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Theorem efgredlemc 16263
Description: The reduced word that forms the base of the sequence in efgsval 16249 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 10915 . 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 5770 . . . . . . . . . . 11  |-  (  _I 
` Word  ( I  X.  2o ) )  C_ Word  ( I  X.  2o )
53, 4eqsstri 3407 . . . . . . . . . 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 16248 . . . . . . . . . . . . . 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 1003 . . . . . . . . . . . . 13  |-  ( A  e.  dom  S  ->  A  e.  (Word  W  \  { (/) } ) )
146, 13syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  (Word  W  \  { (/) } ) )
15 eldifi 3499 . . . . . . . . . . . 12  |-  ( A  e.  (Word  W  \  { (/) } )  ->  A  e. Word  W )
16 wrdf 12261 . . . . . . . . . . . 12  |-  ( A  e. Word  W  ->  A : ( 0..^ (
# `  A )
) --> W )
1714, 15, 163syl 20 . . . . . . . . . . 11  |-  ( ph  ->  A : ( 0..^ ( # `  A
) ) --> W )
18 fzossfz 11591 . . . . . . . . . . . . 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 16258 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( ( # `  A )  -  1 )  e.  NN  /\  ( ( # `  B
)  -  1 )  e.  NN ) )
2524simpld 459 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( # `  A
)  -  1 )  e.  NN )
26 fzo0end 11640 . . . . . . . . . . . . . . 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 2527 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  ( 0..^ ( ( # `  A
)  -  1 ) ) )
2918, 28sseldi 3375 . . . . . . . . . . . 12  |-  ( ph  ->  K  e.  ( 0 ... ( ( # `  A )  -  1 ) ) )
30 lencl 12270 . . . . . . . . . . . . . . 15  |-  ( A  e. Word  W  ->  ( # `
 A )  e. 
NN0 )
3114, 15, 303syl 20 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( # `  A
)  e.  NN0 )
3231nn0zd 10766 . . . . . . . . . . . . 13  |-  ( ph  ->  ( # `  A
)  e.  ZZ )
33 fzoval 11575 . . . . . . . . . . . . 13  |-  ( (
# `  A )  e.  ZZ  ->  ( 0..^ ( # `  A
) )  =  ( 0 ... ( (
# `  A )  -  1 ) ) )
3432, 33syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( 0..^ ( # `  A ) )  =  ( 0 ... (
( # `  A )  -  1 ) ) )
3529, 34eleqtrrd 2520 . . . . . . . . . . 11  |-  ( ph  ->  K  e.  ( 0..^ ( # `  A
) ) )
3617, 35ffvelrnd 5865 . . . . . . . . . 10  |-  ( ph  ->  ( A `  K
)  e.  W )
375, 36sseldi 3375 . . . . . . . . 9  |-  ( ph  ->  ( A `  K
)  e. Word  ( I  X.  2o ) )
38 efgredlemb.p . . . . . . . . . 10  |-  ( ph  ->  P  e.  ( 0 ... ( # `  ( A `  K )
) ) )
39 elfzuz 11470 . . . . . . . . . 10  |-  ( P  e.  ( 0 ... ( # `  ( A `  K )
) )  ->  P  e.  ( ZZ>= `  0 )
)
40 eluzfz1 11479 . . . . . . . . . 10  |-  ( P  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... P
) )
4138, 39, 403syl 20 . . . . . . . . 9  |-  ( ph  ->  0  e.  ( 0 ... P ) )
42 lencl 12270 . . . . . . . . . . . 12  |-  ( ( A `  K )  e. Word  ( I  X.  2o )  ->  ( # `  ( A `  K
) )  e.  NN0 )
4337, 42syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  ( A `  K )
)  e.  NN0 )
44 nn0uz 10916 . . . . . . . . . . 11  |-  NN0  =  ( ZZ>= `  0 )
4543, 44syl6eleq 2533 . . . . . . . . . 10  |-  ( ph  ->  ( # `  ( A `  K )
)  e.  ( ZZ>= ` 
0 ) )
46 eluzfz2 11480 . . . . . . . . . 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 12371 . . . . . . . . 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 >. ) concat  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K )
) >. ) )  =  ( ( A `  K ) substr  <. 0 ,  ( # `  ( A `  K )
) >. ) )
4937, 41, 38, 47, 48syl13anc 1220 . . . . . . . 8  |-  ( ph  ->  ( ( ( A `
 K ) substr  <. 0 ,  P >. ) concat 
( ( A `  K ) substr  <. P , 
( # `  ( A `
 K ) )
>. ) )  =  ( ( A `  K
) substr  <. 0 ,  (
# `  ( A `  K ) ) >.
) )
50 swrdid 12342 . . . . . . . . 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 2475 . . . . . . 7  |-  ( ph  ->  ( ( ( A `
 K ) substr  <. 0 ,  P >. ) concat 
( ( A `  K ) substr  <. P , 
( # `  ( A `
 K ) )
>. ) )  =  ( A `  K ) )
533, 7, 8, 9, 10, 11efgsdm 16248 . . . . . . . . . . . . . 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 1003 . . . . . . . . . . . . 13  |-  ( B  e.  dom  S  ->  B  e.  (Word  W  \  { (/) } ) )
5521, 54syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  (Word  W  \  { (/) } ) )
56 eldifi 3499 . . . . . . . . . . . 12  |-  ( B  e.  (Word  W  \  { (/) } )  ->  B  e. Word  W )
57 wrdf 12261 . . . . . . . . . . . 12  |-  ( B  e. Word  W  ->  B : ( 0..^ (
# `  B )
) --> W )
5855, 56, 573syl 20 . . . . . . . . . . 11  |-  ( ph  ->  B : ( 0..^ ( # `  B
) ) --> W )
59 fzossfz 11591 . . . . . . . . . . . . 13  |-  ( 0..^ ( ( # `  B
)  -  1 ) )  C_  ( 0 ... ( ( # `  B )  -  1 ) )
60 efgredlemb.l . . . . . . . . . . . . . 14  |-  L  =  ( ( ( # `  B )  -  1 )  -  1 )
6124simprd 463 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( # `  B
)  -  1 )  e.  NN )
62 fzo0end 11640 . . . . . . . . . . . . . . 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 2527 . . . . . . . . . . . . 13  |-  ( ph  ->  L  e.  ( 0..^ ( ( # `  B
)  -  1 ) ) )
6559, 64sseldi 3375 . . . . . . . . . . . 12  |-  ( ph  ->  L  e.  ( 0 ... ( ( # `  B )  -  1 ) ) )
66 lencl 12270 . . . . . . . . . . . . . . 15  |-  ( B  e. Word  W  ->  ( # `
 B )  e. 
