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Theorem usg2spthonot0 24754
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 3773 . . . . 5  |-  ( <. S ,  B ,  T >.  e.  ( A ( V 2SPathOnOt  E ) C )  ->  ( A ( V 2SPathOnOt  E ) C )  =/=  (/) )
2 2spontn0vne 24752 . . . . 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 457 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  V USGrph  E )
54adantl 466 . . . . . . . . . 10  |-  ( ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
) )  ->  V USGrph  E )
6 3simpb 993 . . . . . . . . . . . 12  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( A  e.  V  /\  C  e.  V
) )
76adantl 466 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  ->  ( A  e.  V  /\  C  e.  V ) )
87adantl 466 . . . . . . . . . 10  |-  ( ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
) )  ->  ( A  e.  V  /\  C  e.  V )
)
9 simpl 457 . . . . . . . . . 10  |-  ( ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
) )  ->  A  =/=  C )
10 2pthwlkonot 24750 . . . . . . . . . 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 1227 . . . . . . . . 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 2511 . . . . . . . 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 24203 . . . . . . . . . . . 12  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
1413, 6anim12i 566 . . . . . . . . . . 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 466 . . . . . . . . . 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 24739 . . . . . . . . . 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 974 . . . . . . . . 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 261 . . . . . . . 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 253 . . . . . . 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 753 . . . . . . . . . . . . 13  |-  ( ( ( S  =  A  /\  T  =  C )  /\  A  =/= 
C )  ->  S  =  A )
22 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( S  =  A  /\  T  =  C )  ->  T  =  C )
2322adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( S  =  A  /\  T  =  C )  /\  A  =/= 
C )  ->  T  =  C )
24 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( S  =  A  /\  T  =  C )  /\  A  =/= 
C )  ->  A  =/=  C )
2521, 23, 243jca 1175 . . . . . . . . . . . 12  |-  ( ( ( S  =  A  /\  T  =  C )  /\  A  =/= 
C )  ->  ( S  =  A  /\  T  =  C  /\  A  =/=  C ) )
2625ex 434 . . . . . . . . . . 11  |-  ( ( S  =  A  /\  T  =  C )  ->  ( A  =/=  C  ->  ( S  =  A  /\  T  =  C  /\  A  =/=  C
) ) )
2726adantr 465 . . . . . . . . . 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 465 . . . . . . . 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 466 . . . . . . . . . . . 12  |-  ( ( ( S  =  A  /\  T  =  C )  /\  ( A  =/=  C  /\  ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) ) ) )  ->  V USGrph  E )
31 simprrr 764 . . . . . . . . . . . . 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 2513 . . . . . . . . . . . . . . . 16  |-  ( S  =  A  ->  ( S  e.  V  <->  A  e.  V ) )
3332adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( S  =  A  /\  T  =  C )  ->  ( S  e.  V  <->  A  e.  V ) )
34 eleq1 2513 . . . . . . . . . . . . . . . 16  |-  ( T  =  C  ->  ( T  e.  V  <->  C  e.  V ) )
3534adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( S  =  A  /\  T  =  C )  ->  ( T  e.  V  <->  C  e.  V ) )
3633, 353anbi13d 1300 . . . . . . . . . . . . . 14  |-  ( ( S  =  A  /\  T  =  C )  ->  ( ( S  e.  V  /\  B  e.  V  /\  T  e.  V )  <->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) ) )
3736adantr 465 . . . . . . . . . . . . 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 232 . . . . . . . . . . . 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 24748 . . . . . . . . . . . 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 661 . . . . . . . . . . 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 4090 . . . . . . . . . . . . . . . 16  |-  ( S  =  A  ->  { S ,  B }  =  { A ,  B }
)
4241adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( S  =  A  /\  T  =  C )  ->  { S ,  B }  =  { A ,  B } )
4342eleq1d 2510 . . . . . . . . . . . . . 14  |-  ( ( S  =  A  /\  T  =  C )  ->  ( { S ,  B }  e.  ran  E  <->  { A ,  B }  e.  ran  E ) )
44 preq2 4091 . . . . . . . . . . . . . . . 16  |-  ( T  =  C  ->  { B ,  T }  =  { B ,  C }
)
4544adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( S  =  A  /\  T  =  C )  ->  { B ,  T }  =  { B ,  C } )
4645eleq1d 2510 . . . . . . . . . . . . . 14  |-  ( ( S  =  A  /\  T  =  C )  ->  ( { B ,  T }  e.  ran  E  <->  { B ,  C }  e.  ran  E ) )
