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Theorem usg2spthonot0 25696
Description: A simple path of length 2 between two vertices as ordered triple corresponds to two adjacent edges in an undirected simple graph. (Contributed by Alexander van der Vekens, 8-Mar-2018.)
Assertion
Ref Expression
usg2spthonot0  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( <. S ,  B ,  T >.  e.  ( A ( V 2SPathOnOt  E ) C )  <-> 
( ( S  =  A  /\  T  =  C  /\  A  =/= 
C )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) )

Proof of Theorem usg2spthonot0
StepHypRef Expression
1 ne0i 3728 . . . . 5  |-  ( <. S ,  B ,  T >.  e.  ( A ( V 2SPathOnOt  E ) C )  ->  ( A ( V 2SPathOnOt  E ) C )  =/=  (/) )
2 2spontn0vne 25694 . . . . 5  |-  ( ( A ( V 2SPathOnOt  E ) C )  =/=  (/)  ->  A  =/=  C )
31, 2syl 17 . . . 4  |-  ( <. S ,  B ,  T >.  e.  ( A ( V 2SPathOnOt  E ) C )  ->  A  =/=  C )
4 simpl 464 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  V USGrph  E )
54adantl 473 . . . . . . . . . 10  |-  ( ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
) )  ->  V USGrph  E )
6 3simpb 1028 . . . . . . . . . . . 12  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( A  e.  V  /\  C  e.  V
) )
76adantl 473 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  e.  V  /\  C  e.  V ) )
87adantl 473 . . . . . . . . . 10  |-  ( ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
) )  ->  ( A  e.  V  /\  C  e.  V )
)
9 simpl 464 . . . . . . . . . 10  |-  ( ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
) )  ->  A  =/=  C )
10 2pthwlkonot 25692 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V )  /\  A  =/=  C
)  ->  ( A
( V 2SPathOnOt  E ) C )  =  ( A ( V 2WalksOnOt  E ) C ) )
115, 8, 9, 10syl3anc 1292 . . . . . . . . 9  |-  ( ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
) )  ->  ( A ( V 2SPathOnOt  E ) C )  =  ( A ( V 2WalksOnOt  E ) C ) )
1211eleq2d 2534 . . . . . . . 8  |-  ( ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
) )  ->  ( <. S ,  B ,  T >.  e.  ( A ( V 2SPathOnOt  E ) C )  <->  <. S ,  B ,  T >.  e.  ( A ( V 2WalksOnOt  E ) C ) ) )
13 usgrav 25144 . . . . . . . . . . . 12  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
1413, 6anim12i 576 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )
) )
1514adantl 473 . . . . . . . . . 10  |-  ( ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
) )  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
) ) )
16 el2wlkonotot1 25681 . . . . . . . . . 10  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
) )  ->  ( <. S ,  B ,  T >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  ( S  =  A  /\  T  =  C  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) ) ) )
1715, 16syl 17 . . . . . . . . 9  |-  ( ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
) )  ->  ( <. S ,  B ,  T >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  ( S  =  A  /\  T  =  C  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) ) ) )
18 df-3an 1009 . . . . . . . . 9  |-  ( ( S  =  A  /\  T  =  C  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) )  <->  ( ( S  =  A  /\  T  =  C )  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) ) )
1917, 18syl6bb 269 . . . . . . . 8  |-  ( ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
) )  ->  ( <. S ,  B ,  T >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  ( ( S  =  A  /\  T  =  C )  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) ) ) )
2012, 19bitrd 261 . . . . . . 7  |-  ( ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
) )  ->  ( <. S ,  B ,  T >.  e.  ( A ( V 2SPathOnOt  E ) C )  <->  ( ( S  =  A  /\  T  =  C )  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) ) ) )