NN0 )
6755, 56, 663syl 20 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( # `  B
)  e.  NN0 )
6867nn0zd 10766 . . . . . . . . . . . . 13  |-  ( ph  ->  ( # `  B
)  e.  ZZ )
69 fzoval 11575 . . . . . . . . . . . . 13  |-  ( (
# `  B )  e.  ZZ  ->  ( 0..^ ( # `  B
) )  =  ( 0 ... ( (
# `  B )  -  1 ) ) )
7068, 69syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( 0..^ ( # `  B ) )  =  ( 0 ... (
( # `  B )  -  1 ) ) )
7165, 70eleqtrrd 2520 . . . . . . . . . . 11  |-  ( ph  ->  L  e.  ( 0..^ ( # `  B
) ) )
7258, 71ffvelrnd 5865 . . . . . . . . . 10  |-  ( ph  ->  ( B `  L
)  e.  W )
735, 72sseldi 3375 . . . . . . . . 9  |-  ( ph  ->  ( B `  L
)  e. Word  ( I  X.  2o ) )
74 efgredlemb.q . . . . . . . . . 10  |-  ( ph  ->  Q  e.  ( 0 ... ( # `  ( B `  L )
) ) )
75 elfzuz 11470 . . . . . . . . . 10  |-  ( Q  e.  ( 0 ... ( # `  ( B `  L )
) )  ->  Q  e.  ( ZZ>= `  0 )
)
76 eluzfz1 11479 . . . . . . . . . 10  |-  ( Q  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... Q
) )
7774, 75, 763syl 20 . . . . . . . . 9  |-  ( ph  ->  0  e.  ( 0 ... Q ) )
78 lencl 12270 . . . . . . . . . . . 12  |-  ( ( B `  L )  e. Word  ( I  X.  2o )  ->  ( # `  ( B `  L
) )  e.  NN0 )
7973, 78syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  ( B `  L )
)  e.  NN0 )
8079, 44syl6eleq 2533 . . . . . . . . . 10  |-  ( ph  ->  ( # `  ( B `  L )
)  e.  ( ZZ>= ` 
0 ) )
81 eluzfz2 11480 . . . . . . . . . 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 12371 . . . . . . . . 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 >. ) concat  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L )
) >. ) )  =  ( ( B `  L ) substr  <. 0 ,  ( # `  ( B `  L )
) >. ) )
8473, 77, 74, 82, 83syl13anc 1220 . . . . . . . 8  |-  ( ph  ->  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat 
( ( B `  L ) substr  <. Q , 
( # `  ( B `
 L ) )
>. ) )  =  ( ( B `  L
) substr  <. 0 ,  (
# `  ( B `  L ) ) >.
) )
85 swrdid 12342 . . . . . . . . 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 2475 . . . . . . 7  |-  ( ph  ->  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat 
( ( B `  L ) substr  <. Q , 
( # `  ( B `
 L ) )
>. ) )  =  ( B `  L ) )
8852, 87eqeq12d 2457 . . . . . 6  |-  ( ph  ->  ( ( ( ( A `  K ) substr  <. 0 ,  P >. ) concat 
( ( A `  K ) substr  <. P , 
( # `  ( A `
 K ) )
>. ) )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L )
) >. ) )  <->  ( A `  K )  =  ( B `  L ) ) )
892, 88mtbird 301 . . . . 5  |-  ( ph  ->  -.  ( ( ( A `  K ) substr  <. 0 ,  P >. ) concat 
( ( A `  K ) substr  <. P , 
( # `  ( A `
 K ) )
>. ) )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  ( ( 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 16241 . . . . . . . . . . . . . 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 1218 . . . . . . . . . . . . 13  |-  ( ph  ->  ( P ( T `
 ( A `  K ) ) U )  =  ( ( A `  K ) splice  <. P ,  P ,  <" U ( M `
 U ) "> >. ) )
948efgmf 16231 . . . . . . . . . . . . . . . . 17  |-  M :
( I  X.  2o )
--> ( I  X.  2o )
9594ffvelrni 5863 . . . . . . . . . . . . . . . 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 12517 . . . . . . . . . . . . . 14  |-  ( ph  ->  <" U ( M `  U ) ">  e. Word  (
I  X.  2o ) )
98 splval 12414 . . . . . . . . . . . . . 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 >. ) concat  <" U ( M `  U ) "> ) concat  (
( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) ) )
9936, 38, 38, 97, 98syl13anc 1220 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( A `  K ) splice  <. P ,  P ,  <" U
( M `  U
) "> >. )  =  ( ( ( ( A `  K
) substr  <. 0 ,  P >. ) concat  <" U ( M `  U ) "> ) concat  (
( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) ) )
10090, 93, 993eqtrd 2479 . . . . . . . . . . . 12  |-  ( ph  ->  ( S `  A
)  =  ( ( ( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U
( M `  U
) "> ) concat  ( ( 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 16241 . . . . . . . . . . . . . 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 1218 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Q ( T `
 ( B `  L ) ) V )  =  ( ( B `  L ) splice  <. Q ,  Q ,  <" V ( M `
 V ) "> >. ) )
10594ffvelrni 5863 . . . . . . . . . . . . . . . 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 12517 . . . . . . . . . . . . . 14  |-  ( ph  ->  <" V ( M `  V ) ">  e. Word  (
I  X.  2o ) )
108 splval 12414 . . . . . . . . . . . . . 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 >. ) concat  <" V ( M `  V ) "> ) concat  (
( B `  L
) substr  <. Q ,  (
# `  ( B `  L ) ) >.
) ) )
10972, 74, 74, 107, 108syl13anc 1220 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( B `  L ) splice  <. Q ,  Q ,  <" V
( M `  V
) "> >. )  =  ( ( ( ( B `  L
) substr  <. 0 ,  Q >. ) concat  <" V ( M `  V ) "> ) concat  (
( B `  L
) substr  <. Q ,  (
# `  ( B `  L ) ) >.
) ) )
