Theorem eupth2lem3lem6fi 16695

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 1049	. . . . . . 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 1049	. . . . . . 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 16612	. . . . . . . . . 10 |- ( F (Trails ` G ) P -> F : ( 0..^ (♯ ` F ) ) -1-1-> dom I )
8		f1f 5596	. . . . . . . . . 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 5838	. . . . . . . 8 |- ( ph -> ( F ` N ) e. dom I )
12	11	3ad2ant1 1049	. . . . . . 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 1049	. . . . . . 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 1049	. . . . . . 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 2318	. . . . . . . . . 10 |- ( ph -> U e. (Vtx ` Y ) )
19		df-vtx 16238	. . . . . . . . . . 11 |- Vtx = ( g e. _V |-> if ( g e. ( _V X. _V ) , ( 1st ` g ) , ( Base ` g ) ) )
20	19	mptrcl 5785	. . . . . . . . . 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 5406	. . . . . . . . . . 11 |- ( ph -> I Fn dom I )
24		fnressn 5895	. . . . . . . . . . 11 |- ( ( I Fn dom I /\ ( F ` N ) e. dom I ) -> ( I |` { ( F ` N ) } ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
25	23, 11, 24	syl2anc 415	. . . . . . . . . 10 |- ( ph -> ( I |` { ( F ` N ) } ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
26	1, 25	eqtr4d 2274	. . . . . . . . 9 |- ( ph -> (iEdg ` Y ) = ( I |` { ( F ` N ) } ) )
27		eupth2lem3lem6fi.g	. . . . . . . . 9 |- ( ph -> G e. UPGraph )
28	17, 6, 21, 3, 26, 27	upgrspan 16503	. . . . . . . 8 |- ( ph -> Y e. UPGraph )
29	28	3ad2ant1 1049	. . . . . . 7 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> Y e. UPGraph )
30		funfvex 5710	. . . . . . . . 9 |- ( ( Fun I /\ ( F ` N ) e. dom I ) -> ( I ` ( F ` N ) ) e. _V )
31	22, 11, 30	syl2anc 415	. . . . . . . 8 |- ( ph -> ( I ` ( F ` N ) ) e. _V )
32	31	3ad2ant1 1049	. . . . . . 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 3734	. . . . . . . . . . . 12 |- ( ( ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> -. U e. { ( P ` N ) , ( P ` ( N + 1 ) ) } )
39		df-nel 2516	. . . . . . . . . . . 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 2520	. . . . . . . . . . 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 1224	. . . . . . 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 16531	. . . . . 6 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> ( (VtxDeg ` Y ) ` U ) = 0 )
47	46	oveq2d 6095	. . . . 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 16692	. . . . . . . 8 |- ( ph -> ( (VtxDeg ` X ) ` U ) e. NN0 )
53	52	nn0cnd 9605	. . . . . . 7 |- ( ph -> ( (VtxDeg ` X ) ` U ) e. CC )
54	53	addridd 8469	. . . . . 6 |- ( ph -> ( ( (VtxDeg ` X ) ` U ) + 0 ) = ( (VtxDeg ` X ) ` U ) )
55	54	3ad2ant1 1049	. . . . 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 2271	. . . 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 4140	. . 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 677	. 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 5693	. . . . . . . 8 |- ( x = U -> ( (VtxDeg ` X ) ` x ) = ( (VtxDeg ` X ) ` U ) )
60	59	breq2d 4140	. . . . . . 7 |- ( x = U -> ( 2 || ( (VtxDeg ` X ) ` x ) <-> 2 || ( (VtxDeg ` X ) ` U ) ) )
61	60	notbid 677	. . . . . 6 |- ( x = U -> ( -. 2 || ( (VtxDeg ` X ) ` x ) <-> -. 2 || ( (VtxDeg ` X ) ` U ) ) )
62	61	elrab3 2983	. . . . 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 2308	. . . 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 1049	. 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 1051	. . . . . . 7 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> U =/= ( P ` N ) )
69	36	3ad2ant3 1051	. . . . . . 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 2433	. . . . . . 7 |- ( U = ( P ` 0 ) -> ( U =/= ( P ` N ) <-> ( P ` 0 ) =/= ( P ` N ) ) )
72		neeq1 2433	. . . . . . 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 455	. . . 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 2441	. . . . . . 7 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> -. U = ( P ` N ) )
77		biorf 756	. . . . . . 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 740	. . . . . 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 468	. . . 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 2441	. . . . . . 7 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( U =/= ( P ` N ) /\ U =/= ( P ` ( N + 1 ) ) ) ) -> -. U = ( P ` ( N + 1 ) ) )
83		biorf 756	. . . . . . 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 740	. . . . . 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 468	. . . 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 16682	. . . 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 16682	. . . 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 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   e/ wnel 2515   {crab 2532   _Vcvv 2821   (/)c0 3520   ifcif 3638   {csn 3708   {cpr 3709   <.cop 3711   class class class wbr 4128   X. cxp 4770   dom cdm 4772   |` cres 4774   "cima 4775   Fun wfun 5369   Fn wfn 5370   -->wf 5371   -1-1->wf1 5372   ` cfv 5375  (class class class)co 6079   1stc1st 6366   Fincfn 7016   0cc0 8173   1c1 8174   + caddc 8176   2c2 9338   ...cfz 10394  ..^cfzo 10532  ♯chash 11197   || cdvds 12537   Basecbs 13335  Vtxcvtx 16236  iEdgciedg 16237  UPGraphcupgr 16315  VtxDegcvtxdg 16510  Trailsctrls 16604

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-ifp 991  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-1o 6681  df-2o 6682  df-er 6801  df-map 6918  df-en 7017  df-dom 7018  df-fin 7019  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-9 9353  df-n0 9547  df-z 9628  df-dec 9761  df-uz 9905  df-xadd 10158  df-fz 10395  df-fzo 10533  df-ihash 11198  df-word 11288  df-ndx 13338  df-slot 13339  df-base 13341  df-edgf 16229  df-vtx 16238  df-iedg 16239  df-edg 16282  df-uhgrm 16293  df-upgren 16317  df-subgr 16478  df-vtxdg 16511  df-wlks 16542  df-trls 16605

This theorem is referenced by:  eupth2lem3lem7fi  16698