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Theorem ccatswrd 11475
Description: Joining two adjacent subwords makes a longer subword. (Contributed by Stefan O'Rear, 20-Aug-2015.)
Assertion
Ref Expression
ccatswrd  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( S substr  <. X ,  Y >. ) concat 
( S substr  <. Y ,  Z >. ) )  =  ( S substr  <. X ,  Z >. ) )

Proof of Theorem ccatswrd
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 swrdcl 11468 . . . . . 6  |-  ( S  e. Word  A  ->  ( S substr  <. X ,  Y >. )  e. Word  A )
21adantr 451 . . . . 5  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( S substr  <. X ,  Y >. )  e. Word  A
)
3 swrdcl 11468 . . . . . 6  |-  ( S  e. Word  A  ->  ( S substr  <. Y ,  Z >. )  e. Word  A )
43adantr 451 . . . . 5  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( S substr  <. Y ,  Z >. )  e. Word  A
)
5 ccatcl 11445 . . . . 5  |-  ( ( ( S substr  <. X ,  Y >. )  e. Word  A  /\  ( S substr  <. Y ,  Z >. )  e. Word  A
)  ->  ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) )  e. Word  A )
62, 4, 5syl2anc 642 . . . 4  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( S substr  <. X ,  Y >. ) concat 
( S substr  <. Y ,  Z >. ) )  e. Word  A )
7 wrdf 11435 . . . 4  |-  ( ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) )  e. Word  A  -> 
( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) : ( 0..^ ( # `  (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) ) ) --> A )
8 ffn 5405 . . . 4  |-  ( ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) : ( 0..^ ( # `  (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) ) ) --> A  ->  ( ( S substr  <. X ,  Y >. ) concat 
( S substr  <. Y ,  Z >. ) )  Fn  ( 0..^ ( # `  ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) ) ) )
96, 7, 83syl 18 . . 3  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( S substr  <. X ,  Y >. ) concat 
( S substr  <. Y ,  Z >. ) )  Fn  ( 0..^ ( # `  ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) ) ) )
10 ccatlen 11446 . . . . . . 7  |-  ( ( ( S substr  <. X ,  Y >. )  e. Word  A  /\  ( S substr  <. Y ,  Z >. )  e. Word  A
)  ->  ( # `  (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) )  =  ( ( # `  ( S substr  <. X ,  Y >. ) )  +  (
# `  ( S substr  <. Y ,  Z >. ) ) ) )
112, 4, 10syl2anc 642 . . . . . 6  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( # `  (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) )  =  ( ( # `  ( S substr  <. X ,  Y >. ) )  +  (
# `  ( S substr  <. Y ,  Z >. ) ) ) )
12 simpl 443 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  S  e. Word  A
)
13 simpr1 961 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  X  e.  ( 0 ... Y ) )
14 simpr2 962 . . . . . . . . . 10  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  Y  e.  ( 0 ... Z ) )
15 simpr3 963 . . . . . . . . . 10  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  Z  e.  ( 0 ... ( # `  S ) ) )
16 fzass4 10845 . . . . . . . . . . . 12  |-  ( ( Y  e.  ( 0 ... ( # `  S
) )  /\  Z  e.  ( Y ... ( # `
 S ) ) )  <->  ( Y  e.  ( 0 ... Z
)  /\  Z  e.  ( 0 ... ( # `
 S ) ) ) )
1716biimpri 197 . . . . . . . . . . 11  |-  ( ( Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) )  -> 
( Y  e.  ( 0 ... ( # `  S ) )  /\  Z  e.  ( Y ... ( # `  S
) ) ) )
1817simpld 445 . . . . . . . . . 10  |-  ( ( Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) )  ->  Y  e.  ( 0 ... ( # `  S
) ) )
1914, 15, 18syl2anc 642 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  Y  e.  ( 0 ... ( # `  S ) ) )
20 swrdlen 11472 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) )  -> 
( # `  ( S substr  <. X ,  Y >. ) )  =  ( Y  -  X ) )
2112, 13, 19, 20syl3anc 1182 . . . . . . . 8  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( # `  ( S substr  <. X ,  Y >. ) )  =  ( Y  -  X ) )
