Theorem eupth2fi 16333

Description: The only vertices of odd degree in a graph with an Eulerian path are the endpoints, and then only if the endpoints are distinct. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)

Hypotheses

Ref	Expression
eupth2.v	|- V = (Vtx ` G )
eupth2.i	|- I = (iEdg ` G )
eupth2fi.g	|- ( ph -> G e. UMGraph )
eupth2.f	|- ( ph -> Fun I )
eupth2.p	|- ( ph -> F (EulerPaths ` G ) P )
eupth2fi.fi	|- ( ph -> V e. Fin )

Assertion

Ref	Expression
eupth2fi	|- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` G ) ` x ) } = if ( ( P ` 0 ) = ( P ` (♯ ` F ) ) , (/) , { ( P ` 0 ) , ( P ` (♯ ` F ) ) } ) )

Distinct variable groups:   ph, x   x, F   x, I   x, V

Allowed substitution hints:   P( x)   G( x)

Proof of Theorem eupth2fi

Dummy variables n m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eupth2.v	. . . . . . 7 |- V = (Vtx ` G )
2		eupth2.i	. . . . . . 7 |- I = (iEdg ` G )
3		eupth2fi.g	. . . . . . 7 |- ( ph -> G e. UMGraph )
4		eupth2.f	. . . . . . 7 |- ( ph -> Fun I )
5		eupth2.p	. . . . . . 7 |- ( ph -> F (EulerPaths ` G ) P )
6		eqid 2231	. . . . . . 7 |- <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. = <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >.
7	1, 2, 3, 4, 5, 6	eupthvdres 16329	. . . . . 6 |- ( ph -> (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) = (VtxDeg ` G ) )
8	7	fveq1d 5641	. . . . 5 |- ( ph -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) ` x ) = ( (VtxDeg ` G ) ` x ) )
9	8	breq2d 4100	. . . 4 |- ( ph -> ( 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) ` x ) <-> 2 || ( (VtxDeg ` G ) ` x ) ) )
10	9	notbid 673	. . 3 |- ( ph -> ( -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) ` x ) <-> -. 2 || ( (VtxDeg ` G ) ` x ) ) )
11	10	rabbidv 2791	. 2 |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) ` x ) } = { x e. V | -. 2 || ( (VtxDeg ` G ) ` x ) } )
12		eupthiswlk 16309	. . . 4 |- ( F (EulerPaths ` G ) P -> F (Walks ` G ) P )
13		wlkcl 16186	. . . 4 |- ( F (Walks ` G ) P -> (♯ ` F ) e. NN0 )
14	5, 12, 13	3syl 17	. . 3 |- ( ph -> (♯ ` F ) e. NN0 )
15		nn0re 9411	. . . . 5 |- ( (♯ ` F ) e. NN0 -> (♯ ` F ) e. RR )
16	15	leidd 8694	. . . 4 |- ( (♯ ` F ) e. NN0 -> (♯ ` F ) <_ (♯ ` F ) )
17		breq1 4091	. . . . . . 7 |- ( m = 0 -> ( m <_ (♯ ` F ) <-> 0 <_ (♯ ` F ) ) )
18		oveq2 6026	. . . . . . . . . . . . . . . 16 |- ( m = 0 -> ( 0..^ m ) = ( 0..^ 0 ) )
19	18	imaeq2d 5076	. . . . . . . . . . . . . . 15 |- ( m = 0 -> ( F " ( 0..^ m ) ) = ( F " ( 0..^ 0 ) ) )
20	19	reseq2d 5013	. . . . . . . . . . . . . 14 |- ( m = 0 -> ( I |` ( F " ( 0..^ m ) ) ) = ( I |` ( F " ( 0..^ 0 ) ) ) )
21	20	opeq2d 3869	. . . . . . . . . . . . 13 |- ( m = 0 -> <. V , ( I |` ( F " ( 0..^ m ) ) ) >. = <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. )
22	21	fveq2d 5643	. . . . . . . . . . . 12 |- ( m = 0 -> (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) = (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) )
23	22	fveq1d 5641	. . . . . . . . . . 11 |- ( m = 0 -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) = ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) ` x ) )
24	23	breq2d 4100	. . . . . . . . . 10 |- ( m = 0 -> ( 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) <-> 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) ` x ) ) )
