Theorem eupth2lem3lem3fi 16694

Description: Lemma for eupth2lem3fi 16700. 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 5693	. . . . . . . 8 |- ( x = U -> ( (VtxDeg ` X ) ` x ) = ( (VtxDeg ` X ) ` U ) )
3	2	breq2d 4140	. . . . . . 7 |- ( x = U -> ( 2 || ( (VtxDeg ` X ) ` x ) <-> 2 || ( (VtxDeg ` X ) ` U ) ) )
4	3	notbid 677	. . . . . 6 |- ( x = U -> ( -. 2 || ( (VtxDeg ` X ) ` x ) <-> -. 2 || ( (VtxDeg ` X ) ` U ) ) )
5	4	elrab3 2983	. . . . 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 2308	. . . 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 9655	. . . . . 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 16692	. . . . . . 7 |- ( ph -> ( (VtxDeg ` X ) ` U ) e. NN0 )
27	26	nn0zd 9749	. . . . . 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 16693	. . . . . . 7 |- ( ph -> ( (VtxDeg ` Y ) ` U ) e. NN0 )
30	29	nn0zd 9749	. . . . . 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 12664	. . . . . . 7 |- 2 || 2
33	19	ad2antrr 492	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> (Vtx ` Y ) = V )
34	14	trlf1 16612	. . . . . . . . . . 11 |- ( F (Trails ` G ) P -> F : ( 0..^ (♯ ` F ) ) -1-1-> dom I )
35		f1f 5596	. . . . . . . . . . 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 5838	. . . . . . . . 9 |- ( ph -> ( F ` N ) e. dom I )
38	37	ad2antrr 492	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> ( F ` N ) e. dom I )
39	1	ad2antrr 492	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> U e. V )
40	22	ad2antrr 492	. . . . . . . . 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 1002	. . . . . . . . . . . . . 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 3719	. . . . . . . . . . . . 13 |- ( ( P ` N ) = U -> { ( P ` N ) } = { U } )
47	46	eqcoms 2241	. . . . . . . . . . . 12 |- ( U = ( P ` N ) -> { ( P ` N ) } = { U } )
48	45, 47	sylan9eq 2291	. . . . . . . . . . 11 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> ( I ` ( F ` N ) ) = { U } )
49	48	opeq2d 3909	. . . . . . . . . 10 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> <. ( F ` N ) , ( I ` ( F ` N ) ) >. = <. ( F ` N ) , { U } >. )
50	49	sneqd 3721	. . . . . . . . 9 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } = { <. ( F ` N ) , { U } >. } )
51	40, 50	eqtrd 2271	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> (iEdg ` Y ) = { <. ( F ` N ) , { U } >. } )
52	25	ad2antrr 492	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> V e. Fin )
53	33, 38, 39, 51, 52	1loopgrvd2fi 16529	. . . . . . 7 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> ( (VtxDeg ` Y ) ` U ) = 2 )
54	32, 53	breqtrrid 4166	. . . . . 6 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U = ( P ` N ) ) -> 2 || ( (VtxDeg ` Y ) ` U ) )
55		z0even 12661	. . . . . . 7 |- 2 || 0
56	19	ad2antrr 492	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U =/= ( P ` N ) ) -> (Vtx ` Y ) = V )
57	37	ad2antrr 492	. . . . . . . 8 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U =/= ( P ` N ) ) -> ( F ` N ) e. dom I )
58	13, 14, 15, 16, 1, 17	trlsegvdeglem1 16684	. . . . . . . . . 10 |- ( ph -> ( ( P ` N ) e. V /\ ( P ` ( N + 1 ) ) e. V ) )
59	58	simpld 112	. . . . . . . . 9 |- ( ph -> ( P ` N ) e. V )
60	59	ad2antrr 492	. . . . . . . 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 3909	. . . . . . . . . . 11 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> <. ( F ` N ) , ( I ` ( F ` N ) ) >. = <. ( F ` N ) , { ( P ` N ) } >. )
