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Theorem swrdeq 12723
Description: Two subwords of words are equal iff they have the same length and the same symbols at each position. (Contributed by Alexander van der Vekens, 7-Aug-2018.)
Assertion
Ref Expression
swrdeq  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  ->  (
( W substr  <. 0 ,  M >. )  =  ( U substr  <. 0 ,  N >. )  <->  ( M  =  N  /\  A. i  e.  ( 0..^ M ) ( W `  i
)  =  ( U `
 i ) ) ) )
Distinct variable groups:    i, M    i, N    U, i    i, V   
i, W

Proof of Theorem swrdeq
StepHypRef Expression
1 swrdcl 12698 . . . . 5  |-  ( W  e. Word  V  ->  ( W substr  <. 0 ,  M >. )  e. Word  V )
2 swrdcl 12698 . . . . 5  |-  ( U  e. Word  V  ->  ( U substr  <. 0 ,  N >. )  e. Word  V )
31, 2anim12i 564 . . . 4  |-  ( ( W  e. Word  V  /\  U  e. Word  V )  ->  ( ( W substr  <. 0 ,  M >. )  e. Word  V  /\  ( U substr  <. 0 ,  N >. )  e. Word  V
) )
433ad2ant1 1018 . . 3  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  ->  (
( W substr  <. 0 ,  M >. )  e. Word  V  /\  ( U substr  <. 0 ,  N >. )  e. Word  V
) )
5 eqwrd 12633 . . 3  |-  ( ( ( W substr  <. 0 ,  M >. )  e. Word  V  /\  ( U substr  <. 0 ,  N >. )  e. Word  V
)  ->  ( ( W substr  <. 0 ,  M >. )  =  ( U substr  <. 0 ,  N >. )  <-> 
( ( # `  ( W substr  <. 0 ,  M >. ) )  =  (
# `  ( U substr  <.
0 ,  N >. ) )  /\  A. i  e.  ( 0..^ ( # `  ( W substr  <. 0 ,  M >. ) ) ) ( ( W substr  <. 0 ,  M >. ) `  i
)  =  ( ( U substr  <. 0 ,  N >. ) `  i ) ) ) )
64, 5syl 17 . 2  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  ->  (
( W substr  <. 0 ,  M >. )  =  ( U substr  <. 0 ,  N >. )  <->  ( ( # `  ( W substr  <. 0 ,  M >. ) )  =  ( # `  ( U substr  <. 0 ,  N >. ) )  /\  A. i  e.  ( 0..^ ( # `  ( W substr  <. 0 ,  M >. ) ) ) ( ( W substr  <. 0 ,  M >. ) `  i
)  =  ( ( U substr  <. 0 ,  N >. ) `  i ) ) ) )
7 simpl 455 . . . . . 6  |-  ( ( W  e. Word  V  /\  U  e. Word  V )  ->  W  e. Word  V )
873ad2ant1 1018 . . . . 5  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  ->  W  e. Word  V )
9 simpl 455 . . . . . . 7  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  M  e.  NN0 )
1093ad2ant2 1019 . . . . . 6  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  ->  M  e.  NN0 )
11 lencl 12612 . . . . . . . 8  |-  ( W  e. Word  V  ->  ( # `
 W )  e. 
NN0 )
1211adantr 463 . . . . . . 7  |-  ( ( W  e. Word  V  /\  U  e. Word  V )  ->  ( # `  W
)  e.  NN0 )
13123ad2ant1 1018 . . . . . 6  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  ->  ( # `
 W )  e. 