21 simpll 768 . . . . . . . . . . . . 13  |-  ( ( ( S  =  A  /\  T  =  C )  /\  A  =/= 
C )  ->  S  =  A )
22 simpr 468 . . . . . . . . . . . . . 14  |-  ( ( S  =  A  /\  T  =  C )  ->  T  =  C )
2322adantr 472 . . . . . . . . . . . . 13  |-  ( ( ( S  =  A  /\  T  =  C )  /\  A  =/= 
C )  ->  T  =  C )
24 simpr 468 . . . . . . . . . . . . 13  |-  ( ( ( S  =  A  /\  T  =  C )  /\  A  =/= 
C )  ->  A  =/=  C )
2521, 23, 243jca 1210 . . . . . . . . . . . 12  |-  ( ( ( S  =  A  /\  T  =  C )  /\  A  =/= 
C )  ->  ( S  =  A  /\  T  =  C  /\  A  =/=  C ) )
2625ex 441 . . . . . . . . . . 11  |-  ( ( S  =  A  /\  T  =  C )  ->  ( A  =/=  C  ->  ( S  =  A  /\  T  =  C  /\  A  =/=  C
) ) )
2726adantr 472 . . . . . . . . . 10  |-  ( ( ( S  =  A  /\  T  =  C )  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) )  ->  ( A  =/=  C  ->  ( S  =  A  /\  T  =  C  /\  A  =/= 
C ) ) )
2827com12 31 . . . . . . . . 9  |-  ( A  =/=  C  ->  (
( ( S  =  A  /\  T  =  C )  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) )  ->  ( S  =  A  /\  T  =  C  /\  A  =/= 
C ) ) )
2928adantr 472 . . . . . . . 8  |-  ( ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
) )  ->  (
( ( S  =  A  /\  T  =  C )  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) )  ->  ( S  =  A  /\  T  =  C  /\  A  =/= 
C ) ) )
305adantl 473 . . . . . . . . . . . 12  |-  ( ( ( S  =  A  /\  T  =  C )  /\  ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) ) ) )  ->  V USGrph  E )
31 simprrr 783 . . . . . . . . . . . . 13  |-  ( ( ( S  =  A  /\  T  =  C )  /\  ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) ) ) )  ->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )
32 eleq1 2537 . . . . . . . . . . . . . . . 16  |-  ( S  =  A  ->  ( S  e.  V  <->  A  e.  V ) )
3332adantr 472 . . . . . . . . . . . . . . 15  |-  ( ( S  =  A  /\  T  =  C )  ->  ( S  e.  V  <->  A  e.  V ) )
34 eleq1 2537 . . . . . . . . . . . . . . . 16  |-  ( T  =  C  ->  ( T  e.  V  <->  C  e.  V ) )
3534adantl 473 . . . . . . . . . . . . . . 15  |-  ( ( S  =  A  /\  T  =  C )  ->  ( T  e.  V  <->  C  e.  V ) )
3633, 353anbi13d 1367 . . . . . . . . . . . . . 14  |-  ( ( S  =  A  /\  T  =  C )  ->  ( ( S  e.  V  /\  B  e.  V  /\  T  e.  V )  <->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) ) )
3736adantr 472 . . . . . . . . . . . . 13  |-  ( ( ( S  =  A  /\  T  =  C )  /\  ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) ) ) )  ->  ( ( S  e.  V  /\  B  e.  V  /\  T  e.  V )  <->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
) )
3831, 37mpbird 240 . . . . . . . . . . . 12  |-  ( ( ( S  =  A  /\  T  =  C )  /\  ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) ) ) )  ->  ( S  e.  V  /\  B  e.  V  /\  T  e.  V ) )
39 usg2wlkonot 25690 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  ( S  e.  V  /\  B  e.  V  /\  T  e.  V )
)  ->  ( <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T )  <-> 
( { S ,  B }  e.  ran  E  /\  { B ,  T }  e.  ran  E ) ) )
4030, 38, 39syl2anc 673 . . . . . . . . . . 11  |-  ( ( ( S  =  A  /\  T  =  C )  /\  ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) ) ) )  ->  ( <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T )  <-> 
( { S ,  B }  e.  ran  E  /\  { B ,  T }  e.  ran  E ) ) )
41 preq1 4042 . . . . . . . . . . . . . . . 16  |-  ( S  =  A  ->  { S ,  B }  =  { A ,  B }
)
4241adantr 472 . . . . . . . . . . . . . . 15  |-  ( ( S  =  A  /\  T  =  C )  ->  { S ,  B }  =  { A ,  B } )
4342eleq1d 2533 . . . . . . . . . . . . . 14  |-  ( ( S  =  A  /\  T  =  C )  ->  ( { S ,  B }  e.  ran  E  <->  { A ,  B }  e.  ran  E ) )
44 preq2 4043 . . . . . . . . . . . . . . . 16  |-  ( T  =  C  ->  { B ,  T }  =  { B ,  C }
)
4544adantl 473 . . . . . . . . . . . . . . 15  |-  ( ( S  =  A  /\  T  =  C )  ->  { B ,  T }  =  { B ,  C } )
