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Theorem clwlkf1clwwlklem 30522
Description: Lemma for clwlkf1clwwlk 30523. (Contributed by Alexander van der Vekens, 5-Jul-2018.)
Hypotheses
Ref Expression
clwlkfclwwlk.1  |-  A  =  ( 1st `  c
)
clwlkfclwwlk.2  |-  B  =  ( 2nd `  c
)
clwlkfclwwlk.c  |-  C  =  { c  e.  ( V ClWalks  E )  |  (
# `  A )  =  N }
clwlkfclwwlk.f  |-  F  =  ( c  e.  C  |->  ( B substr  <. 0 ,  ( # `  A
) >. ) )
Assertion
Ref Expression
clwlkf1clwwlklem  |-  ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  ->  ( ( ( 2nd `  U ) substr  <. 0 ,  ( # `  ( 1st `  U ) )
>. )  =  (
( 2nd `  W
) substr  <. 0 ,  (
# `  ( 1st `  W ) ) >.
)  ->  A. y  e.  ( 0 ... N
) ( ( 2nd `  U ) `  y
)  =  ( ( 2nd `  W ) `
 y ) ) )
Distinct variable groups:    E, c    N, c    V, c    W, c    C, c    F, c    y, N    U, c, y    y, V    y, W
Allowed substitution hints:    A( y, c)    B( y, c)    C( y)    E( y)    F( y)

Proof of Theorem clwlkf1clwwlklem
StepHypRef Expression
1 clwlkfclwwlk.1 . . . . . . . . . . . . 13  |-  A  =  ( 1st `  c
)
2 clwlkfclwwlk.2 . . . . . . . . . . . . 13  |-  B  =  ( 2nd `  c
)
3 clwlkfclwwlk.c . . . . . . . . . . . . 13  |-  C  =  { c  e.  ( V ClWalks  E )  |  (
# `  A )  =  N }
4 clwlkfclwwlk.f . . . . . . . . . . . . 13  |-  F  =  ( c  e.  C  |->  ( B substr  <. 0 ,  ( # `  A
) >. ) )
51, 2, 3, 4clwlkf1clwwlklem3 30521 . . . . . . . . . . . 12  |-  ( W  e.  C  ->  ( 2nd `  W )  e. Word  V )
61, 2, 3, 4clwlkf1clwwlklem3 30521 . . . . . . . . . . . 12  |-  ( U  e.  C  ->  ( 2nd `  U )  e. Word  V )
75, 6anim12ci 567 . . . . . . . . . . 11  |-  ( ( W  e.  C  /\  U  e.  C )  ->  ( ( 2nd `  U
)  e. Word  V  /\  ( 2nd `  W )  e. Word  V ) )
87adantr 465 . . . . . . . . . 10  |-  ( ( ( W  e.  C  /\  U  e.  C
)  /\  N  e.  NN )  ->  ( ( 2nd `  U )  e. Word  V  /\  ( 2nd `  W )  e. Word  V ) )
9 nnnn0 10586 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  N  e.  NN0 )
109adantl 466 . . . . . . . . . . 11  |-  ( ( ( W  e.  C  /\  U  e.  C
)  /\  N  e.  NN )  ->  N  e. 
