Theorem eupth2lem3lem3fi 16388

Description: Lemma for eupth2lem3fi 16394. If a loop { ( P ` N ) , ( P ` ( N + 1 ) ) } is added to a trail, the degree of the vertices with odd degree remains odd (regarding the subgraphs induced by the involved trails). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 21-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 ) ) ) )
trlsegvdegfi.g	|- ( ph -> G e. UPGraph )
trlsegvdegfi.v	|- ( ph -> V e. Fin )
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 ) ) ) )

Assertion

Ref	Expression
eupth2lem3lem3fi	|- ( ( 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 ) ) } ) ) )

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 eupth2lem3lem3fi

Step	Hyp	Ref	Expression
1		trlsegvdeg.u	. . . . 5 |- ( ph -> U e. V )
2		fveq2 5648	. . . . . . . 8 |- ( x = U -> ( (VtxDeg ` X ) ` x ) = ( (VtxDeg ` X ) ` U ) )
3	2	breq2d 4105	. . . . . . 7 |- ( x = U -> ( 2 || ( (VtxDeg ` X ) ` x ) <-> 2 || ( (VtxDeg ` X ) ` U ) ) )
4	3	notbid 673	. . . . . 6 |- ( x = U -> ( -. 2 || ( (VtxDeg ` X ) ` x ) <-> -. 2 || ( (VtxDeg ` X ) ` U ) ) )
5	4	elrab3 2964	. . . . 5 |- ( U e. V -> ( U e. { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } <-> -. 2 || ( (VtxDeg ` X ) ` U ) ) )
6	1, 5	syl 14	. . . 4 |- ( ph -> ( U e. { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } <-> -. 2 || ( (VtxDeg ` X ) ` U ) ) )
7		eupth2lem3.o	. . . . 5 |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } = if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) )
8	7	eleq2d 2301	. . . 4 |- ( ph -> ( U e. { x e. V | -. 2 || ( (VtxDeg ` X ) ` x ) } <-> U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) )
9	6, 8	bitr3d 190	. . 3 |- ( ph -> ( -. 2 || ( (VtxDeg ` X ) ` U ) <-> U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) )
10	9	adantr 276	. 2 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( -. 2 || ( (VtxDeg ` X ) ` U ) <-> U e. if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) ) )
11		2z 9550	. . . . . 6 |- 2 e. ZZ
12	11	a1i 9	. . . . 5 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> 2 e. ZZ )
13		trlsegvdeg.v	. . . . . . . 8 |- V = (Vtx ` G )
14		trlsegvdeg.i	. . . . . . . 8 |- I = (iEdg ` G )
15		trlsegvdeg.f	. . . . . . . 8 |- ( ph -> Fun I )
16		trlsegvdeg.n	. . . . . . . 8 |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )
17		trlsegvdeg.w	. . . . . . . 8 |- ( ph -> F (Trails ` G ) P )
18		trlsegvdeg.vx	. . . . . . . 8 |- ( ph -> (Vtx ` X ) = V )
19		trlsegvdeg.vy	. . . . . . . 8 |- ( ph -> (Vtx ` Y ) = V )
20		trlsegvdeg.vz	. . . . . . . 8 |- ( ph -> (Vtx ` Z ) = V )
21		trlsegvdeg.ix	. . . . . . . 8 |- ( ph -> (iEdg ` X ) = ( I |` ( F " ( 0..^ N ) ) ) )
22		trlsegvdeg.iy	. . . . . . . 8 |- ( ph -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
23		trlsegvdeg.iz	. . . . . . . 8 |- ( ph -> (iEdg ` Z ) = ( I |` ( F " ( 0 ... N ) ) ) )
24		trlsegvdegfi.g	. . . . . . . 8 |- ( ph -> G e. UPGraph )
25		trlsegvdegfi.v	. . . . . . . 8 |- ( ph -> V e. Fin )
26	13, 14, 15, 16, 1, 17, 18, 19, 20, 21, 22, 23, 24, 25	eupth2lem3lem1fi 16386	. . . . . . 7 |- ( ph -> ( (VtxDeg ` X ) ` U ) e. NN0 )
27	26	nn0zd 9643	. . . . . 6 |- ( ph -> ( (VtxDeg ` X ) ` U ) e. ZZ )
28	27	adantr 276	. . . . 5 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( (VtxDeg ` X ) ` U ) e. ZZ )
