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Theorem usg2spthonot0 28356
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 3634 . . . . 5  |-  ( <. S ,  B ,  T >.  e.  ( A ( V 2SPathOnOt  E ) C )  ->  ( A ( V 2SPathOnOt  E ) C )  =/=  (/) )
2 2spontn0vne 28354 . . . . 5  |-  ( ( A ( V 2SPathOnOt  E ) C )  =/=  (/)  ->  A  =/=  C )
31, 2syl 16 . . . 4  |-  ( <. S ,  B ,  T >.  e.  ( A ( V 2SPathOnOt  E ) C )  ->  A  =/=  C )
4 simpl 444 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  V USGrph  E )
54adantl 453 . . . . . . . . . 10  |-  ( ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
) )  ->  V USGrph  E )
6 3simpb 955 . . . . . . . . . . . 12  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( A  e.  V  /\  C  e.  V
) )
76adantl 453 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  e.  V  /\  C  e.  V ) )
87adantl 453 . . . . . . . . . 10  |-  ( ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
) )  ->  ( A  e.  V  /\  C  e.  V )
)
9 simpl 444 . . . . . . . . . 10  |-  ( ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
) )  ->  A  =/=  C )
10 2pthwlkonot 28352 . . . . . . . . . 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 1184 . . . . . . . . 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 2503 . . . . . . . 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 21371 . . . . . . . . . . . 12  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
1413, 6anim12i 550 . . . . . . . . . . 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 453 . . . . . . . . . 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 28341 . . . . . . . . . 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 16 . . . . . . . . 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 938 . . . . . . . . 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 253 . . . . . . . 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 245 . . . . . . 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 731 . . . . . . . . . . . . 13  |-  ( ( ( S  =  A  /\  T  =  C )  /\  A  =/= 
C )  ->  S  =  A )
22 simpr 448 . . . . . . . . . . . . . 14  |-  ( ( S  =  A  /\  T  =  C )  ->  T  =  C )
2322adantr 452 . . . . . . . . . . . . 13  |-  ( ( ( S  =  A  /\  T  =  C )  /\  A  =/= 
C )  ->  T  =  C )
24 simpr 448 . . . . . . . . . . . . 13  |-  ( ( ( S  =  A  /\  T  =  C )  /\  A  =/= 
C )  ->  A  =/=  C )
2521, 23, 243jca 1134 . . . . . . . . . . . 12  |-  ( ( ( S  =  A  /\  T  =  C )  /\  A  =/= 
C )  ->  ( S  =  A  /\  T  =  C  /\  A  =/=  C ) )
2625ex 424 . . . . . . . . . . 11  |-  ( ( S  =  A  /\  T  =  C )  ->  ( A  =/=  C  ->  ( S  =  A  /\  T  =  C  /\  A  =/=  C
) ) )
2726adantr 452 . . . . . . . . . 10  |-  ( ( ( S  =  A  /\  T  =  C )  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) )  ->  ( A  =/=  C  ->  ( S  =  A  /\  T  =  C  /\  A  =/= 
C ) ) )
2827com12 29 . . . . . . . . 9  |-  ( A  =/=  C  ->  (
( ( S  =  A  /\  T  =  C )  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) )  ->  ( S  =  A  /\  T  =  C  /\  A  =/= 
C ) ) )
2928adantr 452 . . . . . . . 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 453 . . . . . . . . . . . 12  |-  ( ( ( S  =  A  /\  T  =  C )  /\  ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) ) ) )  ->  V USGrph  E )
31 simprrr 742 . . . . . . . . . . . . 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 2496 . . . . . . . . . . . . . . . 16  |-  ( S  =  A  ->  ( S  e.  V  <->  A  e.  V ) )
3332adantr 452 . . . . . . . . . . . . . . 15  |-  ( ( S  =  A  /\  T  =  C )  ->  ( S  e.  V  <->  A  e.  V ) )
34 eleq1 2496 . . . . . . . . . . . . . . . 16  |-  ( T  =  C  ->  ( T  e.  V  <->  C  e.  V ) )
3534adantl 453 . . . . . . . . . . . . . . 15  |-  ( ( S  =  A  /\  T  =  C )  ->  ( T  e.  V  <->  C  e.  V ) )
3633, 353anbi13d 1256 . . . . . . . . . . . . . 14  |-  ( ( S  =  A  /\  T  =  C )  ->  ( ( S  e.  V  /\  B  e.  V  /\  T  e.  V )  <->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) ) )
3736adantr 452 . . . . . . . . . . . . 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 224 . . . . . . . . . . . 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 28350 . . . . . . . . . . . 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 643 . . . . . . . . . . 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 3883 . . . . . . . . . . . . . . . 16  |-  ( S  =  A  ->  { S ,  B }  =  { A ,  B }
)
4241adantr 452 . . . . . . . . . . . . . . 15  |-  ( ( S  =  A  /\  T  =  C )  ->  { S ,  B }  =  { A ,  B } )
4342eleq1d 2502 . . . . . . . . . . . . . 14  |-  ( ( S  =  A  /\  T  =  C )  ->  ( { S ,  B }  e.  ran  E  <->  { A ,  B }  e.  ran  E ) )
44 preq2 3884 . . . . . . . . . . . . . . . 16  |-  ( T  =  C  ->  { B ,  T }  =  { B ,  C }
)
4544adantl 453 . . . . . . . . . . . . . . 15  |-  ( ( S  =  A  /\  T  =  C )  ->  { B ,  T }  =  { B ,  C } )
4645eleq1d 2502 . . . . . . . . . . . . . 14  |-  ( ( S  =  A  /\  T  =  C )  ->  ( { B ,  T }  e.  ran  E  <->  { B ,  C }  e.  ran  E ) )