NN0 )
111, 2, 3, 4clwlkf1clwwlklem1 30519 . . . . . . . . . . . . 13  |-  ( U  e.  C  ->  N  <_  ( # `  ( 2nd `  U ) ) )
1211adantl 466 . . . . . . . . . . . 12  |-  ( ( W  e.  C  /\  U  e.  C )  ->  N  <_  ( # `  ( 2nd `  U ) ) )
1312adantr 465 . . . . . . . . . . 11  |-  ( ( ( W  e.  C  /\  U  e.  C
)  /\  N  e.  NN )  ->  N  <_ 
( # `  ( 2nd `  U ) ) )
141, 2, 3, 4clwlkf1clwwlklem1 30519 . . . . . . . . . . . . 13  |-  ( W  e.  C  ->  N  <_  ( # `  ( 2nd `  W ) ) )
1514adantr 465 . . . . . . . . . . . 12  |-  ( ( W  e.  C  /\  U  e.  C )  ->  N  <_  ( # `  ( 2nd `  W ) ) )
1615adantr 465 . . . . . . . . . . 11  |-  ( ( ( W  e.  C  /\  U  e.  C
)  /\  N  e.  NN )  ->  N  <_ 
( # `  ( 2nd `  W ) ) )
1710, 13, 163jca 1168 . . . . . . . . . 10  |-  ( ( ( W  e.  C  /\  U  e.  C
)  /\  N  e.  NN )  ->  ( N  e.  NN0  /\  N  <_ 
( # `  ( 2nd `  U ) )  /\  N  <_  ( # `  ( 2nd `  W ) ) ) )
188, 17jca 532 . . . . . . . . 9  |-  ( ( ( W  e.  C  /\  U  e.  C
)  /\  N  e.  NN )  ->  ( ( ( 2nd `  U
)  e. Word  V  /\  ( 2nd `  W )  e. Word  V )  /\  ( N  e.  NN0  /\  N  <_  ( # `  ( 2nd `  U ) )  /\  N  <_  ( # `
 ( 2nd `  W
) ) ) ) )
1918exp31 604 . . . . . . . 8  |-  ( W  e.  C  ->  ( U  e.  C  ->  ( N  e.  NN  ->  ( ( ( 2nd `  U
)  e. Word  V  /\  ( 2nd `  W )  e. Word  V )  /\  ( N  e.  NN0  /\  N  <_  ( # `  ( 2nd `  U ) )  /\  N  <_  ( # `
 ( 2nd `  W
) ) ) ) ) ) )
2019com13 80 . . . . . . 7  |-  ( N  e.  NN  ->  ( U  e.  C  ->  ( W  e.  C  -> 
( ( ( 2nd `  U )  e. Word  V  /\  ( 2nd `  W
)  e. Word  V )  /\  ( N  e.  NN0  /\  N  <_  ( # `  ( 2nd `  U ) )  /\  N  <_  ( # `
 ( 2nd `  W
) ) ) ) ) ) )
21203imp 1181 . . . . . 6  |-  ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  ->  ( ( ( 2nd `  U )  e. Word  V  /\  ( 2nd `  W
)  e. Word  V )  /\  ( N  e.  NN0  /\  N  <_  ( # `  ( 2nd `  U ) )  /\  N  <_  ( # `
 ( 2nd `  W
) ) ) ) )
2221adantr 465 . . . . 5  |-  ( ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  /\  ( ( 2nd `  U
) substr  <. 0 ,  (
# `  ( 1st `  U ) ) >.
)  =  ( ( 2nd `  W ) substr  <. 0 ,  ( # `  ( 1st `  W
) ) >. )
)  ->  ( (
( 2nd `  U
)  e. Word  V  /\  ( 2nd `  W )  e. Word  V )  /\  ( N  e.  NN0  /\  N  <_  ( # `  ( 2nd `  U ) )  /\  N  <_  ( # `
 ( 2nd `  W
) ) ) ) )
231, 2, 3, 4clwlkfclwwlk1hashn 30514 . . . . . . . . . 10  |-  ( U  e.  C  ->  ( # `
 ( 1st `  U
) )  =  N )
24233ad2ant2 1010 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  ->  ( # `  ( 1st `  U ) )  =  N )
2524opeq2d 4066 . . . . . . . 8  |-  ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  -> 
<. 0 ,  (
# `  ( 1st `  U ) ) >.  =  <. 0 ,  N >. )
2625oveq2d 6107 . . . . . . 7  |-  ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  ->  ( ( 2nd `  U