4645eleq1d 2533 . . . . . . . . . . . . . 14  |-  ( ( S  =  A  /\  T  =  C )  ->  ( { B ,  T }  e.  ran  E  <->  { B ,  C }  e.  ran  E ) )
4743, 46anbi12d 725 . . . . . . . . . . . . 13  |-  ( ( S  =  A  /\  T  =  C )  ->  ( ( { S ,  B }  e.  ran  E  /\  { B ,  T }  e.  ran  E )  <->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
4847biimpd 212 . . . . . . . . . . . 12  |-  ( ( S  =  A  /\  T  =  C )  ->  ( ( { S ,  B }  e.  ran  E  /\  { B ,  T }  e.  ran  E )  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
4948adantr 472 . . . . . . . . . . 11  |-  ( ( ( S  =  A  /\  T  =  C )  /\  ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) ) ) )  ->  ( ( { S ,  B }  e.  ran  E  /\  { B ,  T }  e.  ran  E )  -> 
( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
5040, 49sylbid 223 . . . . . . . . . 10  |-  ( ( ( S  =  A  /\  T  =  C )  /\  ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) ) ) )  ->  ( <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T )  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
5150impancom 447 . . . . . . . . 9  |-  ( ( ( S  =  A  /\  T  =  C )  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) )  ->  ( ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) ) )  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
5251com12 31 . . . . . . . 8  |-  ( ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
) )  ->  (
( ( S  =  A  /\  T  =  C )  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) )  ->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
5329, 52jcad 542 . . . . . . 7  |-  ( ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
) )  ->  (
( ( S  =  A  /\  T  =  C )  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) )  ->  ( ( S  =  A  /\  T  =  C  /\  A  =/=  C )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) )
5420, 53sylbid 223 . . . . . 6  |-  ( ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
) )  ->  ( <. S ,  B ,  T >.  e.  ( A ( V 2SPathOnOt  E ) C )  ->  (
( S  =  A  /\  T  =  C  /\  A  =/=  C
)  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) )
5554ex 441 . . . . 5  |-  ( A  =/=  C  ->  (
( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  ->  ( <. S ,  B ,  T >.  e.  ( A ( V 2SPathOnOt  E ) C )  ->  (
( S  =  A  /\  T  =  C  /\  A  =/=  C
)  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) ) )
5655com23 80 . . . 4  |-  ( A  =/=  C  ->  ( <. S ,  B ,  T >.  e.  ( A ( V 2SPathOnOt  E ) C )  ->  (
( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  ->  (
( S  =  A  /\  T  =  C  /\  A  =/=  C
)  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) ) )
573, 56mpcom 36 . . 3  |-  ( <. S ,  B ,  T >.  e.  ( A ( V 2SPathOnOt  E ) C )  ->  (
( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  ->  (
( S  =  A  /\  T  =  C  /\  A  =/=  C
)  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) )
5857com12 31 . 2  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( <. S ,  B ,  T >.  e.  ( A ( V 2SPathOnOt  E ) C )  ->  ( ( S  =  A  /\  T  =  C  /\  A  =/= 
C )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) )
59 simpll 768 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  /\  ( S  =  A  /\  T  =  C  /\  A  =/=  C ) )  ->  V USGrph  E )
60 eleq1 2537 . . . . . . . . . . . . . . 15  |-  ( A  =  S  ->  ( A  e.  V  <->  S  e.  V ) )
6160eqcoms 2479 . . . . . . . . . . . . . 14  |-  ( S  =  A  ->  ( A  e.  V  <->  S  e.  V ) )
6261adantr 472 . . . . . . . . . . . . 13  |-  ( ( S  =  A  /\  T  =  C )  ->  ( A  e.  V  <->  S  e.  V ) )
63 eleq1 2537 . . . . . . . . . . . . . . 15  |-  ( C  =  T  ->  ( C  e.  V  <->  T  e.  V ) )
6463eqcoms 2479 . . . . . . . . . . . . . 14  |-  ( T  =  C  ->  ( C  e.  V  <->  T  e.  V ) )