4743, 46anbi12d 692 . . . . . . . . . . . . 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 199 . . . . . . . . . . . 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 452 . . . . . . . . . . 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 207 . . . . . . . . . 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 428 . . . . . . . . 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 29 . . . . . . . 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 520 . . . . . . 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 207 . . . . . 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 424 . . . . 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 74 . . . 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 34 . . 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 29 . 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 731 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  /\  ( S  =  A  /\  T  =  C  /\  A  =/=  C ) )  ->  V USGrph  E )
60 eleq1 2496 . . . . . . . . . . . . . . 15  |-  ( A  =  S  ->  ( A  e.  V  <->  S  e.  V ) )
6160eqcoms 2439 . . . . . . . . . . . . . 14  |-  ( S  =  A  ->  ( A  e.  V  <->  S  e.  V ) )
6261adantr 452 . . . . . . . . . . . . 13  |-  ( ( S  =  A  /\  T  =  C )  ->  ( A  e.  V  <->  S  e.  V ) )
63 eleq1 2496 . . . . . . . . . . . . . . 15  |-  ( C  =  T  ->  ( C  e.  V  <->  T  e.  V ) )
6463eqcoms 2439 . . . . . . . . . . . . . 14  |-  ( T  =  C  ->  ( C  e.  V  <->  T  e.  V ) )
6564adantl 453 . . . . . . . . . . . . 13  |-  ( ( S  =  A  /\  T  =  C )  ->  ( C  e.  V  <->  T  e.  V ) )
6662, 653anbi13d 1256 . . . . . . . . . . . 12  |-  ( ( S  =  A  /\  T  =  C )  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  <->  ( S  e.  V  /\  B  e.  V  /\  T  e.  V ) ) )
6766biimpd 199 . . . . . . . . . . 11  |-  ( ( S  =  A  /\  T  =  C )  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( S  e.  V  /\  B  e.  V  /\  T  e.  V )
) )
6867adantld 454 . . . . . . . . . 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 977 . . . . . . . . 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 420 . . . . . . . 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 643 . . . . . . 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 977 . . . . . . . 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 453 . . . . . . 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 246 . . . . . 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 623 . . . . 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 938 . . . . . . . 8  |-  ( ( S  =  A  /\  T  =  C  /\  A  =/=  C )  <->  ( ( S  =  A  /\  T  =  C )  /\  A  =/=  C
) )
77 ancom 438 . . . . . . . 8  |-  ( ( ( S  =  A  /\  T  =  C )  /\  A  =/= 
C )  <->  ( A  =/=  C  /\  ( S  =  A  /\  T  =  C ) ) )
7876, 77bitri 241 . . . . . . 7  |-  ( ( S  =  A  /\  T  =  C  /\  A  =/=  C )  <->  ( A  =/=  C  /\  ( S  =  A  /\  T  =  C ) ) )
7978anbi1i 677 . . . . . 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 631 . . . . . 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 194 . . . . . . 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 676 . . . . . 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 263 . . . . 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 253 . . . 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 16 . . . . . 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 193 . . . . 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 685 . . . 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 245 . . 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 731 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  /\  A  =/=  C )  ->  V USGrph  E )
907adantr 452 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  /\  A  =/=  C )  ->  ( A  e.  V  /\  C  e.  V )
)
91 simpr 448 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  /\  A  =/=  C )  ->  A  =/=  C )
9210eqcomd 2441 . . . . . . 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 1184 . . . . . 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 2503 . . . . 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 199 . . . 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 587 . . 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 207 . 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 184 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2956   (/)c0 3628   {cpr 3815   <.cotp 3818   class class class wbr 4212   ran crn 4879  (class class class)co 6081   USGrph cusg 21365   2WalksOnOt c2wlkonot 28322   2SPathOnOt c2pthonot 28324
This theorem is referenced by:  usg2spthonot1  28357
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-ot 3824  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-fzo 11136  df-hash 11619  df-word 11723  df-usgra 21367  df-wlk 21516  df-trail 21517  df-pth 21518  df-spth 21519  df-wlkon 21522  df-spthon 21525  df-2wlkonot 28325  df-2spthonot 28327
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