25	24	notbid 673	. . . . . . . . 9 |- ( m = 0 -> ( -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) <-> -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) ` x ) ) )
26	25	rabbidv 2791	. . . . . . . 8 |- ( m = 0 -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) } = { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) ` x ) } )
27		fveq2 5639	. . . . . . . . . 10 |- ( m = 0 -> ( P ` m ) = ( P ` 0 ) )
28	27	eqeq2d 2243	. . . . . . . . 9 |- ( m = 0 -> ( ( P ` 0 ) = ( P ` m ) <-> ( P ` 0 ) = ( P ` 0 ) ) )
29	27	preq2d 3755	. . . . . . . . 9 |- ( m = 0 -> { ( P ` 0 ) , ( P ` m ) } = { ( P ` 0 ) , ( P ` 0 ) } )
30	28, 29	ifbieq2d 3630	. . . . . . . 8 |- ( m = 0 -> if ( ( P ` 0 ) = ( P ` m ) , (/) , { ( P ` 0 ) , ( P ` m ) } ) = if ( ( P ` 0 ) = ( P ` 0 ) , (/) , { ( P ` 0 ) , ( P ` 0 ) } ) )
31	26, 30	eqeq12d 2246	. . . . . . 7 |- ( m = 0 -> ( { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` m ) , (/) , { ( P ` 0 ) , ( P ` m ) } ) <-> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` 0 ) , (/) , { ( P ` 0 ) , ( P ` 0 ) } ) ) )
32	17, 31	imbi12d 234	. . . . . 6 |- ( m = 0 -> ( ( m <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` m ) , (/) , { ( P ` 0 ) , ( P ` m ) } ) ) <-> ( 0 <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` 0 ) , (/) , { ( P ` 0 ) , ( P ` 0 ) } ) ) ) )
33	32	imbi2d 230	. . . . 5 |- ( m = 0 -> ( ( ph -> ( m <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` m ) , (/) , { ( P ` 0 ) , ( P ` m ) } ) ) ) <-> ( ph -> ( 0 <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` 0 ) , (/) , { ( P ` 0 ) , ( P ` 0 ) } ) ) ) ) )
34		breq1 4091	. . . . . . 7 |- ( m = n -> ( m <_ (♯ ` F ) <-> n <_ (♯ ` F ) ) )
35		oveq2 6026	. . . . . . . . . . . . . . . 16 |- ( m = n -> ( 0..^ m ) = ( 0..^ n ) )
36	35	imaeq2d 5076	. . . . . . . . . . . . . . 15 |- ( m = n -> ( F " ( 0..^ m ) ) = ( F " ( 0..^ n ) ) )
37	36	reseq2d 5013	. . . . . . . . . . . . . 14 |- ( m = n -> ( I |` ( F " ( 0..^ m ) ) ) = ( I |` ( F " ( 0..^ n ) ) ) )
38	37	opeq2d 3869	. . . . . . . . . . . . 13 |- ( m = n -> <. V , ( I |` ( F " ( 0..^ m ) ) ) >. = <. V , ( I |` ( F " ( 0..^ n ) ) ) >. )
39	38	fveq2d 5643	. . . . . . . . . . . 12 |- ( m = n -> (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) = (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) )
40	39	fveq1d 5641	. . . . . . . . . . 11 |- ( m = n -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) = ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) )
41	40	breq2d 4100	. . . . . . . . . 10 |- ( m = n -> ( 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) <-> 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) ) )
42	41	notbid 673	. . . . . . . . 9 |- ( m = n -> ( -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) <-> -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) ) )
43	42	rabbidv 2791	. . . . . . . 8 |- ( m = n -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) } = { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } )
44		fveq2 5639	. . . . . . . . . 10 |- ( m = n -> ( P ` m ) = ( P ` n ) )
45	44	eqeq2d 2243	. . . . . . . . 9 |- ( m = n -> ( ( P ` 0 ) = ( P ` m ) <-> ( P ` 0 ) = ( P ` n ) ) )
46	44	preq2d 3755	. . . . . . . . 9 |- ( m = n -> { ( P ` 0 ) , ( P ` m ) } = { ( P ` 0 ) , ( P ` n ) } )
47	45, 46	ifbieq2d 3630	. . . . . . . 8 |- ( m = n -> if ( ( P ` 0 ) = ( P ` m ) , (/) , { ( P ` 0 ) , ( P ` m ) } ) = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) )
