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Theorem efgredlemc 16962
Description: The reduced word that forms the base of the sequence in efgsval 16948 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 11115 . 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 5906 . . . . . . . . . . 11  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
53, 4eqsstri 3519 . . . . . . . . . 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 16947 . . . . . . . . . . . . . 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 1009 . . . . . . . . . . . . 13  |-  ( A  e.  dom  S  ->  A  e.  (Word  W  \  { (/) } ) )
146, 13syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  (Word  W  \  { (/) } ) )
15 eldifi 3612 . . . . . . . . . . . 12  |-  ( A  e.  (Word  W  \  { (/) } )  ->  A  e. Word  W )
16 wrdf 12538 . . . . . . . . . . . 12  |-  ( A  e. Word  W  ->  A : ( 0..^ (
# `  A )
) --> W )
1714, 15, 163syl 20 . . . . . . . . . . 11  |-  ( ph  ->  A : ( 0..^ ( # `  A
) ) --> W )
18 fzossfz 11822 . . . . . . . . . . . . 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 16957 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( ( # `  A )  -  1 )  e.  NN  /\  ( ( # `  B
)  -  1 )  e.  NN ) )
2524simpld 457 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( # `  A
)  -  1 )  e.  NN )
26 fzo0end 11885 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  A
)  -  1 )  e.  NN  ->  (
( ( # `  A
)  -  1 )  -  1 )  e.  ( 0..^ ( (
# `  A )  -  1 ) ) )
2725, 26syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( # `  A )  -  1 )  -  1 )  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) )
2819, 27syl5eqel 2546 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) )
2918, 28sseldi 3487 . . . . . . . . . . . 12  |-  ( ph  ->  K  e.  ( 0 ... ( ( # `  A )  -  1 ) ) )
30 lencl 12549 . . . . . . . . . . . . . . 15  |-  ( A  e. Word  W  ->  ( # `
 A )  e. 
NN0 )
3114, 15, 303syl 20 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( # `  A
)  e.  NN0 )
3231nn0zd 10963 . . . . . . . . . . . . 13  |-  ( ph  ->  ( # `  A
)  e.  ZZ )
33 fzoval 11805 . . . . . . . . . . . . 13  |-  ( (
# `  A )  e.  ZZ  ->  ( 0..^ ( # `  A
) )  =  ( 0 ... ( (
# `  A )  -  1 ) ) )
3432, 33syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( 0..^ ( # `  A ) )  =  ( 0 ... (
( # `  A )  -  1 ) ) )
3529, 34eleqtrrd 2545 . . . . . . . . . . 11  |-  ( ph  ->  K  e.  ( 0..^ ( # `  A
) ) )
3617, 35ffvelrnd 6008 . . . . . . . . . 10  |-  ( ph  ->  ( A `  K
)  e.  W )
375, 36sseldi 3487 . . . . . . . . 9  |-  ( ph  ->  ( A `  K
)  e. Word  ( I  X.  2o ) )
38 efgredlemb.p . . . . . . . . . 10  |-  ( ph  ->  P  e.  ( 0 ... ( # `  ( A `  K )
) ) )
39 elfzuz 11687 . . . . . . . . . 10  |-  ( P  e.  ( 0 ... ( # `  ( A `  K )
) )  ->  P  e.  ( ZZ>= `  0 )
)
40 eluzfz1 11696 . . . . . . . . . 10  |-  ( P  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... P
) )
4138, 39, 403syl 20 . . . . . . . . 9  |-  ( ph  ->  0  e.  ( 0 ... P ) )
42 lencl 12549 . . . . . . . . . . . 12  |-  ( ( A `  K )  e. Word  ( I  X.  2o )  ->  ( # `  ( A `  K
) )  e.  NN0 )
4337, 42syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  ( A `  K )
)  e.  NN0 )
44 nn0uz 11116 . . . . . . . . . . 11  |-  NN0  =  ( ZZ>= `  0 )
4543, 44syl6eleq 2552 . . . . . . . . . 10  |-  ( ph  ->  ( # `  ( A `  K )
)  e.  ( ZZ>= ` 
0 ) )
46 eluzfz2 11697 . . . . . . . . . 10  |-  ( (
# `  ( A `  K ) )  e.  ( ZZ>= `  0 )  ->  ( # `  ( A `  K )
)  e.  ( 0 ... ( # `  ( A `  K )
) ) )
4745, 46syl 16 . . . . . . . . 9  |-  ( ph  ->  ( # `  ( A `  K )
)  e.  ( 0 ... ( # `  ( A `  K )
) ) )
48 ccatswrd 12672 . . . . . . . . 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 1228 . . . . . . . 8  |-  ( ph  ->  ( ( ( A `
 K ) substr  <. 0 ,  P >. ) ++  ( ( A `  K ) substr  <. P , 
( # `  ( A `
 K ) )