22 swrdlen 11472 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) )  -> 
( # `  ( S substr  <. Y ,  Z >. ) )  =  ( Z  -  Y ) )
2312, 14, 15, 22syl3anc 1182 . . . . . . . 8  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( # `  ( S substr  <. Y ,  Z >. ) )  =  ( Z  -  Y ) )
2421, 23oveq12d 5892 . . . . . . 7  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( # `  ( S substr  <. X ,  Y >. ) )  +  ( # `  ( S substr  <. Y ,  Z >. ) ) )  =  ( ( Y  -  X )  +  ( Z  -  Y ) ) )
25 elfzelz 10814 . . . . . . . . . 10  |-  ( Y  e.  ( 0 ... Z )  ->  Y  e.  ZZ )
2614, 25syl 15 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  Y  e.  ZZ )
2726zcnd 10134 . . . . . . . 8  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  Y  e.  CC )
28 elfzelz 10814 . . . . . . . . . 10  |-  ( X  e.  ( 0 ... Y )  ->  X  e.  ZZ )
2913, 28syl 15 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  X  e.  ZZ )
3029zcnd 10134 . . . . . . . 8  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  X  e.  CC )
31 elfzelz 10814 . . . . . . . . . 10  |-  ( Z  e.  ( 0 ... ( # `  S
) )  ->  Z  e.  ZZ )
3215, 31syl 15 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  Z  e.  ZZ )
3332zcnd 10134 . . . . . . . 8  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  Z  e.  CC )
3427, 30, 33npncan3d 9209 . . . . . . 7  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( Y  -  X )  +  ( Z  -  Y
) )  =  ( Z  -  X ) )
3524, 34eqtrd 2328 . . . . . 6  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( # `  ( S substr  <. X ,  Y >. ) )  +  ( # `  ( S substr  <. Y ,  Z >. ) ) )  =  ( Z  -  X
) )
3611, 35eqtrd 2328 . . . . 5  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( # `  (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) )  =  ( Z  -  X ) )
3736oveq2d 5890 . . . 4  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( 0..^ (
# `  ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) ) )  =  ( 0..^ ( Z  -  X
) ) )
3837fneq2d 5352 . . 3  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) )  Fn  ( 0..^ ( # `  ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) ) )  <->  ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) )  Fn  ( 0..^ ( Z  -  X ) ) ) )
399, 38mpbid 201 . 2  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( S substr  <. X ,  Y >. ) concat 
( S substr  <. Y ,  Z >. ) )  Fn  ( 0..^ ( Z  -  X ) ) )
40 swrdcl 11468 . . . . 5  |-  ( S  e. Word  A  ->  ( S substr  <. X ,  Z >. )  e. Word  A )
4140adantr 451 . . . 4  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( S substr  <. X ,  Z >. )  e. Word  A
)
42 wrdf 11435 . . . 4  |-  ( ( S substr  <. X ,  Z >. )  e. Word  A  -> 
( S substr  <. X ,  Z >. ) : ( 0..^ ( # `  ( S substr  <. X ,  Z >. ) ) ) --> A )
43 ffn 5405 . . . 4  |-  ( ( S substr  <. X ,  Z >. ) : ( 0..^ ( # `  ( S substr  <. X ,  Z >. ) ) ) --> A  ->  ( S substr  <. X ,  Z >. )  Fn  (
0..^ ( # `  ( S substr  <. X ,  Z >. ) ) ) )
4441, 42, 433syl 18 . . 3  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( S substr  <. X ,  Z >. )  Fn  (
0..^ ( # `  ( S substr  <. X ,  Z >. ) ) ) )
45 fzass4 10845 . . . . . . . . 9  |-  ( ( X  e.  ( 0 ... Z )  /\  Y  e.  ( X ... Z ) )  <->  ( X  e.  ( 0 ... Y
)  /\  Y  e.  ( 0 ... Z
) ) )
4645biimpri 197 . . . . . . . 8  |-  ( ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z ) )  ->  ( X  e.  ( 0 ... Z
)  /\  Y  e.  ( X ... Z ) ) )
4746simpld 445 . . . . . . 7  |-  ( ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z ) )  ->  X  e.  ( 0 ... Z ) )
4813, 14, 47syl2anc 642 . . . . . 6  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  X  e.  ( 0 ... Z ) )
49 swrdlen 11472 . . . . . 6  |-  ( ( S  e. Word  A  /\  X  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) )  -> 
( # `  ( S substr  <. X ,  Z >. ) )  =  ( Z  -  X ) )