) substr  <. 0 ,  (
# `  ( 1st `  U ) ) >.
)  =  ( ( 2nd `  U ) substr  <. 0 ,  N >. ) )
271, 2, 3, 4clwlkfclwwlk1hashn 30514 . . . . . . . . . 10  |-  ( W  e.  C  ->  ( # `
 ( 1st `  W
) )  =  N )
28273ad2ant3 1011 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  ->  ( # `  ( 1st `  W ) )  =  N )
2928opeq2d 4066 . . . . . . . 8  |-  ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  -> 
<. 0 ,  (
# `  ( 1st `  W ) ) >.  =  <. 0 ,  N >. )
3029oveq2d 6107 . . . . . . 7  |-  ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  ->  ( ( 2nd `  W
) substr  <. 0 ,  (
# `  ( 1st `  W ) ) >.
)  =  ( ( 2nd `  W ) substr  <. 0 ,  N >. ) )
3126, 30eqeq12d 2457 . . . . . 6  |-  ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  ->  ( ( ( 2nd `  U ) substr  <. 0 ,  ( # `  ( 1st `  U ) )
>. )  =  (
( 2nd `  W
) substr  <. 0 ,  (
# `  ( 1st `  W ) ) >.
)  <->  ( ( 2nd `  U ) substr  <. 0 ,  N >. )  =  ( ( 2nd `  W
) substr  <. 0 ,  N >. ) ) )
3231biimpa 484 . . . . 5  |-  ( ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  /\  ( ( 2nd `  U
) substr  <. 0 ,  (
# `  ( 1st `  U ) ) >.
)  =  ( ( 2nd `  W ) substr  <. 0 ,  ( # `  ( 1st `  W
) ) >. )
)  ->  ( ( 2nd `  U ) substr  <. 0 ,  N >. )  =  ( ( 2nd `  W ) substr  <. 0 ,  N >. ) )
33 swrdsymbeq 12341 . . . . 5  |-  ( ( ( ( 2nd `  U
)  e. Word  V  /\  ( 2nd `  W )  e. Word  V )  /\  ( N  e.  NN0  /\  N  <_  ( # `  ( 2nd `  U ) )  /\  N  <_  ( # `
 ( 2nd `  W
) ) ) )  ->  ( ( ( 2nd `  U ) substr  <. 0 ,  N >. )  =  ( ( 2nd `  W ) substr  <. 0 ,  N >. )  ->  A. y  e.  ( 0..^ N ) ( ( 2nd `  U
) `  y )  =  ( ( 2nd `  W ) `  y
) ) )
3422, 32, 33sylc 60 . . . 4  |-  ( ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  /\  ( ( 2nd `  U
) substr  <. 0 ,  (
# `  ( 1st `  U ) ) >.
)  =  ( ( 2nd `  W ) substr  <. 0 ,  ( # `  ( 1st `  W
) ) >. )
)  ->  A. y  e.  ( 0..^ N ) ( ( 2nd `  U
) `  y )  =  ( ( 2nd `  W ) `  y
) )
35 lbfzo0 11586 . . . . . . . . 9  |-  ( 0  e.  ( 0..^ N )  <->  N  e.  NN )
3635biimpri 206 . . . . . . . 8  |-  ( N  e.  NN  ->  0  e.  ( 0..^ N ) )
37363ad2ant1 1009 . . . . . . 7  |-  ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  ->  0  e.  ( 0..^ N ) )
3837adantr 465 . . . . . 6  |-  ( ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  /\  ( ( 2nd `  U