110101, 104, 1093eqtrd 2479 . . . . . . . . . . . 12  |-  ( ph  ->  ( S `  B
)  =  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V
( M `  V
) "> ) concat  ( ( B `  L
) substr  <. Q ,  (
# `  ( B `  L ) ) >.
) ) )
11122, 100, 1103eqtr3d 2483 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U ( M `
 U ) "> ) concat  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K )
) >. ) )  =  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V ( M `
 V ) "> ) concat  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L )
) >. ) ) )
112111adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  P  =  Q )  ->  (
( ( ( A `
 K ) substr  <. 0 ,  P >. ) concat  <" U ( M `
 U ) "> ) concat  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K )
) >. ) )  =  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V ( M `
 V ) "> ) concat  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L )
) >. ) ) )
113 swrdcl 12336 . . . . . . . . . . . . . 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 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  P  =  Q )  ->  (
( A `  K
) substr  <. 0 ,  P >. )  e. Word  ( I  X.  2o ) )
11697adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  P  =  Q )  ->  <" U
( M `  U
) ">  e. Word  ( I  X.  2o ) )
117 ccatcl 12295 . . . . . . . . . . . 12  |-  ( ( ( ( A `  K ) substr  <. 0 ,  P >. )  e. Word  (
I  X.  2o )  /\  <" U ( M `  U ) ">  e. Word  (
I  X.  2o ) )  ->  ( (
( A `  K
) substr  <. 0 ,  P >. ) concat  <" U ( M `  U ) "> )  e. Word 
( I  X.  2o ) )
118115, 116, 117syl2anc 661 . . . . . . . . . . 11  |-  ( (
ph  /\  P  =  Q )  ->  (
( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U
( M `  U
) "> )  e. Word  ( I  X.  2o ) )
119 swrdcl 12336 . . . . . . . . . . . . 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 465 . . . . . . . . . . 11  |-  ( (
ph  /\  P  =  Q )  ->  (
( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
)  e. Word  ( I  X.  2o ) )
122 swrdcl 12336 . . . . . . . . . . . . . 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 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  P  =  Q )  ->  (
( B `  L
) substr  <. 0 ,  Q >. )  e. Word  ( I  X.  2o ) )
125107adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  P  =  Q )  ->  <" V
( M `  V
) ">  e. Word  ( I  X.  2o ) )
126 ccatcl 12295 . . . . . . . . . . . 12  |-  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. )  e. Word  (
I  X.  2o )  /\  <" V ( M `  V ) ">  e. Word  (
I  X.  2o ) )  ->  ( (
( B `  L
) substr  <. 0 ,  Q >. ) concat  <" V ( M `  V ) "> )  e. Word 
( I  X.  2o ) )
127124, 125, 126syl2anc 661 . . . . . . . . . . 11  |-  ( (
ph  /\  P  =  Q )  ->  (
( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V
( M `  V
) "> )  e. Word  ( I  X.  2o ) )
128 swrdcl 12336 . . . . . . . . . . . . 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 465 . . . . . . . . . . 11  |-  ( (
ph  /\  P  =  Q )  ->  (
( B `  L
) substr  <. Q ,  (
# `  ( B `  L ) ) >.
)  e. Word  ( I  X.  2o ) )
131 swrd0len 12339 . . . . . . . . . . . . . . . 16  |-  ( ( ( A `  K
)  e. Word  ( I  X.  2o )  /\  P  e.  ( 0 ... ( # `
 ( A `  K ) ) ) )  ->  ( # `  (
( A `  K
) substr  <. 0 ,  P >. ) )  =  P )
13237, 38, 131syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( # `  (
( A `  K
) substr  <. 0 ,  P >. ) )  =  P )
133 swrd0len 12339 . . . . . . . . . . . . . . . 16  |-  ( ( ( B `  L
)  e. Word  ( I  X.  2o )  /\  Q  e.  ( 0 ... ( # `
 ( B `  L ) ) ) )  ->  ( # `  (
( B `  L
) substr  <. 0 ,  Q >. ) )  =  Q )
13473, 74, 133syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( # `  (
( B `  L
) substr  <. 0 ,  Q >. ) )  =  Q )
135132, 134eqeq12d 2457 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( # `  (
( A `  K
) substr  <. 0 ,  P >. ) )  =  (
# `  ( ( B `  L ) substr  <.
0 ,  Q >. ) )  <->  P  =  Q
) )
136135biimpar 485 . . . . . . . . . . . . 13  |-  ( (
ph  /\  P  =  Q )  ->  ( # `
 ( ( A `
 K ) substr  <. 0 ,  P >. ) )  =  ( # `  ( ( B `  L ) substr  <. 0 ,  Q >. ) ) )
137 s2len 12535 . . . . . . . . . . . . . . 15  |-  ( # `  <" U ( M `  U ) "> )  =  2
138 s2len 12535 . . . . . . . . . . . . . . 15  |-  ( # `  <" V ( M `  V ) "> )  =  2
139137, 138eqtr4i 2466 . . . . . . . . . . . . . 14  |-  ( # `  <" U ( M `  U ) "> )  =  ( # `  <" V ( M `  V ) "> )
140139a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  P  =  Q )  ->  ( # `
 <" U ( M `  U ) "> )  =  ( # `  <" V ( M `  V ) "> ) )
141136, 140oveq12d 6130 . . . . . . . . . . . 12  |-  ( (
ph  /\  P  =  Q )  ->  (
( # `  ( ( A `  K ) substr  <. 0 ,  P >. ) )  +  ( # `  <" U ( M `  U ) "> ) )  =  ( ( # `  ( ( B `  L ) substr  <. 0 ,  Q >. ) )  +  ( # `  <" V ( M `  V ) "> ) ) )
142 ccatlen 12296 . . . . . . . . . . . . 13  |-  ( ( ( ( A `  K ) substr  <. 0 ,  P >. )  e. Word  (
I  X.  2o )  /\  <" U ( M `  U ) ">  e. Word  (
I  X.  2o ) )  ->  ( # `  (
( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U
( M `  U
) "> )
)  =  ( (
# `  ( ( A `  K ) substr  <.
0 ,  P >. ) )  +  ( # `  <" U ( M `  U ) "> ) ) )
143115, 116, 142syl2anc 661 . . . . . . . . . . . 12  |-  ( (
ph  /\  P  =  Q )  ->  ( # `
 ( ( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U ( M `
 U ) "> ) )  =  ( ( # `  (
( A `  K
) substr  <. 0 ,  P >. ) )  +  (
# `  <" U
( M `  U
) "> )
) )
144 ccatlen 12296 . . . . . . . . . . . . 13  |-  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. )  e. Word  (
I  X.  2o )  /\  <" V ( M `  V ) ">  e. Word  (
I  X.  2o ) )  ->  ( # `  (
( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V
( M `  V
) "> )
