Theorem eupth2lem3lem6fi 16483

Description: 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 ) ) ) )
eupth2lem3lem6fi.g	|- ( ph -> G e. UPGraph )
eupth2lem3lem6fi.v	|- ( ph -> V e. Fin )
eupth2lem3lem6fi.o	|- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) )
eupth2lem3lem6fi.e	|- ( ph -> ( I ` ( F ` N ) ) = { ( P ` N ) , ( P ` ( N + 1 ) ) } )

Assertion

Ref	Expression
eupth2lem3lem6fi	|- ( ( 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 eupth2lem3lem6fi

Step	Hyp	Ref	Expression
1		trlsegvdeg.iy	. . . . . . . 8 |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
2	1	3ad2ant1 1045	. . . . . . 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 1045	. . . . . . 7 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> (Vtx ` Y ) = V )
5		trlsegvdeg.w	. . . . . . . . . 10 |- ( ph -> F (Trails ` G ) P )
6		trlsegvdeg.i	. . . . . . . . . . 11 |- I = (iEdg ` G )
7	6	trlf1 16400	. . . . . . . . . 10 |- ( F (Trails ` G ) P -> F : ( 0..^ (♯ ` F ) ) -1-1-> dom I )
8		f1f 5575	. . . . . . . . . 10 |- ( F : ( 0..^ (♯ ` F ) ) -1-1-> dom I -> F : ( 0..^ (♯ ` F ) ) --> dom I )
9	5, 7, 8	3syl 17	. . . . . . . . 9 |- ( ph -> F : ( 0..^ (♯ ` F ) ) --> dom I )
10		trlsegvdeg.n	. . . . . . . . 9 |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )
11	9, 10	ffvelcdmd 5815	. . . . . . . 8 |- ( ph -> ( F ` N ) e. dom I )
12	11	3ad2ant1 1045	. . . . . . 7 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( F ` N ) e. dom I )
13		trlsegvdeg.u	. . . . . . . 8 |- ( ph -> U e. V )
14	13	3ad2ant1 1045	. . . . . . 7 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> U e. V )
15		eupth2lem3lem6fi.v	. . . . . . . 8 |- ( ph -> V e. Fin )
16	15	3ad2ant1 1045	. . . . . . 7 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> V e. Fin )
17		trlsegvdeg.v	. . . . . . . . 9 |- V = (Vtx ` G )
18	13, 3	eleqtrrd 2314	. . . . . . . . . 10 |- ( ph -> U e. (Vtx ` Y ) )
19		df-vtx 16026	. . . . . . . . . . 11 |- Vtx = ( g e. _V |-> if ( g e. ( _V X. _V ) , ( 1st ` g ) , ( Base ` g ) ) )
20	19	mptrcl 5762	. . . . . . . . . 10 |- ( U e. (Vtx ` Y ) -> Y e. _V )
21	18, 20	syl 14	. . . . . . . . 9 |- ( ph -> Y e. _V )
22		trlsegvdeg.f	. . . . . . . . . . . 12 |- ( ph -> Fun I )
23	22	funfnd 5385	. . . . . . . . . . 11 |- ( ph -> I Fn dom I )
24		fnressn 5872	. . . . . . . . . . 11 |- ( ( I Fn dom I /\ ( F ` N ) e. dom I ) -> ( I |` { ( F ` N ) } ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
25	23, 11, 24	syl2anc 411	. . . . . . . . . 10 |- ( ph -> ( I |` { ( F ` N ) } ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
26	1, 25	eqtr4d 2270	. . . . . . . . 9 |- ( ph -> (iEdg ` Y ) = ( I |` { ( F ` N ) } ) )
27		eupth2lem3lem6fi.g	. . . . . . . . 9 |- ( ph -> G e. UPGraph )
28	17, 6, 21, 3, 26, 27	upgrspan 16291	. . . . . . . 8 |- ( ph -> Y e. UPGraph )
29	28	3ad2ant1 1045	. . . . . . 7 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> Y e. UPGraph )
30		funfvex 5689	. . . . . . . . 9 |- ( ( Fun I /\ ( F ` N ) e. dom I ) -> ( I ` ( F ` N ) ) e. _V )
31	22, 11, 30	syl2anc 411	. . . . . . . 8 |- ( ph -> ( I ` ( F ` N ) ) e. _V )
32	31	3ad2ant1 1045	. . . . . . 7 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( I ` ( F ` N ) ) e. _V )
33		eupth2lem3lem6fi.e	. . . . . . . . 9 |- ( ph -> ( I ` ( F ` N ) ) = { ( P ` N ) , ( P ` ( N + 1 ) ) } )
34		simpl 109	. . . . . . . . . . . . . 14 |- ( ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) -> U =/= ( P ` N ) )
35	34	adantl 277	. . . . . . . . . . . . 13 |- ( ( ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> U =/= ( P ` N ) )
36		simpr 110	. . . . . . . . . . . . . 14 |- ( ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) -> U =/= ( P ` ( N + 1 ) ) )
37	36	adantl 277	. . . . . . . . . . . . 13 |- ( ( ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> U =/= ( P ` ( N + 1 ) ) )
