Theorem eupth2lem3lem4 27091

Description: Lemma for eupth2lem3 27096, formerly part of proof of eupth2lem3 27096: If an edge (not a loop) is added to a trail, the degree of the end vertices of this edge remains odd if it was odd before (regarding the subgraphs induced by the involved trails). (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 ) } ) )
eupth2lem3lem3.e	|- ( ph -> if- ( ( P ` N ) = ( P ` ( N + 1 ) ) , ( I ` ( F ` N ) ) = { ( P ` N ) } , { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) )
eupth2lem3lem4.i	|- ( ph -> ( I ` ( F ` N ) ) e. ~P V )

Assertion

Ref	Expression
eupth2lem3lem4	|- ( ( 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 eupth2lem3lem4

Step	Hyp	Ref	Expression
1		fvexd 6203	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( F ` N ) e. _V )
2		trlsegvdeg.u	. . . . . . . . . . . 12 |- ( ph -> U e. V )
3	2	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> U e. V )
4		trlsegvdeg.v	. . . . . . . . . . . . . 14 |- V = (Vtx ` G )
5		trlsegvdeg.i	. . . . . . . . . . . . . 14 |- I = (iEdg ` G )
6		trlsegvdeg.f	. . . . . . . . . . . . . 14 |- ( ph -> Fun I )
7		trlsegvdeg.n	. . . . . . . . . . . . . 14 |- ( ph -> N e. ( 0..^ ( # ` F ) ) )
8		trlsegvdeg.w	. . . . . . . . . . . . . 14 |- ( ph -> F (Trails ` G ) P )
9	4, 5, 6, 7, 2, 8	trlsegvdeglem1 27080	. . . . . . . . . . . . 13 |- ( ph -> ( ( P ` N ) e. V /\ ( P ` ( N + 1 ) ) e. V ) )
10	9	simprd 479	. . . . . . . . . . . 12 |- ( ph -> ( P ` ( N + 1 ) ) e. V )
11	10	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( P ` ( N + 1 ) ) e. V )
12		neeq1 2856	. . . . . . . . . . . . . 14 |- ( ( P ` N ) = U -> ( ( P ` N ) =/= ( P ` ( N + 1 ) ) <-> U =/= ( P ` ( N + 1 ) ) ) )
13	12	biimpcd 239	. . . . . . . . . . . . 13 |- ( ( P ` N ) =/= ( P ` ( N + 1 ) ) -> ( ( P ` N ) = U -> U =/= ( P ` ( N + 1 ) ) ) )
14	13	adantl 482	. . . . . . . . . . . 12 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> ( ( P ` N ) = U -> U =/= ( P ` ( N + 1 ) ) ) )
15	14	imp 445	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> U =/= ( P ` ( N + 1 ) ) )
16		eupth2lem3lem4.i	. . . . . . . . . . . 12 |- ( ph -> ( I ` ( F ` N ) ) e. ~P V )
17	16	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( I ` ( F ` N ) ) e. ~P V )
18		trlsegvdeg.iy	. . . . . . . . . . . 12 |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
19	18	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
20		eupth2lem3lem3.e	. . . . . . . . . . . . . 14 |- ( ph -> if- ( ( P ` N ) = ( P ` ( N + 1 ) ) , ( I ` ( F ` N ) ) = { ( P ` N ) } , { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) )
21	20	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> if- ( ( P ` N ) = ( P ` ( N + 1 ) ) , ( I ` ( F ` N ) ) = { ( P ` N ) } , { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) )
22		df-ne 2795	. . . . . . . . . . . . . . . 16 |- ( ( P ` N ) =/= ( P ` ( N + 1 ) ) <-> -. ( P ` N ) = ( P ` ( N + 1 ) ) )
23		ifpfal 1024	. . . . . . . . . . . . . . . 16 |- ( -. ( P ` N ) = ( P ` ( N + 1 ) ) -> (if- ( ( P ` N ) = ( P ` ( N + 1 ) ) , ( I ` ( F ` N ) ) = { ( P ` N ) } , { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) <-> { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) )
24	22, 23	sylbi 207	. . . . . . . . . . . . . . 15 |- ( ( P ` N ) =/= ( P ` ( N + 1 ) ) -> (if- ( ( P ` N ) = ( P ` ( N + 1 ) ) , ( I ` ( F ` N ) ) = { ( P ` N ) } , { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) <-> { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) )
25	24	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> (if- ( ( P ` N ) = ( P ` ( N + 1 ) ) , ( I ` ( F ` N ) ) = { ( P ` N ) } , { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) <-> { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) )
26		preq1 4268	. . . . . . . . . . . . . . . 16 |- ( ( P ` N ) = U -> { ( P ` N ) , ( P ` ( N + 1 ) ) } = { U , ( P ` ( N + 1 ) ) } )
