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Theorem el2wlkonotot0 25301
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 25298 . 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 1803 . . . . . 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 204 . . . . 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 2908 . . . 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 3081 . . . 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 3081 . . . . 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 1690 . . . 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 273 . . 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 2413 . . . . . . . . . . 11  |-  ( <. A ,  B ,  C >.  =  <. R , 
b ,  S >.  <->  <. R ,  b ,  S >.  =  <. A ,  B ,  C >. )
11 df-ot 3983 . . . . . . . . . . . 12  |-  <. R , 
b ,  S >.  = 
<. <. R ,  b
>. ,  S >.
12 df-ot 3983 . . . . . . . . . . . 12  |-  <. A ,  B ,  C >.  = 
<. <. A ,  B >. ,  C >.
1311, 12eqeq12i 2424 . . . . . . . . . . 11  |-  ( <. R ,  b ,  S >.  =  <. A ,  B ,  C >.  <->  <. <. R ,  b >. ,  S >.  =  <. <. A ,  B >. ,  C >. )
1410, 13bitri 251 . . . . . . . . . 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 4657 . . . . . . . . . . . . . 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 4668 . . . . . . . . . 10  |-  ( (
<. R ,  b >.  e.  _V  /\  S  e.  V )  ->  ( <. <. R ,  b
>. ,  S >.  = 
<. <. A ,  B >. ,  C >.  <->  ( <. R ,  b >.  =  <. A ,  B >.  /\  S  =  C ) ) )
2321, 22syl 17 . . . . . . . . 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 4668 . . . . . . . . . . 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 705 . . . . . . . . 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 281 . . . . . . . 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 2413 . . . . . . . . . . . . . 14  |-  ( R  =  A  <->  A  =  R )
3130biimpi 196 . . . . . . . . . . . . 13  |-  ( R  =  A  ->  A  =  R )
3231adantr 465 . . . . . . . . . . . 12  |-  ( ( R  =  A  /\  b  =  B )  ->  A  =  R )
33 eqcom 2413 . . . . . . . . . . . . 13  |-  ( S  =  C  <->  C  =  S )
3433biimpi 196 . . . . . . . . . . . 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 1005 . . . . . . . . . . 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 simpr2 1006 . . . . . . . . . . 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 )
39 eqtr2 2431 . . . . . . . . . . . . . . . . 17  |-  ( ( R  =  A  /\  R  =  ( p `  0 ) )  ->  A  =  ( p `  0 ) )
4039ex 434 . . . . . . . . . . . . . . . 16  |-  ( R  =  A  ->  ( R  =  ( p `  0 )  ->  A  =  ( p `  0 ) ) )
4140ad2antrr 726 . . . . . . . . . . . . . . 15  |-  ( ( ( R  =  A  /\  b  =  B )  /\  S  =  C )  ->  ( R  =  ( p `  0 )  ->  A  =  ( p `  0 ) ) )
42 eqtr2 2431 . . . . . . . . . . . . . . . . . 18  |-  ( ( b  =  B  /\  b  =  ( p `  1 ) )  ->  B  =  ( p `  1 ) )
4342ex 434 . . . . . . . . . . . . . . . . 17  |-  ( b  =  B  ->  (
b  =  ( p `
 1 )  ->  B  =  ( p `  1 ) ) )
4443adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( R  =  A  /\  b  =  B )  ->  ( b  =  ( p `  1 )  ->  B  =  ( p `  1 ) ) )
4544adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( R  =  A  /\  b  =  B )  /\  S  =  C )  ->  (
b  =  ( p `
 1 )  ->  B  =  ( p `  1 ) ) )
46 eqtr2 2431 . . . . . . . . . . . . . . . . 17  |-  ( ( S  =  C  /\  S  =  ( p `  2 ) )  ->  C  =  ( p `  2 ) )
4746ex 434 . . . . . . . . . . . . . . . 16  |-  ( S  =  C  ->  ( S  =  ( p `  2 )  ->  C  =  ( p `  2 ) ) )
4847adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( ( R  =  A  /\  b  =  B )  /\  S  =  C )  ->  ( S  =  ( p `  2 )  ->  C  =  ( p `  2 ) ) )
4941, 45, 483anim123d 1310 . . . . . . . . . . . . . 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 ) ) ) )
5049com12 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 ) ) ) )