>. ) )  =  ( ( A `  K
) substr  <. 0 ,  (
# `  ( A `  K ) ) >.
) )
50 swrdid 12644 . . . . . . . . 9  |-  ( ( A `  K )  e. Word  ( I  X.  2o )  ->  ( ( A `  K ) substr  <. 0 ,  ( # `  ( A `  K
) ) >. )  =  ( A `  K ) )
5137, 50syl 16 . . . . . . . 8  |-  ( ph  ->  ( ( A `  K ) substr  <. 0 ,  ( # `  ( A `  K )
) >. )  =  ( A `  K ) )
5249, 51eqtrd 2495 . . . . . . 7  |-  ( ph  ->  ( ( ( A `
 K ) substr  <. 0 ,  P >. ) ++  ( ( A `  K ) substr  <. P , 
( # `  ( A `
 K ) )
>. ) )  =  ( A `  K ) )
533, 7, 8, 9, 10, 11efgsdm 16947 . . . . . . . . . . . . . 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 1009 . . . . . . . . . . . . 13  |-  ( B  e.  dom  S  ->  B  e.  (Word  W  \  { (/) } ) )
5521, 54syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  (Word  W  \  { (/) } ) )
56 eldifi 3612 . . . . . . . . . . . 12  |-  ( B  e.  (Word  W  \  { (/) } )  ->  B  e. Word  W )
57 wrdf 12538 . . . . . . . . . . . 12  |-  ( B  e. Word  W  ->  B : ( 0..^ (
# `  B )
) --> W )
5855, 56, 573syl 20 . . . . . . . . . . 11  |-  ( ph  ->  B : ( 0..^ ( # `  B
) ) --> W )
59 fzossfz 11822 . . . . . . . . . . . . 13  |-  ( 0..^ ( ( # `  B
)  -  1 ) )  C_  ( 0 ... ( ( # `  B )  -  1 ) )
60 efgredlemb.l . . . . . . . . . . . . . 14  |-  L  =  ( ( ( # `  B )  -  1 )  -  1 )
6124simprd 461 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( # `  B
)  -  1 )  e.  NN )
62 fzo0end 11885 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  B
)  -  1 )  e.  NN  ->  (
( ( # `  B
)  -  1 )  -  1 )  e.  ( 0..^ ( (
# `  B )  -  1 ) ) )
6361, 62syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( # `  B )  -  1 )  -  1 )  e.  ( 0..^ ( ( # `  B
)  -  1 ) ) )
6460, 63syl5eqel 2546 . . . . . . . . . . . . 13  |-  ( ph  ->  L  e.  ( 0..^ ( ( # `  B
)  -  1 ) ) )
6559, 64sseldi 3487 . . . . . . . . . . . 12  |-  ( ph  ->  L  e.  ( 0 ... ( ( # `  B )  -  1 ) ) )
66 lencl 12549 . . . . . . . . . . . . . . 15  |-  ( B  e. Word  W  ->  ( # `
 B )  e. 
NN0 )
6755, 56, 663syl 20 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( # `  B
)  e.  NN0 )
6867nn0zd 10963 . . . . . . . . . . . . 13  |-  ( ph  ->  ( # `  B
)  e.  ZZ )
69 fzoval 11805 . . . . . . . . . . . . 13  |-  ( (
# `  B )  e.  ZZ  ->  ( 0..^ ( # `  B
) )  =  ( 0 ... ( (
# `  B )  -  1 ) ) )
7068, 69syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( 0..^ ( # `  B ) )  =  ( 0 ... (
( # `  B )  -  1 ) ) )
7165, 70eleqtrrd 2545 . . . . . . . . . . 11  |-  ( ph  ->  L  e.  ( 0..^ ( # `  B
) ) )
7258, 71ffvelrnd 6008 . . . . . . . . . 10  |-  ( ph  ->  ( B `  L
)  e.  W )
735, 72sseldi 3487 . . . . . . . . 9  |-  ( ph  ->  ( B `  L
)  e. Word  ( I  X.  2o ) )
74 efgredlemb.q . . . . . . . . . 10  |-  ( ph  ->  Q  e.  ( 0 ... ( # `  ( B `  L )
) ) )
75 elfzuz 11687 . . . . . . . . . 10  |-  ( Q  e.  ( 0 ... ( # `  ( B `  L )
) )  ->  Q  e.  ( ZZ>= `  0 )
)
76 eluzfz1 11696 . . . . . . . . . 10  |-  ( Q  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... Q
) )
7774, 75, 763syl 20 . . . . . . . . 9  |-  ( ph  ->  0  e.  ( 0 ... Q ) )
78 lencl 12549 . . . . . . . . . . . 12  |-  ( ( B `  L )  e. Word  ( I  X.  2o )  ->  ( # `  ( B `  L
) )  e.  NN0 )
7973, 78syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  ( B `  L )