27	26	sseq1d 3632	. . . . . . . . . . . . . . 15 |- ( ( P ` N ) = U -> ( { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) <-> { U , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) )
28	27	biimpcd 239	. . . . . . . . . . . . . 14 |- ( { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) -> ( ( P ` N ) = U -> { U , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) )
29	25, 28	syl6bi 243	. . . . . . . . . . . . 13 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> (if- ( ( P ` N ) = ( P ` ( N + 1 ) ) , ( I ` ( F ` N ) ) = { ( P ` N ) } , { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) -> ( ( P ` N ) = U -> { U , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) ) )
30	21, 29	mpd 15	. . . . . . . . . . . 12 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> ( ( P ` N ) = U -> { U , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) )
31	30	imp 445	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> { U , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) )
32		trlsegvdeg.vy	. . . . . . . . . . . 12 |- ( ph -> (Vtx ` Y ) = V )
33	32	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> (Vtx ` Y ) = V )
34	1, 3, 11, 15, 17, 19, 31, 33	1hegrvtxdg1 26403	. . . . . . . . . 10 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( (VtxDeg ` Y ) ` U ) = 1 )
35	34	oveq2d 6666	. . . . . . . . 9 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) = ( ( (VtxDeg ` X ) ` U ) + 1 ) )
36	35	breq2d 4665	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) <-> 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) ) )
37	36	notbid 308	. . . . . . 7 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) <-> -. 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) ) )
38		trlsegvdeg.vx	. . . . . . . . . . . . . . 15 |- ( ph -> (Vtx ` X ) = V )
39		trlsegvdeg.vz	. . . . . . . . . . . . . . 15 |- ( ph -> (Vtx ` Z ) = V )
40		trlsegvdeg.ix	. . . . . . . . . . . . . . 15 |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )
41		trlsegvdeg.iz	. . . . . . . . . . . . . . 15 |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )
42	4, 5, 6, 7, 2, 8, 38, 32, 39, 40, 18, 41	eupth2lem3lem1 27088	. . . . . . . . . . . . . 14 |- ( ph -> ( (VtxDeg ` X ) ` U ) e. NN0 )
43	42	nn0zd 11480	. . . . . . . . . . . . 13 |- ( ph -> ( (VtxDeg ` X ) ` U ) e. ZZ )
44		2nn 11185	. . . . . . . . . . . . . 14 |- 2 e. NN
45	44	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> 2 e. NN )
46		1lt2 11194	. . . . . . . . . . . . . 14 |- 1 < 2
47	46	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> 1 < 2 )
48		ndvdsp1 15135	. . . . . . . . . . . . 13 |- ( ( ( (VtxDeg ` X ) ` U ) e. ZZ /\ 2 e. NN /\ 1 < 2 ) -> ( 2 || ( (VtxDeg ` X ) ` U ) -> -. 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) ) )
49	43, 45, 47, 48	syl3anc 1326	. . . . . . . . . . . 12 |- ( ph -> ( 2 || ( (VtxDeg ` X ) ` U ) -> -. 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) ) )
50	49	con2d 129	. . . . . . . . . . 11 |- ( ph -> ( 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) -> -. 2 || ( (VtxDeg ` X ) ` U ) ) )
51		1z 11407	. . . . . . . . . . . . . 14 |- 1 e. ZZ
52		n2dvds1 15104	. . . . . . . . . . . . . 14 |- -. 2 || 1
53		opoe 15087	. . . . . . . . . . . . . 14 |- ( ( ( ( (VtxDeg ` X ) ` U ) e. ZZ /\ -. 2 || ( (VtxDeg ` X ) ` U ) ) /\ ( 1 e. ZZ /\ -. 2 || 1 ) ) -> 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) )
54	51, 52, 53	mpanr12 721	. . . . . . . . . . . . 13 |- ( ( ( (VtxDeg ` X ) ` U ) e. ZZ /\ -. 2 || ( (VtxDeg ` X ) ` U ) ) -> 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) )
55	54	ex 450	. . . . . . . . . . . 12 |- ( ( (VtxDeg ` X ) ` U ) e. ZZ -> ( -. 2 || ( (VtxDeg ` X ) ` U ) -> 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) ) )