29	13, 14, 15, 16, 1, 17, 18, 19, 20, 21, 22, 23, 24, 25	eupth2lem3lem2fi 16387	. . . . . . 7 |- ( ph -> ( (VtxDeg ` Y ) ` U ) e. NN0 )
30	29	nn0zd 9643	. . . . . 6 |- ( ph -> ( (VtxDeg ` Y ) ` U ) e. ZZ )
31	30	adantr 276	. . . . 5 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( (VtxDeg ` Y ) ` U ) e. ZZ )
32		z2even 12536	. . . . . . 7 |- 2 || 2
33	19	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> (Vtx ` Y ) = V )
34	14	trlf1 16306	. . . . . . . . . . 11 |- ( F (Trails ` G ) P -> F : ( 0..^ (♯ ` F ) ) -1-1-> dom I )
35		f1f 5551	. . . . . . . . . . 11 |- ( F : ( 0..^ (♯ ` F ) ) -1-1-> dom I -> F : ( 0..^ (♯ ` F ) ) --> dom I )
36	17, 34, 35	3syl 17	. . . . . . . . . 10 |- ( ph -> F : ( 0..^ (♯ ` F ) ) --> dom I )
37	36, 16	ffvelcdmd 5791	. . . . . . . . 9 |- ( ph -> ( F ` N ) e. dom I )
38	37	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> ( F ` N ) e. dom I )
39	1	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> U e. V )
40	22	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
41		eupth2lem3lem3.e	. . . . . . . . . . . . . 14 |- ( ph -> if- ( ( P ` N ) = ( P ` ( N + 1 ) ) , ( I ` ( F ` N ) ) = { ( P ` N ) } , { ( P ` N ) , ( P ` ( N + 1 ) ) } C_ ( I ` ( F ` N ) ) ) )
42	41	adantr 276	. . . . . . . . . . . . 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 ) ) ) )
43		ifptru 998	. . . . . . . . . . . . . 14 |- ( ( 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 ) ) ) <-> ( I ` ( F ` N ) ) = { ( P ` N ) } ) )
44	43	adantl 277	. . . . . . . . . . . . 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 ) ) ) <-> ( I ` ( F ` N ) ) = { ( P ` N ) } ) )
45	42, 44	mpbid 147	. . . . . . . . . . . 12 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( I ` ( F ` N ) ) = { ( P ` N ) } )
46		sneq 3684	. . . . . . . . . . . . 13 |- ( ( P ` N ) = U -> { ( P ` N ) } = { U } )
47	46	eqcoms 2234	. . . . . . . . . . . 12 |- ( U = ( P ` N ) -> { ( P ` N ) } = { U } )
48	45, 47	sylan9eq 2284	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> ( I ` ( F ` N ) ) = { U } )
49	48	opeq2d 3874	. . . . . . . . . 10 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> <. ( F ` N ) , ( I ` ( F ` N ) ) >. = <. ( F ` N ) , { U } >. )
50	49	sneqd 3686	. . . . . . . . 9 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } = { <. ( F ` N ) , { U } >. } )
51	40, 50	eqtrd 2264	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> (iEdg ` Y ) = { <. ( F ` N ) , { U } >. } )
52	25	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> V e. Fin )
53	33, 38, 39, 51, 52	1loopgrvd2fi 16223	. . . . . . 7 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> ( (VtxDeg ` Y ) ` U ) = 2 )
54	32, 53	breqtrrid 4131	. . . . . 6 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> 2 || ( (VtxDeg ` Y ) ` U ) )
55		z0even 12533	. . . . . . 7 |- 2 || 0
56	19	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U =/= ( P ` N ) ) -> (Vtx ` Y ) = V )
57	37	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U =/= ( P ` N ) ) -> ( F ` N ) e. dom I )
58	13, 14, 15, 16, 1, 17	trlsegvdeglem1 16378	. . . . . . . . . 10 |- ( ph -> ( ( P ` N ) e. V /\ ( P ` ( N + 1 ) ) e. V ) )
59	58	simpld 112	. . . . . . . . 9 |- ( ph -> ( P ` N ) e. V )
