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Theorem el2wlkonotot0 30344
Description: A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
Assertion
Ref Expression
el2wlkonotot0  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( R  e.  V  /\  S  e.  V ) )  -> 
( <. A ,  B ,  C >.  e.  ( R ( V 2WalksOnOt  E ) S )  <->  ( A  =  R  /\  C  =  S  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) )
Distinct variable groups:    A, f, p    B, f, p    C, f, p    f, E, p   
f, V, p    R, f, p    S, f, p   
f, X, p    f, Y, p

Proof of Theorem el2wlkonotot0
Dummy variable  b is distinct from all other variables.
StepHypRef Expression
1 el2wlkonot 30341 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( R  e.  V  /\  S  e.  V ) )  -> 
( <. A ,  B ,  C >.  e.  ( R ( V 2WalksOnOt  E ) S )  <->  E. b  e.  V  ( <. A ,  B ,  C >.  =  <. R ,  b ,  S >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( R  =  ( p `  0
)  /\  b  =  ( p `  1
)  /\  S  =  ( p `  2
) ) ) ) ) )
2 19.42vv 1925 . . . . . 6  |-  ( E. f E. p (
<. A ,  B ,  C >.  =  <. R , 
b ,  S >.  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( R  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  S  =  ( p `  2 ) ) ) )  <->  ( <. A ,  B ,  C >.  =  <. R ,  b ,  S >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( R  =  ( p `  0
)  /\  b  =  ( p `  1
)  /\  S  =  ( p `  2
) ) ) ) )
32bicomi 202 . . . . 5  |-  ( (
<. A ,  B ,  C >.  =  <. R , 
b ,  S >.  /\ 
E. f E. p
( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( R  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  S  =  ( p `  2 ) ) ) )  <->  E. f E. p ( <. A ,  B ,  C >.  = 
<. R ,  b ,  S >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( R  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  S  =  ( p ` 
2 ) ) ) ) )
43rexbii 2735 . . . 4  |-  ( E. b  e.  V  (
<. A ,  B ,  C >.  =  <. R , 
b ,  S >.  /\ 
E. f E. p
( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( R  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  S  =  ( p `  2 ) ) ) )  <->  E. b  e.  V  E. f E. p ( <. A ,  B ,  C >.  = 
<. R ,  b ,  S >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( R  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  S  =  ( p ` 
2 ) ) ) ) )
5 rexcom4 2987 . . . 4  |-  ( E. b  e.  V  E. f E. p ( <. A ,  B ,  C >.  =  <. R , 
b ,  S >.  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( R  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  S  =  ( p `  2 ) ) ) )  <->  E. f E. b  e.  V  E. p ( <. A ,  B ,  C >.  = 
<. R ,  b ,  S >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( R  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  S  =  ( p ` 
2 ) ) ) ) )
6 rexcom4 2987 . . . . 5  |-  ( E. b  e.  V  E. p ( <. A ,  B ,  C >.  = 
<. R ,  b ,  S >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( R  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  S  =  ( p ` 
2 ) ) ) )  <->  E. p E. b  e.  V  ( <. A ,  B ,  C >.  =  <. R ,  b ,  S >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( R  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  S  =  ( p ` 
2 ) ) ) ) )
76exbii 1634 . . . 4  |-  ( E. f E. b  e.  V  E. p (
<. A ,  B ,  C >.  =  <. R , 
b ,  S >.  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( R  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  S  =  ( p `  2 ) ) ) )  <->  E. f E. p E. b  e.  V  ( <. A ,  B ,  C >.  = 
<. R ,  b ,  S >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( R  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  S  =  ( p ` 
2 ) ) ) ) )
84, 5, 73bitri 271 . . 3  |-  ( E. b  e.  V  (
<. A ,  B ,  C >.  =  <. R , 
b ,  S >.  /\ 
E. f E. p
( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( R  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  S  =  ( p `  2 ) ) ) )  <->  E. f E. p E. b  e.  V  ( <. A ,  B ,  C >.  = 
<. R ,  b ,  S >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( R  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  S  =  ( p ` 
2 ) ) ) ) )
98a1i 11 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( R  e.  V  /\  S  e.  V ) )  -> 
( E. b  e.  V  ( <. A ,  B ,  C >.  = 
<. R ,  b ,  S >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( R  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  S  =  ( p `  2 ) ) ) )  <->  E. f E. p E. b  e.  V  ( <. A ,  B ,  C >.  = 
<. R ,  b ,  S >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( R  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  S  =  ( p ` 
2 ) ) ) ) ) )
10 eqcom 2440 . . . . . . . . . . 11  |-  ( <. A ,  B ,  C >.  =  <. R , 
b ,  S >.  <->  <. R ,  b ,  S >.  =  <. A ,  B ,  C >. )
11 df-ot 3881 . . . . . . . . . . . 12  |-  <. R , 
b ,  S >.  = 
<. <. R ,  b
>. ,  S >.