)  =  ( (
# `  ( ( B `  L ) substr  <.
0 ,  Q >. ) )  +  ( # `  <" V ( M `  V ) "> ) ) )
145124, 125, 144syl2anc 661 . . . . . . . . . . . 12  |-  ( (
ph  /\  P  =  Q )  ->  ( # `
 ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V ( M `
 V ) "> ) )  =  ( ( # `  (
( B `  L
) substr  <. 0 ,  Q >. ) )  +  (
# `  <" V
( M `  V
) "> )
) )
146141, 143, 1453eqtr4d 2485 . . . . . . . . . . 11  |-  ( (
ph  /\  P  =  Q )  ->  ( # `
 ( ( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U ( M `
 U ) "> ) )  =  ( # `  (
( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V
( M `  V
) "> )
) )
147 ccatopth 12385 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U ( M `
 U ) "> )  e. Word  (
I  X.  2o )  /\  ( ( A `
 K ) substr  <. P ,  ( # `  ( A `  K )
) >. )  e. Word  (
I  X.  2o ) )  /\  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V
( M `  V
) "> )  e. Word  ( I  X.  2o )  /\  ( ( B `
 L ) substr  <. Q ,  ( # `  ( B `  L )
) >. )  e. Word  (
I  X.  2o ) )  /\  ( # `  ( ( ( A `
 K ) substr  <. 0 ,  P >. ) concat  <" U ( M `
 U ) "> ) )  =  ( # `  (
( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V
( M `  V
) "> )
) )  ->  (
( ( ( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U ( M `
 U ) "> ) concat  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K )
) >. ) )  =  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V ( M `
 V ) "> ) concat  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L )
) >. ) )  <->  ( (
( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U
( M `  U
) "> )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V ( M `
 V ) "> )  /\  (
( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
)  =  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L
) ) >. )
) ) )
148118, 121, 127, 130, 146, 147syl221anc 1229 . . . . . . . . . 10  |-  ( (
ph  /\  P  =  Q )  ->  (
( ( ( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U ( M `
 U ) "> ) concat  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K )
) >. ) )  =  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V ( M `
 V ) "> ) concat  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L )
) >. ) )  <->  ( (
( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U
( M `  U
) "> )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" 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 >. ) concat  <" U ( M `
 U ) "> )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V
( M `  V
) "> )  /\  ( ( A `  K ) substr  <. P , 
( # `  ( A `
 K ) )
>. )  =  (
( B `  L
) substr  <. Q ,  (
# `  ( B `  L ) ) >.
) ) )
150149simpld 459 . . . . . . . 8  |-  ( (
ph  /\  P  =  Q )  ->  (
( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U
( M `  U
) "> )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V ( M `
 V ) "> ) )
151 ccatopth 12385 . . . . . . . . 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 >. ) concat  <" U ( M `  U ) "> )  =  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat  <" 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 1229 . . . . . . . 8  |-  ( (
ph  /\  P  =  Q )  ->  (
( ( ( A `
 K ) substr  <. 0 ,  P >. ) concat  <" U ( M `
 U ) "> )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" 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 459 . . . . . 6  |-  ( (
ph  /\  P  =  Q )  ->  (
( A `  K
) substr  <. 0 ,  P >. )  =  ( ( B `  L ) substr  <. 0 ,  Q >. ) )
155149simprd 463 . . . . . 6  |-  ( (
ph  /\  P  =  Q )  ->  (
( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
)  =  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L
) ) >. )
)
156154, 155oveq12d 6130 . . . . 5  |-  ( (
ph  /\  P  =  Q )  ->  (
( ( A `  K ) substr  <. 0 ,  P >. ) concat  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K )
) >. ) )  =  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat 
( ( B `  L ) substr  <. Q , 
( # `  ( B `
 L ) )
>. ) ) )
15789, 156mtand 659 . . . 4  |-  ( ph  ->  -.  P  =  Q )
158157pm2.21d 106 . . 3  |-  ( ph  ->  ( P  =  Q  ->  ( A ` 
0 )  =  ( B `  0 ) ) )
159 uzp1 10915 . . . 4  |-  ( P  e.  ( ZZ>= `  ( Q  +  1 ) )  ->  ( P  =  ( Q  + 
1 )  \/  P  e.  ( ZZ>= `  ( ( Q  +  1 )  +  1 ) ) ) )
16091s1cld 12315 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  <" U ">  e. Word  ( I  X.  2o ) )
161 ccatcl 12295 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A `  K ) substr  <. 0 ,  P >. )  e. Word  (
I  X.  2o )  /\  <" U ">  e. Word  ( I  X.  2o ) )  ->  (
( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U "> )  e. Word  (
I  X.  2o ) )
162114, 160, 161syl2anc 661 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( ( A `
 K ) substr  <. 0 ,  P >. ) concat  <" U "> )  e. Word  ( I  X.  2o ) )
16396s1cld 12315 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  <" ( M `
 U ) ">  e. Word  ( I  X.  2o ) )
164 ccatass 12307 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( A `
 K ) substr  <. 0 ,  P >. ) concat  <" 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 >. ) concat  <" U "> ) concat  <" ( M `
 U ) "> ) concat  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K )
) >. ) )  =  ( ( ( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U "> ) concat  ( <" ( M `  U ) "> concat  ( ( A `  K ) substr  <. P , 
( # `  ( A `
 K ) )
>. ) ) ) )
165162, 163, 120, 164syl3anc 1218 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( ( ( ( A `  K
) substr  <. 0 ,  P >. ) concat  <" U "> ) concat  <" ( M `  U ) "> ) concat  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K )
) >. ) )  =  ( ( ( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U "> ) concat  ( <" ( M `  U ) "> concat  ( ( A `  K ) substr  <. P , 
( # `  ( A `
 K ) )
>. ) ) ) )
166 ccatass 12307 . . . . . . . . . . . . . . . . . . 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 >. ) concat  <" U "> ) concat  <" ( M `