) substr  <. 0 ,  (
# `  ( 1st `  U ) ) >.
)  =  ( ( 2nd `  W ) substr  <. 0 ,  ( # `  ( 1st `  W
) ) >. )
)  ->  0  e.  ( 0..^ N ) )
39 fveq2 5691 . . . . . . . . 9  |-  ( y  =  0  ->  (
( 2nd `  U
) `  y )  =  ( ( 2nd `  U ) `  0
) )
40 fveq2 5691 . . . . . . . . 9  |-  ( y  =  0  ->  (
( 2nd `  W
) `  y )  =  ( ( 2nd `  W ) `  0
) )
4139, 40eqeq12d 2457 . . . . . . . 8  |-  ( y  =  0  ->  (
( ( 2nd `  U
) `  y )  =  ( ( 2nd `  W ) `  y
)  <->  ( ( 2nd `  U ) `  0
)  =  ( ( 2nd `  W ) `
 0 ) ) )
4241rspcva 3071 . . . . . . 7  |-  ( ( 0  e.  ( 0..^ N )  /\  A. y  e.  ( 0..^ N ) ( ( 2nd `  U ) `
 y )  =  ( ( 2nd `  W
) `  y )
)  ->  ( ( 2nd `  U ) ` 
0 )  =  ( ( 2nd `  W
) `  0 )
)
4342ex 434 . . . . . 6  |-  ( 0  e.  ( 0..^ N )  ->  ( A. y  e.  ( 0..^ N ) ( ( 2nd `  U ) `
 y )  =  ( ( 2nd `  W
) `  y )  ->  ( ( 2nd `  U
) `  0 )  =  ( ( 2nd `  W ) `  0
) ) )
4438, 43syl 16 . . . . 5  |-  ( ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  /\  ( ( 2nd `  U
) substr  <. 0 ,  (
# `  ( 1st `  U ) ) >.
)  =  ( ( 2nd `  W ) substr  <. 0 ,  ( # `  ( 1st `  W
) ) >. )
)  ->  ( A. y  e.  ( 0..^ N ) ( ( 2nd `  U ) `
 y )  =  ( ( 2nd `  W
) `  y )  ->  ( ( 2nd `  U
) `  0 )  =  ( ( 2nd `  W ) `  0
) ) )
451, 2, 3, 4clwlkf1clwwlklem2 30520 . . . . . . . 8  |-  ( U  e.  C  ->  (
( 2nd `  U
) `  0 )  =  ( ( 2nd `  U ) `  N
) )
46453ad2ant2 1010 . . . . . . 7  |-  ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  ->  ( ( 2nd `  U
) `  0 )  =  ( ( 2nd `  U ) `  N
) )
4746adantr 465 . . . . . 6  |-  ( ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  /\  ( ( 2nd `  U
) substr  <. 0 ,  (
# `  ( 1st `  U ) ) >.
)  =  ( ( 2nd `  W ) substr  <. 0 ,  ( # `  ( 1st `  W
) ) >. )
)  ->  ( ( 2nd `  U ) ` 
0 )  =  ( ( 2nd `  U
) `  N )
)
481, 2, 3, 4clwlkf1clwwlklem2 30520 . . . . . . . 8  |-  ( W  e.  C  ->  (
( 2nd `  W
) `  0 )  =  ( ( 2nd `  W ) `  N
) )
49483ad2ant3 1011 . . . . . . 7  |-  ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  ->  ( ( 2nd `  W
) `  0 )  =  ( ( 2nd `  W ) `  N
) )
5049adantr 465 . . . . . 6  |-  ( ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  /\  ( ( 2nd `  U
) substr  <. 0 ,  (
# `  ( 1st `  U ) ) >.
)  =  ( ( 2nd `  W ) substr  <. 0 ,  ( # `  ( 1st `  W
) ) >. )
)  ->  ( ( 2nd `  W ) ` 
0 )  =  ( ( 2nd `  W
) `  N )
)
5147, 50eqeq12d 2457 . . . . 5  |-  ( ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  /\  ( ( 2nd `  U
) substr  <. 0 ,  (
# `  ( 1st `  U ) ) >.
)  =  ( ( 2nd `  W ) substr  <. 0 ,  ( # `  ( 1st `  W
) ) >. )
)  ->  ( (
( 2nd `  U
) `  0 )  =  ( ( 2nd `  W ) `  0
)  <->  ( ( 2nd `  U ) `  N
)  =  ( ( 2nd `  W ) `
 N ) ) )
5244, 51sylibd 214 . . . 4  |-  ( ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  /\  ( ( 2nd `  U