12 df-ot 3881 . . . . . . . . . . . 12  |-  <. A ,  B ,  C >.  = 
<. <. A ,  B >. ,  C >.
1311, 12eqeq12i 2451 . . . . . . . . . . 11  |-  ( <. R ,  b ,  S >.  =  <. A ,  B ,  C >.  <->  <. <. R ,  b >. ,  S >.  =  <. <. A ,  B >. ,  C >. )
1410, 13bitri 249 . . . . . . . . . 10  |-  ( <. A ,  B ,  C >.  =  <. R , 
b ,  S >.  <->  <. <. R ,  b >. ,  S >.  =  <. <. A ,  B >. ,  C >. )
1514a1i 11 . . . . . . . . 9  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( R  e.  V  /\  S  e.  V )
)  /\  b  e.  V )  ->  ( <. A ,  B ,  C >.  =  <. R , 
b ,  S >.  <->  <. <. R ,  b >. ,  S >.  =  <. <. A ,  B >. ,  C >. ) )
16 opex 4551 . . . . . . . . . . . . . 14  |-  <. R , 
b >.  e.  _V
1716a1i 11 . . . . . . . . . . . . 13  |-  ( ( R  e.  V  /\  S  e.  V )  -> 
<. R ,  b >.  e.  _V )
18 simpr 461 . . . . . . . . . . . . 13  |-  ( ( R  e.  V  /\  S  e.  V )  ->  S  e.  V )
1917, 18jca 532 . . . . . . . . . . . 12  |-  ( ( R  e.  V  /\  S  e.  V )  ->  ( <. R ,  b
>.  e.  _V  /\  S  e.  V ) )
2019adantl 466 . . . . . . . . . . 11  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( R  e.  V  /\  S  e.  V ) )  -> 
( <. R ,  b
>.  e.  _V  /\  S  e.  V ) )
2120adantr 465 . . . . . . . . . 10  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( R  e.  V  /\  S  e.  V )
)  /\  b  e.  V )  ->  ( <. R ,  b >.  e.  _V  /\  S  e.  V ) )
22 opthg 4562 . . . . . . . . . 10  |-  ( (
<. R ,  b >.  e.  _V  /\  S  e.  V )  ->  ( <. <. R ,  b
>. ,  S >.  = 
<. <. A ,  B >. ,  C >.  <->  ( <. R ,  b >.  =  <. A ,  B >.  /\  S  =  C ) ) )
2321, 22syl 16 . . . . . . . . 9  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( R  e.  V  /\  S  e.  V )
)  /\  b  e.  V )  ->  ( <. <. R ,  b
>. ,  S >.  = 
<. <. A ,  B >. ,  C >.  <->  ( <. R ,  b >.  =  <. A ,  B >.  /\  S  =  C ) ) )
24 simpl 457 . . . . . . . . . . . 12  |-  ( ( R  e.  V  /\  S  e.  V )  ->  R  e.  V )
2524adantl 466 . . . . . . . . . . 11  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( R  e.  V  /\  S  e.  V ) )  ->  R  e.  V )
26 opthg 4562 . . . . . . . . . . 11  |-  ( ( R  e.  V  /\  b  e.  V )  ->  ( <. R ,  b
>.  =  <. A ,  B >. 
