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Metamath Proof Explorer

Theorem eupth2lem3lem6

Description: Formerly part of proof of eupth2lem3 : If an edge (not a loop) is added to a trail, the degree of vertices not being end vertices of this edge remains odd if it was odd before (regarding the subgraphs induced by the involved trails). Remark: This seems to be not valid for hyperedges joining more vertices than ( P0 ) and ( PN ) : if there is a third vertex in the edge, and this vertex is already contained in the trail, then the degree of this vertex could be affected by this edge! (Contributed by Mario Carneiro, 8-Apr-2015) (Revised by AV, 25-Feb-2021)

Ref Expression
Hypotheses trlsegvdeg.v 
|- V = ( Vtx ` G )
trlsegvdeg.i 
|- I = ( iEdg ` G )
trlsegvdeg.f 
|- ( ph -> Fun I )
trlsegvdeg.n 
|- ( ph -> N e. ( 0 ..^ ( # ` F ) ) )
trlsegvdeg.u 
|- ( ph -> U e. V )
trlsegvdeg.w 
|- ( ph -> F ( Trails ` G ) P )
trlsegvdeg.vx 
|- ( ph -> ( Vtx ` X ) = V )
trlsegvdeg.vy 
|- ( ph -> ( Vtx ` Y ) = V )
trlsegvdeg.vz 
|- ( ph -> ( Vtx ` Z ) = V )
trlsegvdeg.ix 
|- ( ph -> ( iEdg ` X ) = ( I |` ( F " ( 0 ..^ N ) ) ) )
trlsegvdeg.iy 
|- ( ph -> ( iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
trlsegvdeg.iz 
|- ( ph -> ( iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )
eupth2lem3.o 
|- ( ph -> { x e. V | -. 2 || ( ( VtxDeg ` X ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) )
eupth2lem3.e 
|- ( ph -> ( I ` ( F ` N ) ) = { ( P ` N ) , ( P ` ( N + 1 ) ) } )
Assertion eupth2lem3lem6 
|- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( -. 2 || ( ( ( VtxDeg ` X ) ` U ) + ( ( VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )

Proof

Step Hyp Ref Expression
1 trlsegvdeg.v 
 |-  V = ( Vtx ` G )
2 trlsegvdeg.i 
 |-  I = ( iEdg ` G )
3 trlsegvdeg.f 
 |-  ( ph -> Fun I )
4 trlsegvdeg.n 
 |-  ( ph -> N e. ( 0 ..^ ( # ` F ) ) )
5 trlsegvdeg.u 
 |-  ( ph -> U e. V )
6 trlsegvdeg.w 
 |-  ( ph -> F ( Trails ` G ) P )
7 trlsegvdeg.vx 
 |-  ( ph -> ( Vtx ` X ) = V )
8 trlsegvdeg.vy 
 |-  ( ph -> ( Vtx ` Y ) = V )
9 trlsegvdeg.vz 
 |-  ( ph -> ( Vtx ` Z ) = V )
10 trlsegvdeg.ix 
 |-  ( ph -> ( iEdg ` X ) = ( I |` ( F " ( 0 ..^ N ) ) ) )
11 trlsegvdeg.iy 
 |-  ( ph -> ( iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
12 trlsegvdeg.iz 
 |-  ( ph -> ( iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )
13 eupth2lem3.o 
 |-  ( ph -> { x e. V | -. 2 || ( ( VtxDeg ` X ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) )
14 eupth2lem3.e 
 |-  ( ph -> ( I ` ( F ` N ) ) = { ( P ` N ) , ( P ` ( N + 1 ) ) } )
15 11 3ad2ant1 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
16 8 3ad2ant1 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( Vtx ` Y ) = V )
17 fvexd 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( F ` N ) e. _V )
18 5 3ad2ant1 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> U e. V )
19 fvexd 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( I ` ( F ` N ) ) e. _V )
20 simpl 
 |-  ( ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) -> U =/= ( P ` N ) )
21 20 adantl 
 |-  ( ( ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> U =/= ( P ` N ) )
22 simpr 
 |-  ( ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) -> U =/= ( P ` ( N + 1 ) ) )
23 22 adantl 
 |-  ( ( ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> U =/= ( P ` ( N + 1 ) ) )
24 21 23 nelprd 
 |-  ( ( ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> -. U e. { ( P ` N ) , ( P ` ( N + 1 ) ) } )
25 df-nel 
 |-  ( U e/ { ( P ` N ) , ( P ` ( N + 1 ) ) } <-> -. U e. { ( P ` N ) , ( P ` ( N + 1 ) ) } )
26 24 25 sylibr 
 |-  ( ( ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> U e/ { ( P ` N ) , ( P ` ( N + 1 ) ) } )
27 neleq2 
 |-  ( ( I ` ( F ` N ) ) = { ( P ` N ) , ( P ` ( N + 1 ) ) } -> ( U e/ ( I ` ( F ` N ) ) <-> U e/ { ( P ` N ) , ( P ` ( N + 1 ) ) } ) )
28 26 27 imbitrrid 
 |-  ( ( I ` ( F ` N ) ) = { ( P ` N ) , ( P ` ( N + 1 ) ) } -> ( ( ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> U e/ ( I ` ( F ` N ) ) ) )
29 28 expd 
 |-  ( ( I ` ( F ` N ) ) = { ( P ` N ) , ( P ` ( N + 1 ) ) } -> ( ( P ` N ) =/= ( P ` ( N + 1 ) ) -> ( ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) -> U e/ ( I ` ( F ` N ) ) ) ) )
30 14 29 syl 
 |-  ( ph -> ( ( P ` N ) =/= ( P ` ( N + 1 ) ) -> ( ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) -> U e/ ( I ` ( F ` N ) ) ) ) )
31 30 3imp 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> U e/ ( I ` ( F ` N ) ) )
32 15 16 17 18 19 31 1hevtxdg0 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( ( VtxDeg ` Y ) ` U ) = 0 )
33 32 oveq2d 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( ( ( VtxDeg ` X ) ` U ) + ( ( VtxDeg ` Y ) ` U ) ) = ( ( ( VtxDeg ` X ) ` U ) + 0 ) )
34 1 2 3 4 5 6 7 8 9 10 11 12 eupth2lem3lem1 
 |-  ( ph -> ( ( VtxDeg ` X ) ` U ) e. NN0 )
35 34 nn0cnd 
 |-  ( ph -> ( ( VtxDeg ` X ) ` U ) e. CC )
36 35 addridd 
 |-  ( ph -> ( ( ( VtxDeg ` X ) ` U ) + 0 ) = ( ( VtxDeg ` X ) ` U ) )
37 36 3ad2ant1 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( ( ( VtxDeg ` X ) ` U ) + 0 ) = ( ( VtxDeg ` X ) ` U ) )
38 33 37 eqtrd 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( ( ( VtxDeg ` X ) ` U ) + ( ( VtxDeg ` Y ) ` U ) ) = ( ( VtxDeg ` X ) ` U ) )
39 38 breq2d 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( 2 || ( ( ( VtxDeg ` X ) ` U ) + ( ( VtxDeg ` Y ) ` U ) ) <-> 2 || ( ( VtxDeg ` X ) ` U ) ) )
40 39 notbid 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( -. 2 || ( ( ( VtxDeg ` X ) ` U ) + ( ( VtxDeg ` Y ) ` U ) ) <-> -. 2 || ( ( VtxDeg ` X ) ` U ) ) )
41 fveq2 
 |-  ( x = U -> ( ( VtxDeg ` X ) ` x ) = ( ( VtxDeg ` X ) ` U ) )
42 41 breq2d 
 |-  ( x = U -> ( 2 || ( ( VtxDeg ` X ) ` x ) <-> 2 || ( ( VtxDeg ` X ) ` U ) ) )
43 42 notbid 
 |-  ( x = U -> ( -. 2 || ( ( VtxDeg ` X ) ` x ) <-> -. 2 || ( ( VtxDeg ` X ) ` U ) ) )
44 43 elrab3 