)  e.  NN0 )
8079, 44syl6eleq 2552 . . . . . . . . . 10  |-  ( ph  ->  ( # `  ( B `  L )
)  e.  ( ZZ>= ` 
0 ) )
81 eluzfz2 11697 . . . . . . . . . 10  |-  ( (
# `  ( B `  L ) )  e.  ( ZZ>= `  0 )  ->  ( # `  ( B `  L )
)  e.  ( 0 ... ( # `  ( B `  L )
) ) )
8280, 81syl 16 . . . . . . . . 9  |-  ( ph  ->  ( # `  ( B `  L )
)  e.  ( 0 ... ( # `  ( B `  L )
) ) )
83 ccatswrd 12672 . . . . . . . . 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 1228 . . . . . . . 8  |-  ( ph  ->  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( B `  L ) substr  <. Q , 
( # `  ( B `
 L ) )
>. ) )  =  ( ( B `  L
) substr  <. 0 ,  (
# `  ( B `  L ) ) >.
) )
85 swrdid 12644 . . . . . . . . 9  |-  ( ( B `  L )  e. Word  ( I  X.  2o )  ->  ( ( B `  L ) substr  <. 0 ,  ( # `  ( B `  L
) ) >. )  =  ( B `  L ) )
8673, 85syl 16 . . . . . . . 8  |-  ( ph  ->  ( ( B `  L ) substr  <. 0 ,  ( # `  ( B `  L )
) >. )  =  ( B `  L ) )
8784, 86eqtrd 2495 . . . . . . 7  |-  ( ph  ->  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( B `  L ) substr  <. Q , 
( # `  ( B `
 L ) )
>. ) )  =  ( B `  L ) )
8852, 87eqeq12d 2476 . . . . . 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 299 . . . . 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 16940 . . . . . . . . . . . . . 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 1226 . . . . . . . . . . . . 13  |-  ( ph  ->  ( P ( T `
 ( A `  K ) ) U )  =  ( ( A `  K ) splice  <. P ,  P ,  <" U ( M `
 U ) "> >. ) )
948efgmf 16930 . . . . . . . . . . . . . . . . 17  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
9594ffvelrni 6006 . . . . . . . . . . . . . . . 16  |-  ( U  e.  ( I  X.  2o )  ->  ( M `
 U )  e.  ( I  X.  2o ) )
9691, 95syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( M `  U
)  e.  ( I  X.  2o ) )
9791, 96s2cld 12825 . . . . . . . . . . . . . 14  |-  ( ph  ->  <" U ( M `  U ) ">  e. Word  (
I  X.  2o ) )
98 splval 12718 . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . 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 2499 . . . . . . . . . . . 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 16940 . . . . . . . . . . . . . 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 1226 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Q ( T `
 ( B `  L ) ) V )  =  ( ( B `  L ) splice  <. Q ,  Q ,  <" V ( M `
 V ) "> >. ) )
10594ffvelrni 6006 . . . . . . . . . . . . . . . 16  |-  ( V  e.  ( I  X.  2o )  ->  ( M `
 V )  e.  ( I  X.  2o ) )
106102, 105syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( M `  V
)  e.  ( I  X.  2o ) )
107102, 106s2cld 12825 . . . . . . . . . . . . . 14  |-  ( ph  ->  <" V ( M `  V ) ">  e. Word  (
I  X.  2o ) )
108 splval 12718 . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . 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 2499 . . . . . . . . . . . 12  |-  ( ph  ->  ( S `  B
)  =  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V
( M `  V
) "> ) ++  ( ( B `  L ) substr  <. Q , 
( # `  ( B `
 L ) )
>. ) ) )
11122, 100, 1103eqtr3d 2503 . . . . . . . . . . 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 463 . . . . . . . . . 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 12635 . . . . . . . . . . . . . 14  |-  ( ( A `  K )  e. Word  ( I  X.  2o )  ->  ( ( A `  K ) substr  <. 0 ,  P >. )  e. Word  ( I  X.  2o ) )
