) )  =  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( B `  L ) substr  <. Q , 
( # `  ( B `
 L ) )
>. ) ) )
243223, 227, 2423eqtrd 2509 . . . . . . 7  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( A `  K ) substr  <. 0 ,  P >. ) ++  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K
) ) >. )
)  =  ( ( ( B `  L
) substr  <. 0 ,  Q >. ) ++  ( ( B `
 L ) substr  <. Q ,  ( # `  ( B `  L )
) >. ) ) )
24489, 243mtand 671 . . . . . 6  |-  ( ph  ->  -.  P  =  ( Q  +  1 ) )
245244pm2.21d 109 . . . . 5  |-  ( ph  ->  ( P  =  ( Q  +  1 )  ->  ( A ` 
0 )  =  ( B `  0 ) ) )
246 elfzelz 11826 . . . . . . . . . . . 12  |-  ( Q  e.  ( 0 ... ( # `  ( B `  L )
) )  ->  Q  e.  ZZ )
24774, 246syl 17 . . . . . . . . . . 11  |-  ( ph  ->  Q  e.  ZZ )
248247zcnd 11064 . . . . . . . . . 10  |-  ( ph  ->  Q  e.  CC )
249 1cnd 9677 . . . . . . . . . 10  |-  ( ph  ->  1  e.  CC )
250248, 249, 249addassd 9683 . . . . . . . . 9  |-  ( ph  ->  ( ( Q  + 
1 )  +  1 )  =  ( Q  +  ( 1  +  1 ) ) )
251 df-2 10690 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
252251oveq2i 6319 . . . . . . . . 9  |-  ( Q  +  2 )  =  ( Q  +  ( 1  +  1 ) )
253250, 252syl6eqr 2523 . . . . . . . 8  |-  ( ph  ->  ( ( Q  + 
1 )  +  1 )  =  ( Q  +  2 ) )
254253fveq2d 5883 . . . . . . 7  |-  ( ph  ->  ( ZZ>= `  ( ( Q  +  1 )  +  1 ) )  =  ( ZZ>= `  ( Q  +  2 ) ) )
255254eleq2d 2534 . . . . . 6  |-  ( ph  ->  ( P  e.  (
ZZ>= `  ( ( Q  +  1 )  +  1 ) )  <->  P  e.  ( ZZ>= `  ( Q  +  2 ) ) ) )
2563, 7, 8, 9, 10, 11efgsfo 17467 . . . . . . . . . 10  |-  S : dom  S -onto-> W
257 swrdcl 12829 . . . . . . . . . . . . 13  |-  ( ( A `  K )  e. Word  ( I  X.  2o )  ->  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K
) ) >. )  e. Word  ( I  X.  2o ) )
25837, 257syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. )  e. Word  (
I  X.  2o ) )
259 ccatcl 12771 . . . . . . . . . . . 12  |-  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. )  e. Word  (
I  X.  2o )  /\  ( ( A `
 K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. )  e. Word  (
I  X.  2o ) )  ->  ( (
( B `  L
) substr  <. 0 ,  Q >. ) ++  ( ( A `
 K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  e. Word 
( I  X.  2o ) )
260123, 258, 259syl2anc 673 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  e. Word 
( I  X.  2o ) )
2613efgrcl 17443 . . . . . . . . . . . . 13  |-  ( ( A `  K )  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
26236, 261syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
263262simprd 470 . . . . . . . . . . 11  |-  ( ph  ->  W  = Word  ( I  X.  2o ) )
264260, 263eleqtrrd 2552 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  e.  W )
265 foelrn 6056 . . . . . . . . . 10  |-  ( ( S : dom  S -onto-> W  /\  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  e.  W )  ->  E. c  e.  dom  S ( ( ( B `  L
) substr  <. 0 ,  Q >. ) ++  ( ( A `
 K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) )
266256, 264, 265sylancr 676 . . . . . . . . 9  |-  ( ph  ->  E. c  e.  dom  S ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) )
267266adantr 472 . . . . . . . 8  |-  ( (
ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  ->  E. c  e.  dom  S ( ( ( B `  L
) substr  <. 0 ,  Q >. ) ++  ( ( A `
 K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) )
26820ad2antrr 740 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  ->  A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a )
)  <  ( # `  ( S `  A )
)  ->  ( ( S `  a )  =  ( S `  b )  ->  (
a `  0 )  =  ( b ` 
0 ) ) ) )
2696ad2antrr 740 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  ->  A  e.  dom  S )
