Theorem eupth2lem3lem6 27093

Description: Formerly part of proof of eupth2lem3 27096: 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 ( P ` 0 ) and ( P ` N ): 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.)

Hypotheses

Ref	Expression
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

Ref	Expression
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 ) ) } ) ) )

Distinct variable groups:   x, U   x, V   x, X

Allowed substitution hints:   ph( x)   P( x)   F( x)   G( x)   I( x)   N( x)   Y( x)   Z( x)

Proof of Theorem eupth2lem3lem6

Step	Hyp	Ref	Expression
1		trlsegvdeg.iy	. . . . . . . 8 |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
2	1	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
3		trlsegvdeg.vy	. . . . . . . 8 |- ( ph -> (Vtx ` Y ) = V )
4	3	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> (Vtx ` Y ) = V )
5		fvexd 6203	. . . . . . 7 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( F ` N ) e. _V )
6		trlsegvdeg.u	. . . . . . . 8 |- ( ph -> U e. V )
7	6	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> U e. V )
8		fvexd 6203	. . . . . . 7 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( I ` ( F ` N ) ) e. _V )
9		eupth2lem3.e	. . . . . . . . 9 |- ( ph -> ( I ` ( F ` N ) ) = { ( P ` N ) , ( P ` ( N + 1 ) ) } )
10		simpl 473	. . . . . . . . . . . . . 14 |- ( ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) -> U =/= ( P ` N ) )
11	10	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> U =/= ( P ` N ) )
12		simpr 477	. . . . . . . . . . . . . 14 |- ( ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) -> U =/= ( P ` ( N + 1 ) ) )
13	12	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> U =/= ( P ` ( N + 1 ) ) )
14	11, 13	nelprd 4203	. . . . . . . . . . . 12 |- ( ( ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> -. U e. { ( P ` N ) , ( P ` ( N + 1 ) ) } )
15		df-nel 2898	. . . . . . . . . . . 12 |- ( U e/ { ( P ` N ) , ( P ` ( N + 1 ) ) } <-> -. U e. { ( P ` N ) , ( P ` ( N + 1 ) ) } )
16	14, 15	sylibr 224	. . . . . . . . . . 11 |- ( ( ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> U e/ { ( P ` N ) , ( P ` ( N + 1 ) ) } )
17		neleq2 2903	. . . . . . . . . . 11 |- ( ( I ` ( F ` N ) ) = { ( P ` N ) , ( P ` ( N + 1 ) ) } -> ( U e/ ( I ` ( F ` N ) ) <-> U e/ { ( P ` N ) , ( P ` ( N + 1 ) ) } ) )
18	16, 17	syl5ibr 236	. . . . . . . . . 10 |- ( ( 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 ) ) ) )
19	18	expd 452	. . . . . . . . 9 |- ( ( 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 ) ) ) ) )
20	9, 19	syl 17	. . . . . . . 8 |- ( ph -> ( ( P ` N ) =/= ( P ` ( N + 1 ) ) -> ( ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) -> U e/ ( I ` ( F ` N ) ) ) ) )
21	20	3imp 1256	. . . . . . 7 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> U e/ ( I ` ( F ` N ) ) )
22	2, 4, 5, 7, 8, 21	1hevtxdg0 26401	. . . . . 6 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( (VtxDeg ` Y ) ` U ) = 0 )
23	22	oveq2d 6666	. . . . 5 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) = ( ( (VtxDeg ` X ) ` U ) + 0 ) )
24		trlsegvdeg.v	. . . . . . . . 9 |- V = (Vtx ` G )
25		trlsegvdeg.i	. . . . . . . . 9 |- I = (iEdg ` G )
26		trlsegvdeg.f	. . . . . . . . 9 |- ( ph -> Fun I )
27		trlsegvdeg.n	. . . . . . . . 9 |- ( ph -> N e. ( 0..^ ( # ` F ) ) )
28		trlsegvdeg.w	. . . . . . . . 9 |- ( ph -> F (Trails ` G ) P )
29		trlsegvdeg.vx	. . . . . . . . 9 |- ( ph -> (Vtx ` X ) = V )
30		trlsegvdeg.vz	. . . . . . . . 9 |- ( ph -> (Vtx ` Z ) = V )
31		trlsegvdeg.ix	. . . . . . . . 9 |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )
32		trlsegvdeg.iz	. . . . . . . . 9 |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )
33	24, 25, 26, 27, 6, 28, 29, 3, 30, 31, 1, 32	eupth2lem3lem1 27088	. . . . . . . 8 |- ( ph -> ( (VtxDeg ` X ) ` U ) e. NN0 )
34	33	nn0cnd 11353	. . . . . . 7 |- ( ph -> ( (VtxDeg ` X ) ` U ) e. CC )
35	34	addid1d 10236	. . . . . 6 |- ( ph -> ( ( (VtxDeg ` X ) ` U ) + 0 ) = ( (VtxDeg ` X ) ` U ) )
36	35	3ad2ant1 1082	. . . . 5 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( ( (VtxDeg ` X ) ` U ) + 0 ) = ( (VtxDeg ` X ) ` U ) )
37	23, 36	eqtrd 2656	. . . 4 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) = ( (VtxDeg ` X ) ` U ) )
38	37	breq2d 4665	. . 3 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) <-> 2 || ( (VtxDeg ` X ) ` U ) ) )