5012, 48, 15, 49syl3anc 1182 . . . . 5  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( # `  ( S substr  <. X ,  Z >. ) )  =  ( Z  -  X ) )
5150oveq2d 5890 . . . 4  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( 0..^ (
# `  ( S substr  <. X ,  Z >. ) ) )  =  ( 0..^ ( Z  -  X ) ) )
5251fneq2d 5352 . . 3  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( S substr  <. X ,  Z >. )  Fn  ( 0..^ (
# `  ( S substr  <. X ,  Z >. ) ) )  <->  ( S substr  <. X ,  Z >. )  Fn  ( 0..^ ( Z  -  X ) ) ) )
5344, 52mpbid 201 . 2  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( S substr  <. X ,  Z >. )  Fn  (
0..^ ( Z  -  X ) ) )
54 simpr 447 . . . . 5  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Z  -  X ) ) )  ->  x  e.  ( 0..^ ( Z  -  X ) ) )
5526, 29zsubcld 10138 . . . . . 6  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( Y  -  X )  e.  ZZ )
5655adantr 451 . . . . 5  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Z  -  X ) ) )  ->  ( Y  -  X )  e.  ZZ )
57 fzospliti 10914 . . . . 5  |-  ( ( x  e.  ( 0..^ ( Z  -  X
) )  /\  ( Y  -  X )  e.  ZZ )  ->  (
x  e.  ( 0..^ ( Y  -  X
) )  \/  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) ) )
5854, 56, 57syl2anc 642 . . . 4  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Z  -  X ) ) )  ->  ( x  e.  ( 0..^ ( Y  -  X ) )  \/  x  e.  ( ( Y  -  X
)..^ ( Z  -  X ) ) ) )
592adantr 451 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  ( S substr  <. X ,  Y >. )  e. Word  A )
604adantr 451 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  ( S substr  <. Y ,  Z >. )  e. Word  A )
6121oveq2d 5890 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( 0..^ (
# `  ( S substr  <. X ,  Y >. ) ) )  =  ( 0..^ ( Y  -  X ) ) )
6261eleq2d 2363 . . . . . . . 8  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( x  e.  ( 0..^ ( # `  ( S substr  <. X ,  Y >. ) ) )  <-> 
x  e.  ( 0..^ ( Y  -  X
) ) ) )
6362biimpar 471 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  x  e.  ( 0..^ ( # `  ( S substr  <. X ,  Y >. ) ) ) )
64 ccatval1 11447 . . . . . . 7  |-  ( ( ( S substr  <. X ,  Y >. )  e. Word  A  /\  ( S substr  <. Y ,  Z >. )  e. Word  A  /\  x  e.  (
0..^ ( # `  ( S substr  <. X ,  Y >. ) ) ) )  ->  ( ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) `  x )  =  ( ( S substr  <. X ,  Y >. ) `  x
) )
6559, 60, 63, 64syl3anc 1182 . . . . . 6  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  ( (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) `  x )  =  ( ( S substr  <. X ,  Y >. ) `
 x ) )
66 simpll 730 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  S  e. Word  A )
67 simplr1 997 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  X  e.  ( 0 ... Y
) )
6819adantr 451 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  Y  e.  ( 0 ... ( # `
 S ) ) )
69 simpr 447 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  x  e.  ( 0..^ ( Y  -  X ) ) )
70 swrdfv 11473 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) )  /\  x  e.  ( 0..^ ( Y  -  X
) ) )  -> 
( ( S substr  <. X ,  Y >. ) `  x
)  =  ( S `
 ( x  +  X ) ) )
7166, 67, 68, 69, 70syl31anc 1185 . . . . . 6  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  ( ( S substr  <. X ,  Y >. ) `  x )  =  ( S `  ( x  +  X
) ) )
7265, 71eqtrd 2328 . . . . 5  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Y  -  X ) ) )  ->  ( (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) `  x )  =  ( S `  ( x  +  X
) ) )
732adantr 451 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( S substr  <. X ,  Y >. )  e. Word  A
)
744adantr 451 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( S substr  <. Y ,  Z >. )  e. Word  A
)
7521, 35oveq12d 5892 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( # `  ( S substr  <. X ,  Y >. ) )..^ ( ( # `  ( S substr  <. X ,  Y >. ) )  +  (