 |-  ( U e. V -> ( U e. { x e. V | -. 2 || ( ( VtxDeg ` X ) ` x ) } <-> -. 2 || ( ( VtxDeg ` X ) ` U ) ) )
45 5 44 syl 
 |-  ( ph -> ( U e. { x e. V | -. 2 || ( ( VtxDeg ` X ) ` x ) } <-> -. 2 || ( ( VtxDeg ` X ) ` U ) ) )
46 13 eleq2d 
 |-  ( ph -> ( U e. { x e. V | -. 2 || ( ( VtxDeg ` X ) ` x ) } <-> U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) )
47 45 46 bitr3d 
 |-  ( ph -> ( -. 2 || ( ( VtxDeg ` X ) ` U ) <-> U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) )
48 47 3ad2ant1 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( -. 2 || ( ( VtxDeg ` X ) ` U ) <-> U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) )
49 20 3ad2ant3 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> U =/= ( P ` N ) )
50 22 3ad2ant3 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> U =/= ( P ` ( N + 1 ) ) )
51 49 50 2thd 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( U =/= ( P ` N ) <-> U =/= ( P ` ( N + 1 ) ) ) )
52 neeq1 
 |-  ( U = ( P ` 0 ) -> ( U =/= ( P ` N ) <-> ( P ` 0 ) =/= ( P ` N ) ) )
53 neeq1 
 |-  ( U = ( P ` 0 ) -> ( U =/= ( P ` ( N + 1 ) ) <-> ( P ` 0 ) =/= ( P ` ( N + 1 ) ) ) )
54 52 53 bibi12d 
 |-  ( U = ( P ` 0 ) -> ( ( U =/= ( P ` N ) <-> U =/= ( P ` ( N + 1 ) ) ) <-> ( ( P ` 0 ) =/= ( P ` N ) <-> ( P ` 0 ) =/= ( P ` ( N + 1 ) ) ) ) )
55 51 54 syl5ibcom 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( U = ( P ` 0 ) -> ( ( P ` 0 ) =/= ( P ` N ) <-> ( P ` 0 ) =/= ( P ` ( N + 1 ) ) ) ) )
56 55 pm5.32rd 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( ( ( P ` 0 ) =/= ( P ` N ) /\ U = ( P ` 0 ) ) <-> ( ( P ` 0 ) =/= ( P ` ( N + 1 ) ) /\ U = ( P ` 0 ) ) ) )
57 49 neneqd 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> -. U = ( P ` N ) )
58 biorf 
 |-  ( -. U = ( P ` N ) -> ( U = ( P ` 0 ) <-> ( U = ( P ` N ) \/ U = ( P ` 0 ) ) ) )
59 57 58 syl 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( U = ( P ` 0 ) <-> ( U = ( P ` N ) \/ U = ( P ` 0 ) ) ) )
60 orcom 
 |-  ( ( U = ( P ` N ) \/ U = ( P ` 0 ) ) <-> ( U = ( P ` 0 ) \/ U = ( P ` N ) ) )
61 59 60 bitrdi 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( U = ( P ` 0 ) <-> ( U = ( P ` 0 ) \/ U = ( P ` N ) ) ) )
62 61 anbi2d 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( ( ( P ` 0 ) =/= ( P ` N ) /\ U = ( P ` 0 ) ) <-> ( ( P ` 0 ) =/= ( P ` N ) /\ ( U = ( P ` 0 ) \/ U = ( P ` N ) ) ) ) )
63 50 neneqd 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> -. U = ( P ` ( N + 1 ) ) )
64 biorf 
 |-  ( -. U = ( P ` ( N + 1 ) ) -> ( U = ( P ` 0 ) <-> ( U = ( P ` ( N + 1 ) ) \/ U = ( P ` 0 ) ) ) )
65 63 64 syl 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( U = ( P ` 0 ) <-> ( U = ( P ` ( N + 1 ) ) \/ U = ( P ` 0 ) ) ) )
66 orcom 
 |-  ( ( U = ( P ` ( N + 1 ) ) \/ U = ( P ` 0 ) ) <-> ( U = ( P ` 0 ) \/ U = ( P ` ( N + 1 ) ) ) )
67 65 66 bitrdi 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( U = ( P ` 0 ) <-> ( U = ( P ` 0 ) \/ U = ( P ` ( N + 1 ) ) ) ) )
68 67 anbi2d 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( ( ( P ` 0 ) =/= ( P ` ( N + 1 ) ) /\ U = ( P ` 0 ) ) <-> ( ( P ` 0 ) =/= ( P ` ( N + 1 ) ) /\ ( U = ( P ` 0 ) \/ U = ( P ` ( N + 1 ) ) ) ) ) )
69 56 62 68 3bitr3d 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( ( ( P ` 0 ) =/= ( P ` N ) /\ ( U = ( P ` 0 ) \/ U = ( P ` N ) ) ) <-> ( ( P ` 0 ) =/= ( P ` ( N + 1 ) ) /\ ( U = ( P ` 0 ) \/ U = ( P ` ( N + 1 ) ) ) ) ) )
70 eupth2lem1 
 |-  ( U e. V -> ( U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) <-> ( ( P ` 0 ) =/= ( P ` N ) /\ ( U = ( P ` 0 ) \/ U = ( P ` N ) ) ) ) )
71 18 70 syl 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) <-> ( ( P ` 0 ) =/= ( P ` N ) /\ ( U = ( P ` 0 ) \/ U = ( P ` N ) ) ) ) )
72 eupth2lem1 
 |-  ( U e. V -> ( U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) <-> ( ( P ` 0 ) =/= ( P ` ( N + 1 ) ) /\ ( U = ( P ` 0 ) \/ U = ( P ` ( N + 1 ) ) ) ) ) )
73 18 72 syl 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) <-> ( ( P ` 0 ) =/= ( P ` ( N + 1 ) ) /\ ( U = ( P ` 0 ) \/ U = ( P ` ( N + 1 ) ) ) ) ) )
74 69 71 73 3bitr4d 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )
75 40 48 74 3bitrd 
 |-  ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( -. 2 || ( ( ( VtxDeg ` X ) ` U ) + ( ( VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )