Theorem eupth2fi 16720

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 2238	. . . . . . 7 |- <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. = <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >.
7	1, 2, 3, 4, 5, 6	eupthvdres 16716	. . . . . 6 |- ( ph -> (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) = (VtxDeg ` G ) )
8	7	fveq1d 5697	. . . . 5 |- ( ph -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) ` x ) = ( (VtxDeg ` G ) ` x ) )
9	8	breq2d 4142	. . . 4 |- ( ph -> ( 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) ` x ) <-> 2 || ( (VtxDeg ` G ) ` x ) ) )
10	9	notbid 677	. . 3 |- ( ph -> ( -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) ` x ) <-> -. 2 || ( (VtxDeg ` G ) ` x ) ) )
11	10	rabbidv 2810	. 2 |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) ` x ) } = { x e. V | -. 2 || ( (VtxDeg ` G ) ` x ) } )
12		eupthiswlk 16696	. . . 4 |- ( F (EulerPaths ` G ) P -> F (Walks ` G ) P )
13		wlkcl 16573	. . . 4 |- ( F (Walks ` G ) P -> (♯ ` F ) e. NN0 )
14	5, 12, 13	3syl 17	. . 3 |- ( ph -> (♯ ` F ) e. NN0 )
15		nn0re 9572	. . . . 5 |- ( (♯ ` F ) e. NN0 -> (♯ ` F ) e. RR )
16	15	leidd 8842	. . . 4 |- ( (♯ ` F ) e. NN0 -> (♯ ` F ) <_ (♯ ` F ) )
17		breq1 4133	. . . . . . 7 |- ( m = 0 -> ( m <_ (♯ ` F ) <-> 0 <_ (♯ ` F ) ) )
18		oveq2 6093	. . . . . . . . . . . . . . . 16 |- ( m = 0 -> ( 0..^ m ) = ( 0..^ 0 ) )
19	18	imaeq2d 5126	. . . . . . . . . . . . . . 15 |- ( m = 0 -> ( F " ( 0..^ m ) ) = ( F " ( 0..^ 0 ) ) )
20	19	reseq2d 5063	. . . . . . . . . . . . . 14 |- ( m = 0 -> ( I |` ( F " ( 0..^ m ) ) ) = ( I |` ( F " ( 0..^ 0 ) ) ) )
21	20	opeq2d 3911	. . . . . . . . . . . . 13 |- ( m = 0 -> <. V , ( I |` ( F " ( 0..^ m ) ) ) >. = <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. )
22	21	fveq2d 5699	. . . . . . . . . . . 12 |- ( m = 0 -> (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) = (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) )
23	22	fveq1d 5697	. . . . . . . . . . 11 |- ( m = 0 -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) = ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) ` x ) )
24	23	breq2d 4142	. . . . . . . . . 10 |- ( m = 0 -> ( 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) <-> 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) ` x ) ) )
25	24	notbid 677	. . . . . . . . 9 |- ( m = 0 -> ( -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) <-> -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) ` x ) ) )
26	25	rabbidv 2810	. . . . . . . 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 5695	. . . . . . . . . 10 |- ( m = 0 -> ( P ` m ) = ( P ` 0 ) )
28	27	eqeq2d 2250	. . . . . . . . 9 |- ( m = 0 -> ( ( P ` 0 ) = ( P ` m ) <-> ( P ` 0 ) = ( P ` 0 ) ) )
29	27	preq2d 3795	. . . . . . . . 9 |- ( m = 0 -> { ( P ` 0 ) , ( P ` m ) } = { ( P ` 0 ) , ( P ` 0 ) } )
30	28, 29	ifbieq2d 3665	. . . . . . . 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 2253	. . . . . . 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 4133	. . . . . . 7 |- ( m = n -> ( m <_ (♯ ` F ) <-> n <_ (♯ ` F ) ) )
35		oveq2 6093	. . . . . . . . . . . . . . . 16 |- ( m = n -> ( 0..^ m ) = ( 0..^ n ) )
36	35	imaeq2d 5126	. . . . . . . . . . . . . . 15 |- ( m = n -> ( F " ( 0..^ m ) ) = ( F " ( 0..^ n ) ) )
37	36	reseq2d 5063	. . . . . . . . . . . . . 14 |- ( m = n -> ( I |` ( F " ( 0..^ m ) ) ) = ( I |` ( F " ( 0..^ n ) ) ) )
38	37	opeq2d 3911	. . . . . . . . . . . . 13 |- ( m = n -> <. V , ( I |` ( F " ( 0..^ m ) ) ) >. = <. V , ( I |` ( F " ( 0..^ n ) ) ) >. )