# `  ( S substr  <. Y ,  Z >. ) ) ) )  =  ( ( Y  -  X )..^ ( Z  -  X ) ) )
7675eleq2d 2363 . . . . . . . 8  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( x  e.  ( ( # `  ( S substr  <. X ,  Y >. ) )..^ ( (
# `  ( S substr  <. X ,  Y >. ) )  +  ( # `  ( S substr  <. Y ,  Z >. ) ) ) )  <->  x  e.  (
( Y  -  X
)..^ ( Z  -  X ) ) ) )
7776biimpar 471 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  x  e.  ( ( # `  ( S substr  <. X ,  Y >. ) )..^ ( (
# `  ( S substr  <. X ,  Y >. ) )  +  ( # `  ( S substr  <. Y ,  Z >. ) ) ) ) )
78 ccatval2 11448 . . . . . . 7  |-  ( ( ( S substr  <. X ,  Y >. )  e. Word  A  /\  ( S substr  <. Y ,  Z >. )  e. Word  A  /\  x  e.  (
( # `  ( S substr  <. X ,  Y >. ) )..^ ( ( # `  ( S substr  <. X ,  Y >. ) )  +  ( # `  ( S substr  <. Y ,  Z >. ) ) ) ) )  ->  ( (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) `  x )  =  ( ( S substr  <. Y ,  Z >. ) `
 ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) ) ) )
7973, 74, 77, 78syl3anc 1182 . . . . . 6  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) `  x )  =  ( ( S substr  <. Y ,  Z >. ) `  (
x  -  ( # `  ( S substr  <. X ,  Y >. ) ) ) ) )
80 simpll 730 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  S  e. Word  A
)
81 simplr2 998 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  Y  e.  ( 0 ... Z ) )
82 simplr3 999 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  Z  e.  ( 0 ... ( # `  S ) ) )
8321oveq2d 5890 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) )  =  ( x  -  ( Y  -  X ) ) )
8483adantr 451 . . . . . . . 8  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) )  =  ( x  -  ( Y  -  X ) ) )
8534oveq2d 5890 . . . . . . . . . . 11  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( Y  -  X )..^ ( ( Y  -  X
)  +  ( Z  -  Y ) ) )  =  ( ( Y  -  X )..^ ( Z  -  X
) ) )
8685eleq2d 2363 . . . . . . . . . 10  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( x  e.  ( ( Y  -  X )..^ ( ( Y  -  X )  +  ( Z  -  Y
) ) )  <->  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) ) )
8786biimpar 471 . . . . . . . . 9  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  x  e.  ( ( Y  -  X
)..^ ( ( Y  -  X )  +  ( Z  -  Y
) ) ) )
8832, 26zsubcld 10138 . . . . . . . . . 10  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( Z  -  Y )  e.  ZZ )
8988adantr 451 . . . . . . . . 9  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( Z  -  Y )  e.  ZZ )
90 fzosubel3 10926 . . . . . . . . 9  |-  ( ( x  e.  ( ( Y  -  X )..^ ( ( Y  -  X )  +  ( Z  -  Y ) ) )  /\  ( Z  -  Y )  e.  ZZ )  ->  (
x  -  ( Y  -  X ) )  e.  ( 0..^ ( Z  -  Y ) ) )
9187, 89, 90syl2anc 642 . . . . . . . 8  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( x  -  ( Y  -  X
) )  e.  ( 0..^ ( Z  -  Y ) ) )
9284, 91eqeltrd 2370 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) )  e.  ( 0..^ ( Z  -  Y ) ) )
93 swrdfv 11473 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  Y  e.  (
0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) )  /\  ( x  -  ( # `
 ( S substr  <. X ,  Y >. ) ) )  e.  ( 0..^ ( Z  -  Y ) ) )  ->  (
( S substr  <. Y ,  Z >. ) `  (
x  -  ( # `  ( S substr  <. X ,  Y >. ) ) ) )  =  ( S `
 ( ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) )  +  Y ) ) )
9480, 81, 82, 92, 93syl31anc 1185 . . . . . 6  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( ( S substr  <. Y ,  Z >. ) `
 ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) ) )  =  ( S `  (
( x  -  ( # `
 ( S substr  <. X ,  Y >. ) ) )  +  Y ) ) )
9583oveq1d 5889 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) )  +  Y )  =  ( ( x  -  ( Y  -  X )
)  +  Y ) )
9695adantr 451 . . . . . . . 8  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) )  +  Y )  =  ( ( x  -  ( Y  -  X )
)  +  Y ) )
97 elfzoelz 10891 . . . . . . . . . . 11  |-  ( x  e.  ( ( Y  -  X )..^ ( Z  -  X ) )  ->  x  e.  ZZ )