51503ad2ant3 1022 . . . . . . . . . . . 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 ) ) ) )
5251impcom 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 ) ) )
5337, 38, 523jca 1179 . . . . . . . . . 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 ) ) ) )
5436, 53jca 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 ) ) ) ) )
5554ex 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 ) ) ) ) ) )
5629, 55syl6bi 230 . . . . . . 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
) ) ) ) ) ) )
5756impd 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 ) ) ) ) ) )
5857rexlimdva 2898 . . . . 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 ) ) ) ) ) )
59 el2wlkonotlem 25291 . . . . . . . . . . . . . . . . 17  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2 )  ->  (
p `  1 )  e.  V )
6059adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2 )  /\  B  =  ( p `  1 ) )  ->  ( p ` 
1 )  e.  V
)
61 eleq1 2476 . . . . . . . . . . . . . . . . 17  |-  ( B  =  ( p ` 
1 )  ->  ( B  e.  V  <->  ( p `  1 )  e.  V ) )
6261adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2 )  /\  B  =  ( p `  1 ) )  ->  ( B  e.  V  <->  ( p ` 
1 )  e.  V
) )
6360, 62mpbird 234 . . . . . . . . . . . . . . 15  |-  ( ( ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2 )  /\  B  =  ( p `  1 ) )  ->  B  e.  V
)
6463a1d 26 . . . . . . . . . . . . . 14  |-  ( ( ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2 )  /\  B  =  ( p `  1 ) )  ->  ( ( A  =  R  /\  C  =  S )  ->  B  e.  V ) )
6564ex 434 . . . . . . . . . . . . 13  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2 )  ->  ( B  =  ( p `  1 )  -> 
( ( A  =  R  /\  C  =  S )  ->  B  e.  V ) ) )
6665ex 434 . . . . . . . . . . . 12  |-  ( f ( V Walks  E ) p  ->  ( ( # `
 f )  =  2  ->  ( B  =  ( p ` 
1 )  ->  (
( A  =  R  /\  C  =  S )  ->  B  e.  V ) ) ) )
6766com13 82 . . . . . . . . . . 11  |-  ( B  =  ( p ` 
1 )  ->  (
( # `  f )  =  2  ->  (
f ( V Walks  E
) p  ->  (
( A  =  R  /\  C  =  S )  ->  B  e.  V ) ) ) )
68673ad2ant2 1021 . . . . . . . . . 10  |-  ( ( A  =  ( p `
 0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( # `  f )  =  2  ->  ( f ( V Walks  E ) p  ->  ( ( A  =  R  /\  C  =  S )  ->  B  e.  V ) ) ) )
6968com13 82 . . . . . . . . 9  |-  ( f ( V Walks  E ) p  ->  ( ( # `
 f )  =  2  ->  ( ( A  =  ( p `  0 )  /\  B  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( A  =  R  /\  C  =  S )  ->  B  e.  V ) ) ) )
70693imp 1193 . . . . . . . 8  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  B  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( ( A  =  R  /\  C  =  S )  ->  B  e.  V ) )
7170impcom 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
)
72 simpl 457 . . . . . . . . . . 11  |-  ( ( A  =  R  /\  C  =  S )  ->  A  =  R )
7372ad2antrr 726 . . . . . . . . . 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 )
74 eqcom 2413 . . . . . . . . . . . 12  |-  ( b  =  B  <->  B  =  b )
7574biimpi 196 . . . . . . . . . . 11  |-  ( b  =  B  ->  B  =  b )
7675adantl 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 )
77 simpr 461 . . . . . . . . . . 11  |-  ( ( A  =  R  /\  C  =  S )  ->  C  =  S )
7877ad2antrr 726 . . . . . . . . . 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 )
7973, 76, 78oteq123d 4176 . . . . . . . . 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 >. )
80 simpr1 1005 . . . . . . . . . . 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 )
8180adantr 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 )
82 simplr2 1042 . . . . . . . . . 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 )
83 eqtr2 2431 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  =  R  /\  A  =  ( p `  0 ) )  ->  R  =  ( p `  0 ) )
8483ex 434 . . . . . . . . . . . . . . . . . 18  |-  ( A  =  R  ->  ( A  =  ( p `  0 )  ->  R  =  ( p `  0 ) ) )