38	35, 37	nelprd 3717	. . . . . . . . . . . 12 |- ( ( ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> -. U e. { ( P ` N ) , ( P ` ( N + 1 ) ) } )
39		df-nel 2510	. . . . . . . . . . . 12 |- ( U e/ { ( P ` N ) , ( P ` ( N + 1 ) ) } <-> -. U e. { ( P ` N ) , ( P ` ( N + 1 ) ) } )
40	38, 39	sylibr 134	. . . . . . . . . . 11 |- ( ( ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> U e/ { ( P ` N ) , ( P ` ( N + 1 ) ) } )
41		neleq2 2514	. . . . . . . . . . 11 |- ( ( I ` ( F ` N ) ) = { ( P ` N ) , ( P ` ( N + 1 ) ) } -> ( U e/ ( I ` ( F ` N ) ) <-> U e/ { ( P ` N ) , ( P ` ( N + 1 ) ) } ) )
42	40, 41	imbitrrid 156	. . . . . . . . . 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 ) ) ) )
43	42	expd 258	. . . . . . . . 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 ) ) ) ) )
44	33, 43	syl 14	. . . . . . . 8 |- ( ph -> ( ( P ` N ) =/= ( P ` ( N + 1 ) ) -> ( ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) -> U e/ ( I ` ( F ` N ) ) ) ) )
45	44	3imp 1220	. . . . . . 7 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> U e/ ( I ` ( F ` N ) ) )
46	2, 4, 12, 14, 16, 29, 32, 45	1hevtxdg0fi 16319	. . . . . 6 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( (VtxDeg ` Y ) ` U ) = 0 )
47	46	oveq2d 6068	. . . . 5 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) = ( ( (VtxDeg ` X ) ` U ) + 0 ) )
48		trlsegvdeg.vx	. . . . . . . . 9 |- ( ph -> (Vtx ` X ) = V )
49		trlsegvdeg.vz	. . . . . . . . 9 |- ( ph -> (Vtx ` Z ) = V )
50		trlsegvdeg.ix	. . . . . . . . 9 |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )
51		trlsegvdeg.iz	. . . . . . . . 9 |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )
52	17, 6, 22, 10, 13, 5, 48, 3, 49, 50, 1, 51, 27, 15	eupth2lem3lem1fi 16480	. . . . . . . 8 |- ( ph -> ( (VtxDeg ` X ) ` U ) e. NN0 )
53	52	nn0cnd 9557	. . . . . . 7 |- ( ph -> ( (VtxDeg ` X ) ` U ) e. CC )
54	53	addridd 8424	. . . . . 6 |- ( ph -> ( ( (VtxDeg ` X ) ` U ) + 0 ) = ( (VtxDeg ` X ) ` U ) )
55	54	3ad2ant1 1045	. . . . 5 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( ( (VtxDeg ` X ) ` U ) + 0 ) = ( (VtxDeg ` X ) ` U ) )
56	47, 55	eqtrd 2267	. . . 4 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) = ( (VtxDeg ` X ) ` U ) )
57	56	breq2d 4123	. . 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 ) ) )
58	57	notbid 673	. 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 ) ) )
59		fveq2 5672	. . . . . . . 8 |- ( x = U -> ( (VtxDeg ` X ) ` x ) = ( (VtxDeg ` X ) ` U ) )
60	59	breq2d 4123	. . . . . . 7 |- ( x = U -> ( 2 || ( (VtxDeg ` X ) ` x ) <-> 2 || ( (VtxDeg ` X ) ` U ) ) )
61	60	notbid 673	. . . . . 6 |- ( x = U -> ( -. 2 || ( (VtxDeg ` X ) ` x ) <-> -. 2 || ( (VtxDeg ` X ) ` U ) ) )
62	61	elrab3 2976	. . . . 5 |- ( U e. V -> ( U e. { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } <-> -. 2 || ( (VtxDeg ` X ) ` U ) ) )
63	13, 62	syl 14	. . . 4 |- ( ph -> ( U e. { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } <-> -. 2 || ( (VtxDeg ` X ) ` U ) ) )
64		eupth2lem3lem6fi.o	. . . . 5 |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) )
65	64	eleq2d 2304	. . . 4 |- ( ph -> ( U e. { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } <-> U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) )
66	63, 65	bitr3d 190	. . 3 |- ( ph -> ( -. 2 || ( (VtxDeg ` X ) ` U ) <-> U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) )
67	66	3ad2ant1 1045	. 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 ) } ) ) )
68	34	3ad2ant3 1047	. . . . . . 7 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> U =/= ( P ` N ) )
69	36	3ad2ant3 1047	. . . . . . 7 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> U =/= ( P ` ( N + 1 ) ) )
70	68, 69	2thd 175	. . . . . 6 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( U =/= ( P ` N ) <-> U =/= ( P ` ( N + 1 ) ) ) )
71		neeq1 2427	. . . . . . 7 |- ( U = ( P ` 0 ) -> ( U =/= ( P ` N ) <-> ( P ` 0 ) =/= ( P ` N ) ) )