<->  ( R  =  A  /\  b  =  B ) ) )
2725, 26sylan 471 . . . . . . . . . 10  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( R  e.  V  /\  S  e.  V )
)  /\  b  e.  V )  ->  ( <. R ,  b >.  =  <. A ,  B >.  <-> 
( R  =  A  /\  b  =  B ) ) )
2827anbi1d 704 . . . . . . . . 9  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( R  e.  V  /\  S  e.  V )
)  /\  b  e.  V )  ->  (
( <. R ,  b
>.  =  <. A ,  B >.  /\  S  =  C )  <->  ( ( R  =  A  /\  b  =  B )  /\  S  =  C
) ) )
2915, 23, 283bitrd 279 . . . . . . . 8  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( R  e.  V  /\  S  e.  V )
)  /\  b  e.  V )  ->  ( <. A ,  B ,  C >.  =  <. R , 
b ,  S >.  <->  (
( R  =  A  /\  b  =  B )  /\  S  =  C ) ) )
30 eqcom 2440 . . . . . . . . . . . . . 14  |-  ( R  =  A  <->  A  =  R )
3130biimpi 194 . . . . . . . . . . . . 13  |-  ( R  =  A  ->  A  =  R )
3231adantr 465 . . . . . . . . . . . 12  |-  ( ( R  =  A  /\  b  =  B )  ->  A  =  R )
33 eqcom 2440 . . . . . . . . . . . . 13  |-  ( S  =  C  <->  C  =  S )
3433biimpi 194 . . . . . . . . . . . 12  |-  ( S  =  C  ->  C  =  S )
3532, 34anim12i 566 . . . . . . . . . . 11  |-  ( ( ( R  =  A  /\  b  =  B )  /\  S  =  C )  ->  ( A  =  R  /\  C  =  S )
)
3635adantr 465 . . . . . . . . . 10  |-  ( ( ( ( R  =  A  /\  b  =  B )  /\  S  =  C )  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( R  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  S  =  ( p ` 
2 ) ) ) )  ->  ( A  =  R  /\  C  =  S ) )
37 simpr1 994 . . . . . . . . . . 11  |-  ( ( ( ( R  =  A  /\  b  =  B )  /\  S  =  C )  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( R  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  S  =  ( p ` 
2 ) ) ) )  ->  f ( V Walks  E ) p )
38 simp2 989 . . . . . . . . . . . 12  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( R  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  S  =  ( p ` 
2 ) ) )  ->  ( # `  f
)  =  2 )
3938adantl 466 . . . . . . . . . . 11  |-  ( ( ( ( R  =  A  /\  b  =  B )  /\  S  =  C )  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( R  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  S  =  ( p ` 
2 ) ) ) )  ->  ( # `  f
)  =  2 )
40 eqtr2 2456 . . . . . . . . . . . . . . . . . 18  |-  ( ( R  =  A  /\  R  =  ( p `  0 ) )  ->  A  =  ( p `  0 ) )
4140ex 434 . . . . . . . . . . . . . . . . 17  |-  ( R  =  A  ->  ( R  =  ( p `  0 )  ->  A  =  ( p `  0 ) ) )
4241adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( R  =  A  /\  b  =  B )  ->  ( R  =  ( p `  0 )  ->  A  =  ( p `  0 ) ) )
4342adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( R  =  A  /\  b  =  B )  /\  S  =  C )  ->  ( R  =  ( p `  0 )  ->  A  =  ( p `  0 ) ) )
44 eqtr2 2456 . . . . . . . . . . . . . . . . . 18  |-  ( ( b  =  B  /\  b  =  ( p `  1 ) )  ->  B  =  ( p `  1 ) )
4544ex 434 . . . . . . . . . . . . . . . . 17  |-  ( b  =  B  ->  (
b  =  ( p `
 1 )  ->  B  =  ( p `  1 ) ) )
4645adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( R  =  A  /\  b  =  B )  ->  ( b  =  ( p `  1 )  ->  B  =  ( p `  1 ) ) )
4746adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( R  =  A  /\  b  =  B )  /\  S  =  C )  ->  (
b  =  ( p `
 1 )  ->  B  =  ( p `  1 ) ) )
48 eqtr2 2456 . . . . . . . . . . . . . . . . 17  |-  ( ( S  =  C  /\  S  =  ( p `  2 ) )  ->  C  =  ( p `  2 ) )
4948ex 434 . . . . . . . . . . . . . . . 16  |-  ( S  =  C  ->  ( S  =  ( p `  2 )  ->  C  =  ( p `  2 ) ) )
5049adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( ( R  =  A  /\  b  =  B )  /\  S  =  C )  ->  ( S  =  ( p `  2 )  ->  C  =  ( p `  2 ) ) )
5143, 47, 503anim123d 1296 . . . . . . . . . . . . . 14  |-  ( ( ( R  =  A  /\  b  =  B )  /\  S  =  C )  ->  (
