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Theorem swrdccatin12 12493
Description: The subword of a concatenation of two words within both of the concatenated words. (Contributed by Alexander van der Vekens, 5-Apr-2018.) (Revised by Alexander van der Vekens, 27-May-2018.)
Hypothesis
Ref Expression
swrdccatin12.l  |-  L  =  ( # `  A
)
Assertion
Ref Expression
swrdccatin12  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) )  ->  ( ( A concat  B ) substr  <. M ,  N >. )  =  ( ( A substr  <. M ,  L >. ) concat  ( B substr  <.
0 ,  ( N  -  L ) >.
) ) ) )

Proof of Theorem swrdccatin12
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 ccatcl 12385 . . . . 5  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( A concat  B )  e. Word  V )
21adantr 465 . . . 4  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( A concat  B )  e. Word  V )
3 elfz0fzfz0 11595 . . . . 5  |-  ( ( M  e.  ( 0 ... L )  /\  N  e.  ( L ... ( L  +  (
# `  B )
) ) )  ->  M  e.  ( 0 ... N ) )
43adantl 466 . . . 4  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  M  e.  ( 0 ... N
) )
5 elfzuz2 11566 . . . . . . . . 9  |-  ( M  e.  ( 0 ... L )  ->  L  e.  ( ZZ>= `  0 )
)
65adantl 466 . . . . . . . 8  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  M  e.  ( 0 ... L ) )  ->  L  e.  ( ZZ>= `  0 )
)
7 fzss1 11607 . . . . . . . 8  |-  ( L  e.  ( ZZ>= `  0
)  ->  ( L ... ( L  +  (
# `  B )
) )  C_  (
0 ... ( L  +  ( # `  B ) ) ) )
86, 7syl 16 . . . . . . 7  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  M  e.  ( 0 ... L ) )  ->  ( L ... ( L  +  (
# `  B )
) )  C_  (
0 ... ( L  +  ( # `  B ) ) ) )
9 ccatlen 12386 . . . . . . . . . 10  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( # `  ( A concat  B ) )  =  ( ( # `  A
)  +  ( # `  B ) ) )
10 swrdccatin12.l . . . . . . . . . . . 12  |-  L  =  ( # `  A
)
1110eqcomi 2464 . . . . . . . . . . 11  |-  ( # `  A )  =  L
1211oveq1i 6203 . . . . . . . . . 10  |-  ( (
# `  A )  +  ( # `  B
) )  =  ( L  +  ( # `  B ) )
139, 12syl6eq 2508 . . . . . . . . 9  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( # `  ( A concat  B ) )  =  ( L  +  (
# `  B )
) )
1413adantr 465 . . . . . . . 8  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  M  e.  ( 0 ... L ) )  ->  ( # `  ( A concat  B ) )  =  ( L  +  (
# `  B )
) )
1514oveq2d 6209 . . . . . . 7  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  M  e.  ( 0 ... L ) )  ->  ( 0 ... ( # `  ( A concat  B ) ) )  =  ( 0 ... ( L  +  (
# `  B )
) ) )
168, 15sseqtr4d 3494 . . . . . 6  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  M  e.  ( 0 ... L ) )  ->  ( L ... ( L  +  (
# `  B )
) )  C_  (
0 ... ( # `  ( A concat  B ) ) ) )
1716sseld 3456 . . . . 5  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  M  e.  ( 0 ... L ) )  ->  ( N  e.  ( L ... ( L  +  ( # `  B
) ) )  ->  N  e.  ( 0 ... ( # `  ( A concat  B ) ) ) ) )
1817impr 619 . . . 4  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  N  e.  ( 0 ... ( # `
 ( A concat  B
) ) ) )
19 swrdvalfn 12443 . . . 4  |-  ( ( ( A concat  B )  e. Word  V  /\  M  e.  ( 0 ... N
)  /\  N  e.  ( 0 ... ( # `
 ( A concat  B
) ) ) )  ->  ( ( A concat  B ) substr  <. M ,  N >. )  Fn  (
0..^ ( N  -  M ) ) )
202, 4, 18, 19syl3anc 1219 . . 3  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( ( A concat  B ) substr  <. M ,  N >. )  Fn  (
0..^ ( N  -  M ) ) )
21 swrdcl 12426 . . . . . . 7  |-  ( A  e. Word  V  ->  ( A substr  <. M ,  L >. )  e. Word  V )
22 swrdcl 12426 . . . . . . 7  |-  ( B  e. Word  V  ->  ( B substr  <. 0 ,  ( N  -  L )
>. )  e. Word  V )
2321, 22anim12i 566 . . . . . 6  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( ( A substr  <. M ,  L >. )  e. Word  V  /\  ( B substr  <. 0 ,  ( N  -  L ) >. )  e. Word  V ) )
2423adantr 465 . . . . 5  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( ( A substr  <. M ,  L >. )  e. Word  V  /\  ( B substr  <. 0 ,  ( N  -  L
) >. )  e. Word  V
) )
25 ccatvalfn 12391 . . . . 5  |-  ( ( ( A substr  <. M ,  L >. )  e. Word  V  /\  ( B substr  <. 0 ,  ( N  -  L ) >. )  e. Word  V )  ->  (
( A substr  <. M ,  L >. ) concat  ( B substr  <.