60	59	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U =/= ( P ` N ) ) -> ( P ` N ) e. V )
61	22	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> (iEdg ` Y ) = { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } )
62	45	opeq2d 3874	. . . . . . . . . . 11 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> <. ( F ` N ) , ( I ` ( F ` N ) ) >. = <. ( F ` N ) , { ( P ` N ) } >. )
63	62	sneqd 3686	. . . . . . . . . 10 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } = { <. ( F ` N ) , { ( P ` N ) } >. } )
64	61, 63	eqtrd 2264	. . . . . . . . 9 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> (iEdg ` Y ) = { <. ( F ` N ) , { ( P ` N ) } >. } )
65	64	adantr 276	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U =/= ( P ` N ) ) -> (iEdg ` Y ) = { <. ( F ` N ) , { ( P ` N ) } >. } )
66	25	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U =/= ( P ` N ) ) -> V e. Fin )
67	1	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> U e. V )
68	67	anim1i 340	. . . . . . . . 9 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U =/= ( P ` N ) ) -> ( U e. V /\ U =/= ( P ` N ) ) )
69		eldifsn 3804	. . . . . . . . 9 |- ( U e. ( V \ { ( P ` N ) } ) <-> ( U e. V /\ U =/= ( P ` N ) ) )
70	68, 69	sylibr 134	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U =/= ( P ` N ) ) -> U e. ( V \ { ( P ` N ) } ) )
71	56, 57, 60, 65, 66, 70	1loopgrvd0fi 16224	. . . . . . 7 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U =/= ( P ` N ) ) -> ( (VtxDeg ` Y ) ` U ) = 0 )
72	55, 71	breqtrrid 4131	. . . . . 6 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U =/= ( P ` N ) ) -> 2 || ( (VtxDeg ` Y ) ` U ) )
73	25	adantr 276	. . . . . . . 8 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> V e. Fin )
74		trliswlk 16304	. . . . . . . . . . 11 |- ( F (Trails ` G ) P -> F (Walks ` G ) P )
75	13	wlkp 16252	. . . . . . . . . . 11 |- ( F (Walks ` G ) P -> P : ( 0 ... (♯ ` F ) ) --> V )
76	17, 74, 75	3syl 17	. . . . . . . . . 10 |- ( ph -> P : ( 0 ... (♯ ` F ) ) --> V )
77		elfzofz 10441	. . . . . . . . . . 11 |- ( N e. ( 0..^ (♯ ` F ) ) -> N e. ( 0 ... (♯ ` F ) ) )
78	16, 77	syl 14	. . . . . . . . . 10 |- ( ph -> N e. ( 0 ... (♯ ` F ) ) )
79	76, 78	ffvelcdmd 5791	. . . . . . . . 9 |- ( ph -> ( P ` N ) e. V )
80	79	adantr 276	. . . . . . . 8 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( P ` N ) e. V )
81		fidceq 7099	. . . . . . . 8 |- ( ( V e. Fin /\ U e. V /\ ( P ` N ) e. V ) -> DECID U = ( P ` N ) )
82	73, 67, 80, 81	syl3anc 1274	. . . . . . 7 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> DECID U = ( P ` N ) )
83		dcne 2414	. . . . . . 7 |- (DECID U = ( P ` N ) <-> ( U = ( P ` N ) \/ U =/= ( P ` N ) ) )
84	82, 83	sylib 122	. . . . . 6 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( U = ( P ` N ) \/ U =/= ( P ` N ) ) )
85	54, 72, 84	mpjaodan 806	. . . . 5 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> 2 || ( (VtxDeg ` Y ) ` U ) )
86		dvdsadd2b 12462	. . . . 5 |- ( ( 2 e. ZZ /\ ( (VtxDeg ` X ) ` U ) e. ZZ /\ ( ( (VtxDeg ` Y ) ` U ) e. ZZ /\ 2 || ( (VtxDeg ` Y ) ` U ) ) ) -> ( 2 || ( (VtxDeg ` X ) ` U ) <-> 2 || ( ( (VtxDeg ` Y ) ` U ) + ( (VtxDeg ` X ) ` U ) ) ) )
87	12, 28, 31, 85, 86	syl112anc 1278	. . . 4 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( 2 || ( (VtxDeg ` X ) ` U ) <-> 2 || ( ( (VtxDeg ` Y ) ` U ) + ( (VtxDeg ` X ) ` U ) ) ) )