 U ) "> )  =  ( ( ( A `  K ) substr  <. 0 ,  P >. ) concat  ( <" U "> concat  <" ( M `  U ) "> ) ) )
167114, 160, 163, 166syl3anc 1218 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( ( ( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U "> ) concat  <" ( M `
 U ) "> )  =  ( ( ( A `  K ) substr  <. 0 ,  P >. ) concat  ( <" U "> concat  <" ( M `  U ) "> ) ) )
168 df-s2 12496 . . . . . . . . . . . . . . . . . . 19  |-  <" U
( M `  U
) ">  =  ( <" U "> concat 
<" ( M `  U ) "> )
169168oveq2i 6123 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A `  K
) substr  <. 0 ,  P >. ) concat  <" U ( M `  U ) "> )  =  ( ( ( A `
 K ) substr  <. 0 ,  P >. ) concat 
( <" U "> concat 
<" ( M `  U ) "> ) )
170167, 169syl6eqr 2493 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( ( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U "> ) concat  <" ( M `
 U ) "> )  =  ( ( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U
( M `  U
) "> )
)
171170oveq1d 6127 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( ( ( ( A `  K
) substr  <. 0 ,  P >. ) concat  <" U "> ) concat  <" ( M `  U ) "> ) concat  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K )
) >. ) )  =  ( ( ( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U ( M `
 U ) "> ) concat  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K )
) >. ) ) )
172102s1cld 12315 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  <" V ">  e. Word  ( I  X.  2o ) )
173106s1cld 12315 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  <" ( M `
 V ) ">  e. Word  ( I  X.  2o ) )
174 ccatass 12307 . . . . . . . . . . . . . . . . . . 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 >. ) concat  <" V "> ) concat  <" ( M `
 V ) "> )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  ( <" V "> concat  <" ( M `  V ) "> ) ) )
175123, 172, 173, 174syl3anc 1218 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V "> ) concat  <" ( M `
 V ) "> )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  ( <" V "> concat  <" ( M `  V ) "> ) ) )
176 df-s2 12496 . . . . . . . . . . . . . . . . . . 19  |-  <" V
( M `  V
) ">  =  ( <" V "> concat 
<" ( M `  V ) "> )
177176oveq2i 6123 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( B `  L
) substr  <. 0 ,  Q >. ) concat  <" V ( M `  V ) "> )  =  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat 
( <" V "> concat 
<" ( M `  V ) "> ) )
178175, 177syl6eqr 2493 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V "> ) concat  <" ( M `
 V ) "> )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V
( M `  V
) "> )
)
179178oveq1d 6127 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( ( ( ( B `  L
) substr  <. 0 ,  Q >. ) concat  <" V "> ) concat  <" ( M `  V ) "> ) concat  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L )
) >. ) )  =  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V ( M `
 V ) "> ) concat  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L )
) >. ) ) )
180111, 171, 1793eqtr4d 2485 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( ( ( ( A `  K
) substr  <. 0 ,  P >. ) concat  <" U "> ) concat  <" ( M `  U ) "> ) concat  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K )
) >. ) )  =  ( ( ( ( ( B `  L
) substr  <. 0 ,  Q >. ) concat  <" V "> ) concat  <" ( M `  V ) "> ) concat  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L )
) >. ) ) )
181165, 180eqtr3d 2477 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U "> ) concat  ( <" ( M `  U ) "> concat  ( ( A `  K ) substr  <. P , 
( # `  ( A `
 K ) )
>. ) ) )  =  ( ( ( ( ( B `  L
) substr  <. 0 ,  Q >. ) concat  <" V "> ) concat  <" ( M `  V ) "> ) concat  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L )
) >. ) ) )
182181adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( ( A `
 K ) substr  <. 0 ,  P >. ) concat  <" U "> ) concat  ( <" ( M `  U ) "> concat  ( ( A `  K ) substr  <. P , 
( # `  ( A `
 K ) )
>. ) ) )  =  ( ( ( ( ( B `  L
) substr  <. 0 ,  Q >. ) concat  <" V "> ) concat  <" ( M `  V ) "> ) concat  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L )
) >. ) ) )
183162adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U "> )  e. Word  (
I  X.  2o ) )
184163adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  <" ( M `  U ) ">  e. Word  ( I  X.  2o ) )
185120adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
)  e. Word  ( I  X.  2o ) )
186 ccatcl 12295 . . . . . . . . . . . . . . 15  |-  ( (
<" ( M `  U ) ">  e. Word  ( I  X.  2o )  /\  ( ( A `
 K ) substr  <. P ,  ( # `  ( A `  K )
) >. )  e. Word  (
I  X.  2o ) )  ->  ( <" ( M `  U
) "> concat  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K
) ) >. )
)  e. Word  ( I  X.  2o ) )
187184, 185, 186syl2anc 661 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( <" ( M `  U ) "> concat  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) )  e. Word  (
I  X.  2o ) )
188 ccatcl 12295 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. )  e. Word  (
I  X.  2o )  /\  <" V ">  e. Word  ( I  X.  2o ) )  ->  (
( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V "> )  e. Word  (
I  X.  2o ) )
189123, 172, 188syl2anc 661 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat  <" V "> )  e. Word  ( I  X.  2o ) )
190189adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V "> )  e. Word  (
I  X.  2o ) )
191173adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  <" ( M `  V ) ">  e. Word  ( I  X.  2o ) )
192 ccatcl 12295 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat  <" V "> )  e. Word  ( I  X.  2o )  /\  <" ( M `  V
) ">  e. Word  ( I  X.  2o ) )  ->  ( (
( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V "> ) concat  <" ( M `  V ) "> )  e. Word  (
I  X.  2o ) )
193190, 191, 192syl2anc 661 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat  <" V "> ) concat  <" ( M `
 V ) "> )  e. Word  (
I  X.  2o ) )