4743, 46anbi12d 710 . . . . . . . . . . . . 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 207 . . . . . . . . . . . 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 465 . . . . . . . . . . 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 215 . . . . . . . . . 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 440 . . . . . . . . 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 533 . . . . . . 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 215 . . . . . 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 434 . . . . 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 78 . . . 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 753 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  /\  ( S  =  A  /\  T  =  C  /\  A  =/=  C ) )  ->  V USGrph  E )
60 eleq1 2513 . . . . . . . . . . . . . . 15  |-  ( A  =  S  ->  ( A  e.  V  <->  S  e.  V ) )
6160eqcoms 2453 . . . . . . . . . . . . . 14  |-  ( S  =  A  ->  ( A  e.  V  <->  S  e.  V ) )
6261adantr 465 . . . . . . . . . . . . 13  |-  ( ( S  =  A  /\  T  =  C )  ->  ( A  e.  V  <->  S  e.  V ) )
63 eleq1 2513 . . . . . . . . . . . . . . 15  |-  ( C  =  T  ->  ( C  e.  V  <->  T  e.  V ) )
6463eqcoms 2453 . . . . . . . . . . . . . 14  |-  ( T  =  C  ->  ( C  e.  V  <->  T  e.  V ) )
6564adantl 466 . . . . . . . . . . . . 13  |-  ( ( S  =  A  /\  T  =  C )  ->  ( C  e.  V  <->  T  e.  V ) )
6662, 653anbi13d 1300 . . . . . . . . . . . 12  |-  ( ( S  =  A  /\  T  =  C )  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  <->  ( S  e.  V  /\  B  e.  V  /\  T  e.  V ) ) )
6766biimpd 207 . . . . . . . . . . 11  |-  ( ( S  =  A  /\  T  =  C )  ->  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( S  e.  V  /\  B  e.  V  /\  T  e.  V )
) )
6867adantld 467 . . . . . . . . . 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 1015 . . . . . . . . 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 430 . . . . . . . 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 661 . . . . . . 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 1015 . . . . . . . 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 466 . . . . . . 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 254 . . . . . 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 641 . . . . 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 974 . . . . . . . 8  |-  ( ( S  =  A  /\  T  =  C  /\  A  =/=  C )  <->  ( ( S  =  A  /\  T  =  C )  /\  A  =/=  C
) )
77 ancom 450 . . . . . . . 8  |-  ( ( ( S  =  A  /\  T  =  C )  /\  A  =/= 
C )  <->  ( A  =/=  C  /\  ( S  =  A  /\  T  =  C ) ) )
7876, 77bitri 249 . . . . . . 7  |-  ( ( S  =  A  /\  T  =  C  /\  A  =/=  C )  <->  ( A  =/=  C  /\  ( S  =  A  /\  T  =  C ) ) )
7978anbi1i 695 . . . . . 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 649 . . . . . 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 202 . . . . . . 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 694 . . . . . 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 271 . . . . 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 261 . . . 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 201 . . . . 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 703 . . . 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 253 . . 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 753 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  /\  A  =/=  C )  ->  V USGrph  E )
907adantr 465 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  /\  A  =/=  C )  ->  ( A  e.  V  /\  C  e.  V )
)
91 simpr 461 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V
) )  /\  A  =/=  C )  ->  A  =/=  C )
9210eqcomd 2449 . . . . . . 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 1227 . . . . . 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 2511 . . . . 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 207 . . . 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 603 . . 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 215 . 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 191 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 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802    =/= wne 2636   _Vcvv 3093   (/)c0 3767   {cpr 4012   <.cotp 4018   class class class wbr 4433   ran crn 4986  (class class class)co 6277   USGrph cusg 24195   2WalksOnOt c2wlkonot 24720   2SPathOnOt c2pthonot 24722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-ot 4019  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-map 7420  df-pm 7421  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-card 8318  df-cda 8546  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-n0 10797  df-z 10866  df-uz 11086  df-fz 11677  df-fzo 11799  df-hash 12380  df-word 12516  df-usgra 24198  df-wlk 24373  df-trail 24374  df-pth 24375  df-spth 24376  df-wlkon 24379  df-spthon 24382  df-2wlkonot 24723  df-2spthonot 24725
This theorem is referenced by:  usg2spthonot1  24755
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