) substr  <. 0 ,  (
# `  ( 1st `  U ) ) >.
)  =  ( ( 2nd `  W ) substr  <. 0 ,  ( # `  ( 1st `  W
) ) >. )
)  ->  ( A. y  e.  ( 0..^ N ) ( ( 2nd `  U ) `
 y )  =  ( ( 2nd `  W
) `  y )  ->  ( ( 2nd `  U
) `  N )  =  ( ( 2nd `  W ) `  N
) ) )
5334, 52jcai 536 . . 3  |-  ( ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  /\  ( ( 2nd `  U
) substr  <. 0 ,  (
# `  ( 1st `  U ) ) >.
)  =  ( ( 2nd `  W ) substr  <. 0 ,  ( # `  ( 1st `  W
) ) >. )
)  ->  ( A. y  e.  ( 0..^ N ) ( ( 2nd `  U ) `
 y )  =  ( ( 2nd `  W
) `  y )  /\  ( ( 2nd `  U
) `  N )  =  ( ( 2nd `  W ) `  N
) ) )
54 elnn0uz 10898 . . . . . . . . 9  |-  ( N  e.  NN0  <->  N  e.  ( ZZ>=
`  0 ) )
559, 54sylib 196 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  ( ZZ>= `  0 )
)
56553ad2ant1 1009 . . . . . . 7  |-  ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  ->  N  e.  ( ZZ>= ` 
0 ) )
5756adantr 465 . . . . . 6  |-  ( ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  /\  ( ( 2nd `  U
) substr  <. 0 ,  (
# `  ( 1st `  U ) ) >.
)  =  ( ( 2nd `  W ) substr  <. 0 ,  ( # `  ( 1st `  W
) ) >. )
)  ->  N  e.  ( ZZ>= `  0 )
)
58 fzisfzounsn 30211 . . . . . 6  |-  ( N  e.  ( ZZ>= `  0
)  ->  ( 0 ... N )  =  ( ( 0..^ N )  u.  { N } ) )
5957, 58syl 16 . . . . 5  |-  ( ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  /\  ( ( 2nd `  U
) substr  <. 0 ,  (
# `  ( 1st `  U ) ) >.
)  =  ( ( 2nd `  W ) substr  <. 0 ,  ( # `  ( 1st `  W
) ) >. )
)  ->  ( 0 ... N )  =  ( ( 0..^ N )  u.  { N } ) )
6059raleqdv 2923 . . . 4  |-  ( ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  /\  ( ( 2nd `  U
) substr  <. 0 ,  (
# `  ( 1st `  U ) ) >.
)  =  ( ( 2nd `  W ) substr  <. 0 ,  ( # `  ( 1st `  W
) ) >. )
)  ->  ( A. y  e.  ( 0 ... N ) ( ( 2nd `  U
) `  y )  =  ( ( 2nd `  W ) `  y
)  <->  A. y  e.  ( ( 0..^ N )  u.  { N }
) ( ( 2nd `  U ) `  y
)  =  ( ( 2nd `  W ) `
 y ) ) )
61 simpl1 991 . . . . 5  |-  ( ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  /\  ( ( 2nd `  U