27021ad2antrr 740 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  ->  B  e.  dom  S )
27122ad2antrr 740 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  -> 
( S `  A
)  =  ( S `
 B ) )
27223ad2antrr 740 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  ->  -.  ( A `  0
)  =  ( B `
 0 ) )
27338ad2antrr 740 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  ->  P  e.  ( 0 ... ( # `  ( A `  K )
) ) )
27474ad2antrr 740 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  ->  Q  e.  ( 0 ... ( # `  ( B `  L )
) ) )
27591ad2antrr 740 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  ->  U  e.  ( I  X.  2o ) )
276102ad2antrr 740 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  ->  V  e.  ( I  X.  2o ) )
27790ad2antrr 740 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  -> 
( S `  A
)  =  ( P ( T `  ( A `  K )
) U ) )
278101ad2antrr 740 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  -> 
( S `  B
)  =  ( Q ( T `  ( B `  L )
) V ) )
2792ad2antrr 740 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  ->  -.  ( A `  K
)  =  ( B `
 L ) )
280 simplr 770 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  ->  P  e.  ( ZZ>= `  ( Q  +  2
) ) )
281 simprl 772 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  -> 
c  e.  dom  S
)
282 simprr 774 . . . . . . . . . 10  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  -> 
( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) )
283282eqcomd 2477 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  -> 
( S `  c
)  =  ( ( ( B `  L
) substr  <. 0 ,  Q >. ) ++  ( ( A `
 K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) ) )
2843, 7, 8, 9, 10, 11, 268, 269, 270, 271, 272, 19, 60, 273, 274, 275, 276, 277, 278, 279, 280, 281, 283efgredlemd 17472 . . . . . . . 8  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  -> 
( A `  0
)  =  ( B `
 0 ) )
285267, 284rexlimddv 2875 . . . . . . 7  |-  ( (
ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  ->  ( A `  0 )  =  ( B `  0
) )
286285ex 441 . . . . . 6  |-  ( ph  ->  ( P  e.  (
ZZ>= `  ( Q  + 
2 ) )  -> 
( A `  0
)  =  ( B `
 0 ) ) )
287255, 286sylbid 223 . . . . 5  |-  ( ph  ->  ( P  e.  (
ZZ>= `  ( ( Q  +  1 )  +  1 ) )  -> 
( A `  0
)  =  ( B `
 0 ) ) )
288245, 287jaod 387 . . . 4  |-  ( ph  ->  ( ( P  =  ( Q  +  1 )  \/  P  e.  ( ZZ>= `  ( ( Q  +  1 )  +  1 ) ) )  ->  ( A `  0 )  =  ( B `  0
) ) )
289159, 288syl5 32 . . 3  |-  ( ph  ->  ( P  e.  (
ZZ>= `  ( Q  + 
1 ) )  -> 
( A `  0
)  =  ( B `
 0 ) ) )
290158, 289jaod 387 . 2  |-  ( ph  ->  ( ( P  =  Q  \/  P  e.  ( ZZ>= `  ( Q  +  1 ) ) )  ->  ( A `  0 )  =  ( B `  0
) ) )
2911, 290syl5 32 1  |-  ( ph  ->  ( P  e.  (
ZZ>= `  Q )  -> 
( A `  0
)  =  ( B `
 0 ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3031    \ cdif 3387   (/)c0 3722   {csn 3959   <.cop 3965   <.cotp 3967   U_ciun 4269   class class class wbr 4395    |-> cmpt 4454    _I cid 4749    X. cxp 4837   dom cdm 4839   ran crn 4840   -->wf 5585   -onto->wfo 5587   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   1oc1o 7193   2oc2o 7194   0cc0 9557   1c1 9558    + caddc 9560    < clt 9693    - cmin 9880   NNcn 10631   2c2 10681   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810  ..^cfzo 11942   #chash 12553  Word cword 12703   ++ cconcat 12705   <"cs1 12706   substr csubstr 12707   splice csplice 12708   <"cs2 12996   ~FG cefg 17434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-ot 3968  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711  df-concat 12713  df-s1 12714  df-substr 12715  df-splice 12716  df-s2 13003
This theorem is referenced by:  efgredlemb  17474
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