88	29	nn0cnd 9500	. . . . . . 7 |- ( ph -> ( (VtxDeg ` Y ) ` U ) e. CC )
89	26	nn0cnd 9500	. . . . . . 7 |- ( ph -> ( (VtxDeg ` X ) ` U ) e. CC )
90	88, 89	addcomd 8373	. . . . . 6 |- ( ph -> ( ( (VtxDeg ` Y ) ` U ) + ( (VtxDeg ` X ) ` U ) ) = ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) )
91	90	breq2d 4105	. . . . 5 |- ( ph -> ( 2 || ( ( (VtxDeg ` Y ) ` U ) + ( (VtxDeg ` X ) ` U ) ) <-> 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) ) )
92	91	adantr 276	. . . 4 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( 2 || ( ( (VtxDeg ` Y ) ` U ) + ( (VtxDeg ` X ) ` U ) ) <-> 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) ) )
93	87, 92	bitrd 188	. . 3 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( 2 || ( (VtxDeg ` X ) ` U ) <-> 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) ) )
94	93	notbid 673	. 2 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( -. 2 || ( (VtxDeg ` X ) ` U ) <-> -. 2 || ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) ) )
95		simpr 110	. . . . 5 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( P ` N ) = ( P ` ( N + 1 ) ) )
96	95	eqeq2d 2243	. . . 4 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( ( P ` 0 ) = ( P ` N ) <-> ( P ` 0 ) = ( P ` ( N + 1 ) ) ) )
97	95	preq2d 3759	. . . 4 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> { ( P ` 0 ) , ( P ` N ) } = { ( P ` 0 ) , ( P ` ( N + 1 ) ) } )
98	96, 97	ifbieq2d 3634	. . 3 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> if ( ( P ` 0 ) = ( P ` N ) , (/) , { ( P ` 0 ) , ( P ` N ) } ) = if ( ( P ` 0 ) = ( P ` ( N + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( N + 1 ) ) } ) )
99	98	eleq2d 2301	. 2 |- ( ( ph /\ ( P ` N ) = ( 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 ) ) } ) ) )
100	10, 94, 99	3bitr3d 218	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 ) ) } ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716  DECID wdc 842  if-wif 986   = wceq 1398   e. wcel 2202   =/= wne 2403   {crab 2515   \ cdif 3198   C_ wss 3201   (/)c0 3496   ifcif 3607   {csn 3673   {cpr 3674   <.cop 3676   class class class wbr 4093   dom cdm 4731   |` cres 4733   "cima 4734   Fun wfun 5327   -->wf 5329   -1-1->wf1 5330   ` cfv 5333  (class class class)co 6028   Fincfn 6952   0cc0 8075   1c1 8076   + caddc 8078   2c2 9237   ZZcz 9522   ...cfz 10286  ..^cfzo 10420  ♯chash 11081   || cdvds 12409  Vtxcvtx 15930  iEdgciedg 15931  UPGraphcupgr 16009  VtxDegcvtxdg 16204  Walkscwlks 16235  Trailsctrls 16298

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-2o 6626  df-er 6745  df-map 6862  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-inn 9187  df-2 9245  df-3 9246  df-4 9247  df-5 9248  df-6 9249  df-7 9250  df-8 9251  df-9 9252  df-n0 9446  df-z 9523  df-dec 9655  df-uz 9799  df-xadd 10051  df-fz 10287  df-fzo 10421  df-ihash 11082  df-word 11161  df-dvds 12410  df-ndx 13146  df-slot 13147  df-base 13149  df-edgf 15923  df-vtx 15932  df-iedg 15933  df-edg 15976  df-uhgrm 15987  df-ushgrm 15988  df-upgren 16011  df-uspgren 16073  df-subgr 16172  df-vtxdg 16205  df-wlks 16236  df-trls 16299

This theorem is referenced by:  eupth2lem3lem7fi  16392