11437, 113syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( A `  K ) substr  <. 0 ,  P >. )  e. Word  (
I  X.  2o ) )
115114adantr 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  P  =  Q )  ->  (
( A `  K
) substr  <. 0 ,  P >. )  e. Word  ( I  X.  2o ) )
11697adantr 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  P  =  Q )  ->  <" U
( M `  U
) ">  e. Word  ( I  X.  2o ) )
117 ccatcl 12582 . . . . . . . . . . . 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 659 . . . . . . . . . . 11  |-  ( (
ph  /\  P  =  Q )  ->  (
( ( A `  K ) substr  <. 0 ,  P >. ) ++  <" U
( M `  U
) "> )  e. Word  ( I  X.  2o ) )
119 swrdcl 12635 . . . . . . . . . . . . 13  |-  ( ( A `  K )  e. Word  ( I  X.  2o )  ->  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K
) ) >. )  e. Word  ( I  X.  2o ) )
12037, 119syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( A `  K ) substr  <. P , 
( # `  ( A `
 K ) )
>. )  e. Word  ( I  X.  2o ) )
121120adantr 463 . . . . . . . . . . 11  |-  ( (
ph  /\  P  =  Q )  ->  (
( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
)  e. Word  ( I  X.  2o ) )
122 swrdcl 12635 . . . . . . . . . . . . . 14  |-  ( ( B `  L )  e. Word  ( I  X.  2o )  ->  ( ( B `  L ) substr  <. 0 ,  Q >. )  e. Word  ( I  X.  2o ) )
12373, 122syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( B `  L ) substr  <. 0 ,  Q >. )  e. Word  (
I  X.  2o ) )
124123adantr 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  P  =  Q )  ->  (
( B `  L
) substr  <. 0 ,  Q >. )  e. Word  ( I  X.  2o ) )
125107adantr 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  P  =  Q )  ->  <" V
( M `  V
) ">  e. Word  ( I  X.  2o ) )
126 ccatcl 12582 . . . . . . . . . . . 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 659 . . . . . . . . . . 11  |-  ( (
ph  /\  P  =  Q )  ->  (
( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V
( M `  V
) "> )  e. Word  ( I  X.  2o ) )
128 swrdcl 12635 . . . . . . . . . . . . 13  |-  ( ( B `  L )  e. Word  ( I  X.  2o )  ->  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L
) ) >. )  e. Word  ( I  X.  2o ) )
12973, 128syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( B `  L ) substr  <. Q , 
( # `  ( B `
 L ) )
>. )  e. Word  ( I  X.  2o ) )
130129adantr 463 . . . . . . . . . . 11  |-  ( (
ph  /\  P  =  Q )  ->  (
( B `  L
) substr  <. Q ,  (
# `  ( B `  L ) ) >.
)  e. Word  ( I  X.  2o ) )
131 swrd0len 12638 . . . . . . . . . . . . . . . 16  |-  ( ( ( A `  K
)  e. Word  ( I  X.  2o )  /\  P  e.  ( 0 ... ( # `
 ( A `  K ) ) ) )  ->  ( # `  (
( A `  K
) substr  <. 0 ,  P >. ) )  =  P )
13237, 38, 131syl2anc 659 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( # `  (
( A `  K
) substr  <. 0 ,  P >. ) )  =  P )
133 swrd0len 12638 . . . . . . . . . . . . . . . 16  |-  ( ( ( B `  L
)  e. Word  ( I  X.  2o )  /\  Q  e.  ( 0 ... ( # `
 ( B `  L ) ) ) )  ->  ( # `  (
( B `  L
) substr  <. 0 ,  Q >. ) )  =  Q )
13473, 74, 133syl2anc 659 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( # `  (
( B `  L
) substr  <. 0 ,  Q >. ) )  =  Q )
135132, 134eqeq12d 2476 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( # `  (