72		neeq1 2427	. . . . . . 7 |- ( U = ( P ` 0 ) -> ( U =/= ( P ` ( N + 1 ) ) <-> ( P ` 0 ) =/= ( P ` ( N + 1 ) ) ) )
73	71, 72	bibi12d 235	. . . . . 6 |- ( U = ( P ` 0 ) -> ( ( U =/= ( P ` N ) <-> U =/= ( P ` ( N + 1 ) ) ) <-> ( ( P ` 0 ) =/= ( P ` N ) <-> ( P ` 0 ) =/= ( P ` ( N + 1 ) ) ) ) )
74	70, 73	syl5ibcom 155	. . . . 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 ) ) ) ) )
75	74	pm5.32rd 451	. . . 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 ) ) ) )
76	68	neneqd 2435	. . . . . . 7 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> -. U = ( P ` N ) )
77		biorf 752	. . . . . . 7 |- ( -. U = ( P ` N ) -> ( U = ( P ` 0 ) <-> ( U = ( P ` N ) \/ U = ( P ` 0 ) ) ) )
78	76, 77	syl 14	. . . . . 6 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( U = ( P ` 0 ) <-> ( U = ( P ` N ) \/ U = ( P ` 0 ) ) ) )
79		orcom 736	. . . . . 6 |- ( ( U = ( P ` N ) \/ U = ( P ` 0 ) ) <-> ( U = ( P ` 0 ) \/ U = ( P ` N ) ) )
80	78, 79	bitrdi 196	. . . . 5 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( U = ( P ` 0 ) <-> ( U = ( P ` 0 ) \/ U = ( P ` N ) ) ) )
81	80	anbi2d 464	. . . 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 ) ) ) ) )
82	69	neneqd 2435	. . . . . . 7 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> -. U = ( P ` ( N + 1 ) ) )
83		biorf 752	. . . . . . 7 |- ( -. U = ( P ` ( N + 1 ) ) -> ( U = ( P ` 0 ) <-> ( U = ( P ` ( N + 1 ) ) \/ U = ( P ` 0 ) ) ) )
84	82, 83	syl 14	. . . . . 6 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( U = ( P ` 0 ) <-> ( U = ( P ` ( N + 1 ) ) \/ U = ( P ` 0 ) ) ) )
85		orcom 736	. . . . . 6 |- ( ( U = ( P ` ( N + 1 ) ) \/ U = ( P ` 0 ) ) <-> ( U = ( P ` 0 ) \/ U = ( P ` ( N + 1 ) ) ) )
86	84, 85	bitrdi 196	. . . . 5 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( U = ( P ` 0 ) <-> ( U = ( P ` 0 ) \/ U = ( P ` ( N + 1 ) ) ) ) )
87	86	anbi2d 464	. . . 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 ) ) ) ) ) )
88	75, 81, 87	3bitr3d 218	. . 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 ) ) ) ) ) )
89		eupth2lem1 16470	. . . 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 ) ) ) ) )
90	14, 89	syl 14	. . 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 ) ) ) ) )
91		eupth2lem1 16470	. . . 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 ) ) ) ) ) )
92	14, 91	syl 14	. . 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 ) ) ) ) ) )
93	88, 90, 92	3bitr4d 220	. 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 ) ) } ) ) )
94	58, 67, 93	3bitrd 214	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   /\ wa 104   <-> wb 105   \/ wo 716   /\ w3a 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   e/ wnel 2509   {crab 2526   _Vcvv 2815   (/)c0 3510   ifcif 3622   {csn 3691   {cpr 3692   <.cop 3694   class class class wbr 4111   X. cxp 4749   dom cdm 4751   |` cres 4753   "cima 4754   Fun wfun 5348   Fn wfn 5349   -->wf 5350   -1-1->wf1 5351   ` cfv 5354  (class class class)co 6052   1stc1st 6334   Fincfn 6977   0cc0 8129   1c1 8130   + caddc 8132   2c2 9290   ...cfz 10345  ..^cfzo 10480  ♯chash 11142   || cdvds 12477   Basecbs 13229  Vtxcvtx 16024  iEdgciedg 16025  UPGraphcupgr 16103  VtxDegcvtxdg 16298  Trailsctrls 16392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-dc 843  df-ifp 987  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-1o 6649  df-2o 6650  df-er 6769  df-map 6886  df-en 6978  df-dom 6979  df-fin 6980  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-5 9301  df-6 9302  df-7 9303  df-8 9304  df-9 9305  df-n0 9499  df-z 9580  df-dec 9713  df-uz 9857  df-xadd 10109  df-fz 10346  df-fzo 10481  df-ihash 11143  df-word 11229  df-ndx 13232  df-slot 13233  df-base 13235  df-edgf 16017  df-vtx 16026  df-iedg 16027  df-edg 16070  df-uhgrm 16081  df-upgren 16105  df-subgr 16266  df-vtxdg 16299  df-wlks 16330  df-trls 16393

This theorem is referenced by:  eupth2lem3lem7fi  16486