194129adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( B `  L
) substr  <. Q ,  (
# `  ( B `  L ) ) >.
)  e. Word  ( I  X.  2o ) )
195 ccatlen 12296 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. )  e. Word  (
I  X.  2o )  /\  <" V ">  e. Word  ( I  X.  2o ) )  ->  ( # `
 ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V "> ) )  =  ( ( # `  (
( B `  L
) substr  <. 0 ,  Q >. ) )  +  (
# `  <" V "> ) ) )
196123, 172, 195syl2anc 661 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( # `  (
( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V "> ) )  =  ( ( # `  (
( B `  L
) substr  <. 0 ,  Q >. ) )  +  (
# `  <" V "> ) ) )
197 s1len 12317 . . . . . . . . . . . . . . . . . . . . 21  |-  ( # `  <" V "> )  =  1
198197a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( # `  <" V "> )  =  1 )
199134, 198oveq12d 6130 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( ( # `  (
( B `  L
) substr  <. 0 ,  Q >. ) )  +  (
# `  <" V "> ) )  =  ( Q  +  1 ) )
200196, 199eqtrd 2475 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( # `  (
( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V "> ) )  =  ( Q  +  1 ) )
201132, 200eqeq12d 2457 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( # `  (
( A `  K
) substr  <. 0 ,  P >. ) )  =  (
# `  ( (
( B `  L
) substr  <. 0 ,  Q >. ) concat  <" V "> ) )  <->  P  =  ( Q  +  1
) ) )
202201biimpar 485 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( # `
 ( ( A `
 K ) substr  <. 0 ,  P >. ) )  =  ( # `  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat  <" V "> ) ) )
203 s1len 12317 . . . . . . . . . . . . . . . . . 18  |-  ( # `  <" U "> )  =  1
204 s1len 12317 . . . . . . . . . . . . . . . . . 18  |-  ( # `  <" ( M `
 V ) "> )  =  1
205203, 204eqtr4i 2466 . . . . . . . . . . . . . . . . 17  |-  ( # `  <" U "> )  =  ( # `
 <" ( M `
 V ) "> )
206205a1i 11 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( # `
 <" U "> )  =  ( # `
 <" ( M `
 V ) "> ) )
207202, 206oveq12d 6130 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( # `  ( ( A `  K ) substr  <. 0 ,  P >. ) )  +  ( # `  <" U "> ) )  =  ( ( # `  (
( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V "> ) )  +  ( # `  <" ( M `  V
) "> )
) )
208114adantr 465 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( A `  K
) substr  <. 0 ,  P >. )  e. Word  ( I  X.  2o ) )
209160adantr 465 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  <" U ">  e. Word  ( I  X.  2o ) )
210 ccatlen 12296 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A `  K ) substr  <. 0 ,  P >. )  e. Word  (
I  X.  2o )  /\  <" U ">  e. Word  ( I  X.  2o ) )  ->  ( # `
 ( ( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U "> ) )  =  ( ( # `  (
( A `  K
) substr  <. 0 ,  P >. ) )  +  (
# `  <" U "> ) ) )
211208, 209, 210syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( # `
 ( ( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U "> ) )  =  ( ( # `  (
( A `  K
) substr  <. 0 ,  P >. ) )  +  (
# `  <" U "> ) ) )
212 ccatlen 12296 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat  <" V "> )  e. Word  ( I  X.  2o )  /\  <" ( M `  V
) ">  e. Word  ( I  X.  2o ) )  ->  ( # `  (
( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat  <" V "> ) concat  <" ( M `
 V ) "> ) )  =  ( ( # `  (
( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V "> ) )  +  ( # `  <" ( M `  V
) "> )
) )
213190, 191, 212syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( # `
 ( ( ( ( B `  L
) substr  <. 0 ,  Q >. ) concat  <" V "> ) concat  <" ( M `  V ) "> ) )  =  ( ( # `  (
( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V "> ) )  +  ( # `  <" ( M `  V
) "> )
) )
214207, 211, 2133eqtr4d 2485 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( # `
 ( ( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U "> ) )  =  (
# `  ( (
( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V "> ) concat  <" ( M `  V ) "> ) ) )
215 ccatopth 12385 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U "> )  e. Word  ( I  X.  2o )  /\  ( <" ( M `  U ) "> concat  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) )  e. Word  (
I  X.  2o ) )  /\  ( ( ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat  <" V "> ) concat  <" ( M `
 V ) "> )  e. Word  (
I  X.  2o )  /\  ( ( B `
 L ) substr  <. Q ,  ( # `  ( B `  L )
) >. )  e. Word  (
I  X.  2o ) )  /\  ( # `  ( ( ( A `
 K ) substr  <. 0 ,  P >. ) concat  <" U "> ) )  =  (
# `  ( (
( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V "> ) concat  <" ( M `  V ) "> ) ) )  ->  ( ( ( ( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U "> ) concat  ( <" ( M `  U
) "> concat  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K
) ) >. )
) )  =  ( ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V "> ) concat  <" ( M `
 V ) "> ) concat  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L )
) >. ) )  <->  ( (
( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U "> )  =  ( ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat  <" V "> ) concat  <" ( M `
 V ) "> )  /\  ( <" ( M `  U ) "> concat  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) )  =  ( ( B `  L
) substr  <. Q ,  (
# `  ( B `  L ) ) >.
) ) ) )
216183, 187, 193, 194, 214, 215syl221anc 1229 . . . . . . . . . . . . 13  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( ( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U "> ) concat  ( <" ( M `  U ) "> concat  ( ( A `  K ) substr  <. P , 
( # `  ( A `
 K ) )
>. ) ) )  =  ( ( ( ( ( B `  L
) substr  <. 0 ,  Q >. ) concat  <" V "> ) concat  <" ( M `  V ) "> ) concat  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L )
) >. ) )  <->  ( (
( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U "> )  =  ( ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat  <" V "> ) concat  <" ( M `
 V ) "> )  /\  ( <" ( M `  U ) "> concat  ( ( 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 >. ) concat  <" U "> )  =  ( (
( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V "> ) concat  <" ( M `  V ) "> )  /\  ( <" ( M `  U ) "> concat  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) )  =  ( ( B `  L