6564adantl 473 . . . . . . . . . . . . 13  |-  ( ( S  =  A  /\  T  =  C )  ->  ( C  e.  V  <->  T  e.  V ) )
6662, 653anbi13d 1367 . . . . . . . . . . . 12  |-  ( ( S  =  A  /\  T  =  C )  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  <->  ( S  e.  V  /\  B  e.  V  /\  T  e.  V ) ) )
6766biimpd 212 . . . . . . . . . . 11  |-  ( ( S  =  A  /\  T  =  C )  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( S  e.  V  /\  B  e.  V  /\  T  e.  V )
) )
6867adantld 474 . . . . . . . . . 10  |-  ( ( S  =  A  /\  T  =  C )  ->  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  ->  ( S  e.  V  /\  B  e.  V  /\  T  e.  V )
) )
69683adant3 1050 . . . . . . . . 9  |-  ( ( S  =  A  /\  T  =  C  /\  A  =/=  C )  -> 
( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  ->  ( S  e.  V  /\  B  e.  V  /\  T  e.  V )
) )
7069impcom 437 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  /\  ( S  =  A  /\  T  =  C  /\  A  =/=  C ) )  ->  ( S  e.  V  /\  B  e.  V  /\  T  e.  V ) )
7159, 70, 39syl2anc 673 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  /\  ( S  =  A  /\  T  =  C  /\  A  =/=  C ) )  ->  ( <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T )  <-> 
( { S ,  B }  e.  ran  E  /\  { B ,  T }  e.  ran  E ) ) )
72473adant3 1050 . . . . . . . 8  |-  ( ( S  =  A  /\  T  =  C  /\  A  =/=  C )  -> 
( ( { S ,  B }  e.  ran  E  /\  { B ,  T }  e.  ran  E )  <->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
7372adantl 473 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  /\  ( S  =  A  /\  T  =  C  /\  A  =/=  C ) )  ->  ( ( { S ,  B }  e.  ran  E  /\  { B ,  T }  e.  ran  E )  <->  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
7471, 73bitr2d 262 . . . . . 6  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  /\  ( S  =  A  /\  T  =  C  /\  A  =/=  C ) )  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  <->  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) ) )
7574pm5.32da 653 . . . . 5  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
( S  =  A  /\  T  =  C  /\  A  =/=  C
)  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  <-> 
( ( S  =  A  /\  T  =  C  /\  A  =/= 
C )  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) ) ) )
76 df-3an 1009 . . . . . . . 8  |-  ( ( S  =  A  /\  T  =  C  /\  A  =/=  C )  <->  ( ( S  =  A  /\  T  =  C )  /\  A  =/=  C
) )
77 ancom 457 . . . . . . . 8  |-  ( ( ( S  =  A  /\  T  =  C )  /\  A  =/= 
C )  <->  ( A  =/=  C  /\  ( S  =  A  /\  T  =  C ) ) )
7876, 77bitri 257 . . . . . . 7  |-  ( ( S  =  A  /\  T  =  C  /\  A  =/=  C )  <->  ( A  =/=  C  /\  ( S  =  A  /\  T  =  C ) ) )
7978anbi1i 709 . . . . . 6  |-  ( ( ( S  =  A  /\  T  =  C  /\  A  =/=  C
)  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) )  <->  ( ( A  =/=  C  /\  ( S  =  A  /\  T  =  C )
)  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) ) )
80 anass 661 . . . . . 6  |-  ( ( ( A  =/=  C  /\  ( S  =  A  /\  T  =  C ) )  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) )  <->  ( A  =/= 
C  /\  ( ( S  =  A  /\  T  =  C )  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) ) ) )
8118bicomi 207 . . . . . . 7  |-  ( ( ( S  =  A  /\  T  =  C )  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) )  <->  ( S  =  A  /\  T  =  C  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) ) )
8281anbi2i 708 . . . . . 6  |-  ( ( A  =/=  C  /\  ( ( S  =  A  /\  T  =  C )  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) ) )  <->  ( A  =/=  C  /\  ( S  =  A  /\  T  =  C  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) ) ) )
8379, 80, 823bitri 279 . . . . 5  |-  ( ( ( S  =  A  /\  T  =  C  /\  A  =/=  C
)  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) )  <->  ( A  =/= 