0 ,  ( N  -  L ) >.
) )  Fn  (
0..^ ( ( # `  ( A substr  <. M ,  L >. ) )  +  ( # `  ( B substr  <. 0 ,  ( N  -  L )
>. ) ) ) ) )
2624, 25syl 16 . . . 4  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( ( A substr  <. M ,  L >. ) concat  ( B substr  <. 0 ,  ( N  -  L ) >. )
)  Fn  ( 0..^ ( ( # `  ( A substr  <. M ,  L >. ) )  +  (
# `  ( B substr  <.
0 ,  ( N  -  L ) >.
) ) ) ) )
27 simpll 753 . . . . . . . . 9  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  A  e. Word  V )
28 simprl 755 . . . . . . . . 9  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  M  e.  ( 0 ... L
) )
29 lencl 12360 . . . . . . . . . . . 12  |-  ( A  e. Word  V  ->  ( # `
 A )  e. 
NN0 )
30 elnn0uz 11002 . . . . . . . . . . . . . 14  |-  ( (
# `  A )  e.  NN0  <->  ( # `  A
)  e.  ( ZZ>= ` 
0 ) )
31 eluzfz2 11569 . . . . . . . . . . . . . 14  |-  ( (
# `  A )  e.  ( ZZ>= `  0 )  ->  ( # `  A
)  e.  ( 0 ... ( # `  A
) ) )
3230, 31sylbi 195 . . . . . . . . . . . . 13  |-  ( (
# `  A )  e.  NN0  ->  ( # `  A
)  e.  ( 0 ... ( # `  A
) ) )
3310, 32syl5eqel 2543 . . . . . . . . . . . 12  |-  ( (
# `  A )  e.  NN0  ->  L  e.  ( 0 ... ( # `
 A ) ) )
3429, 33syl 16 . . . . . . . . . . 11  |-  ( A  e. Word  V  ->  L  e.  ( 0 ... ( # `
 A ) ) )
3534adantr 465 . . . . . . . . . 10  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  L  e.  ( 0 ... ( # `  A
) ) )
3635adantr 465 . . . . . . . . 9  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  L  e.  ( 0 ... ( # `
 A ) ) )
37 swrdlen 12430 . . . . . . . . 9  |-  ( ( A  e. Word  V  /\  M  e.  ( 0 ... L )  /\  L  e.  ( 0 ... ( # `  A
) ) )  -> 
( # `  ( A substr  <. M ,  L >. ) )  =  ( L  -  M ) )
3827, 28, 36, 37syl3anc 1219 . . . . . . . 8  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( # `  ( A substr  <. M ,  L >. ) )  =  ( L  -  M ) )
39 simpr 461 . . . . . . . . . 10  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  B  e. Word  V )
4039adantr 465 . . . . . . . . 9  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  B  e. Word  V )
41 lencl 12360 . . . . . . . . . . . . 13  |-  ( B  e. Word  V  ->  ( # `
 B )  e. 
NN0 )
4241nn0zd 10849 . . . . . . . . . . . 12  |-  ( B  e. Word  V  ->  ( # `
 B )  e.  ZZ )
4342adantl 466 . . . . . . . . . . 11  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( # `  B
)  e.  ZZ )
44 simpr 461 . . . . . . . . . . 11  |-  ( ( M  e.  ( 0 ... L )  /\  N  e.  ( L ... ( L  +  (
# `  B )
) ) )  ->  N  e.  ( L ... ( L  +  (
# `  B )
) ) )
4543, 44anim12i 566 . . . . . . . . . 10  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( ( # `
 B )  e.  ZZ  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )
46 elfzmlbp 11601 . . . . . . . . . 10  |-  ( ( ( # `  B
)  e.  ZZ  /\  N  e.  ( L ... ( L  +  (
# `  B )
) ) )  -> 
( N  -  L
)  e.  ( 0 ... ( # `  B
) ) )
4745, 46syl 16 . . . . . . . . 9  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( N  -  L )  e.  ( 0 ... ( # `  B ) ) )
48 swrd0len 12429 . . . . . . . . 9  |-  ( ( B  e. Word  V  /\  ( N  -  L
)  e.  ( 0 ... ( # `  B
) ) )  -> 
( # `  ( B substr  <. 0 ,  ( N  -  L ) >.
) )  =  ( N  -  L ) )
4940, 47, 48syl2anc 661 . . . . . . . 8  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( # `  ( B substr  <. 0 ,  ( N  -  L )
>. ) )  =  ( N  -  L ) )
5038, 49oveq12d 6211 . . . . . . 7  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( ( # `
 ( A substr  <. M ,  L >. ) )  +  ( # `  ( B substr  <. 0 ,  ( N  -  L )
>. ) ) )  =  ( ( L  -  M )  +  ( N  -  L ) ) )
51 elfz2nn0 11590 . . . . . . . . . . 11  |-  ( M  e.  ( 0 ... L )  <->  ( M  e.  NN0  /\  L  e. 