39	38	fveq2d 5699	. . . . . . . . . . . 12 |- ( m = n -> (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) = (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) )
40	39	fveq1d 5697	. . . . . . . . . . 11 |- ( m = n -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) = ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) )
41	40	breq2d 4142	. . . . . . . . . 10 |- ( m = n -> ( 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) <-> 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) ) )
42	41	notbid 677	. . . . . . . . 9 |- ( m = n -> ( -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) <-> -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) ) )
43	42	rabbidv 2810	. . . . . . . 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 5695	. . . . . . . . . 10 |- ( m = n -> ( P ` m ) = ( P ` n ) )
45	44	eqeq2d 2250	. . . . . . . . 9 |- ( m = n -> ( ( P ` 0 ) = ( P ` m ) <-> ( P ` 0 ) = ( P ` n ) ) )
46	44	preq2d 3795	. . . . . . . . 9 |- ( m = n -> { ( P ` 0 ) , ( P ` m ) } = { ( P ` 0 ) , ( P ` n ) } )
47	45, 46	ifbieq2d 3665	. . . . . . . 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 2253	. . . . . . 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 4133	. . . . . . 7 |- ( m = ( n + 1 ) -> ( m <_ (♯ ` F ) <-> ( n + 1 ) <_ (♯ ` F ) ) )
52		oveq2 6093	. . . . . . . . . . . . . . . 16 |- ( m = ( n + 1 ) -> ( 0..^ m ) = ( 0..^ ( n + 1 ) ) )
53	52	imaeq2d 5126	. . . . . . . . . . . . . . 15 |- ( m = ( n + 1 ) -> ( F " ( 0..^ m ) ) = ( F " ( 0..^ ( n + 1 ) ) ) )
54	53	reseq2d 5063	. . . . . . . . . . . . . 14 |- ( m = ( n + 1 ) -> ( I |` ( F " ( 0..^ m ) ) ) = ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) )
55	54	opeq2d 3911	. . . . . . . . . . . . 13 |- ( m = ( n + 1 ) -> <. V , ( I |` ( F " ( 0..^ m ) ) ) >. = <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. )
56	55	fveq2d 5699	. . . . . . . . . . . 12 |- ( m = ( n + 1 ) -> (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) = (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) )
57	56	fveq1d 5697	. . . . . . . . . . 11 |- ( m = ( n + 1 ) -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) = ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) )
58	57	breq2d 4142	. . . . . . . . . 10 |- ( m = ( n + 1 ) -> ( 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) <-> 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) ) )
59	58	notbid 677	. . . . . . . . 9 |- ( m = ( n + 1 ) -> ( -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) <-> -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) ) )
60	59	rabbidv 2810	. . . . . . . 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 5695	. . . . . . . . . 10 |- ( m = ( n + 1 ) -> ( P ` m ) = ( P ` ( n + 1 ) ) )
62	61	eqeq2d 2250	. . . . . . . . 9 |- ( m = ( n + 1 ) -> ( ( P ` 0 ) = ( P ` m ) <-> ( P ` 0 ) = ( P ` ( n + 1 ) ) ) )
63	61	preq2d 3795	. . . . . . . . 9 |- ( m = ( n + 1 ) -> { ( P ` 0 ) , ( P ` m ) } = { ( P ` 0 ) , ( P ` ( n + 1 ) ) } )
64	62, 63	ifbieq2d 3665	. . . . . . . 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 2253	. . . . . . 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 4133	. . . . . . 7 |- ( m = (♯ ` F ) -> ( m <_ (♯ ` F ) <-> (♯ ` F ) <_ (♯ ` F ) ) )
69		oveq2 6093	. . . . . . . . . . . . . . . 16 |- ( m = (♯ ` F ) -> ( 0..^ m ) = ( 0..^ (♯ ` F ) ) )
70	69	imaeq2d 5126	. . . . . . . . . . . . . . 15 |- ( m = (♯ ` F ) -> ( F " ( 0..^ m ) ) = ( F " ( 0..^ (♯ ` F ) ) ) )
71	70	reseq2d 5063	. . . . . . . . . . . . . 14 |- ( m = (♯ ` F ) -> ( I |` ( F " ( 0..^ m ) ) ) = ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) )