56	43, 55	syl 17	. . . . . . . . . . 11 |- ( ph -> ( -. 2 || ( (VtxDeg ` X ) ` U ) -> 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) ) )
57	50, 56	impbid 202	. . . . . . . . . 10 |- ( ph -> ( 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) <-> -. 2 || ( (VtxDeg ` X ) ` U ) ) )
58		fveq2 6191	. . . . . . . . . . . . . 14 |- ( x = U -> ( (VtxDeg ` X ) ` x ) = ( (VtxDeg ` X ) ` U ) )
59	58	breq2d 4665	. . . . . . . . . . . . 13 |- ( x = U -> ( 2 || ( (VtxDeg ` X ) ` x ) <-> 2 || ( (VtxDeg ` X ) ` U ) ) )
60	59	notbid 308	. . . . . . . . . . . 12 |- ( x = U -> ( -. 2 || ( (VtxDeg ` X ) ` x ) <-> -. 2 || ( (VtxDeg ` X ) ` U ) ) )
61	60	elrab3 3364	. . . . . . . . . . 11 |- ( U e. V -> ( U e. { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } <-> -. 2 || ( (VtxDeg ` X ) ` U ) ) )
62	2, 61	syl 17	. . . . . . . . . 10 |- ( ph -> ( U e. { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } <-> -. 2 || ( (VtxDeg ` X ) ` U ) ) )
63		eupth2lem3.o	. . . . . . . . . . 11 |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) )
64	63	eleq2d 2687	. . . . . . . . . 10 |- ( ph -> ( U e. { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } <-> U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) )
65	57, 62, 64	3bitr2d 296	. . . . . . . . 9 |- ( ph -> ( 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) <-> U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) )
66	65	notbid 308	. . . . . . . 8 |- ( ph -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) <-> -. U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) )
67	66	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) <-> -. U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) )
68		fvex 6201	. . . . . . . . 9 |- ( P ` N ) e. _V
69	68	eupth2lem2 27079	. . . . . . . 8 |- ( ( ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( P ` N ) = U ) -> ( -. U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )
70	69	adantll 750	. . . . . . 7 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( -. U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )
71	37, 67, 70	3bitrd 294	. . . . . 6 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` N ) = U ) -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )
72	71	expcom 451	. . . . 5 |- ( ( P ` N ) = U -> ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) ) )
73	72	eqcoms 2630	. . . 4 |- ( U = ( P ` N ) -> ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) ) )
74		fvexd 6203	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( F ` N ) e. _V )
75	9	simpld 475	. . . . . . . . . . . 12 |- ( ph -> ( P ` N ) e. V )
76	75	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( P ` N ) e. V )
77	2	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> U e. V )
78		neeq2 2857	. . . . . . . . . . . . . 14 |- ( ( P ` ( N + 1 ) ) = U -> ( ( P ` N ) =/= ( P ` ( N + 1 ) ) <-> ( P ` N ) =/= U ) )
79	78	biimpcd 239	. . . . . . . . . . . . 13 |- ( ( P ` N ) =/= ( P ` ( N + 1 ) ) -> ( ( P ` ( N + 1 ) ) = U -> ( P ` N ) =/= U ) )
80	79	adantl 482	. . . . . . . . . . . 12 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> ( ( P ` ( N + 1 ) ) = U -> ( P ` N ) =/= U ) )
81	80	imp 445	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( P ` N ) =/= U )
82	16	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( I ` ( F ` N ) ) e. ~P V )
83	18	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
84		preq2 4269	. . . . . . . . . . . . . . . 16 |- ( ( P ` ( N + 1 ) ) = U -> { ( P ` N ) , ( P ` ( N + 1 ) ) } = { ( P ` N ) , U } )
85	84	sseq1d 3632	. . . . . . . . . . . . . . 15 |- ( ( P ` ( N + 1 ) ) = U -> ( { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) <-> { ( P ` N ) , U } C_ ( I ` ( F ` N ) ) ) )
86	85	biimpcd 239	. . . . . . . . . . . . . 14 |- ( { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) -> ( ( P ` ( N + 1 ) ) = U -> { ( P ` N ) , U } C_ ( I ` ( F ` N ) ) ) )