( A `  K
) substr  <. 0 ,  P >. ) )  =  (
# `  ( ( B `  L ) substr  <.
0 ,  Q >. ) )  <->  P  =  Q
) )
136135biimpar 483 . . . . . . . . . . . . 13  |-  ( (
ph  /\  P  =  Q )  ->  ( # `
 ( ( A `
 K ) substr  <. 0 ,  P >. ) )  =  ( # `  ( ( B `  L ) substr  <. 0 ,  Q >. ) ) )
137 s2len 12843 . . . . . . . . . . . . . . 15  |-  ( # `  <" U ( M `  U ) "> )  =  2
138 s2len 12843 . . . . . . . . . . . . . . 15  |-  ( # `  <" V ( M `  V ) "> )  =  2
139137, 138eqtr4i 2486 . . . . . . . . . . . . . 14  |-  ( # `  <" U ( M `  U ) "> )  =  ( # `  <" V ( M `  V ) "> )
140139a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  P  =  Q )  ->  ( # `
 <" U ( M `  U ) "> )  =  ( # `  <" V ( M `  V ) "> ) )
141136, 140oveq12d 6288 . . . . . . . . . . . 12  |-  ( (
ph  /\  P  =  Q )  ->  (
( # `  ( ( A `  K ) substr  <. 0 ,  P >. ) )  +  ( # `  <" U ( M `  U ) "> ) )  =  ( ( # `  ( ( B `  L ) substr  <. 0 ,  Q >. ) )  +  ( # `  <" V ( M `  V ) "> ) ) )
142 ccatlen 12583 . . . . . . . . . . . . 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 659 . . . . . . . . . . . 12  |-  ( (
ph  /\  P  =  Q )  ->  ( # `
 ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++ 
<" U ( M `
 U ) "> ) )  =  ( ( # `  (
( A `  K
) substr  <. 0 ,  P >. ) )  +  (
# `  <" U
( M `  U
) "> )
) )
144 ccatlen 12583 . . . . . . . . . . . . 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 659 . . . . . . . . . . . 12  |-  ( (
ph  /\  P  =  Q )  ->  ( # `
 ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++ 
<" V ( M `
 V ) "> ) )  =  ( ( # `  (
( B `  L
) substr  <. 0 ,  Q >. ) )  +  (
# `  <" V
( M `  V
) "> )
) )
146141, 143, 1453eqtr4d 2505 . . . . . . . . . . 11  |-  ( (
ph  /\  P  =  Q )  ->  ( # `
 ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++ 
<" U ( M `
 U ) "> ) )  =  ( # `  (
( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V
( M `  V
) "> )
) )
147 ccatopth 12686 . . . . . . . . . . 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 1237 . . . . . . . . . 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 210 . . . . . . . . 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 457 . . . . . . . 8  |-  ( (
ph  /\  P  =  Q )  ->  (
( ( A `  K ) substr  <. 0 ,  P >. ) ++  <" U
( M `  U
) "> )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++ 
<" V ( M `
 V ) "> ) )
151 ccatopth 12686 . . . . . . . . 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 1237 . . . . . . . 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 210 . . . . . . 7  |-  ( (
ph  /\  P  =  Q )  ->  (
( ( A `  K ) substr  <. 0 ,  P >. )  =  ( ( B `  L
) substr  <. 0 ,  Q >. )  /\  <" U
( M `  U
) ">  =  <" V ( M `
 V ) "> ) )
154153simpld 457 . . . . . 6  |-  ( (
ph  /\  P  =  Q )  ->  (
( A `  K
) substr  <. 0 ,  P >. )  =  ( ( B `  L ) substr  <. 0 ,  Q >. ) )
155149simprd 461 . . . . . 6  |-  ( (
ph  /\  P  =  Q )  ->  (
( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
)  =  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L
) ) >. )
)
156154, 155oveq12d 6288 . . . . 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 657 . . . 4  |-  ( ph  ->  -.  P  =  Q )
158157pm2.21d 106 . . 3  |-  ( ph  ->  ( P  =  Q  ->  ( A ` 
0 )  =  ( B `  0 ) ) )
159 uzp1 11115 . . . 4  |-  ( P  e.  ( ZZ>= `  ( Q  +  1 ) )  ->  ( P  =  ( Q  + 
1 )  \/  P  e.  ( ZZ>= `  ( ( Q  +  1 )  +  1 ) ) ) )
16091s1cld 12604 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  <" U ">  e. Word  ( I  X.  2o ) )
161 ccatcl 12582 . . . . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( ( A `
 K ) substr  <. 0 ,  P >. ) ++ 
<" U "> )  e. Word  ( I  X.  2o ) )
16396s1cld 12604 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  <" ( M `
 U ) ">  e. Word  ( I  X.  2o ) )
164 ccatass 12594 . . . . . . . . . . . . . . . 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 1226 . . . . . . . . . . . . . . 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 12594 . . . . . . . . . . . . . . . . . . 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 1226 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++ 
<" U "> ) ++  <" ( M `
 U ) "> )  =  ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++  ( <" U "> ++  <" ( M `  U ) "> ) ) )
168 df-s2 12804 . . . . . . . . . . . . . . . . . . 19  |-  <" U
( M `  U
) ">  =  ( <" U "> ++  <" ( M `
 U ) "> )
169168oveq2i 6281 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A `  K
) substr  <. 0 ,  P >. ) ++  <" U ( M `  U ) "> )  =  ( ( ( A `
 K ) substr  <. 0 ,  P >. ) ++  ( <" U "> ++  <" ( M `