NN0  /\  M  <_  L ) )
52 elfzelz 11563 . . . . . . . . . . . . . 14  |-  ( N  e.  ( L ... ( L  +  ( # `
 B ) ) )  ->  N  e.  ZZ )
53 nn0cn 10693 . . . . . . . . . . . . . . . . . 18  |-  ( L  e.  NN0  ->  L  e.  CC )
5453adantl 466 . . . . . . . . . . . . . . . . 17  |-  ( ( M  e.  NN0  /\  L  e.  NN0 )  ->  L  e.  CC )
5554adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  ZZ  /\  ( M  e.  NN0  /\  L  e.  NN0 )
)  ->  L  e.  CC )
56 nn0cn 10693 . . . . . . . . . . . . . . . . 17  |-  ( M  e.  NN0  ->  M  e.  CC )
5756ad2antrl 727 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  ZZ  /\  ( M  e.  NN0  /\  L  e.  NN0 )
)  ->  M  e.  CC )
58 zcn 10755 . . . . . . . . . . . . . . . . 17  |-  ( N  e.  ZZ  ->  N  e.  CC )
5958adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  ZZ  /\  ( M  e.  NN0  /\  L  e.  NN0 )
)  ->  N  e.  CC )
6055, 57, 593jca 1168 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  ZZ  /\  ( M  e.  NN0  /\  L  e.  NN0 )
)  ->  ( L  e.  CC  /\  M  e.  CC  /\  N  e.  CC ) )
6160ex 434 . . . . . . . . . . . . . 14  |-  ( N  e.  ZZ  ->  (
( M  e.  NN0  /\  L  e.  NN0 )  ->  ( L  e.  CC  /\  M  e.  CC  /\  N  e.  CC )
) )
6252, 61syl 16 . . . . . . . . . . . . 13  |-  ( N  e.  ( L ... ( L  +  ( # `
 B ) ) )  ->  ( ( M  e.  NN0  /\  L  e.  NN0 )  ->  ( L  e.  CC  /\  M  e.  CC  /\  N  e.  CC ) ) )
6362com12 31 . . . . . . . . . . . 12  |-  ( ( M  e.  NN0  /\  L  e.  NN0 )  -> 
( N  e.  ( L ... ( L  +  ( # `  B
) ) )  -> 
( L  e.  CC  /\  M  e.  CC  /\  N  e.  CC )
) )
64633adant3 1008 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  L  e.  NN0  /\  M  <_  L )  ->  ( N  e.  ( L ... ( L  +  (
# `  B )
) )  ->  ( L  e.  CC  /\  M  e.  CC  /\  N  e.  CC ) ) )
6551, 64sylbi 195 . . . . . . . . . 10  |-  ( M  e.  ( 0 ... L )  ->  ( N  e.  ( L ... ( L  +  (
# `  B )
) )  ->  ( L  e.  CC  /\  M  e.  CC  /\  N  e.  CC ) ) )
6665imp 429 . . . . . . . . 9  |-  ( ( M  e.  ( 0 ... L )  /\  N  e.  ( L ... ( L  +  (
# `  B )
) ) )  -> 
( L  e.  CC  /\  M  e.  CC  /\  N  e.  CC )
)
6766adantl 466 . . . . . . . 8  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( L  e.  CC  /\  M  e.  CC  /\  N  e.  CC ) )
68 npncan3 9751 . . . . . . . 8  |-  ( ( L  e.  CC  /\  M  e.  CC  /\  N  e.  CC )  ->  (
( L  -  M
)  +  ( N  -  L ) )  =  ( N  -  M ) )
6967, 68syl 16 . . . . . . 7  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( ( L  -  M )  +  ( N  -  L ) )  =  ( N  -  M
) )
7050, 69eqtr2d 2493 . . . . . 6  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( N  -  M )  =  ( ( # `  ( A substr  <. M ,  L >. ) )  +  (
# `  ( B substr  <.
0 ,  ( N  -  L ) >.
) ) ) )
7170oveq2d 6209 . . . . 5  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( 0..^ ( N  -  M
) )  =  ( 0..^ ( ( # `  ( A substr  <. M ,  L >. ) )  +  ( # `  ( B substr  <. 0 ,  ( N  -  L )
>. ) ) ) ) )
7271fneq2d 5603 . . . 4  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( (
( A substr  <. M ,  L >. ) concat  ( B substr  <.
0 ,  ( N  -  L ) >.
) )  Fn  (
0..^ ( N  -  M ) )  <->  ( ( A substr  <. M ,  L >. ) concat  ( B substr  <. 0 ,  ( N  -  L ) >. )
)  Fn  ( 0..^ ( ( # `  ( A substr  <. M ,  L >. ) )  +  (
# `  ( B substr  <.
0 ,  ( N  -  L ) >.