( R  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  S  =  ( p `  2 ) )  ->  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )
5251com12 31 . . . . . . . . . . . . 13  |-  ( ( R  =  ( p `
 0 )  /\  b  =  ( p `  1 )  /\  S  =  ( p `  2 ) )  ->  ( ( ( R  =  A  /\  b  =  B )  /\  S  =  C
)  ->  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )
53523ad2ant3 1011 . . . . . . . . . . . 12  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( R  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  S  =  ( p ` 
2 ) ) )  ->  ( ( ( R  =  A  /\  b  =  B )  /\  S  =  C
)  ->  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )
5453impcom 430 . . . . . . . . . . 11  |-  ( ( ( ( R  =  A  /\  b  =  B )  /\  S  =  C )  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( R  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  S  =  ( p ` 
2 ) ) ) )  ->  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )
5537, 39, 543jca 1168 . . . . . . . . . 10  |-  ( ( ( ( R  =  A  /\  b  =  B )  /\  S  =  C )  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( R  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  S  =  ( p ` 
2 ) ) ) )  ->  ( f
( V Walks  E )
p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )
5636, 55jca 532 . . . . . . . . 9  |-  ( ( ( ( R  =  A  /\  b  =  B )  /\  S  =  C )  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( R  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  S  =  ( p ` 
2 ) ) ) )  ->  ( ( A  =  R  /\  C  =  S )  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )
5756ex 434 . . . . . . . 8  |-  ( ( ( R  =  A  /\  b  =  B )  /\  S  =  C )  ->  (
( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( R  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  S  =  ( p `  2 ) ) )  ->  ( ( A  =  R  /\  C  =  S )  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) )
5829, 57syl6bi 228 . . . . . . 7  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( R  e.  V  /\  S  e.  V )
)  /\  b  e.  V )  ->  ( <. A ,  B ,  C >.  =  <. R , 
b ,  S >.  -> 
( ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( R  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  S  =  ( p `  2 ) ) )  ->  (
( A  =  R  /\  C  =  S )  /\  ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) ) ) ) )
5958impd 431 . . . . . 6  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( R  e.  V  /\  S  e.  V )
)  /\  b  e.  V )  ->  (
( <. A ,  B ,  C >.  =  <. R ,  b ,  S >.  /\  ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( R  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  S  =  ( p `  2 ) ) ) )  -> 
( ( A  =  R  /\  C  =  S )  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) ) )
6059rexlimdva 2836 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( R  e.  V  /\  S  e.  V ) )  -> 
( E. b  e.  V  ( <. A ,  B ,  C >.  = 
<. R ,  b ,  S >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( R  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  S  =  ( p ` 
2 ) ) ) )  ->  ( ( A  =  R  /\  C  =  S )  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) )
61 el2wlkonotlem 30334 . . . . . . . . . . . . . . . . 17  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2 )  ->  (
p `  1 )  e.  V )
6261adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2 )  /\  B  =  ( p `  1 ) )  ->  ( p ` 
1 )  e.  V
)
63 eleq1 2498 . . . . . . . . . . . . . . . . 17  |-  ( B  =  ( p ` 
1 )  ->  ( B  e.  V  <->  ( p `  1 )  e.  V ) )
6463adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2 )  /\  B  =  ( p `  1 ) )  ->  ( B  e.  V  <->  ( p ` 
1 )  e.  V
) )