72	71	opeq2d 3911	. . . . . . . . . . . . 13 |- ( m = (♯ ` F ) -> <. V , ( I |` ( F " ( 0..^ m ) ) ) >. = <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. )
73	72	fveq2d 5699	. . . . . . . . . . . 12 |- ( m = (♯ ` F ) -> (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) = (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) )
74	73	fveq1d 5697	. . . . . . . . . . 11 |- ( m = (♯ ` F ) -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) = ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) ` x ) )
75	74	breq2d 4142	. . . . . . . . . 10 |- ( m = (♯ ` F ) -> ( 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) <-> 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) ` x ) ) )
76	75	notbid 677	. . . . . . . . 9 |- ( m = (♯ ` F ) -> ( -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) <-> -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) ` x ) ) )
77	76	rabbidv 2810	. . . . . . . 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 5695	. . . . . . . . . 10 |- ( m = (♯ ` F ) -> ( P ` m ) = ( P ` (♯ ` F ) ) )
79	78	eqeq2d 2250	. . . . . . . . 9 |- ( m = (♯ ` F ) -> ( ( P ` 0 ) = ( P ` m ) <-> ( P ` 0 ) = ( P ` (♯ ` F ) ) ) )
80	78	preq2d 3795	. . . . . . . . 9 |- ( m = (♯ ` F ) -> { ( P ` 0 ) , ( P ` m ) } = { ( P ` 0 ) , ( P ` (♯ ` F ) ) } )
81	79, 80	ifbieq2d 3665	. . . . . . . 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 2253	. . . . . . 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 16718	. . . . . . 7 |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) ` x ) } = (/) )
87		eqid 2238	. . . . . . . 8 |- ( P ` 0 ) = ( P ` 0 )
88	87	iftruei 3646	. . . . . . 7 |- if ( ( P ` 0 ) = ( P ` 0 ) , (/) , { ( P ` 0 ) , ( P ` 0 ) } ) = (/)
89	86, 88	eqtr4di 2289	. . . . . 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 16719	. . . . . . 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 9760	. . . 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 2273	1 |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` G ) ` x ) } = if ( ( P ` 0 ) = ( P ` (♯ ` F ) ) , (/) , { ( P ` 0 ) , ( P ` (♯ ` F ) ) } ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   = wceq 1402   e. wcel 2209   {crab 2532   (/)c0 3520   ifcif 3638   {cpr 3710   <.cop 3712   class class class wbr 4130   |` cres 4776   "cima 4777   Fun wfun 5371   ` cfv 5377  (class class class)co 6085   Fincfn 7022   0cc0 8179   1c1 8180   + caddc 8182   <_ cle 8361   2c2 9355   NN0cn0 9563  ..^cfzo 10549  ♯chash 11214   || cdvds 12554  Vtxcvtx 16253  iEdgciedg 16254  UMGraphcumgr 16333  VtxDegcvtxdg 16527  Walkscwlks 16558  EulerPathsceupth 16683

This proof depends on 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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-ifp 991  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-xor 1425  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-rmo 2536  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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6807  df-map 6924  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-z 9645  df-dec 9778  df-uz 9922  df-q 10020  df-rp 10055  df-xadd 10175  df-fz 10412  df-fzo 10550  df-fl 10705  df-mod 10760  df-seqfrec 10885  df-exp 10976  df-ihash 11215  df-word 11305  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-dvds 12555  df-ndx 13355  df-slot 13356  df-base 13358  df-edgf 16246  df-vtx 16255  df-iedg 16256  df-edg 16299  df-uhgrm 16310  df-ushgrm 16311  df-upgren 16334  df-umgren 16335  df-uspgren 16396  df-subgr 16495  df-vtxdg 16528  df-wlks 16559  df-trls 16622  df-eupth 16684

This theorem is used by:  eulerpathprum  16721