) ) ) ) ) )
7326, 72mpbird 232 . . 3  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( ( A substr  <. M ,  L >. ) concat  ( B substr  <. 0 ,  ( N  -  L ) >. )
)  Fn  ( 0..^ ( N  -  M
) ) )
74 simprl 755 . . . . . 6  |-  ( ( k  e.  ( 0..^ ( L  -  M
) )  /\  (
( ( A  e. Word  V  /\  B  e. Word  V
)  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  /\  k  e.  ( 0..^ ( N  -  M ) ) ) )  ->  (
( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) ) )
75 simpr 461 . . . . . . . 8  |-  ( ( ( ( A  e. Word  V  /\  B  e. Word  V
)  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  /\  k  e.  ( 0..^ ( N  -  M ) ) )  ->  k  e.  ( 0..^ ( N  -  M ) ) )
7675anim2i 569 . . . . . . 7  |-  ( ( k  e.  ( 0..^ ( L  -  M
) )  /\  (
( ( A  e. Word  V  /\  B  e. Word  V
)  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  /\  k  e.  ( 0..^ ( N  -  M ) ) ) )  ->  (
k  e.  ( 0..^ ( L  -  M
) )  /\  k  e.  ( 0..^ ( N  -  M ) ) ) )
7776ancomd 451 . . . . . 6  |-  ( ( k  e.  ( 0..^ ( L  -  M
) )  /\  (
( ( A  e. Word  V  /\  B  e. Word  V
)  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  /\  k  e.  ( 0..^ ( N  -  M ) ) ) )  ->  (
k  e.  ( 0..^ ( N  -  M
) )  /\  k  e.  ( 0..^ ( L  -  M ) ) ) )
7810swrdccatin12lem3 12492 . . . . . 6  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( (
k  e.  ( 0..^ ( N  -  M
) )  /\  k  e.  ( 0..^ ( L  -  M ) ) )  ->  ( (
( A concat  B ) substr  <. M ,  N >. ) `
 k )  =  ( ( A substr  <. M ,  L >. ) `  k
) ) )
7974, 77, 78sylc 60 . . . . 5  |-  ( ( k  e.  ( 0..^ ( L  -  M
) )  /\  (
( ( A  e. Word  V  /\  B  e. Word  V
)  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  /\  k  e.  ( 0..^ ( N  -  M ) ) ) )  ->  (
( ( A concat  B
) substr  <. M ,  N >. ) `  k )  =  ( ( A substr  <. M ,  L >. ) `
 k ) )
8024ad2antrl 727 . . . . . . 7  |-  ( ( k  e.  ( 0..^ ( L  -  M
) )  /\  (
( ( A  e. Word  V  /\  B  e. Word  V
)  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  /\  k  e.  ( 0..^ ( N  -  M ) ) ) )  ->  (
( A substr  <. M ,  L >. )  e. Word  V  /\  ( B substr  <. 0 ,  ( N  -  L ) >. )  e. Word  V ) )
81 simpl 457 . . . . . . . 8  |-  ( ( k  e.  ( 0..^ ( L  -  M
) )  /\  (
( ( A  e. Word  V  /\  B  e. Word  V
)  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  /\  k  e.  ( 0..^ ( N  -  M ) ) ) )  ->  k  e.  ( 0..^ ( L  -  M ) ) )
82 nn0fz0 11635 . . . . . . . . . . . . . . . 16  |-  ( (
# `  A )  e.  NN0  <->  ( # `  A
)  e.  ( 0 ... ( # `  A
) ) )
8329, 82sylib 196 . . . . . . . . . . . . . . 15  |-  ( A  e. Word  V  ->  ( # `
 A )  e.  ( 0 ... ( # `
 A ) ) )
8410, 83syl5eqel 2543 . . . . . . . . . . . . . 14  |-  ( A  e. Word  V  ->  L  e.  ( 0 ... ( # `
 A ) ) )
8584adantr 465 . . . . . . . . . . . . 13  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  L  e.  ( 0 ... ( # `  A
) ) )
8685adantr 465 . . . . . . . . . . . 12  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  L  e.  ( 0 ... ( # `
 A ) ) )
8727, 28, 863jca 1168 . . . . . . . . . . 11  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( A  e. Word  V  /\  M  e.  ( 0 ... L
)  /\  L  e.  ( 0 ... ( # `
 A ) ) ) )
8887ad2antrl 727 . . . . . . . . . 10  |-  ( ( k  e.  ( 0..^ ( L  -  M
) )  /\  (
( ( A  e. Word  V  /\  B  e. Word  V
)  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  /\  k  e.  ( 0..^ ( N  -  M ) ) ) )  ->  ( A  e. Word  V  /\  M  e.  ( 0 ... L
)  /\  L  e.  ( 0 ... ( # `
 A ) ) ) )
8988, 37syl 16 . . . . . . . . 9  |-  ( ( k  e.  ( 0..^ ( L  -  M