48	43, 47	eqeq12d 2246	. . . . . . 7 |- ( m = n -> ( { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` m ) , (/) , { ( P ` 0 ) , ( P ` m ) } ) <-> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) )
49	34, 48	imbi12d 234	. . . . . 6 |- ( m = n -> ( ( m <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` m ) , (/) , { ( P ` 0 ) , ( P ` m ) } ) ) <-> ( n <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) )
50	49	imbi2d 230	. . . . 5 |- ( m = n -> ( ( ph -> ( m <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` m ) , (/) , { ( P ` 0 ) , ( P ` m ) } ) ) ) <-> ( ph -> ( n <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) ) )
51		breq1 4091	. . . . . . 7 |- ( m = ( n + 1 ) -> ( m <_ (♯ ` F ) <-> ( n + 1 ) <_ (♯ ` F ) ) )
52		oveq2 6026	. . . . . . . . . . . . . . . 16 |- ( m = ( n + 1 ) -> ( 0..^ m ) = ( 0..^ ( n + 1 ) ) )
53	52	imaeq2d 5076	. . . . . . . . . . . . . . 15 |- ( m = ( n + 1 ) -> ( F " ( 0..^ m ) ) = ( F " ( 0..^ ( n + 1 ) ) ) )
54	53	reseq2d 5013	. . . . . . . . . . . . . 14 |- ( m = ( n + 1 ) -> ( I |` ( F " ( 0..^ m ) ) ) = ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) )
55	54	opeq2d 3869	. . . . . . . . . . . . 13 |- ( m = ( n + 1 ) -> <. V , ( I |` ( F " ( 0..^ m ) ) ) >. = <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. )
56	55	fveq2d 5643	. . . . . . . . . . . 12 |- ( m = ( n + 1 ) -> (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) = (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) )
57	56	fveq1d 5641	. . . . . . . . . . 11 |- ( m = ( n + 1 ) -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) = ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) )
58	57	breq2d 4100	. . . . . . . . . 10 |- ( m = ( n + 1 ) -> ( 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) <-> 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) ) )
59	58	notbid 673	. . . . . . . . 9 |- ( m = ( n + 1 ) -> ( -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) <-> -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) ) )
60	59	rabbidv 2791	. . . . . . . 8 |- ( m = ( n + 1 ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) } = { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) } )
61		fveq2 5639	. . . . . . . . . 10 |- ( m = ( n + 1 ) -> ( P ` m ) = ( P ` ( n + 1 ) ) )
62	61	eqeq2d 2243	. . . . . . . . 9 |- ( m = ( n + 1 ) -> ( ( P ` 0 ) = ( P ` m ) <-> ( P ` 0 ) = ( P ` ( n + 1 ) ) ) )
63	61	preq2d 3755	. . . . . . . . 9 |- ( m = ( n + 1 ) -> { ( P ` 0 ) , ( P ` m ) } = { ( P ` 0 ) , ( P ` ( n + 1 ) ) } )
64	62, 63	ifbieq2d 3630	. . . . . . . 8 |- ( m = ( n + 1 ) -> if ( ( P ` 0 ) = ( P ` m ) , (/) , { ( P ` 0 ) , ( P ` m ) } ) = if ( ( P ` 0 ) = ( P ` ( n + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( n + 1 ) ) } ) )
65	60, 64	eqeq12d 2246	. . . . . . 7 |- ( m = ( n + 1 ) -> ( { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` m ) , (/) , { ( P ` 0 ) , ( P ` m ) } ) <-> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` ( n + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( n + 1 ) ) } ) ) )
66	51, 65	imbi12d 234	. . . . . 6 |- ( m = ( n + 1 ) -> ( ( m <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` m ) , (/) , { ( P ` 0 ) , ( P ` m ) } ) ) <-> ( ( n + 1 ) <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` ( n + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( n + 1 ) ) } ) ) ) )