 U ) "> ) )
170167, 169syl6eqr 2513 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++ 
<" U "> ) ++  <" ( M `
 U ) "> )  =  ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++  <" U
( M `  U
) "> )
)
171170oveq1d 6285 . . . . . . . . . . . . . . . 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 12604 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  <" V ">  e. Word  ( I  X.  2o ) )
173106s1cld 12604 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  <" ( M `
 V ) ">  e. Word  ( I  X.  2o ) )
174 ccatass 12594 . . . . . . . . . . . . . . . . . . 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 1226 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++ 
<" V "> ) ++  <" ( M `
 V ) "> )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++  ( <" V "> ++  <" ( M `  V ) "> ) ) )
176 df-s2 12804 . . . . . . . . . . . . . . . . . . 19  |-  <" V
( M `  V
) ">  =  ( <" V "> ++  <" ( M `
 V ) "> )
177176oveq2i 6281 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( B `  L
) substr  <. 0 ,  Q >. ) ++  <" V ( M `  V ) "> )  =  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( <" V "> ++  <" ( M `
 V ) "> ) )
178175, 177syl6eqr 2513 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++ 
<" V "> ) ++  <" ( M `
 V ) "> )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V
( M `  V
) "> )
)
179178oveq1d 6285 . . . . . . . . . . . . . . . 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 2505 . . . . . . . . . . . . . . 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 2497 . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( A `  K ) substr  <. 0 ,  P >. ) ++  <" U "> )  e. Word  (
I  X.  2o ) )
184163adantr 463 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  <" ( M `  U ) ">  e. Word  ( I  X.  2o ) )
185120adantr 463 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
)  e. Word  ( I  X.  2o ) )
186 ccatcl 12582 . . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( <" ( M `  U ) "> ++  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) )  e. Word  (
I  X.  2o ) )
188 ccatcl 12582 . . . . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++ 
<" V "> )  e. Word  ( I  X.  2o ) )
190189adantr 463 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V "> )  e. Word  (
I  X.  2o ) )
191173adantr 463 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  <" ( M `  V ) ">  e. Word  ( I  X.  2o ) )
192 ccatcl 12582 . . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++ 
<" V "> ) ++  <" ( M `
 V ) "> )  e. Word  (
I  X.  2o ) )
194129adantr 463 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( B `  L
) substr  <. Q ,  (
# `  ( B `  L ) ) >.
)  e. Word  ( I  X.  2o ) )
195 ccatlen 12583 . . . . . . . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( # `  (
( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V "> ) )  =  ( ( # `  (
( B `  L
) substr  <. 0 ,  Q >. ) )  +  (
# `  <" V "> ) ) )
197 s1len 12606 . . . . . . . . . . . . . . . . . . . . 21  |-  ( # `  <" V "> )  =  1
198197a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( # `  <" V "> )  =  1 )
199134, 198oveq12d 6288 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( ( # `  (
( B `  L
) substr  <. 0 ,  Q >. ) )  +  (
# `  <" V "> ) )  =  ( Q  +  1 ) )
200196, 199eqtrd 2495 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( # `  (
( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V "> ) )  =  ( Q  +  1 ) )
201132, 200eqeq12d 2476 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( # `  (
( A `  K
) substr  <. 0 ,  P >. ) )  =  (
# `  ( (
( B `  L
) substr  <. 0 ,  Q >. ) ++  <" V "> ) )  <->  P  =  ( Q  +  1
) ) )
202201biimpar 483 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( # `
 ( ( A `
 K ) substr  <. 0 ,  P >. ) )  =  ( # `  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++ 
<" V "> ) ) )
203 s1len 12606 . . . . . . . . . . . . . . . . . 18  |-  ( # `  <" U "> )  =  1
204 s1len 12606 . . . . . . . . . . . . . . . . . 18  |-  ( # `  <" ( M `
 V ) "> )  =  1
205203, 204eqtr4i 2486 . . . . . . . . . . . . . . . . 17  |-  ( # `  <" U "> )  =  ( # `
 <" ( M `
 V ) "> )
206205a1i 11 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( # `
 <" U "> )  =  ( # `
 <" ( M `
 V ) "> ) )
207202, 206oveq12d 6288 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( # `  ( ( A `  K ) substr  <. 0 ,  P >. ) )  +  ( # `  <" U "> ) )  =  ( ( # `  (
( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V "> ) )  +  ( # `  <" ( M `  V
) "> )
) )