) )  /\  (
( ( A  e. Word  V  /\  B  e. Word  V
)  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  /\  k  e.  ( 0..^ ( N  -  M ) ) ) )  ->  ( # `
 ( A substr  <. M ,  L >. ) )  =  ( L  -  M
) )
9089oveq2d 6209 . . . . . . . 8  |-  ( ( k  e.  ( 0..^ ( L  -  M
) )  /\  (
( ( A  e. Word  V  /\  B  e. Word  V
)  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  /\  k  e.  ( 0..^ ( N  -  M ) ) ) )  ->  (
0..^ ( # `  ( A substr  <. M ,  L >. ) ) )  =  ( 0..^ ( L  -  M ) ) )
9181, 90eleqtrrd 2542 . . . . . . 7  |-  ( ( k  e.  ( 0..^ ( L  -  M
) )  /\  (
( ( A  e. Word  V  /\  B  e. Word  V
)  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  /\  k  e.  ( 0..^ ( N  -  M ) ) ) )  ->  k  e.  ( 0..^ ( # `  ( A substr  <. M ,  L >. ) ) ) )
92 df-3an 967 . . . . . . 7  |-  ( ( ( A substr  <. M ,  L >. )  e. Word  V  /\  ( B substr  <. 0 ,  ( N  -  L ) >. )  e. Word  V  /\  k  e.  ( 0..^ ( # `  ( A substr  <. M ,  L >. ) ) ) )  <->  ( ( ( A substr  <. M ,  L >. )  e. Word  V  /\  ( B substr  <. 0 ,  ( N  -  L
) >. )  e. Word  V
)  /\  k  e.  ( 0..^ ( # `  ( A substr  <. M ,  L >. ) ) ) ) )
9380, 91, 92sylanbrc 664 . . . . . 6  |-  ( ( k  e.  ( 0..^ ( L  -  M
) )  /\  (
( ( A  e. Word  V  /\  B  e. Word  V
)  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  /\  k  e.  ( 0..^ ( N  -  M ) ) ) )  ->  (
( A substr  <. M ,  L >. )  e. Word  V  /\  ( B substr  <. 0 ,  ( N  -  L ) >. )  e. Word  V  /\  k  e.  ( 0..^ ( # `  ( A substr  <. M ,  L >. ) ) ) ) )
94 ccatval1 12387 . . . . . 6  |-  ( ( ( A substr  <. M ,  L >. )  e. Word  V  /\  ( B substr  <. 0 ,  ( N  -  L ) >. )  e. Word  V  /\  k  e.  ( 0..^ ( # `  ( A substr  <. M ,  L >. ) ) ) )  ->  ( (
( A substr  <. M ,  L >. ) concat  ( B substr  <.
0 ,  ( N  -  L ) >.
) ) `  k
)  =  ( ( A substr  <. M ,  L >. ) `  k ) )
9593, 94syl 16 . . . . 5  |-  ( ( k  e.  ( 0..^ ( L  -  M
) )  /\  (
( ( A  e. Word  V  /\  B  e. Word  V
)  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  /\  k  e.  ( 0..^ ( N  -  M ) ) ) )  ->  (
( ( A substr  <. M ,  L >. ) concat  ( B substr  <.
0 ,  ( N  -  L ) >.
) ) `  k
)  =  ( ( A substr  <. M ,  L >. ) `  k ) )
9679, 95eqtr4d 2495 . . . 4  |-  ( ( k  e.  ( 0..^ ( L  -  M
) )  /\  (
( ( A  e. Word  V  /\  B  e. Word  V
)  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  /\  k  e.  ( 0..^ ( N  -  M ) ) ) )  ->  (
( ( A concat  B
) substr  <. M ,  N >. ) `  k )  =  ( ( ( A substr  <. M ,  L >. ) concat  ( B substr  <. 0 ,  ( N  -  L ) >. )
) `  k )
)
97 simprl 755 . . . . . 6  |-  ( ( -.  k  e.  ( 0..^ ( L  -  M ) )  /\  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L )  /\  N  e.  ( L ... ( L  +  (
# `  B )
) ) ) )  /\  k  e.  ( 0..^ ( N  -  M ) ) ) )  ->  ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L )  /\  N  e.  ( L ... ( L  +  (
# `  B )
) ) ) ) )
9875anim2i 569 . . . . . . 7  |-  ( ( -.  k  e.  ( 0..^ ( L  -  M ) )  /\  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L )  /\  N  e.  ( L ... ( L  +  (
# `  B )
) ) ) )  /\  k  e.  ( 0..^ ( N  -  M ) ) ) )  ->  ( -.  k  e.  ( 0..^ ( L  -  M
) )  /\  k  e.  ( 0..^ ( N  -  M ) ) ) )
9998ancomd 451 . . . . . 6  |-  ( ( -.  k  e.  ( 0..^ ( L  -  M ) )  /\  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L )  /\  N  e.  ( L ... ( L  +  (
# `  B )
) ) ) )  /\  k  e.  ( 0..^ ( N  -  M ) ) ) )  ->  ( k  e.  ( 0..^ ( N  -  M ) )  /\  -.  k  e.  ( 0..^ ( L  -  M ) ) ) )