67	66	imbi2d 230	. . . . 5 |- ( m = ( n + 1 ) -> ( ( ph -> ( m <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` m ) , (/) , { ( P ` 0 ) , ( P ` m ) } ) ) ) <-> ( ph -> ( ( n + 1 ) <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` ( n + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( n + 1 ) ) } ) ) ) ) )
68		breq1 4091	. . . . . . 7 |- ( m = (♯ ` F ) -> ( m <_ (♯ ` F ) <-> (♯ ` F ) <_ (♯ ` F ) ) )
69		oveq2 6026	. . . . . . . . . . . . . . . 16 |- ( m = (♯ ` F ) -> ( 0..^ m ) = ( 0..^ (♯ ` F ) ) )
70	69	imaeq2d 5076	. . . . . . . . . . . . . . 15 |- ( m = (♯ ` F ) -> ( F " ( 0..^ m ) ) = ( F " ( 0..^ (♯ ` F ) ) ) )
71	70	reseq2d 5013	. . . . . . . . . . . . . 14 |- ( m = (♯ ` F ) -> ( I |` ( F " ( 0..^ m ) ) ) = ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) )
72	71	opeq2d 3869	. . . . . . . . . . . . 13 |- ( m = (♯ ` F ) -> <. V , ( I |` ( F " ( 0..^ m ) ) ) >. = <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. )
73	72	fveq2d 5643	. . . . . . . . . . . 12 |- ( m = (♯ ` F ) -> (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) = (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) )
74	73	fveq1d 5641	. . . . . . . . . . 11 |- ( m = (♯ ` F ) -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) = ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) ` x ) )
75	74	breq2d 4100	. . . . . . . . . 10 |- ( m = (♯ ` F ) -> ( 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) <-> 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) ` x ) ) )
76	75	notbid 673	. . . . . . . . 9 |- ( m = (♯ ` F ) -> ( -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) <-> -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) ` x ) ) )
77	76	rabbidv 2791	. . . . . . . 8 |- ( m = (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) } = { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) ` x ) } )
78		fveq2 5639	. . . . . . . . . 10 |- ( m = (♯ ` F ) -> ( P ` m ) = ( P ` (♯ ` F ) ) )
79	78	eqeq2d 2243	. . . . . . . . 9 |- ( m = (♯ ` F ) -> ( ( P ` 0 ) = ( P ` m ) <-> ( P ` 0 ) = ( P ` (♯ ` F ) ) ) )
80	78	preq2d 3755	. . . . . . . . 9 |- ( m = (♯ ` F ) -> { ( P ` 0 ) , ( P ` m ) } = { ( P ` 0 ) , ( P ` (♯ ` F ) ) } )
81	79, 80	ifbieq2d 3630	. . . . . . . 8 |- ( m = (♯ ` F ) -> if ( ( P ` 0 ) = ( P ` m ) , (/) , { ( P ` 0 ) , ( P ` m ) } ) = if ( ( P ` 0 ) = ( P ` (♯ ` F ) ) , (/) , { ( P ` 0 ) , ( P ` (♯ ` F ) ) } ) )
82	77, 81	eqeq12d 2246	. . . . . . 7 |- ( m = (♯ ` F ) -> ( { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` m ) , (/) , { ( P ` 0 ) , ( P ` m ) } ) <-> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` (♯ ` F ) ) , (/) , { ( P ` 0 ) , ( P ` (♯ ` F ) ) } ) ) )
83	68, 82	imbi12d 234	. . . . . 6 |- ( m = (♯ ` F ) -> ( ( m <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` m ) , (/) , { ( P ` 0 ) , ( P ` m ) } ) ) <-> ( (♯ ` F ) <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` (♯ ` F ) ) , (/) , { ( P ` 0 ) , ( P ` (♯ ` F ) ) } ) ) ) )
84	83	imbi2d 230	. . . . 5 |- ( m = (♯ ` F ) -> ( ( ph -> ( m <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` m ) , (/) , { ( P ` 0 ) , ( P ` m ) } ) ) ) <-> ( ph -> ( (♯ ` F ) <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` (♯ ` F ) ) , (/) , { ( P ` 0 ) , ( P ` (♯ ` F ) ) } ) ) ) ) )
85		eupth2fi.fi	. . . . . . . 8 |- ( ph -> V e. Fin )
86	1, 2, 3, 4, 5, 85	eupth2lembfi 16331	. . . . . . 7 |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) ` x ) } = (/) )