) substr  <. 0 ,  (
# `  ( 1st `  U ) ) >.
)  =  ( ( 2nd `  W ) substr  <. 0 ,  ( # `  ( 1st `  W
) ) >. )
)  ->  N  e.  NN )
62 fveq2 5691 . . . . . . 7  |-  ( y  =  N  ->  (
( 2nd `  U
) `  y )  =  ( ( 2nd `  U ) `  N
) )
63 fveq2 5691 . . . . . . 7  |-  ( y  =  N  ->  (
( 2nd `  W
) `  y )  =  ( ( 2nd `  W ) `  N
) )
6462, 63eqeq12d 2457 . . . . . 6  |-  ( y  =  N  ->  (
( ( 2nd `  U
) `  y )  =  ( ( 2nd `  W ) `  y
)  <->  ( ( 2nd `  U ) `  N
)  =  ( ( 2nd `  W ) `
 N ) ) )
6564ralunsn 4079 . . . . 5  |-  ( N  e.  NN  ->  ( A. y  e.  (
( 0..^ N )  u.  { N }
) ( ( 2nd `  U ) `  y
)  =  ( ( 2nd `  W ) `
 y )  <->  ( A. y  e.  ( 0..^ N ) ( ( 2nd `  U ) `
 y )  =  ( ( 2nd `  W
) `  y )  /\  ( ( 2nd `  U
) `  N )  =  ( ( 2nd `  W ) `  N
) ) ) )
6661, 65syl 16 . . . 4  |-  ( ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  /\  ( ( 2nd `  U
) substr  <. 0 ,  (
# `  ( 1st `  U ) ) >.
)  =  ( ( 2nd `  W ) substr  <. 0 ,  ( # `  ( 1st `  W
) ) >. )
)  ->  ( A. y  e.  ( (
0..^ N )  u. 
{ N } ) ( ( 2nd `  U
) `  y )  =  ( ( 2nd `  W ) `  y
)  <->  ( A. y  e.  ( 0..^ N ) ( ( 2nd `  U
) `  y )  =  ( ( 2nd `  W ) `  y
)  /\  ( ( 2nd `  U ) `  N )  =  ( ( 2nd `  W
) `  N )
) ) )
6760, 66bitrd 253 . . 3  |-  ( ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  /\  ( ( 2nd `  U
) substr  <. 0 ,  (
# `  ( 1st `  U ) ) >.
)  =  ( ( 2nd `  W ) substr  <. 0 ,  ( # `  ( 1st `  W
) ) >. )
)  ->  ( A. y  e.  ( 0 ... N ) ( ( 2nd `  U
) `  y )  =  ( ( 2nd `  W ) `  y
)  <->  ( A. y  e.  ( 0..^ N ) ( ( 2nd `  U
) `  y )  =  ( ( 2nd `  W ) `  y
)  /\  ( ( 2nd `  U ) `  N )  =  ( ( 2nd `  W
) `  N )
) ) )
6853, 67mpbird 232 . 2  |-  ( ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  /\  ( ( 2nd `  U
) substr  <. 0 ,  (
# `  ( 1st `  U ) ) >.
)  =  ( ( 2nd `  W ) substr  <. 0 ,  ( # `  ( 1st `  W
) ) >. )
)  ->  A. y  e.  ( 0 ... N
) ( ( 2nd `  U ) `  y
)  =  ( ( 2nd `  W ) `
 y ) )
6968ex 434 1  |-  ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  ->  ( ( ( 2nd `  U ) substr  <. 0 ,  ( # `  ( 1st `  U ) )
>. )  =  (
( 2nd `  W
) substr  <. 0 ,  (
# `  ( 1st `  W ) ) >.
)  ->  A. y  e.  ( 0 ... N
) ( ( 2nd `  U ) `  y
)  =  ( ( 2nd `  W ) `
 y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2715   {crab 2719    u. cun 3326   {csn 3877   <.cop 3883   class class class wbr 4292    e. cmpt 4350   ` cfv 5418  (class class class)co 6091   1stc1st 6575   2ndc2nd 6576   0cc0 9282    <_ cle 9419   NNcn 10322   NN0cn0 10579   ZZ>=cuz 10861   ...cfz 11437  ..^cfzo 11548   #chash 12103  Word cword 12221   substr csubstr 12225   ClWalks cclwlk 30412
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
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 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-fzo 11549  df-hash 12104  df-word 12229  df-substr 12233  df-wlk 23415  df-clwlk 30415
This theorem is referenced by:  clwlkf1clwwlk  30523
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