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Theorem ccatswrd 11763
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 11756 . . . . . 6  |-  ( S  e. Word  A  ->  ( S substr  <. X ,  Y >. )  e. Word  A )
21adantr 452 . . . . 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 11756 . . . . . 6  |-  ( S  e. Word  A  ->  ( S substr  <. Y ,  Z >. )  e. Word  A )
43adantr 452 . . . . 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 11733 . . . . 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 643 . . . 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 11723 . . . 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 5583 . . . 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 19 . . 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 11734 . . . . . . 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 643 . . . . . 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 444 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  S  e. Word  A
)
13 simpr1 963 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  X  e.  ( 0 ... Y ) )
14 simpr2 964 . . . . . . . . . 10  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  Y  e.  ( 0 ... Z ) )
15 simpr3 965 . . . . . . . . . 10  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  Z  e.  ( 0 ... ( # `  S ) ) )
16 fzass4 11080 . . . . . . . . . . . 12  |-  ( ( Y  e.  ( 0 ... ( # `  S
) )  /\  Z  e.  ( Y ... ( # `
 S ) ) )  <->  ( Y  e.  ( 0 ... Z
)  /\  Z  e.  ( 0 ... ( # `
 S ) ) ) )
1716biimpri 198 . . . . . . . . . . 11  |-  ( ( Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) )  -> 
( Y  e.  ( 0 ... ( # `  S ) )  /\  Z  e.  ( Y ... ( # `  S
) ) ) )
1817simpld 446 . . . . . . . . . 10  |-  ( ( Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) )  ->  Y  e.  ( 0 ... ( # `  S
) ) )
1914, 15, 18syl2anc 643 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  Y  e.  ( 0 ... ( # `  S ) ) )
20 swrdlen 11760 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) )  -> 
( # `  ( S substr  <. X ,  Y >. ) )  =  ( Y  -  X ) )
2112, 13, 19, 20syl3anc 1184 . . . . . . . 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 11760 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) )  -> 
( # `  ( S substr  <. Y ,  Z >. ) )  =  ( Z  -  Y ) )
2312, 14, 15, 22syl3anc 1184 . . . . . . . 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 6091 . . . . . . 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 11049 . . . . . . . . . 10  |-  ( Y  e.  ( 0 ... Z )  ->  Y  e.  ZZ )
2614, 25syl 16 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  Y  e.  ZZ )
2726zcnd 10366 . . . . . . . 8  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  Y  e.  CC )
28 elfzelz 11049 . . . . . . . . . 10  |-  ( X  e.  ( 0 ... Y )  ->  X  e.  ZZ )
2913, 28syl 16 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  X  e.  ZZ )
3029zcnd 10366 . . . . . . . 8  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  X  e.  CC )
31 elfzelz 11049 . . . . . . . . . 10  |-  ( Z  e.  ( 0 ... ( # `  S
) )  ->  Z  e.  ZZ )
3215, 31syl 16 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  Z  e.  ZZ )
3332zcnd 10366 . . . . . . . 8  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  Z  e.  CC )
3427, 30, 33npncan3d 9437 . . . . . . 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 2467 . . . . . 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 2467 . . . . 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 6089 . . . 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 5529 . . 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 202 . 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 11756 . . . . 5  |-  ( S  e. Word  A  ->  ( S substr  <. X ,  Z >. )  e. Word  A )
4140adantr 452 . . . 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 11723 . . . 4  |-  ( ( S substr  <. X ,  Z >. )  e. Word  A  -> 
( S substr  <. X ,  Z >. ) : ( 0..^ ( # `  ( S substr  <. X ,  Z >. ) ) ) --> A )
43 ffn 5583 . . . 4  |-  ( ( S substr  <. X ,  Z >. ) : ( 0..^ ( # `  ( S substr  <. X ,  Z >. ) ) ) --> A  ->  ( S substr  <. X ,  Z >. )  Fn  (
0..^ ( # `  ( S substr  <. X ,  Z >. ) ) ) )
4441, 42, 433syl 19 . . 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 11080 . . . . . . . . 9  |-  ( ( X  e.  ( 0 ... Z )  /\  Y  e.  ( X ... Z ) )  <->  ( X  e.  ( 0 ... Y
)  /\  Y  e.  ( 0 ... Z
) ) )
4645biimpri 198 . . . . . . . 8  |-  ( ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z ) )  ->  ( X  e.  ( 0 ... Z
)  /\  Y  e.  ( X ... Z ) ) )
4746simpld 446 . . . . . . 7  |-  ( ( X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... Z ) )  ->  X  e.  ( 0 ... Z ) )
4813, 14, 47syl2anc 643 . . . . . 6  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  X  e.  ( 0 ... Z ) )
49 swrdlen 11760 . . . . . 6  |-  ( ( S  e. Word  A  /\  X  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) )  -> 
( # `  ( S substr  <. X ,  Z >. ) )  =  ( Z  -  X ) )
5012, 48, 15, 49syl3anc 1184 . . . . 5  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( # `  ( S substr  <. X ,  Z >. ) )  =  ( Z  -  X ) )
5150oveq2d 6089 . . . 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 5529 . . 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 202 . 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 448 . . . . 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 10370 . . . . . 6  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( Y  -  X )  e.  ZZ )
5655adantr 452 . . . . 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 11155 . . . . 5  |-  ( ( x  e.  ( 0..^ ( Z  -  X
) )  /\  ( Y  -  X )  e.  ZZ )  ->  (
x  e.  ( 0..^ ( Y  -  X