39	38	notbid 308	. 2 |- ( ( 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		fveq2 6191	. . . . . . . 8 |- ( x = U -> ( (VtxDeg ` X ) ` x ) = ( (VtxDeg ` X ) ` U ) )
41	40	breq2d 4665	. . . . . . 7 |- ( x = U -> ( 2 || ( (VtxDeg ` X ) ` x ) <-> 2 || ( (VtxDeg ` X ) ` U ) ) )
42	41	notbid 308	. . . . . 6 |- ( x = U -> ( -. 2 || ( (VtxDeg ` X ) ` x ) <-> -. 2 || ( (VtxDeg ` X ) ` U ) ) )
43	42	elrab3 3364	. . . . 5 |- ( U e. V -> ( U e. { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } <-> -. 2 || ( (VtxDeg ` X ) ` U ) ) )
44	6, 43	syl 17	. . . 4 |- ( ph -> ( U e. { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } <-> -. 2 || ( (VtxDeg ` X ) ` U ) ) )
45		eupth2lem3.o	. . . . 5 |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) )
46	45	eleq2d 2687	. . . 4 |- ( ph -> ( U e. { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } <-> U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) )
47	44, 46	bitr3d 270	. . 3 |- ( ph -> ( -. 2 || ( (VtxDeg ` X ) ` U ) <-> U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) )
48	47	3ad2ant1 1082	. 2 |- ( ( 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	10	3ad2ant3 1084	. . . . . . 7 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> U =/= ( P ` N ) )
50	12	3ad2ant3 1084	. . . . . . 7 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> U =/= ( P ` ( N + 1 ) ) )
51	49, 50	2thd 255	. . . . . 6 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( U =/= ( P ` N ) <-> U =/= ( P ` ( N + 1 ) ) ) )
52		neeq1 2856	. . . . . . 7 |- ( U = ( P ` 0 ) -> ( U =/= ( P ` N ) <-> ( P ` 0 ) =/= ( P ` N ) ) )
53		neeq1 2856	. . . . . . 7 |- ( U = ( P ` 0 ) -> ( U =/= ( P ` ( N + 1 ) ) <-> ( P ` 0 ) =/= ( P ` ( N + 1 ) ) ) )
54	52, 53	bibi12d 335	. . . . . 6 |- ( U = ( P ` 0 ) -> ( ( U =/= ( P ` N ) <-> U =/= ( P ` ( N + 1 ) ) ) <-> ( ( P ` 0 ) =/= ( P ` N ) <-> ( P ` 0 ) =/= ( P ` ( N + 1 ) ) ) ) )
55	51, 54	syl5ibcom 235	. . . . 5 |- ( ( 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 672	. . . 4 |- ( ( 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 2799	. . . . . . 7 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> -. U = ( P ` N ) )
58		biorf 420	. . . . . . 7 |- ( -. U = ( P ` N ) -> ( U = ( P ` 0 ) <-> ( U = ( P ` N ) \/ U = ( P ` 0 ) ) ) )
59	57, 58	syl 17	. . . . . 6 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( U = ( P ` 0 ) <-> ( U = ( P ` N ) \/ U = ( P ` 0 ) ) ) )
60		orcom 402	. . . . . 6 |- ( ( U = ( P ` N ) \/ U = ( P ` 0 ) ) <-> ( U = ( P ` 0 ) \/ U = ( P ` N ) ) )
61	59, 60	syl6bb 276	. . . . 5 |- ( ( 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 740	. . . 4 |- ( ( 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 2799	. . . . . . 7 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> -. U = ( P ` ( N + 1 ) ) )
64		biorf 420	. . . . . . 7 |- ( -. U = ( P ` ( N + 1 ) ) -> ( U = ( P ` 0 ) <-> ( U = ( P ` ( N + 1 ) ) \/ U = ( P ` 0 ) ) ) )
65	63, 64	syl 17	. . . . . 6 |- ( ( 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 402	. . . . . 6 |- ( ( U = ( P ` ( N + 1 ) ) \/ U = ( P ` 0 ) ) <-> ( U = ( P ` 0 ) \/ U = ( P ` ( N + 1 ) ) ) )
67	65, 66	syl6bb 276	. . . . 5 |- ( ( 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 740	. . . 4 |- ( ( 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 298	. . 3 |- ( ( 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 27078	. . . 4 |- ( 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	7, 70	syl 17	. . 3 |- ( ( 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 27078	. . . 4 |- ( 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	7, 72	syl 17	. . 3 |- ( ( 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 300	. 2 |- ( ( 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	39, 48, 74	3bitrd 294	1 |- ( ( 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 ) ) } ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   e/ wnel 2897   {crab 2916   _Vcvv 3200   (/)c0 3915   ifcif 4086   {csn 4177   {cpr 4179   <.cop 4183   class class class wbr 4653   |` cres 5116   "cima 5117   Fun wfun 5882   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   2c2 11070   ...cfz 12326  ..^cfzo 12465   #chash 13117   || cdvds 14983  Vtxcvtx 25874  iEdgciedg 25875  VtxDegcvtxdg 26361  Trailsctrls 26587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-xadd 11947  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-vtxdg 26362  df-wlks 26495  df-trls 26589

This theorem is referenced by:  eupth2lem3lem7  27094