87	25, 86	syl6bi 243	. . . . . . . . . . . . 13 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> (if- ( ( P ` N ) = ( P ` ( N + 1 ) ) , ( I ` ( F ` N ) ) = { ( P ` N ) } , { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) -> ( ( P ` ( N + 1 ) ) = U -> { ( P ` N ) , U } C_ ( I ` ( F ` N ) ) ) ) )
88	21, 87	mpd 15	. . . . . . . . . . . 12 |- ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> ( ( P ` ( N + 1 ) ) = U -> { ( P ` N ) , U } C_ ( I ` ( F ` N ) ) ) )
89	88	imp 445	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> { ( P ` N ) , U } C_ ( I ` ( F ` N ) ) )
90	32	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> (Vtx ` Y ) = V )
91	74, 76, 77, 81, 82, 83, 89, 90	1hegrvtxdg1r 26404	. . . . . . . . . 10 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( (VtxDeg ` Y ) ` U ) = 1 )
92	91	oveq2d 6666	. . . . . . . . 9 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) = ( ( (VtxDeg ` X ) ` U ) + 1 ) )
93	92	breq2d 4665	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) <-> 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) ) )
94	93	notbid 308	. . . . . . 7 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) <-> -. 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) ) )
95	66	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + 1 ) <-> -. U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) )
96		necom 2847	. . . . . . . . . 10 |- ( ( P ` N ) =/= ( P ` ( N + 1 ) ) <-> ( P ` ( N + 1 ) ) =/= ( P ` N ) )
97		fvex 6201	. . . . . . . . . . 11 |- ( P ` ( N + 1 ) ) e. _V
98	97	eupth2lem2 27079	. . . . . . . . . 10 |- ( ( ( P ` ( N + 1 ) ) =/= ( P ` N ) /\ ( P ` ( N + 1 ) ) = U ) -> ( -. U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) <-> U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) )
99	96, 98	sylanb 489	. . . . . . . . 9 |- ( ( ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( -. U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) <-> U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) )
100	99	con1bid 345	. . . . . . . 8 |- ( ( ( P ` N ) =/= ( P ` ( N + 1 ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( -. U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )
101	100	adantll 750	. . . . . . 7 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( -. U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )
102	94, 95, 101	3bitrd 294	. . . . . 6 |- ( ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) /\ ( P ` ( N + 1 ) ) = U ) -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) )
103	102	expcom 451	. . . . 5 |- ( ( P ` ( N + 1 ) ) = U -> ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) ) )
104	103	eqcoms 2630	. . . 4 |- ( U = ( P ` ( N + 1 ) ) -> ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) ) )
105	73, 104	jaoi 394	. . 3 |- ( ( U = ( P ` N ) \/ U = ( P ` ( N + 1 ) ) ) -> ( ( ph /\ ( P ` N ) =/= ( P ` ( N + 1 ) ) ) -> ( -. 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) <-> U e. if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) ) ) )
106	105	com12 32	. 2 |- ( ( 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 ) ) } ) ) ) )
107	106	3impia 1261	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  if-wif 1012   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ifcif 4086   ~Pcpw 4158   {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   < clt 10074   NNcn 11020   2c2 11070   ZZcz 11377   ...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  ax-pre-sup 10014

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-rmo 2920  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-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-xadd 11947  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-word 13299  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-vtxdg 26362  df-wlks 26495  df-trls 26589

This theorem is referenced by:  eupth2lem3lem7  27094