) )  \/  x  e.  ( ( Y  -  X )..^ ( Z  -  X ) ) ) )
5854, 56, 57syl2anc 643 . . . 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 452 . . . . . . 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 452 . . . . . . 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 6089 . . . . . . . . 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 2502 . . . . . . . 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 472 . . . . . . 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 11735 . . . . . . 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 1184 . . . . . 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 731 . . . . . . 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 999 . . . . . . 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 452 . . . . . . 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 448 . . . . . . 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 11761 . . . . . . 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 1187 . . . . . 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 2467 . . . . 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 452 . . . . . . 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 452 . . . . . . 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 6091 . . . . . . . . 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 2502 . . . . . . . 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 472 . . . . . . 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 11736 . . . . . . 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 1184 . . . . . 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 731 . . . . . . 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 1000 . . . . . . 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 1001 . . . . . . 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 6089 . . . . . . . . 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 452 . . . . . . . 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 6089 . . . . . . . . . . 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 2502 . . . . . . . . . 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 472 . . . . . . . . 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 10370 . . . . . . . . . 10  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( Z  -  Y )  e.  ZZ )
8988adantr 452 . . . . . . . . 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 11169 . . . . . . . . 9  |-  ( ( x  e.  ( ( Y  -  X )..^ ( ( Y  -  X )  +  ( Z  -  Y ) ) )  /\  ( Z  -  Y )  e.  ZZ )  ->  (
x  -  ( Y  -  X ) )  e.  ( 0..^ ( Z  -  Y ) ) )
9187, 89, 90syl2anc 643 . . . . . . . 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 2509 . . . . . . 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 11761 . . . . . . 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 1187 . . . . . 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 6088 . . . . . . . . 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 452 . . . . . . . 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 11130 . . . . . . . . . . 11  |-  ( x  e.  ( ( Y  -  X )..^ ( Z  -  X ) )  ->  x  e.  ZZ )
9897zcnd 10366 . . . . . . . . . 10  |-  ( x  e.  ( ( Y  -  X )..^ ( Z  -  X ) )  ->  x  e.  CC )
9998adantl 453 . . . . . . . . 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 9401 . . . . . . . . . 10  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( Y  -  X )  e.  CC )
101100adantr 452 . . . . . . . . 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 452 . . . . . . . . 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 9423 . . . . . . . 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 9406 . . . . . . . . . 10  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( Y  -  ( Y  -  X
) )  =  X )
105104oveq2d 6089 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... Z )  /\  Z  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( x  +  ( Y  -  ( Y  -  X )
) )  =  ( x  +  X ) )
106105adantr 452 . . . . . . . 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 2471 . . . . . . 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 5724 . . . . . 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 2471 . . . . 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 761 . . . 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 457 . . 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 731 . . . 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 452 . . . 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 1001 . . . 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 11761 . . . 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 1187 . . 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 2470 . 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 5822 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 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   <.cop 3809    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   CCcc 8978   0cc0 8980    + caddc 8983    - cmin 9281   ZZcz 10272   ...cfz 11033  ..^cfzo 11125   #chash 11608  Word cword 11707   concat cconcat 11708   substr csubstr 11710
This theorem is referenced by:  splid  11772  splval2  11776  wrdeqcats1  11778  wrdeqs1cat  11779  swrds2  11870  efgredleme  15365  efgredlemc  15367  efgcpbllemb  15377  frgpuplem  15394  2cshw1lem3  28180  2cshw2lem3  28184
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7816  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-n0 10212  df-z 10273  df-uz 10479  df-fz 11034  df-fzo 11126  df-hash 11609  df-word 11713  df-concat 11714  df-substr 11716
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