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Theorem usg2spthonot0 25617
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 3737 . . . . 5  |-  ( <. S ,  B ,  T >.  e.  ( A ( V 2SPathOnOt  E ) C )  ->  ( A ( V 2SPathOnOt  E ) C )  =/=  (/) )
2 2spontn0vne 25615 . . . . 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 459 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  V USGrph  E )
54adantl 468 . . . . . . . . . 10  |-  ( ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
) )  ->  V USGrph  E )
6 3simpb 1006 . . . . . . . . . . . 12  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( A  e.  V  /\  C  e.  V
) )
76adantl 468 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  e.  V  /\  C  e.  V ) )
87adantl 468 . . . . . . . . . 10  |-  ( ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
) )  ->  ( A  e.  V  /\  C  e.  V )
)
9 simpl 459 . . . . . . . . . 10  |-  ( ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
) )  ->  A  =/=  C )
10 2pthwlkonot 25613 . . . . . . . . . 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 1268 . . . . . . . . 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 2514 . . . . . . . 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 25065 . . . . . . . . . . . 12  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
1413, 6anim12i 570 . . . . . . . . . . 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 468 . . . . . . . . . 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 25602 . . . . . . . . . 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 987 . . . . . . . . 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 265 . . . . . . . 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 257 . . . . . . 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 760 . . . . . . . . . . . . 13  |-  ( ( ( S  =  A  /\  T  =  C )  /\  A  =/= 
C )  ->  S  =  A )
22 simpr 463 . . . . . . . . . . . . . 14  |-  ( ( S  =  A  /\  T  =  C )  ->  T  =  C )
2322adantr 467 . . . . . . . . . . . . 13  |-  ( ( ( S  =  A  /\  T  =  C )  /\  A  =/= 
C )  ->  T  =  C )
24 simpr 463 . . . . . . . . . . . . 13  |-  ( ( ( S  =  A  /\  T  =  C )  /\  A  =/= 
C )  ->  A  =/=  C )
2521, 23, 243jca 1188 . . . . . . . . . . . 12  |-  ( ( ( S  =  A  /\  T  =  C )  /\  A  =/= 
C )  ->  ( S  =  A  /\  T  =  C  /\  A  =/=  C ) )
2625ex 436 . . . . . . . . . . 11  |-  ( ( S  =  A  /\  T  =  C )  ->  ( A  =/=  C  ->  ( S  =  A  /\  T  =  C  /\  A  =/=  C
) ) )
2726adantr 467 . . . . . . . . . 10  |-  ( ( ( S  =  A  /\  T  =  C )  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) )  ->  ( A  =/=  C  ->  ( S  =  A  /\  T  =  C  /\  A  =/= 
C ) ) )
2827com12 32 . . . . . . . . 9  |-  ( A  =/=  C  ->  (
( ( S  =  A  /\  T  =  C )  /\  <. S ,  B ,  T >.  e.  ( S ( V 2WalksOnOt  E ) T ) )  ->  ( S  =  A  /\  T  =  C  /\  A  =/= 
C ) ) )
2928adantr 467 . . . . . . . 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 468 . . . . . . . . . . . 12  |-  ( ( ( S  =  A  /\  T  =  C )  /\  ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) ) ) )  ->  V USGrph  E )
31 simprrr 775 . . . . . . . . . . . . 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 2517 . . . . . . . . . . . . . . . 16  |-  ( S  =  A  ->  ( S  e.  V  <->  A  e.  V ) )
3332adantr 467 . . . . . . . . . . . . . . 15  |-  ( ( S  =  A  /\  T  =  C )  ->  ( S  e.  V  <->  A  e.  V ) )
34 eleq1 2517 . . . . . . . . . . . . . . . 16  |-  ( T  =  C  ->  ( T  e.  V  <->  C  e.  V ) )
3534adantl 468 . . . . . . . . . . . . . . 15  |-  ( ( S  =  A  /\  T  =  C )  ->  ( T  e.  V  <->  C  e.  V ) )
3633, 353anbi13d 1341 . . . . . . . . . . . . . 14  |-  ( ( S  =  A  /\  T  =  C )  ->  ( ( S  e.  V  /\  B  e.  V  /\  T  e.  V )  <->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) ) )
3736adantr 467 . . . . . . . . . . . . 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 236 . . . . . . . . . . . 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 25611 . . . . . . . . . . . 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 667 . . . . . . . . . . 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 4051 . . . . . . . . . . . . . . . 16  |-  ( S  =  A  ->  { S ,  B }  =  { A ,  B }
)
4241adantr 467 . . . . . . . . . . . . . . 15  |-  ( ( S  =  A  /\  T  =  C )  ->  { S ,  B }  =  { A ,  B } )
4342eleq1d 2513 . . . . . . . . . . . . . 14  |-  ( ( S  =  A  /\  T  =  C )  ->  ( { S ,  B }  e.  ran  E  <->  { A ,  B }  e.  ran  E ) )
44 preq2 4052 . . . . . . . . . . . . . . . 16  |-  ( T  =  C  ->  { B ,  T }  =  { B ,  C }
)
4544adantl 468 . . . . . . . . . . . . . . 15  |-  ( ( S  =  A  /\  T  =  C )  ->  { B ,  T }  =  { B ,  C } )
4645eleq1d 2513 . . . . . . . . . . . . . 14  |-  ( ( S  =  A  /\  T  =  C )  ->  ( { B ,  T }  e.  ran  E  <->  { B ,  C }  e.  ran  E ) )