) substr  <. Q ,  (
# `  ( B `  L ) ) >.
) ) )
218217simpld 459 . . . . . . . . . . 11  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( A `  K ) substr  <. 0 ,  P >. ) concat  <" U "> )  =  ( ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat  <" V "> ) concat  <" ( M `
 V ) "> ) )
219 ccatopth 12385 . . . . . . . . . . . 12  |-  ( ( ( ( ( A `
 K ) substr  <. 0 ,  P >. )  e. Word  ( I  X.  2o )  /\  <" U ">  e. Word  ( I  X.  2o ) )  /\  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V "> )  e. Word  ( I  X.  2o )  /\  <" ( M `  V
) ">  e. Word  ( I  X.  2o ) )  /\  ( # `  ( ( A `  K ) substr  <. 0 ,  P >. ) )  =  ( # `  (
( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V "> ) ) )  ->  ( ( ( ( A `  K
) substr  <. 0 ,  P >. ) concat  <" U "> )  =  (
( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat  <" V "> ) concat  <" ( M `
 V ) "> )  <->  ( (
( A `  K
) substr  <. 0 ,  P >. )  =  ( ( ( B `  L
) substr  <. 0 ,  Q >. ) concat  <" V "> )  /\  <" U ">  =  <" ( M `  V ) "> ) ) )
220208, 209, 190, 191, 202, 219syl221anc 1229 . . . . . . . . . . 11  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( ( A `
 K ) substr  <. 0 ,  P >. ) concat  <" U "> )  =  ( (
( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V "> ) concat  <" ( M `  V ) "> )  <->  ( (
( A `  K
) substr  <. 0 ,  P >. )  =  ( ( ( B `  L
) substr  <. 0 ,  Q >. ) concat  <" V "> )  /\  <" U ">  =  <" ( M `  V ) "> ) ) )
221218, 220mpbid 210 . . . . . . . . . 10  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( A `  K ) substr  <. 0 ,  P >. )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V "> )  /\  <" U ">  =  <" ( M `  V ) "> ) )
222221simpld 459 . . . . . . . . 9  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( A `  K
) substr  <. 0 ,  P >. )  =  ( ( ( B `  L
) substr  <. 0 ,  Q >. ) concat  <" V "> ) )
223222oveq1d 6127 . . . . . . . 8  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( A `  K ) substr  <. 0 ,  P >. ) concat  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K )
) >. ) )  =  ( ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  <" V "> ) concat  ( ( A `  K ) substr  <. P , 
( # `  ( A `
 K ) )
>. ) ) )
224123adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( B `  L
) substr  <. 0 ,  Q >. )  e. Word  ( I  X.  2o ) )
225172adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  <" V ">  e. Word  ( I  X.  2o ) )
226 ccatass 12307 . . . . . . . . 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 >. ) concat  <" V "> ) concat  ( ( A `  K ) substr  <. P , 
( # `  ( A `
 K ) )
>. ) )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  ( <" V "> concat  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K
) ) >. )
) ) )
227224, 225, 185, 226syl3anc 1218 . . . . . . . 8  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat  <" V "> ) concat  ( ( A `  K ) substr  <. P , 
( # `  ( A `
 K ) )
>. ) )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  ( <" V "> concat  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K
) ) >. )
) ) )
228221simprd 463 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  <" U ">  =  <" ( M `  V ) "> )
229 s111 12323 . . . . . . . . . . . . . . . . 17  |-  ( ( U  e.  ( I  X.  2o )  /\  ( M `  V )  e.  ( I  X.  2o ) )  ->  ( <" U ">  =  <" ( M `
 V ) ">  <->  U  =  ( M `  V )
) )
23091, 106, 229syl2anc 661 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( <" U ">  =  <" ( M `  V ) ">  <->  U  =  ( M `  V )
) )
231230adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( <" U ">  =  <" ( M `
 V ) ">  <->  U  =  ( M `  V )
) )
232228, 231mpbid 210 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  U  =  ( M `  V ) )
233232fveq2d 5716 . . . . . . . . . . . . 13  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( M `  U )  =  ( M `  ( M `  V ) ) )
2348efgmnvl 16232 . . . . . . . . . . . . . . 15  |-  ( V  e.  ( I  X.  2o )  ->  ( M `
 ( M `  V ) )  =  V )
235102, 234syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M `  ( M `  V )
)  =  V )
236235adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( M `  ( M `  V ) )  =  V )
237233, 236eqtrd 2475 . . . . . . . . . . . 12  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( M `  U )  =  V )
238237s1eqd 12313 . . . . . . . . . . 11  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  <" ( M `  U ) ">  =  <" V "> )
239238oveq1d 6127 . . . . . . . . . 10  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( <" ( M `  U ) "> concat  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) )  =  (
<" V "> concat  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) ) )
240217simprd 463 . . . . . . . . . 10  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( <" ( M `  U ) "> concat  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) )  =  ( ( B `  L
) substr  <. Q ,  (
# `  ( B `  L ) ) >.
) )
241239, 240eqtr3d 2477 . . . . . . . . 9  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  ( <" V "> concat  ( ( A `  K
) substr  <. P ,  (
# `  ( A `  K ) ) >.
) )  =  ( ( B `  L
) substr  <. Q ,  (
# `  ( B `  L ) ) >.
) )
242241oveq2d 6128 . . . . . . . 8  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( B `  L ) substr  <. 0 ,  Q >. ) concat  ( <" V "> concat  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K
) ) >. )
) )  =  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  ( ( B `  L ) substr  <. Q ,  ( # `  ( B `  L )
) >. ) ) )
243223, 227, 2423eqtrd 2479 . . . . . . 7  |-  ( (
ph  /\  P  =  ( Q  +  1
) )  ->  (
( ( A `  K ) substr  <. 0 ,  P >. ) concat  ( ( A `  K ) substr  <. P ,  ( # `  ( A `  K )
) >. ) )  =  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat 
( ( B `  L ) substr  <. Q , 
( # `  ( B `
 L ) )
>. ) ) )
24489, 243mtand 659 . . . . . 6  |-  ( ph  ->  -.  P  =  ( Q  +  1 ) )
245244pm2.21d 106 . . . . 5  |-  ( ph  ->  ( P  =  ( Q  +  1 )  ->  ( A ` 
0 )  =  ( B `  0 ) ) )
246 elfzelz 11474 . . . . . . . . . . . 12  |-  ( Q  e.  ( 0 ... ( # `  ( B `  L )
) )  ->  Q  e.  ZZ )
24774, 246syl 16 . . . . . . . . . . 11  |-  ( ph  ->  Q  e.  ZZ )
248247zcnd 10769 . . . . . . . . . 10  |-  ( ph  ->  Q  e.  CC )
249 ax-1cn 9361 . . . . . . . . . . 11  |-  1  e.  CC
250249a1i 11 . . . . . . . . . 10  |-  ( ph  ->  1  e.  CC )
251248, 250, 250addassd 9429 . . . . . . . . 9  |-  ( ph  ->  ( ( Q  + 
1 )  +  1 )  =  ( Q  +  ( 1  +  1 ) ) )
252 df-2 10401 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
253252oveq2i 6123 . . . . . . . . 9  |-  ( Q  +  2 )  =  ( Q  +  ( 1  +  1 ) )
254251, 253syl6eqr 2493 . . . . . . . 8  |-  ( ph  ->  ( ( Q  + 
1 )  +  1 )  =  ( Q  +  2 ) )
255254fveq2d 5716 . . . . . . 7  |-  ( ph  ->  ( ZZ>= `  ( ( Q  +  1 )  +  1 ) )  =  ( ZZ>= `  ( Q  +  2 ) ) )
256255eleq2d 2510 . . . . . 6  |-  ( ph  ->  ( P  e.  (
ZZ>= `  ( ( Q  +  1 )  +  1 ) )  <->  P  e.  ( ZZ>= `  ( Q  +  2 ) ) ) )
2573, 7, 8, 9, 10, 11efgsfo 16257 . . . . . . . . . 10  |-  S : dom  S -onto-> W