10010swrdccatin12lem2 12491 . . . . . 6  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( (
k  e.  ( 0..^ ( N  -  M
) )  /\  -.  k  e.  ( 0..^ ( L  -  M
) ) )  -> 
( ( ( A concat  B ) substr  <. M ,  N >. ) `  k
)  =  ( ( B substr  <. 0 ,  ( N  -  L )
>. ) `  ( k  -  ( # `  ( A substr  <. M ,  L >. ) ) ) ) ) )
10197, 99, 100sylc 60 . . . . 5  |-  ( ( -.  k  e.  ( 0..^ ( L  -  M ) )  /\  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L )  /\  N  e.  ( L ... ( L  +  (
# `  B )
) ) ) )  /\  k  e.  ( 0..^ ( N  -  M ) ) ) )  ->  ( (
( A concat  B ) substr  <. M ,  N >. ) `
 k )  =  ( ( B substr  <. 0 ,  ( N  -  L ) >. ) `  ( k  -  ( # `
 ( A substr  <. M ,  L >. ) ) ) ) )
10224ad2antrl 727 . . . . . . 7  |-  ( ( -.  k  e.  ( 0..^ ( L  -  M ) )  /\  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L )  /\  N  e.  ( L ... ( L  +  (
# `  B )
) ) ) )  /\  k  e.  ( 0..^ ( N  -  M ) ) ) )  ->  ( ( A substr  <. M ,  L >. )  e. Word  V  /\  ( B substr  <. 0 ,  ( N  -  L
) >. )  e. Word  V
) )
103 elfzuz 11559 . . . . . . . . . . . . 13  |-  ( N  e.  ( L ... ( L  +  ( # `
 B ) ) )  ->  N  e.  ( ZZ>= `  L )
)
104 eluzelz 10974 . . . . . . . . . . . . . 14  |-  ( N  e.  ( ZZ>= `  L
)  ->  N  e.  ZZ )
105 simpll 753 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( L  e.  NN0  /\  M  e.  NN0 )  /\  N  e.  ZZ )  ->  L  e.  NN0 )
106 simpr 461 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( L  e.  NN0  /\  M  e.  NN0 )  ->  M  e.  NN0 )
107106adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( L  e.  NN0  /\  M  e.  NN0 )  /\  N  e.  ZZ )  ->  M  e.  NN0 )
108 simpr 461 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( L  e.  NN0  /\  M  e.  NN0 )  /\  N  e.  ZZ )  ->  N  e.  ZZ )
109105, 107, 1083jca 1168 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( L  e.  NN0  /\  M  e.  NN0 )  /\  N  e.  ZZ )  ->  ( L  e. 
NN0  /\  M  e.  NN0 
/\  N  e.  ZZ ) )
110109ex 434 . . . . . . . . . . . . . . . . 17  |-  ( ( L  e.  NN0  /\  M  e.  NN0 )  -> 
( N  e.  ZZ  ->  ( L  e.  NN0  /\  M  e.  NN0  /\  N  e.  ZZ )
) )
111110ancoms 453 . . . . . . . . . . . . . . . 16  |-  ( ( M  e.  NN0  /\  L  e.  NN0 )  -> 
( N  e.  ZZ  ->  ( L  e.  NN0  /\  M  e.  NN0  /\  N  e.  ZZ )
) )
1121113adant3 1008 . . . . . . . . . . . . . . 15  |-  ( ( M  e.  NN0  /\  L  e.  NN0  /\  M  <_  L )  ->  ( N  e.  ZZ  ->  ( L  e.  NN0  /\  M  e.  NN0  /\  N  e.  ZZ ) ) )
11351, 112sylbi 195 . . . . . . . . . . . . . 14  |-  ( M  e.  ( 0 ... L )  ->  ( N  e.  ZZ  ->  ( L  e.  NN0  /\  M  e.  NN0  /\  N  e.  ZZ ) ) )
114104, 113syl5com 30 . . . . . . . . . . . . 13  |-  ( N  e.  ( ZZ>= `  L
)  ->  ( M  e.  ( 0 ... L
)  ->  ( L  e.  NN0  /\  M  e. 
NN0  /\  N  e.  ZZ ) ) )
115103, 114syl 16 . . . . . . . . . . . 12  |-  ( N  e.  ( L ... ( L  +  ( # `
 B ) ) )  ->  ( M  e.  ( 0 ... L
)  ->  ( L  e.  NN0  /\  M  e. 
NN0  /\  N  e.  ZZ ) ) )
116115impcom 430 . . . . . . . . . . 11  |-  ( ( M  e.  ( 0 ... L )  /\  N  e.  ( L ... ( L  +  (
# `  B )
) ) )  -> 
( L  e.  NN0  /\  M  e.  NN0  /\  N  e.  ZZ )
)
117116adantl 466 . . . . . . . . . 10  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( L  e.  NN0  /\  M  e. 
NN0  /\  N  e.  ZZ ) )
118117ad2antrl 727 . . . . . . . . 9  |-  ( ( -.  k  e.  ( 0..^ ( L  -  M ) )  /\  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L )  /\  N  e.  ( L ... ( L  +  (
# `  B )
) ) ) )  /\  k  e.  ( 0..^ ( N  -  M ) ) ) )  ->  ( L  e.  NN0  /\  M  e. 