4743, 46anbi12d 717 . . . . . . . . . . . . 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 211 . . . . . . . . . . . 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 467 . . . . . . . . . . 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 219 . . . . . . . . . 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 442 . . . . . . . . 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 32 . . . . . . . 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 536 . . . . . . 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 219 . . . . . 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 436 . . . . 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 81 . . . 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 37 . . 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 32 . 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 760 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  /\  ( S  =  A  /\  T  =  C  /\  A  =/=  C ) )  ->  V USGrph  E )
60 eleq1 2517 . . . . . . . . . . . . . . 15  |-  ( A  =  S  ->  ( A  e.  V  <->  S  e.  V ) )
6160eqcoms 2459 . . . . . . . . . . . . . 14  |-  ( S  =  A  ->  ( A  e.  V  <->  S  e.  V ) )
6261adantr 467 . . . . . . . . . . . . 13  |-  ( ( S  =  A  /\  T  =  C )  ->  ( A  e.  V  <->  S  e.  V ) )
63 eleq1 2517 . . . . . . . . . . . . . . 15  |-  ( C  =  T  ->  ( C  e.  V  <->  T  e.  V ) )
6463eqcoms 2459 . . . . . . . . . . . . . 14  |-  ( T  =  C  ->  ( C  e.  V  <->  T  e.  V ) )
6564adantl 468 . . . . . . . . . . . . 13  |-  ( ( S  =  A  /\  T  =  C )  ->  ( C  e.  V  <->  T  e.  V ) )
6662, 653anbi13d 1341 . . . . . . . . . . . 12  |-  ( ( S  =  A  /\  T  =  C )  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  <->  ( S  e.  V  /\  B  e.  V  /\  T  e.  V ) ) )
6766biimpd 211 . . . . . . . . . . 11  |-  ( ( S  =  A  /\  T  =  C )  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( S  e.  V  /\  B  e.  V  /\  T  e.  V )
) )
6867adantld 469 . . . . . . . . . 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 1028 . . . . . . . . 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 432 . . . . . . . 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 667 . . . . . . 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 1028 . . . . . . . 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 468 . . . . . . 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 258 . . . . . 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 647 . . . . 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 987 . . . . . . . 8  |-  ( ( S  =  A  /\  T  =  C  /\  A  =/=  C )  <->  ( ( S  =  A  /\  T  =  C )  /\  A  =/=  C
) )
77 ancom 452 . . . . . . . 8  |-  ( ( ( S  =  A  /\  T  =  C )  /\  A  =/= 
C )  <->  ( A  =/=  C  /\  ( S  =  A  /\  T  =  C ) ) )
7876, 77bitri 253 . . . . . . 7  |-  ( ( S  =  A  /\  T  =  C  /\  A  =/=  C )  <->  ( A  =/=  C  /\  ( S  =  A  /\  T  =  C ) ) )
7978anbi1i 701 . . . . . 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 655 . . . . . 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 206 . . . . . . 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 700 . . . . . 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 275 . . . . 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 265 . . . 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 205 . . . . 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 710 . . . 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 257 . . 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 760 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  /\  A  =/=  C )  ->  V USGrph  E )
907adantr 467 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  /\  A  =/=  C )  ->  ( A  e.  V  /\  C  e.  V )
)
91 simpr 463 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  /\  A  =/=  C )  ->  A  =/=  C )
9210eqcomd 2457 . . . . . . 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 1268 . . . . . 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 2514 . . . . 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 211 . . . 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 608 . . 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 219 . 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 194 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 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   _Vcvv 3045   (/)c0 3731   {cpr 3970   <.cotp 3976   class class class wbr 4402   ran crn 4835  (class class class)co 6290   USGrph cusg 25057   2WalksOnOt c2wlkonot 25583   2SPathOnOt c2pthonot 25585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-ot 3977  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-hash 12516  df-word 12664  df-usgra 25060  df-wlk 25236  df-trail 25237  df-pth 25238  df-spth 25239  df-wlkon 25242  df-spthon 25245  df-2wlkonot 25586  df-2spthonot 25588
This theorem is referenced by:  usg2spthonot1  25618
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