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Theorem clwlkf1clwwlklem 24976
Description: Lemma for clwlkf1clwwlk 24977. (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 24975 . . . . . . . . . . . 12  |-  ( W  e.  C  ->  ( 2nd `  W )  e. Word  V )
61, 2, 3, 4clwlkf1clwwlklem3 24975 . . . . . . . . . . . 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 10823 . . . . . . . . . . . 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 24973 . . . . . . . . . . . . 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 24973 . . . . . . . . . . . . 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 1176 . . . . . . . . . 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 1190 . . . . . 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 24968 . . . . . . . . . 10  |-  ( U  e.  C  ->  ( # `
 ( 1st `  U
) )  =  N )
24233ad2ant2 1018 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  ->  ( # `  ( 1st `  U ) )  =  N )
2524opeq2d 4226 . . . . . . . 8  |-  ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  -> 
<. 0 ,  (
# `  ( 1st `  U ) ) >.  =  <. 0 ,  N >. )
2625oveq2d 6312 . . . . . . 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 24968 . . . . . . . . . 10  |-  ( W  e.  C  ->  ( # `
 ( 1st `  W
) )  =  N )
28273ad2ant3 1019 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  ->  ( # `  ( 1st `  W ) )  =  N )
2928opeq2d 4226 . . . . . . . 8  |-  ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  -> 
<. 0 ,  (
# `  ( 1st `  W ) ) >.  =  <. 0 ,  N >. )
3029oveq2d 6312 . . . . . . 7  |-  ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  ->  ( ( 2nd `  W
) substr  <. 0 ,  (
# `  ( 1st `  W ) ) >.
)  =  ( ( 2nd `  W ) substr  <. 0 ,  N >. ) )
3126, 30eqeq12d 2479 . . . . . 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 12681 . . . . 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 11861 . . . . . . . . 9  |-  ( 0  e.  ( 0..^ N )  <->  N  e.  NN )
3635biimpri 206 . . . . . . . 8  |-  ( N  e.  NN  ->  0  e.  ( 0..^ N ) )
37363ad2ant1 1017 . . . . . . 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 5872 . . . . . . . 8  |-  ( y  =  0  ->  (
( 2nd `  U
) `  y )  =  ( ( 2nd `  U ) `  0
) )
40 fveq2 5872 . . . . . . . 8  |-  ( y  =  0  ->  (
( 2nd `  W
) `  y )  =  ( ( 2nd `  W ) `  0
) )
4139, 40eqeq12d 2479 . . . . . . 7  |-  ( y  =  0  ->  (
( ( 2nd `  U
) `  y )  =  ( ( 2nd `  W ) `  y
)  <->  ( ( 2nd `  U ) `  0
)  =  ( ( 2nd `  W ) `
 0 ) ) )
4241rspcv 3206 . . . . . 6  |-  ( 0  e.  ( 0..^ N )  ->  ( A. y  e.  ( 0..^ N ) ( ( 2nd `  U ) `
 y )  =  ( ( 2nd `  W
) `  y )  ->  ( ( 2nd `  U
) `  0 )  =  ( ( 2nd `  W ) `  0
) ) )
4338, 42syl 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
) ) )
441, 2, 3, 4clwlkf1clwwlklem2 24974 . . . . . . . 8  |-  ( U  e.  C  ->  (
( 2nd `  U
) `  0 )  =  ( ( 2nd `  U ) `  N
) )
45443ad2ant2 1018 . . . . . . 7  |-  ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  ->  ( ( 2nd `  U
) `  0 )  =  ( ( 2nd `  U ) `  N
) )
4645adantr 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 )
)
471, 2, 3, 4clwlkf1clwwlklem2 24974 . . . . . . . 8  |-  ( W  e.  C  ->  (
( 2nd `  W
) `  0 )  =  ( ( 2nd `  W ) `  N
) )
48473ad2ant3 1019 . . . . . . 7  |-  ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  ->  ( ( 2nd `  W
) `  0 )  =  ( ( 2nd `  W ) `  N
) )
4948adantr 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 )
)
5046, 49eqeq12d 2479 . . . . 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 ) ) )
5143, 50sylibd 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
) ) )
5234, 51jcai 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
) ) )
53 elnn0uz 11143 . . . . . . . . 9  |-  ( N  e.  NN0  <->  N  e.  ( ZZ>=
`  0 ) )
549, 53sylib 196 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  ( ZZ>= `  0 )
)
55543ad2ant1 1017 . . . . . . 7  |-  ( ( N  e.  NN  /\  U  e.  C  /\  W  e.  C )  ->  N  e.  ( ZZ>= ` 
0 ) )
5655adantr 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 )
)
57 fzisfzounsn 11924 . . . . . 6  |-  ( N  e.  ( ZZ>= `  0
)  ->  ( 0 ... N )  =  ( ( 0..^ N )  u.  { N } ) )
5856, 57syl 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 } ) )
5958raleqdv 3060 . . . 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 ) ) )
60 simpl1 999 . . . . 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 )
61 fveq2 5872 . . . . . . 7  |-  ( y  =  N  ->  (
( 2nd `  U
) `  y )  =  ( ( 2nd `  U ) `  N
) )
62 fveq2 5872 . . . . . . 7  |-  ( y  =  N  ->  (
( 2nd `  W
) `  y )  =  ( ( 2nd `  W ) `  N
) )
6361, 62eqeq12d 2479 . . . . . 6  |-  ( y  =  N  ->  (
( ( 2nd `  U
) `  y )  =  ( ( 2nd `  W ) `  y
)  <->  ( ( 2nd `  U ) `  N
)  =  ( ( 2nd `  W ) `
 N ) ) )
6463ralunsn 4239 . . . . 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
) ) ) )
6560, 64syl 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 )
) ) )
6659, 65bitrd 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 )
) ) )
6752, 66mpbird 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 ) )
6867ex 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 973    = wceq 1395    e. wcel 1819   A.wral 2807   {crab 2811    u. cun 3469   {csn 4032   <.cop 4038   class class class wbr 4456    |-> cmpt 4515   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798   0cc0 9509    <_ cle 9646   NNcn 10556   NN0cn0 10816   ZZ>=cuz 11106   ...cfz 11697  ..^cfzo 11821   #chash 12408  Word cword 12538   substr csubstr 12542   ClWalks cclwlk 24874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-substr 12550  df-wlk 24635  df-clwlk 24877
This theorem is referenced by:  clwlkf1clwwlk  24977
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