NN0  /\  N  e.  ZZ ) )
119 swrdccatin12lem1 12486 . . . . . . . . 9  |-  ( ( L  e.  NN0  /\  M  e.  NN0  /\  N  e.  ZZ )  ->  (
( k  e.  ( 0..^ ( N  -  M ) )  /\  -.  k  e.  (
0..^ ( L  -  M ) ) )  ->  k  e.  ( ( L  -  M
)..^ ( ( L  -  M )  +  ( N  -  L
) ) ) ) )
120118, 99, 119sylc 60 . . . . . . . 8  |-  ( ( -.  k  e.  ( 0..^ ( L  -  M ) )  /\  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L )  /\  N  e.  ( L ... ( L  +  (
# `  B )
) ) ) )  /\  k  e.  ( 0..^ ( N  -  M ) ) ) )  ->  k  e.  ( ( L  -  M )..^ ( ( L  -  M )  +  ( N  -  L
) ) ) )
12127, 28, 86, 37syl3anc 1219 . . . . . . . . . 10  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( # `  ( A substr  <. M ,  L >. ) )  =  ( L  -  M ) )
12239adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  ( L ... ( L  +  ( # `  B ) ) )  /\  ( A  e. Word  V  /\  B  e. Word  V ) )  ->  B  e. Word  V )
12343adantl 466 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  ( L ... ( L  +  ( # `  B ) ) )  /\  ( A  e. Word  V  /\  B  e. Word  V ) )  -> 
( # `  B )  e.  ZZ )
124 simpl 457 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  ( L ... ( L  +  ( # `  B ) ) )  /\  ( A  e. Word  V  /\  B  e. Word  V ) )  ->  N  e.  ( L ... ( L  +  (
# `  B )
) ) )
125123, 124, 46syl2anc 661 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  ( L ... ( L  +  ( # `  B ) ) )  /\  ( A  e. Word  V  /\  B  e. Word  V ) )  -> 
( N  -  L
)  e.  ( 0 ... ( # `  B
) ) )
126122, 125jca 532 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  ( L ... ( L  +  ( # `  B ) ) )  /\  ( A  e. Word  V  /\  B  e. Word  V ) )  -> 
( B  e. Word  V  /\  ( N  -  L
)  e.  ( 0 ... ( # `  B
) ) ) )
127126ex 434 . . . . . . . . . . . . . 14  |-  ( N  e.  ( L ... ( L  +  ( # `
 B ) ) )  ->  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( B  e. Word  V  /\  ( N  -  L )  e.  ( 0 ... ( # `
 B ) ) ) ) )
128127adantl 466 . . . . . . . . . . . . 13  |-  ( ( M  e.  ( 0 ... L )  /\  N  e.  ( L ... ( L  +  (
# `  B )
) ) )  -> 
( ( A  e. Word  V  /\  B  e. Word  V
)  ->  ( B  e. Word  V  /\  ( N  -  L )  e.  ( 0 ... ( # `
 B ) ) ) ) )
129128impcom 430 . . . . . . . . . . . 12  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( B  e. Word  V  /\  ( N  -  L )  e.  ( 0 ... ( # `
 B ) ) ) )
130129, 48syl 16 . . . . . . . . . . 11  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( # `  ( B substr  <. 0 ,  ( N  -  L )
>. ) )  =  ( N  -  L ) )
131121, 130oveq12d 6211 . . . . . . . . . 10  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( ( # `
 ( A substr  <. M ,  L >. ) )  +  ( # `  ( B substr  <. 0 ,  ( N  -  L )
>. ) ) )  =  ( ( L  -  M )  +  ( N  -  L ) ) )
132121, 131oveq12d 6211 . . . . . . . . 9  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( ( # `
 ( A substr  <. M ,  L >. ) )..^ ( ( # `  ( A substr  <. M ,  L >. ) )  +  (
# `  ( B substr  <.
0 ,  ( N  -  L ) >.
) ) ) )  =  ( ( L  -  M )..^ ( ( L  -  M
)  +  ( N  -  L ) ) ) )
133132ad2antrl 727 . . . . . . . 8  |-  ( ( -.  k  e.  ( 0..^ ( L  -  M ) )  /\  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L )  /\  N  e.  ( L ... ( L  +  (
# `  B )
) ) ) )  /\  k  e.  ( 0..^ ( N  -  M ) ) ) )  ->  ( ( # `
 ( A substr  <. M ,  L >. ) )..^ ( ( # `  ( A substr  <. M ,  L >. ) )  +  (
# `  ( B substr  <.
0 ,  ( N  -  L ) >.