208114adantr 463 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( A `  K
) substr  <. 0 ,  P >. )  e. Word  ( I  X.  2o ) )
209160adantr 463 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  <" U ">  e. Word  ( I  X.  2o ) )
210 ccatlen 12583 . . . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( # `
 ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++ 
<" U "> ) )  =  ( ( # `  (
( A `  K
) substr  <. 0 ,  P >. ) )  +  (
# `  <" U "> ) ) )
212 ccatlen 12583 . . . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( # `
 ( ( ( ( B `  L
) substr  <. 0 ,  Q >. ) ++  <" V "> ) ++  <" ( M `  V ) "> ) )  =  ( ( # `  (
( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V "> ) )  +  ( # `  <" ( M `  V
) "> )
) )
214207, 211, 2133eqtr4d 2505 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( # `
 ( ( ( A `  K ) substr  <. 0 ,  P >. ) ++ 
<" U "> ) )  =  (
# `  ( (
( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V "> ) ++  <" ( M `  V ) "> ) ) )
215 ccatopth 12686 . . . . . . . . . . . . . 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 1237 . . . . . . . . . . . . 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 210 . . . . . . . . . . . 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 457 . . . . . . . . . . 11  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( A `  K ) substr  <. 0 ,  P >. ) ++  <" U "> )  =  ( ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++ 
<" V "> ) ++  <" ( M `
 V ) "> ) )
219 ccatopth 12686 . . . . . . . . . . . 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 1237 . . . . . . . . . . 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 210 . . . . . . . . . 10  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( A `  K ) substr  <. 0 ,  P >. )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) ++  <" V "> )  /\  <" U ">  =  <" ( M `  V ) "> ) )
222221simpld 457 . . . . . . . . 9  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( A `  K
) substr  <. 0 ,  P >. )  =  ( ( ( B `  L
) substr  <. 0 ,  Q >. ) ++  <" V "> ) )
223222oveq1d 6285 . . . . . . . 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 463 . . . . . . . . 9  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( B `  L
) substr  <. 0 ,  Q >. )  e. Word  ( I  X.  2o ) )
225172adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  <" V ">  e. Word  ( I  X.  2o ) )
226 ccatass 12594 . . . . . . . . 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 1226 . . . . . . . 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 461 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  <" U ">  =  <" ( M `  V ) "> )
229 s111 12612 . . . . . . . . . . . . . . . . 17  |-  ( ( U  e.  ( I  X.  2o )  /\  ( M `  V )  e.  ( I  X.  2o ) )  ->  ( <" U ">  =  <" ( M `
 V ) ">  <->  U  =  ( M `  V )
) )
23091, 106, 229syl2anc 659 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( <" U ">  =  <" ( M `  V ) ">  <->  U  =  ( M `  V )
) )
231230adantr 463 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( <" U ">  =  <" ( M `
 V ) ">  <->  U  =  ( M `  V )
) )
232228, 231mpbid 210 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  U  =  ( M `  V ) )
233232fveq2d 5852 . . . . . . . . . . . . 13  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( M `  U )  =  ( M `  ( M `  V ) ) )
2348efgmnvl 16931 . . . . . . . . . . . . . . 15  |-  ( V  e.  ( I  X.  2o )  ->  ( M `
 ( M `  V ) )  =  V )
235102, 234syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M `  ( M `  V )
)  =  V )
236235adantr 463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( M `  ( M `  V ) )  =  V )
237233, 236eqtrd 2495 . . . . . . . . . . . 12  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( M `  U )  =  V )
238237s1eqd 12602 . . . . . . . . . . 11  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  <" ( M `  U ) ">  =  <" V "> )
239238oveq1d 6285 . . . . . . . . . 10  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( <" ( M `  U ) "> ++  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) )  =  (
<" V "> ++  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) ) )
240217simprd 461 . . . . . . . . . 10  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( <" ( M `  U ) "> ++  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) )  =  ( ( B `  L
) substr  <. Q ,  (
# `  ( B `  L ) ) >.
) )
241239, 240eqtr3d 2497 . . . . . . . . 9  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( <" V "> ++  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) )  =  ( ( B `  L