258 swrdcl 12336 . . . . . . . . . . . . 13  |-  ( ( A `  K )  e. Word  ( I  X.  2o )  ->  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K
) ) >. )  e. Word  ( I  X.  2o ) )
25937, 258syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. )  e. Word  (
I  X.  2o ) )
260 ccatcl 12295 . . . . . . . . . . . 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 >. ) concat  ( ( A `
 K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  e. Word 
( I  X.  2o ) )
261123, 259, 260syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat 
( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  e. Word 
( I  X.  2o ) )
2623efgrcl 16233 . . . . . . . . . . . . 13  |-  ( ( A `  K )  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
26336, 262syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
264263simprd 463 . . . . . . . . . . 11  |-  ( ph  ->  W  = Word  ( I  X.  2o ) )
265261, 264eleqtrrd 2520 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat 
( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  e.  W )
266 foelrn 5883 . . . . . . . . . 10  |-  ( ( S : dom  S -onto-> W  /\  ( ( ( B `  L ) substr  <. 0 ,  Q >. ) concat 
( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  e.  W )  ->  E. c  e.  dom  S ( ( ( B `  L
) substr  <. 0 ,  Q >. ) concat  ( ( A `
 K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) )
267257, 265, 266sylancr 663 . . . . . . . . 9  |-  ( ph  ->  E. c  e.  dom  S ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat 
( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) )
268267adantr 465 . . . . . . . 8  |-  ( (
ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  ->  E. c  e.  dom  S ( ( ( B `  L
) substr  <. 0 ,  Q >. ) concat  ( ( A `
 K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) )
26920ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat 
( ( 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 ) ) ) )
2706ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat 
( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  ->  A  e.  dom  S )
27121ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat 
( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  ->  B  e.  dom  S )
27222ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat 
( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  -> 
( S `  A
)  =  ( S `
 B ) )
27323ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat 
( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  ->  -.  ( A `  0
)  =  ( B `
 0 ) )
27438ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat 
( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  ->  P  e.  ( 0 ... ( # `  ( A `  K )
) ) )
27574ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat 
( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  ->  Q  e.  ( 0 ... ( # `  ( B `  L )
) ) )
27691ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat 
( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  ->  U  e.  ( I  X.  2o ) )
277102ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat 
( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  ->  V  e.  ( I  X.  2o ) )
27890ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat 
( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  -> 
( S `  A
)  =  ( P ( T `  ( A `  K )
) U ) )
279101ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat 
( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  -> 
( S `  B
)  =  ( Q ( T `  ( B `  L )
) V ) )
2802ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat 
( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  ->  -.  ( A `  K
)  =  ( B `
 L ) )
281 simplr 754 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat 
( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  ->  P  e.  ( ZZ>= `  ( Q  +  2
) ) )
282 simprl 755 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat 
( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  -> 
c  e.  dom  S
)
283 simprr 756 . . . . . . . . . 10  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat 
( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  -> 
( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat 
( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) )
284283eqcomd 2448 . . . . . . . . 9  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat 
( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  -> 
( S `  c
)  =  ( ( ( B `  L
) substr  <. 0 ,  Q >. ) concat  ( ( A `
 K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) ) )
2853, 7, 8, 9, 10, 11, 269, 270, 271, 272, 273, 19, 60, 274, 275, 276, 277, 278, 279, 280, 281, 282, 284efgredlemd 16262 . . . . . . . 8  |-  ( ( ( ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  /\  ( c  e.  dom  S  /\  ( ( ( B `
 L ) substr  <. 0 ,  Q >. ) concat 
( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K )
) >. ) )  =  ( S `  c
) ) )  -> 
( A `  0
)  =  ( B `
 0 ) )
286268, 285rexlimddv 2866 . . . . . . 7  |-  ( (
ph  /\  P  e.  ( ZZ>= `  ( Q  +  2 ) ) )  ->  ( A `  0 )  =  ( B `  0
) )
287286ex 434 . . . . . 6  |-  ( ph  ->  ( P  e.  (
ZZ>= `  ( Q  + 
2 ) )  -> 
( A `  0
)  =  ( B `
 0 ) ) )
288256, 287sylbid 215 . . . . 5  |-  ( ph  ->  ( P  e.  (
ZZ>= `  ( ( Q  +  1 )  +  1 ) )  -> 
( A `  0
)  =  ( B `
 0 ) ) )
289245, 288jaod 380 . . . 4  |-  ( ph  ->  ( ( P  =  ( Q  +  1 )  \/  P  e.  ( ZZ>= `  ( ( Q  +  1 )  +  1 ) ) )  ->  ( A `  0 )  =  ( B `  0
) ) )
290159, 289syl5 32 . . 3  |-  ( ph  ->  ( P  e.  (
ZZ>= `  ( Q  + 
1 ) )  -> 
( A `  0
)  =  ( B `
 0 ) ) )
291158, 290jaod 380 . 2  |-  ( ph  ->  ( ( P  =  Q  \/  P  e.  ( ZZ>= `  ( Q  +  1 ) ) )  ->  ( A `  0 )  =  ( B `  0
) ) )
2921, 291syl5 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 368    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2736   E.wrex 2737   {crab 2740   _Vcvv 2993    \ cdif 3346   (/)c0 3658   {csn 3898   <.cop 3904   <.cotp 3906   U_ciun 4192   class class class wbr 4313    e. cmpt 4371    _I cid 4652    X. cxp 4859   dom cdm 4861   ran crn 4862   -->wf 5435   -onto->wfo 5437   ` cfv 5439  (class class class)co 6112    e. cmpt2 6114   1oc1o 6934   2oc2o 6935   CCcc 9301   0cc0 9303   1c1 9304    + caddc 9306    < clt 9439    - cmin 9616   NNcn 10343   2c2 10392   NN0cn0 10600   ZZcz 10667   ZZ>=cuz 10882   ...cfz 11458  ..^cfzo 11569   #chash 12124  Word cword 12242   concat cconcat 12244   <"cs1 12245   substr csubstr 12246   splice csplice 12247   <"cs2 12489   ~FG cefg 16224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-ot 3907  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-2o 6942  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-card 8130  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-n0 10601  df-z 10668  df-uz 10883  df-rp 11013  df-fz 11459  df-fzo 11570  df-hash 12125  df-word 12250  df-concat 12252  df-s1 12253  df-substr 12254  df-splice 12255  df-s2 12496
This theorem is referenced by:  efgredlemb  16264
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