) ) ) )  =  ( ( L  -  M )..^ ( ( L  -  M
)  +  ( N  -  L ) ) ) )
134120, 133eleqtrrd 2542 . . . . . . 7  |-  ( ( -.  k  e.  ( 0..^ ( L  -  M ) )  /\  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L )  /\  N  e.  ( L ... ( L  +  (
# `  B )
) ) ) )  /\  k  e.  ( 0..^ ( N  -  M ) ) ) )  ->  k  e.  ( ( # `  ( A substr  <. M ,  L >. ) )..^ ( (
# `  ( A substr  <. M ,  L >. ) )  +  ( # `  ( B substr  <. 0 ,  ( N  -  L ) >. )
) ) ) )
135 df-3an 967 . . . . . . 7  |-  ( ( ( A substr  <. M ,  L >. )  e. Word  V  /\  ( B substr  <. 0 ,  ( N  -  L ) >. )  e. Word  V  /\  k  e.  ( ( # `  ( A substr  <. M ,  L >. ) )..^ ( (
# `  ( A substr  <. M ,  L >. ) )  +  ( # `  ( B substr  <. 0 ,  ( N  -  L ) >. )
) ) ) )  <-> 
( ( ( A substr  <. M ,  L >. )  e. Word  V  /\  ( B substr  <. 0 ,  ( N  -  L )
>. )  e. Word  V )  /\  k  e.  ( ( # `  ( A substr  <. M ,  L >. ) )..^ ( (
# `  ( A substr  <. M ,  L >. ) )  +  ( # `  ( B substr  <. 0 ,  ( N  -  L ) >. )
) ) ) ) )
136102, 134, 135sylanbrc 664 . . . . . 6  |-  ( ( -.  k  e.  ( 0..^ ( L  -  M ) )  /\  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L )  /\  N  e.  ( L ... ( L  +  (
# `  B )
) ) ) )  /\  k  e.  ( 0..^ ( N  -  M ) ) ) )  ->  ( ( A substr  <. M ,  L >. )  e. Word  V  /\  ( B substr  <. 0 ,  ( N  -  L
) >. )  e. Word  V  /\  k  e.  (
( # `  ( A substr  <. M ,  L >. ) )..^ ( ( # `  ( A substr  <. M ,  L >. ) )  +  ( # `  ( B substr  <. 0 ,  ( N  -  L )
>. ) ) ) ) ) )
137 ccatval2 12388 . . . . . 6  |-  ( ( ( A substr  <. M ,  L >. )  e. Word  V  /\  ( B substr  <. 0 ,  ( N  -  L ) >. )  e. Word  V  /\  k  e.  ( ( # `  ( A substr  <. M ,  L >. ) )..^ ( (
# `  ( A substr  <. M ,  L >. ) )  +  ( # `  ( B substr  <. 0 ,  ( N  -  L ) >. )
) ) ) )  ->  ( ( ( A substr  <. M ,  L >. ) concat  ( B substr  <. 0 ,  ( N  -  L ) >. )
) `  k )  =  ( ( B substr  <. 0 ,  ( N  -  L ) >.
) `  ( k  -  ( # `  ( A substr  <. M ,  L >. ) ) ) ) )
138136, 137syl 16 . . . . 5  |-  ( ( -.  k  e.  ( 0..^ ( L  -  M ) )  /\  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L )  /\  N  e.  ( L ... ( L  +  (
# `  B )
) ) ) )  /\  k  e.  ( 0..^ ( N  -  M ) ) ) )  ->  ( (
( A substr  <. M ,  L >. ) concat  ( B substr  <.
0 ,  ( N  -  L ) >.
) ) `  k
)  =  ( ( B substr  <. 0 ,  ( N  -  L )
>. ) `  ( k  -  ( # `  ( A substr  <. M ,  L >. ) ) ) ) )
139101, 138eqtr4d 2495 . . . 4  |-  ( ( -.  k  e.  ( 0..^ ( L  -  M ) )  /\  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L )  /\  N  e.  ( L ... ( L  +  (
# `  B )
) ) ) )  /\  k  e.  ( 0..^ ( N  -  M ) ) ) )  ->  ( (
( A concat  B ) substr  <. M ,  N >. ) `
 k )  =  ( ( ( A substr  <. M ,  L >. ) concat 
( B substr  <. 0 ,  ( N  -  L
) >. ) ) `  k ) )
14096, 139pm2.61ian 788 . . 3  |-  ( ( ( ( A  e. Word  V  /\  B  e. Word  V
)  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  /\  k  e.  ( 0..^ ( N  -  M ) ) )  ->  ( (
( A concat  B ) substr  <. M ,  N >. ) `
 k )  =  ( ( ( A substr  <. M ,  L >. ) concat 
( B substr  <. 0 ,  ( N  -  L
) >. ) ) `  k ) )
14120, 73, 140eqfnfvd 5902 . 2  |-  ( ( ( A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) ) )  ->  ( ( A concat  B ) substr  <. M ,  N >. )  =  ( ( A substr  <. M ,  L >. ) concat  ( B substr  <.
0 ,  ( N  -  L ) >.
) ) )
142141ex 434 1  |-  ( ( A  e. Word  V  /\  B  e. Word  V )  ->  ( ( M  e.  ( 0 ... L
)  /\  N  e.  ( L ... ( L  +  ( # `  B
) ) ) )  ->  ( ( A concat  B ) substr  <. M ,  N >. )  =  ( ( A substr  <. M ,  L >. ) concat  ( B substr  <.
0 ,  ( N  -  L ) >.
) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    C_ wss 3429   <.cop 3984   class class class wbr 4393    Fn wfn 5514   ` cfv 5519  (class class class)co 6193   CCcc 9384   0cc0 9386    + caddc 9389    <_ cle 9523    - cmin 9699   NN0cn0 10683   ZZcz 10750   ZZ>=cuz 10965   ...cfz 11547  ..^cfzo 11658   #chash 12213  Word cword 12332   concat cconcat 12334   substr csubstr 12336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-n0 10684  df-z 10751  df-uz 10966  df-fz 11548  df-fzo 11659  df-hash 12214  df-word 12340  df-concat 12342  df-substr 12344
This theorem is referenced by:  swrdccat3  12494  swrdccatin12d  12503
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