87		eqid 2231	. . . . . . . 8 |- ( P ` 0 ) = ( P ` 0 )
88	87	iftruei 3611	. . . . . . 7 |- if ( ( P ` 0 ) = ( P ` 0 ) , (/) , { ( P ` 0 ) , ( P ` 0 ) } ) = (/)
89	86, 88	eqtr4di 2282	. . . . . 6 |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` 0 ) , (/) , { ( P ` 0 ) , ( P ` 0 ) } ) )
90	89	a1d 22	. . . . 5 |- ( ph -> ( 0 <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` 0 ) , (/) , { ( P ` 0 ) , ( P ` 0 ) } ) ) )
91	1, 2, 3, 4, 5, 85	eupth2lemsfi 16332	. . . . . . 7 |- ( ( ph /\ n e. NN0 ) -> ( ( n <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) -> ( ( n + 1 ) <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` ( n + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( n + 1 ) ) } ) ) ) )
92	91	expcom 116	. . . . . 6 |- ( n e. NN0 -> ( ph -> ( ( n <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) -> ( ( n + 1 ) <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` ( n + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( n + 1 ) ) } ) ) ) ) )
93	92	a2d 26	. . . . 5 |- ( n e. NN0 -> ( ( ph -> ( n <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) -> ( ph -> ( ( n + 1 ) <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` ( n + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( n + 1 ) ) } ) ) ) ) )
94	33, 50, 67, 84, 90, 93	nn0ind 9594	. . . 4 |- ( (♯ ` F ) e. NN0 -> ( ph -> ( (♯ ` F ) <_ (♯ ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` (♯ ` F ) ) , (/) , { ( P ` 0 ) , ( P ` (♯ ` F ) ) } ) ) ) )
95	16, 94	mpid 42	. . 3 |- ( (♯ ` F ) e. NN0 -> ( ph -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` (♯ ` F ) ) , (/) , { ( P ` 0 ) , ( P ` (♯ ` F ) ) } ) ) )
96	14, 95	mpcom 36	. 2 |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` (♯ ` F ) ) , (/) , { ( P ` 0 ) , ( P ` (♯ ` F ) ) } ) )
97	11, 96	eqtr3d 2266	1 |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` G ) ` x ) } = if ( ( P ` 0 ) = ( P ` (♯ ` F ) ) , (/) , { ( P ` 0 ) , ( P ` (♯ ` F ) ) } ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1397   e. wcel 2202   {crab 2514   (/)c0 3494   ifcif 3605   {cpr 3670   <.cop 3672   class class class wbr 4088   |` cres 4727   "cima 4728   Fun wfun 5320   ` cfv 5326  (class class class)co 6018   Fincfn 6909   0cc0 8032   1c1 8033   + caddc 8035   <_ cle 8215   2c2 9194   NN0cn0 9402  ..^cfzo 10377  ♯chash 11038   || cdvds 12350  Vtxcvtx 15866  iEdgciedg 15867  UMGraphcumgr 15946  VtxDegcvtxdg 16140  Walkscwlks 16171  EulerPathsceupth 16296

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-ifp 986  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-xor 1420  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-frec 6557  df-1o 6582  df-2o 6583  df-oadd 6586  df-er 6702  df-map 6819  df-en 6910  df-dom 6911  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-z 9480  df-dec 9612  df-uz 9756  df-q 9854  df-rp 9889  df-xadd 10008  df-fz 10244  df-fzo 10378  df-fl 10531  df-mod 10586  df-seqfrec 10711  df-exp 10802  df-ihash 11039  df-word 11115  df-cj 11404  df-re 11405  df-im 11406  df-rsqrt 11560  df-abs 11561  df-dvds 12351  df-ndx 13087  df-slot 13088  df-base 13090  df-edgf 15859  df-vtx 15868  df-iedg 15869  df-edg 15912  df-uhgrm 15923  df-ushgrm 15924  df-upgren 15947  df-umgren 15948  df-uspgren 16009  df-subgr 16108  df-vtxdg 16141  df-wlks 16172  df-trls 16235  df-eupth 16297

This theorem is referenced by: (None)