63	62	sneqd 3721	. . . . . . . . . 10 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> { <. ( F ` N ) , ( I ` ( F ` N ) ) >. } = { <. ( F ` N ) , { ( P ` N ) } >. } )
64	61, 63	eqtrd 2271	. . . . . . . . 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 492	. . . . . . . 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 3839	. . . . . . . . 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 16530	. . . . . . 7 |- ( ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) /\ U =/= ( P ` N ) ) -> ( (VtxDeg ` Y ) ` U ) = 0 )
72	55, 71	breqtrrid 4166	. . . . . 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 16610	. . . . . . . . . . 11 |- ( F (Trails ` G ) P -> F (Walks ` G ) P )
75	13	wlkp 16558	. . . . . . . . . . 11 |- ( F (Walks ` G ) P -> P : ( 0 ... (♯ ` F ) ) --> V )
76	17, 74, 75	3syl 17	. . . . . . . . . 10 |- ( ph -> P : ( 0 ... (♯ ` F ) ) --> V )
77		elfzofz 10553	. . . . . . . . . . 11 |- ( N e. ( 0..^ (♯ ` F ) ) -> N e. ( 0 ... (♯ ` F ) ) )
78	16, 77	syl 14	. . . . . . . . . 10 |- ( ph -> N e. ( 0 ... (♯ ` F ) ) )
79	76, 78	ffvelcdmd 5838	. . . . . . . . 9 |- ( ph -> ( P ` N ) e. V )
80	79	adantr 276	. . . . . . . 8 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( P ` N ) e. V )
81		fidceq 7165	. . . . . . . 8 |- ( ( V e. Fin /\ U e. V /\ ( P ` N ) e. V ) -> DECID U = ( P ` N ) )
82	73, 67, 80, 81	syl3anc 1278	. . . . . . 7 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> DECID U = ( P ` N ) )
83		dcne 2431	. . . . . . 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 810	. . . . 5 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> 2 || ( (VtxDeg ` Y ) ` U ) )
86		dvdsadd2b 12590	. . . . 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 1282	. . . 4 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( 2 || ( (VtxDeg ` X ) ` U ) <-> 2 || ( ( (VtxDeg ` Y ) ` U ) + ( (VtxDeg ` X ) ` U ) ) ) )
88	29	nn0cnd 9605	. . . . . . 7 |- ( ph -> ( (VtxDeg ` Y ) ` U ) e. CC )
89	26	nn0cnd 9605	. . . . . . 7 |- ( ph -> ( (VtxDeg ` X ) ` U ) e. CC )
90	88, 89	addcomd 8471	. . . . . 6 |- ( ph -> ( ( (VtxDeg ` Y ) ` U ) + ( (VtxDeg ` X ) ` U ) ) = ( ( (VtxDeg ` X ) ` U ) + ( (VtxDeg ` Y ) ` U ) ) )
91	90	breq2d 4140	. . . . 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 677	. 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 2250	. . . 4 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> ( ( P ` 0 ) = ( P ` N ) <-> ( P ` 0 ) = ( P ` ( N + 1 ) ) ) )
97	95	preq2d 3794	. . . 4 |- ( ( ph /\ ( P ` N ) = ( P ` ( N + 1 ) ) ) -> { ( P ` 0 ) , ( P ` N ) } = { ( P ` 0 ) , ( P ` ( N + 1 ) ) } )
98	96, 97	ifbieq2d 3665	. . 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 2308	. 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 720  DECID wdc 846  if-wif 990   = wceq 1402   e. wcel 2209   =/= wne 2420   {crab 2532   \ cdif 3217   C_ wss 3220   (/)c0 3520   ifcif 3638   {csn 3708   {cpr 3709   <.cop 3711   class class class wbr 4128   dom cdm 4772   |` cres 4774   "cima 4775   Fun wfun 5369   -->wf 5371   -1-1->wf1 5372   ` cfv 5375  (class class class)co 6079   Fincfn 7016   0cc0 8173   1c1 8174   + caddc 8176   2c2 9338   ZZcz 9627   ...cfz 10394  ..^cfzo 10532  ♯chash 11197   || cdvds 12537  Vtxcvtx 16236  iEdgciedg 16237  UPGraphcupgr 16315  VtxDegcvtxdg 16510  Walkscwlks 16541  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-dvds 12538  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-ushgrm 16294  df-upgren 16317  df-uspgren 16379  df-subgr 16478  df-vtxdg 16511  df-wlks 16542  df-trls 16605

This theorem is referenced by:  eupth2lem3lem7fi  16698