C  /\  ( S  =  A  /\  T  =  C  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) ) ) )
8475, 83syl6bb 269 . . . 4  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
( S  =  A  /\  T  =  C  /\  A  =/=  C
)  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  <-> 
( A  =/=  C  /\  ( S  =  A  /\  T  =  C  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) ) ) ) )
8514, 16syl 17 . . . . . 6  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( <. S ,  B ,  T >.  e.  ( A ( V 2WalksOnOt  E ) C )  <-> 
( S  =  A  /\  T  =  C  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) ) ) )
8685bicomd 206 . . . . 5  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( S  =  A  /\  T  =  C  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) )  <->  <. S ,  B ,  T >.  e.  ( A ( V 2WalksOnOt  E ) C ) ) )
8786anbi2d 718 . . . 4  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( A  =/=  C  /\  ( S  =  A  /\  T  =  C  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) ) )  <-> 
( A  =/=  C  /\  <. S ,  B ,  T >.  e.  ( A ( V 2WalksOnOt  E ) C ) ) ) )
8884, 87bitrd 261 . . 3  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
( S  =  A  /\  T  =  C  /\  A  =/=  C
)  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  <-> 
( A  =/=  C  /\  <. S ,  B ,  T >.  e.  ( A ( V 2WalksOnOt  E ) C ) ) ) )
89 simpll 768 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  /\  A  =/=  C )  ->  V USGrph  E )
907adantr 472 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  /\  A  =/=  C )  ->  ( A  e.  V  /\  C  e.  V )
)
91 simpr 468 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  /\  A  =/=  C )  ->  A  =/=  C )
9210eqcomd 2477 . . . . . . 7  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V )  /\  A  =/=  C
)  ->  ( A
( V 2WalksOnOt  E ) C )  =  ( A ( V 2SPathOnOt  E ) C ) )
9389, 90, 91, 92syl3anc 1292 . . . . . 6  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  /\  A  =/=  C )  ->  ( A ( V 2WalksOnOt  E ) C )  =  ( A ( V 2SPathOnOt  E ) C ) )
9493eleq2d 2534 . . . . 5  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  /\  A  =/=  C )  ->  ( <. S ,  B ,  T >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  <. S ,  B ,  T >.  e.  ( A ( V 2SPathOnOt  E ) C ) ) )
9594biimpd 212 . . . 4  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  /\  A  =/=  C )  ->  ( <. S ,  B ,  T >.  e.  ( A ( V 2WalksOnOt  E ) C )  ->  <. S ,  B ,  T >.  e.  ( A ( V 2SPathOnOt  E ) C ) ) )
9695expimpd 614 . . 3  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( ( A  =/=  C  /\  <. S ,  B ,  T >.  e.  ( A ( V 2WalksOnOt  E ) C ) )  ->  <. S ,  B ,  T >.  e.  ( A ( V 2SPathOnOt  E ) C ) ) )
9788, 96sylbid 223 . 2  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( (
( S  =  A  /\  T  =  C  /\  A  =/=  C
)  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) )  ->  <. S ,  B ,  T >.  e.  ( A ( V 2SPathOnOt  E ) C ) ) )
9858, 97impbid 195 1  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( <. S ,  B ,  T >.  e.  ( A ( V 2SPathOnOt  E ) C )  <-> 
( ( S  =  A  /\  T  =  C  /\  A  =/= 
C )  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   _Vcvv 3031   (/)c0 3722   {cpr 3961   <.cotp 3967   class class class wbr 4395   ran crn 4840  (class class class)co 6308   USGrph cusg 25136   2WalksOnOt c2wlkonot 25662   2SPathOnOt c2pthonot 25664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-ot 3968  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711  df-usgra 25139  df-wlk 25315  df-trail 25316  df-pth 25317  df-spth 25318  df-wlkon 25321  df-spthon 25324  df-2wlkonot 25665  df-2spthonot 25667
This theorem is referenced by:  usg2spthonot1  25697
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