NN0 )
14 simpl 455 . . . . . . 7  |-  ( ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
)  ->  M  <_  (
# `  W )
)
15143ad2ant3 1020 . . . . . 6  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  ->  M  <_  ( # `  W
) )
16 elfz2nn0 11822 . . . . . 6  |-  ( M  e.  ( 0 ... ( # `  W
) )  <->  ( M  e.  NN0  /\  ( # `  W )  e.  NN0  /\  M  <_  ( # `  W
) ) )
1710, 13, 15, 16syl3anbrc 1181 . . . . 5  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  ->  M  e.  ( 0 ... ( # `
 W ) ) )
18 swrd0len 12701 . . . . 5  |-  ( ( W  e. Word  V  /\  M  e.  ( 0 ... ( # `  W
) ) )  -> 
( # `  ( W substr  <. 0 ,  M >. ) )  =  M )
198, 17, 18syl2anc 659 . . . 4  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  ->  ( # `
 ( W substr  <. 0 ,  M >. ) )  =  M )
20 simpr 459 . . . . . 6  |-  ( ( W  e. Word  V  /\  U  e. Word  V )  ->  U  e. Word  V )
21203ad2ant1 1018 . . . . 5  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  ->  U  e. Word  V )
22 simpr 459 . . . . . . 7  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  N  e.  NN0 )
23223ad2ant2 1019 . . . . . 6  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  ->  N  e.  NN0 )
24 lencl 12612 . . . . . . . 8  |-  ( U  e. Word  V  ->  ( # `
 U )  e. 
NN0 )
2524adantl 464 . . . . . . 7  |-  ( ( W  e. Word  V  /\  U  e. Word  V )  ->  ( # `  U
)  e.  NN0 )
26253ad2ant1 1018 . . . . . 6  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  ->  ( # `
 U )  e. 
NN0 )
27 simpr 459 . . . . . . 7  |-  ( ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
)  ->  N  <_  (
# `  U )
)
28273ad2ant3 1020 . . . . . 6  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  ->  N  <_  ( # `  U
) )
29 elfz2nn0 11822 . . . . . 6  |-  ( N  e.  ( 0 ... ( # `  U
) )  <->  ( N  e.  NN0  /\  ( # `  U )  e.  NN0  /\  N  <_  ( # `  U
) ) )
3023, 26, 28, 29syl3anbrc 1181 . . . . 5  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  ->  N  e.  ( 0 ... ( # `
 U ) ) )
31 swrd0len 12701 . . . . 5  |-  ( ( U  e. Word  V  /\  N  e.  ( 0 ... ( # `  U
) ) )  -> 
( # `  ( U substr  <. 0 ,  N >. ) )  =  N )
3221, 30, 31syl2anc 659 . . . 4  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  ->  ( # `
 ( U substr  <. 0 ,  N >. ) )  =  N )
3319, 32eqeq12d 2424 . . 3  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  ->  (
( # `  ( W substr  <. 0 ,  M >. ) )  =  ( # `  ( U substr  <. 0 ,  N >. ) )  <->  M  =  N ) )
3433anbi1d 703 . 2  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  ->  (
( ( # `  ( W substr  <. 0 ,  M >. ) )  =  (
# `  ( U substr  <.
0 ,  N >. ) )  /\  A. i  e.  ( 0..^ ( # `  ( W substr  <. 0 ,  M >. ) ) ) ( ( W substr  <. 0 ,  M >. ) `  i
)  =  ( ( U substr  <. 0 ,  N >. ) `  i ) )  <->  ( M  =  N  /\  A. i  e.  ( 0..^ ( # `  ( W substr  <. 0 ,  M >. ) ) ) ( ( W substr  <. 0 ,  M >. ) `  i
)  =  ( ( U substr  <. 0 ,  N >. ) `  i ) ) ) )
358adantr 463 . . . . . . 7  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( M  e.  NN0  /\  N  e. 
NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  /\  M  =  N )  ->  W  e. Word  V )
3617adantr 463 . . . . . . 7  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( M  e.  NN0  /\  N  e. 
NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  /\  M  =  N )  ->  M  e.  ( 0 ... ( # `
 W ) ) )
3735, 36, 18syl2anc 659 . . . . . 6  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( M  e.  NN0  /\  N  e. 
NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  /\  M  =  N )  ->  ( # `
 ( W substr  <. 0 ,  M >. ) )  =  M )
3837oveq2d 6293 . . . . 5  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( M  e.  NN0  /\  N  e. 
NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  /\  M  =  N )  ->  (
0..^ ( # `  ( W substr  <. 0 ,  M >. ) ) )  =  ( 0..^ M ) )
3938raleqdv 3009 . . . 4  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( M  e.  NN0  /\  N  e. 
NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  /\  M  =  N )  ->  ( A. i  e.  (
0..^ ( # `  ( W substr  <. 0 ,  M >. ) ) ) ( ( W substr  <. 0 ,  M >. ) `  i
)  =  ( ( U substr  <. 0 ,  N >. ) `  i )  <->  A. i  e.  (
0..^ M ) ( ( W substr  <. 0 ,  M >. ) `  i
)  =  ( ( U substr  <. 0 ,  N >. ) `  i ) ) )
4035adantr 463 . . . . . . 7  |-  ( ( ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  /\  M  =  N )  /\  i  e.  ( 0..^ M ) )  ->  W  e. Word  V )
4136adantr 463 . . . . . . 7  |-  ( ( ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  /\  M  =  N )  /\  i  e.  ( 0..^ M ) )  ->  M  e.  ( 0 ... ( # `
 W ) ) )
42 simpr 459 . . . . . . 7  |-  ( ( ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  /\  M  =  N )  /\  i  e.  ( 0..^ M ) )  ->  i  e.  ( 0..^ M ) )
43 swrd0fv 12718 . . . . . . 7  |-  ( ( W  e. Word  V  /\  M  e.  ( 0 ... ( # `  W
) )  /\  i  e.  ( 0..^ M ) )  ->  ( ( W substr  <. 0 ,  M >. ) `  i )  =  ( W `  i ) )
4440, 41, 42, 43syl3anc 1230 . . . . . 6  |-  ( ( ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  /\  M  =  N )  /\  i  e.  ( 0..^ M ) )  ->  ( ( W substr  <. 0 ,  M >. ) `  i )  =  ( W `  i ) )
45 oveq2 6285 . . . . . . . . . . 11  |-  ( M  =  N  ->  (
0..^ M )  =  ( 0..^ N ) )
4645eleq2d 2472 . . . . . . . . . 10  |-  ( M  =  N  ->  (
i  e.  ( 0..^ M )  <->  i  e.  ( 0..^ N ) ) )
4746adantl 464 . . . . . . . . 9  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( M  e.  NN0  /\  N  e. 
NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  /\  M  =  N )  ->  (
i  e.  ( 0..^ M )  <->  i  e.  ( 0..^ N ) ) )
4821adantr 463 . . . . . . . . . . . 12  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( M  e.  NN0  /\  N  e. 
NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  /\  i  e.  ( 0..^ N ) )  ->  U  e. Word  V )
4930adantr 463 . . . . . . . . . . . 12  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( M  e.  NN0  /\  N  e. 
NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  /\  i  e.  ( 0..^ N ) )  ->  N  e.  ( 0 ... ( # `
 U ) ) )
50 simpr 459 . . . . . . . . . . . 12  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( M  e.  NN0  /\  N  e. 
NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  /\  i  e.  ( 0..^ N ) )  ->  i  e.  ( 0..^ N ) )
5148, 49, 503jca 1177 . . . . . . . . . . 11  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( M  e.  NN0  /\  N  e. 
NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  /\  i  e.  ( 0..^ N ) )  ->  ( U  e. Word  V  /\  N  e.  ( 0 ... ( # `
 U ) )  /\  i  e.  ( 0..^ N ) ) )
5251ex 432 . . . . . . . . . 10  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  ->  (
i  e.  ( 0..^ N )  ->  ( U  e. Word  V  /\  N  e.  ( 0 ... ( # `
 U ) )  /\  i  e.  ( 0..^ N ) ) ) )
5352adantr 463 . . . . . . . . 9  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( M  e.  NN0  /\  N  e. 
NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  /\  M  =  N )  ->  (
i  e.  ( 0..^ N )  ->  ( U  e. Word  V  /\  N  e.  ( 0 ... ( # `
 U ) )  /\  i  e.  ( 0..^ N ) ) ) )
5447, 53sylbid 215 . . . . . . . 8  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( M  e.  NN0  /\  N  e. 
NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  /\  M  =  N )  ->  (
i  e.  ( 0..^ M )  ->  ( U  e. Word  V  /\  N  e.  ( 0 ... ( # `
 U ) )  /\  i  e.  ( 0..^ N ) ) ) )
5554imp 427 . . . . . . 7  |-  ( ( ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  /\  M  =  N )  /\  i  e.  ( 0..^ M ) )  ->  ( U  e. Word  V  /\  N  e.  ( 0 ... ( # `
 U ) )  /\  i  e.  ( 0..^ N ) ) )
56 swrd0fv 12718 . . . . . . 7  |-  ( ( U  e. Word  V  /\  N  e.  ( 0 ... ( # `  U
) )  /\  i  e.  ( 0..^ N ) )  ->  ( ( U substr  <. 0 ,  N >. ) `  i )  =  ( U `  i ) )
5755, 56syl 17 . . . . . 6  |-  ( ( ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  /\  M  =  N )  /\  i  e.  ( 0..^ M ) )  ->  ( ( U substr  <. 0 ,  N >. ) `  i )  =  ( U `  i ) )
5844, 57eqeq12d 2424 . . . . 5  |-  ( ( ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  /\  M  =  N )  /\  i  e.  ( 0..^ M ) )  ->  ( (
( W substr  <. 0 ,  M >. ) `  i
)  =  ( ( U substr  <. 0 ,  N >. ) `  i )  <-> 
( W `  i
)  =  ( U `
 i ) ) )
5958ralbidva 2839 . . . 4  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( M  e.  NN0  /\  N  e. 
NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  /\  M  =  N )  ->  ( A. i  e.  (
0..^ M ) ( ( W substr  <. 0 ,  M >. ) `  i
)  =  ( ( U substr  <. 0 ,  N >. ) `  i )  <->  A. i  e.  (
0..^ M ) ( W `  i )  =  ( U `  i ) ) )
6039, 59bitrd 253 . . 3  |-  ( ( ( ( W  e. Word  V  /\  U  e. Word  V
)  /\  ( M  e.  NN0  /\  N  e. 
NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  /\  M  =  N )  ->  ( A. i  e.  (
0..^ ( # `  ( W substr  <. 0 ,  M >. ) ) ) ( ( W substr  <. 0 ,  M >. ) `  i
)  =  ( ( U substr  <. 0 ,  N >. ) `  i )  <->  A. i  e.  (
0..^ M ) ( W `  i )  =  ( U `  i ) ) )
6160pm5.32da 639 . 2  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  ->  (
( M  =  N  /\  A. i  e.  ( 0..^ ( # `  ( W substr  <. 0 ,  M >. ) ) ) ( ( W substr  <. 0 ,  M >. ) `  i
)  =  ( ( U substr  <. 0 ,  N >. ) `  i ) )  <->  ( M  =  N  /\  A. i  e.  ( 0..^ M ) ( W `  i
)  =  ( U `
 i ) ) ) )
626, 34, 613bitrd 279 1  |-  ( ( ( W  e. Word  V  /\  U  e. Word  V )  /\  ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( M  <_  ( # `  W
)  /\  N  <_  (
# `  U )
) )  ->  (
( W substr  <. 0 ,  M >. )  =  ( U substr  <. 0 ,  N >. )  <->  ( M  =  N  /\  A. i  e.  ( 0..^ M ) ( W `  i
)  =  ( U `
 i ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2753   <.cop 3977   class class class wbr 4394   ` cfv 5568  (class class class)co 6277   0cc0 9521    <_ cle 9658   NN0cn0 10835   ...cfz 11724  ..^cfzo 11852   #chash 12450  Word cword 12581   substr csubstr 12585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-n0 10836  df-z 10905  df-uz 11127  df-fz 11725  df-fzo 11853  df-hash 12451  df-word 12589  df-substr 12593
This theorem is referenced by:  clwlkf1clwwlklem  25253
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