8584adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( A  =  R  /\  C  =  S )  ->  ( A  =  ( p `  0 )  ->  R  =  ( p `  0 ) ) )
8685adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( b  =  B  /\  ( A  =  R  /\  C  =  S
) )  ->  ( A  =  ( p `  0 )  ->  R  =  ( p `  0 ) ) )
87 eqtr2 2431 . . . . . . . . . . . . . . . . . . 19  |-  ( ( B  =  b  /\  B  =  ( p `  1 ) )  ->  b  =  ( p `  1 ) )
8887ex 434 . . . . . . . . . . . . . . . . . 18  |-  ( B  =  b  ->  ( B  =  ( p `  1 )  -> 
b  =  ( p `
 1 ) ) )
8988eqcoms 2416 . . . . . . . . . . . . . . . . 17  |-  ( b  =  B  ->  ( B  =  ( p `  1 )  -> 
b  =  ( p `
 1 ) ) )
9089adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( b  =  B  /\  ( A  =  R  /\  C  =  S
) )  ->  ( B  =  ( p `  1 )  -> 
b  =  ( p `
 1 ) ) )
91 eqtr2 2431 . . . . . . . . . . . . . . . . . . 19  |-  ( ( C  =  S  /\  C  =  ( p `  2 ) )  ->  S  =  ( p `  2 ) )
9291ex 434 . . . . . . . . . . . . . . . . . 18  |-  ( C  =  S  ->  ( C  =  ( p `  2 )  ->  S  =  ( p `  2 ) ) )
9392adantl 466 . . . . . . . . . . . . . . . . 17  |-  ( ( A  =  R  /\  C  =  S )  ->  ( C  =  ( p `  2 )  ->  S  =  ( p `  2 ) ) )
9493adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( b  =  B  /\  ( A  =  R  /\  C  =  S
) )  ->  ( C  =  ( p `  2 )  ->  S  =  ( p `  2 ) ) )
9586, 90, 943anim123d 1310 . . . . . . . . . . . . . . 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 ) ) ) )
9695ex 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
) ) ) ) )
9796com13 82 . . . . . . . . . . . . 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 ) ) ) ) )
98973ad2ant3 1022 . . . . . . . . . . . 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 ) ) ) ) )
9998impcom 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 ) ) ) )
10099imp 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 ) ) )
10181, 82, 1003jca 1179 . . . . . . . . 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 ) ) ) )
10279, 101jca 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 ) ) ) ) )
103102a1d 26 . . . . . . 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 ) ) ) ) ) )
10471, 103rspcimedv 3164 . . . . . 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 ) ) ) ) ) )
105104com12 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 ) ) ) ) ) )
10658, 105impbid 192 . . . 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 ) ) ) ) ) )
1071062exbidv 1739 . . 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 ) ) ) ) ) )
108 df-3an 978 . . . 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
) ) ) ) )
109 19.42vv 1803 . . . . 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
) ) ) ) )
110109bicomi 204 . . . 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 ) ) ) ) )
111108, 110bitri 251 . . 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 ) ) ) ) )
112107, 111syl6bbr 265 . 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 ) ) ) ) ) )
1131, 9, 1123bitrd 281 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 186    /\ wa 369    /\ w3a 976    = wceq 1407   E.wex 1635    e. wcel 1844   E.wrex 2757   _Vcvv 3061   <.cop 3980   <.cotp 3982   class class class wbr 4397   ` cfv 5571  (class class class)co 6280   0cc0 9524   1c1 9525   2c2 10628   #chash 12454   Walks cwalk 24927   2WalksOnOt c2wlkonot 25284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-ot 3983  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-map 7461  df-pm 7462  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-card 8354  df-cda 8582  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-2 10637  df-n0 10839  df-z 10908  df-uz 11130  df-fz 11729  df-fzo 11857  df-hash 12455  df-word 12593  df-wlk 24937  df-wlkon 24943  df-2wlkonot 25287
This theorem is referenced by:  el2wlkonotot  25302  el2wlkonotot1  25303  el2wlksotot  25311
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