6562, 64mpbird 232 . . . . . . . . . . . . . . 15  |-  ( ( ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2 )  /\  B  =  ( p `  1 ) )  ->  B  e.  V
)
6665a1d 25 . . . . . . . . . . . . . 14  |-  ( ( ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2 )  /\  B  =  ( p `  1 ) )  ->  ( ( A  =  R  /\  C  =  S )  ->  B  e.  V ) )
6766ex 434 . . . . . . . . . . . . 13  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2 )  ->  ( B  =  ( p `  1 )  -> 
( ( A  =  R  /\  C  =  S )  ->  B  e.  V ) ) )
6867ex 434 . . . . . . . . . . . 12  |-  ( f ( V Walks  E ) p  ->  ( ( # `
 f )  =  2  ->  ( B  =  ( p ` 
1 )  ->  (
( A  =  R  /\  C  =  S )  ->  B  e.  V ) ) ) )
6968com13 80 . . . . . . . . . . 11  |-  ( B  =  ( p ` 
1 )  ->  (
( # `  f )  =  2  ->  (
f ( V Walks  E
) p  ->  (
( A  =  R  /\  C  =  S )  ->  B  e.  V ) ) ) )
70693ad2ant2 1010 . . . . . . . . . 10  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( # `  f )  =  2  ->  ( f ( V Walks  E ) p  ->  ( ( A  =  R  /\  C  =  S )  ->  B  e.  V ) ) ) )
7170com13 80 . . . . . . . . 9  |-  ( f ( V Walks  E ) p  ->  ( ( # `
 f )  =  2  ->  ( ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( A  =  R  /\  C  =  S )  ->  B  e.  V ) ) ) )
72713imp 1181 . . . . . . . 8  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( ( A  =  R  /\  C  =  S )  ->  B  e.  V ) )
7372impcom 430 . . . . . . 7  |-  ( ( ( A  =  R  /\  C  =  S )  /\  ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) )  ->  B  e.  V
)
74 simpl 457 . . . . . . . . . . . 12  |-  ( ( A  =  R  /\  C  =  S )  ->  A  =  R )
7574adantr 465 . . . . . . . . . . 11  |-  ( ( ( A  =  R  /\  C  =  S )  /\  ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) )  ->  A  =  R )
7675adantr 465 . . . . . . . . . 10  |-  ( ( ( ( A  =  R  /\  C  =  S )  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  /\  b  =  B )  ->  A  =  R )
77 eqcom 2440 . . . . . . . . . . . 12  |-  ( b  =  B  <->  B  =  b )
7877biimpi 194 . . . . . . . . . . 11  |-  ( b  =  B  ->  B  =  b )
7978adantl 466 . . . . . . . . . 10  |-  ( ( ( ( A  =  R  /\  C  =  S )  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  /\  b  =  B )  ->  B  =  b )
80 simpr 461 . . . . . . . . . . . 12  |-  ( ( A  =  R  /\  C  =  S )  ->  C  =  S )
8180adantr 465 . . . . . . . . . . 11  |-  ( ( ( A  =  R  /\  C  =  S )  /\  ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) )  ->  C  =  S )
8281adantr 465 . . . . . . . . . 10  |-  ( ( ( ( A  =  R  /\  C  =  S )  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  /\  b  =  B )  ->  C  =  S )
8376, 79, 82oteq123d 4069 . . . . . . . . 9  |-  ( ( ( ( A  =  R  /\  C  =  S )  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  /\  b  =  B )  ->  <. A ,  B ,  C >.  = 
<. R ,  b ,  S >. )
84 simpr1 994 . . . . . . . . . . 11  |-  ( ( ( A  =  R  /\  C  =  S )  /\  ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) )  ->  f ( V Walks 
E ) p )
8584adantr 465 . . . . . . . . . 10  |-  ( ( ( ( A  =  R  /\  C  =  S )  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  /\  b  =  B )  ->  f
( V Walks  E )
p )
86 simplr2 1031 . . . . . . . . . 10  |-  ( ( ( ( A  =  R  /\  C  =  S )  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  /\  b  =  B )  ->  ( # `
 f )  =  2 )
87 eqtr2 2456 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  =  R  /\  A  =  ( p `  0 ) )  ->  R  =  ( p `  0 ) )
8887ex 434 . . . . . . . . . . . . . . . . . 18  |-  ( A  =  R  ->  ( A  =  ( p `  0 )  ->  R  =  ( p `  0 ) ) )
8988adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( A  =  R  /\  C  =  S )  ->  ( A  =  ( p `  0 )  ->  R  =  ( p `  0 ) ) )