) substr  <. Q ,  (
# `  ( B `  L ) ) >.
) )
242241oveq2d 6286 . . . . . . . 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 2499 . . . . . . 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 657 . . . . . 6  |-  ( ph  ->  -.  P  =  ( Q  +  1 ) )
245244pm2.21d 106 . . . . 5  |-  ( ph  ->  ( P  =  ( Q  +  1 )  ->  ( A ` 
0 )  =  ( B `  0 ) ) )
246 elfzelz 11691 . . . . . . . . . . . 12  |-  ( Q  e.  ( 0 ... ( # `  ( B `  L )
) )  ->  Q  e.  ZZ )
24774, 246syl 16 . . . . . . . . . . 11  |-  ( ph  ->  Q  e.  ZZ )
248247zcnd 10966 . . . . . . . . . 10  |-  ( ph  ->  Q  e.  CC )
249 1cnd 9601 . . . . . . . . . 10  |-  ( ph  ->  1  e.  CC )
250248, 249, 249addassd 9607 . . . . . . . . 9  |-  ( ph  ->  ( ( Q  + 
1 )  +  1 )  =  ( Q  +  ( 1  +  1 ) ) )
251 df-2 10590 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
252251oveq2i 6281 . . . . . . . . 9  |-  ( Q  +  2 )  =  ( Q  +  ( 1  +  1 ) )
253250, 252syl6eqr 2513 . . . . . . . 8  |-  ( ph  ->  ( ( Q  + 
1 )  +  1 )  =  ( Q  +  2 ) )
254253fveq2d 5852 . . . . . . 7  |-  ( ph  ->  ( ZZ>= `  ( ( Q  +  1 )  +  1 ) )  =  ( ZZ>= `  ( Q  +  2 ) ) )
255254eleq2d 2524 . . . . . 6  |-  ( ph  ->  ( P  e.  (
ZZ>= `  ( ( Q  +  1 )  +  1 ) )  <->  P  e.  ( ZZ>= `  ( Q  +  2 ) ) ) )
2563, 7, 8, 9, 10, 11efgsfo 16956 . . . . . . . . . 10  |-  S : dom  S -onto-> W
257 swrdcl 12635 . . . . . . . . . . . . 13  |-  ( ( A `  K )  e. Word  ( I  X.  2o )  ->  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K
) ) >. )  e. Word  ( I  X.  2o ) )
25837, 257syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. )  e. Word  (
I  X.  2o ) )
259 ccatcl 12582 . . . . . . . . . . . 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 659 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  e. Word 
( I  X.  2o ) )
2613efgrcl 16932 . . . . . . . . . . . . 13  |-  ( ( A `  K )  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
26236, 261syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
263262simprd 461 . . . . . . . . . . 11  |-  ( ph  ->  W  = Word  ( I  X.  2o ) )
264260, 263eleqtrrd 2545 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  e.  W )
265 foelrn 6026 . . . . . . . . . 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 661 . . . . . . . . 9  |-  ( ph  ->  E. c  e.  dom  S ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) ++  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) )
267266adantr 463 . . . . . . . 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 723 . . . . . . . . 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 723 . . . . . . . . 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 723 . . . . . . . . 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 723 . . . . . . . . 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 723 . . . . . . . . 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 723 . . . . . . . . 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 723 . . . . . . . . 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 723 . . . . . . . . 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 723 . . . . . . . . 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 723 . . . . . . . . 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 723 . . . . . . . . 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 723 . . . . . . . . 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 753 . . . . . . . . 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 754 . . . . . . . . 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 755 . . . . . . . . . 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 2462 . . . . . . . . 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 16961 . . . . . . . 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 2950 . . . . . . 7  |-  ( (
ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  ->  ( A `  0 )  =  ( B `  0
) )
286285ex 432 . . . . . 6  |-  ( ph  ->  ( P  e.  (
ZZ>= `  ( Q  + 
2 ) )  -> 
( A `  0
)  =  ( B `
 0 ) ) )
287255, 286sylbid 215 . . . . 5  |-  ( ph  ->  ( P  e.  (
ZZ>= `  ( ( Q  +  1 )  +  1 ) )  -> 
( A `  0
)  =  ( B `
 0 ) ) )
288245, 287jaod 378 . . . 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 378 . 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 184    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   {crab 2808   _Vcvv 3106    \ cdif 3458   (/)c0 3783   {csn 4016   <.cop 4022   <.cotp 4024   U_ciun 4315   class class class wbr 4439    |-> cmpt 4497    _I cid 4779    X. cxp 4986   dom cdm 4988   ran crn 4989   -->wf 5566   -onto->wfo 5568   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   1oc1o 7115   2oc2o 7116   0cc0 9481   1c1 9482    + caddc 9484    < clt 9617    - cmin 9796   NNcn 10531   2c2 10581   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11082   ...cfz 11675  ..^cfzo 11799   #chash 12387  Word cword 12518   ++ cconcat 12520   <"cs1 12521   substr csubstr 12522   splice csplice 12523   <"cs2 12797   ~FG cefg 16923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-ot 4025  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fz 11676  df-fzo 11800  df-hash 12388  df-word 12526  df-concat 12528  df-s1 12529  df-substr 12530  df-splice 12531  df-s2 12804
This theorem is referenced by:  efgredlemb  16963
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