9897zcnd 10134 . . . . . . . . . 10  |-  ( x  e.  ( ( Y  -  X )..^ ( Z  -  X ) )  ->  x  e.  CC )
9998adantl 452 . . . . . . . . 9  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  x  e.  CC )
10027, 30subcld 9173 . . . . . . . . . 10  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( Y  -  X )  e.  CC )
101100adantr 451 . . . . . . . . 9  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( Y  -  X )  e.  CC )
10227adantr 451 . . . . . . . . 9  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  Y  e.  CC )
10399, 101, 102subadd23d 9195 . . . . . . . 8  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( ( x  -  ( Y  -  X ) )  +  Y )  =  ( x  +  ( Y  -  ( Y  -  X ) ) ) )
10427, 30nncand 9178 . . . . . . . . . 10  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( Y  -  ( Y  -  X
) )  =  X )
105104oveq2d 5890 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( x  +  ( Y  -  ( Y  -  X )
) )  =  ( x  +  X ) )
106105adantr 451 . . . . . . . 8  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( x  +  ( Y  -  ( Y  -  X )
) )  =  ( x  +  X ) )
10796, 103, 1063eqtrd 2332 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) )  +  Y )  =  ( x  +  X ) )
108107fveq2d 5545 . . . . . 6  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( S `  ( ( x  -  ( # `  ( S substr  <. X ,  Y >. ) ) )  +  Y
) )  =  ( S `  ( x  +  X ) ) )
10979, 94, 1083eqtrd 2332 . . . . 5  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) )  ->  ( ( ( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) `  x )  =  ( S `  ( x  +  X ) ) )
11072, 109jaodan 760 . . . 4  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  ( x  e.  ( 0..^ ( Y  -  X ) )  \/  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) ) )  ->  ( (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) `  x )  =  ( S `  ( x  +  X
) ) )
11158, 110syldan 456 . . 3  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Z  -  X ) ) )  ->  ( (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) `  x )  =  ( S `  ( x  +  X
) ) )
112 simpll 730 . . . 4  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Z  -  X ) ) )  ->  S  e. Word  A )
11348adantr 451 . . . 4  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Z  -  X ) ) )  ->  X  e.  ( 0 ... Z
) )
114 simplr3 999 . . . 4  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Z  -  X ) ) )  ->  Z  e.  ( 0 ... ( # `
 S ) ) )
115 swrdfv 11473 . . . 4  |-  ( ( ( S  e. Word  A  /\  X  e.  (
0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) )  /\  x  e.  ( 0..^ ( Z  -  X
) ) )  -> 
( ( S substr  <. X ,  Z >. ) `  x
)  =  ( S `
 ( x  +  X ) ) )
116112, 113, 114, 54, 115syl31anc 1185 . . 3  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Z  -  X ) ) )  ->  ( ( S substr  <. X ,  Z >. ) `  x )  =  ( S `  ( x  +  X
) ) )
117111, 116eqtr4d 2331 . 2  |-  ( ( ( S  e. Word  A  /\  ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S ) ) ) )  /\  x  e.  ( 0..^ ( Z  -  X ) ) )  ->  ( (
( S substr  <. X ,  Y >. ) concat  ( S substr  <. Y ,  Z >. ) ) `  x )  =  ( ( S substr  <. X ,  Z >. ) `
 x ) )
11839, 53, 117eqfnfvd 5641 1  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( S substr  <. X ,  Y >. ) concat 
( S substr  <. Y ,  Z >. ) )  =  ( S substr  <. X ,  Z >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   <.cop 3656    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753    + caddc 8756    - cmin 9053   ZZcz 10040   ...cfz 10798  ..^cfzo 10886   #chash 11353  Word cword 11419   concat cconcat 11420   substr csubstr 11422
This theorem is referenced by:  splid  11484  splval2  11488  wrdeqcats1  11490  wrdeqs1cat  11491  swrds2  11576  efgredleme  15068  efgredlemc  15070  efgcpbllemb  15080  frgpuplem  15097
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-hash 11354  df-word 11425  df-concat 11426  df-substr 11428
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