9089adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( b  =  B  /\  ( A  =  R  /\  C  =  S
) )  ->  ( A  =  ( p `  0 )  ->  R  =  ( p `  0 ) ) )
91 eqtr2 2456 . . . . . . . . . . . . . . . . . . 19  |-  ( ( B  =  b  /\  B  =  ( p `  1 ) )  ->  b  =  ( p `  1 ) )
9291ex 434 . . . . . . . . . . . . . . . . . 18  |-  ( B  =  b  ->  ( B  =  ( p `  1 )  -> 
b  =  ( p `
 1 ) ) )
9392eqcoms 2441 . . . . . . . . . . . . . . . . 17  |-  ( b  =  B  ->  ( B  =  ( p `  1 )  -> 
b  =  ( p `
 1 ) ) )
9493adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( b  =  B  /\  ( A  =  R  /\  C  =  S
) )  ->  ( B  =  ( p `  1 )  -> 
b  =  ( p `
 1 ) ) )
95 eqtr2 2456 . . . . . . . . . . . . . . . . . . 19  |-  ( ( C  =  S  /\  C  =  ( p `  2 ) )  ->  S  =  ( p `  2 ) )
9695ex 434 . . . . . . . . . . . . . . . . . 18  |-  ( C  =  S  ->  ( C  =  ( p `  2 )  ->  S  =  ( p `  2 ) ) )
9796adantl 466 . . . . . . . . . . . . . . . . 17  |-  ( ( A  =  R  /\  C  =  S )  ->  ( C  =  ( p `  2 )  ->  S  =  ( p `  2 ) ) )
9897adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( b  =  B  /\  ( A  =  R  /\  C  =  S
) )  ->  ( C  =  ( p `  2 )  ->  S  =  ( p `  2 ) ) )
9990, 94, 983anim123d 1296 . . . . . . . . . . . . . . 15  |-  ( ( b  =  B  /\  ( A  =  R  /\  C  =  S
) )  ->  (
( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( R  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  S  =  ( p ` 
2 ) ) ) )
10099ex 434 . . . . . . . . . . . . . 14  |-  ( b  =  B  ->  (
( A  =  R  /\  C  =  S )  ->  ( ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( R  =  ( p `  0
)  /\  b  =  ( p `  1
)  /\  S  =  ( p `  2
) ) ) ) )
101100com13 80 . . . . . . . . . . . . 13  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( A  =  R  /\  C  =  S )  ->  (
b  =  B  -> 
( R  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  S  =  ( p `  2 ) ) ) ) )
1021013ad2ant3 1011 . . . . . . . . . . . 12  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( ( A  =  R  /\  C  =  S )  ->  (
b  =  B  -> 
( R  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  S  =  ( p `  2 ) ) ) ) )
103102impcom 430 . . . . . . . . . . 11  |-  ( ( ( A  =  R  /\  C  =  S )  /\  ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) )  ->  ( b  =  B  ->  ( R  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  S  =  ( p ` 
2 ) ) ) )
104103imp 429 . . . . . . . . . 10  |-  ( ( ( ( A  =  R  /\  C  =  S )  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  /\  b  =  B )  ->  ( R  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  S  =  ( p `  2 ) ) )
10585, 86, 1043jca 1168 . . . . . . . . 9  |-  ( ( ( ( A  =  R  /\  C  =  S )  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  /\  b  =  B )  ->  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( R  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  S  =  ( p ` 
2 ) ) ) )
10683, 105jca 532 . . . . . . . 8  |-  ( ( ( ( A  =  R  /\  C  =  S )  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  /\  b  =  B )  ->  ( <. A ,  B ,  C >.  =  <. R , 
b ,  S >.  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( R  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  S  =  ( p `  2 ) ) ) ) )
107106a1d 25 . . . . . . 7  |-  ( ( ( ( A  =  R  /\  C  =  S )  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  /\  b  =  B )  ->  (
( ( V  e.  X  /\  E  e.  Y )  /\  ( R  e.  V  /\  S  e.  V )
)  ->  ( <. A ,  B ,  C >.  =  <. R ,  b ,  S >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( R  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  S  =  ( p ` 
2 ) ) ) ) ) )
10873, 107rspcimedv 3070 . . . . . 6  |-  ( ( ( A  =  R  /\  C  =  S )  /\  ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) )  ->  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( R  e.  V  /\  S  e.  V
) )  ->  E. b  e.  V  ( <. A ,  B ,  C >.  =  <. R ,  b ,  S >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( R  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  S  =  ( p ` 
2 ) ) ) ) ) )
109108com12 31 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( R  e.  V  /\  S  e.  V ) )  -> 
( ( ( A  =  R  /\  C  =  S )  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  ->  E. b  e.  V  ( <. A ,  B ,  C >.  =  <. R ,  b ,  S >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( R  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  S  =  ( p ` 
2 ) ) ) ) ) )
11060, 109impbid 191 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( R  e.  V  /\  S  e.  V ) )  -> 
( E. b  e.  V  ( <. A ,  B ,  C >.  = 
<. R ,  b ,  S >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( R  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  S  =  ( p ` 
2 ) ) ) )  <->  ( ( A  =  R  /\  C  =  S )  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) ) )
1111102exbidv 1682 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( R  e.  V  /\  S  e.  V ) )  -> 
( E. f E. p E. b  e.  V  ( <. A ,  B ,  C >.  = 
<. R ,  b ,  S >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( R  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  S  =  ( p ` 
2 ) ) ) )  <->  E. f E. p
( ( A  =  R  /\  C  =  S )  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) ) )
112 df-3an 967 . . . 4  |-  ( ( A  =  R  /\  C  =  S  /\  E. f E. p ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  <->  ( ( A  =  R  /\  C  =  S )  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) ) )
113 19.42vv 1925 . . . . 5  |-  ( E. f E. p ( ( A  =  R  /\  C  =  S )  /\  ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) )  <-> 
( ( A  =  R  /\  C  =  S )  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  B  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) ) )
114113bicomi 202 . . . 4  |-  ( ( ( A  =  R  /\  C  =  S )  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )  <->  E. f E. p ( ( A  =  R  /\  C  =  S )  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) )
115112, 114bitri 249 . . 3  |-  ( ( A  =  R  /\  C  =  S  /\  E. f E. p ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  <->  E. f E. p
( ( A  =  R  /\  C  =  S )  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) )
116111, 115syl6bbr 263 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( R  e.  V  /\  S  e.  V ) )  -> 
( E. f E. p E. b  e.  V  ( <. A ,  B ,  C >.  = 
<. R ,  b ,  S >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( R  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  S  =  ( p ` 
2 ) ) ) )  <->  ( A  =  R  /\  C  =  S  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) )
1171, 9, 1163bitrd 279 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( R  e.  V  /\  S  e.  V ) )  -> 
( <. A ,  B ,  C >.  e.  ( R ( V 2WalksOnOt  E ) S )  <->  ( A  =  R  /\  C  =  S  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756   E.wrex 2711   _Vcvv 2967   <.cop 3878   <.cotp 3880   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   0cc0 9274   1c1 9275   2c2 10363   #chash 12095   Walks cwalk 23356   2WalksOnOt c2wlkonot 30327
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-ot 3881  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-fzo 11541  df-word 12221  df-wlk 23366  df-wlkon 23372  df-2wlkonot 30330
This theorem is referenced by:  el2wlkonotot  30345  el2wlkonotot1  30346  el2wlksotot  30354
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