Theorem eupth2 27099

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 )
eupth2.g	|- ( ph -> G e. UPGraph )
eupth2.f	|- ( ph -> Fun I )
eupth2.p	|- ( ph -> F (EulerPaths ` G ) P )

Assertion

Ref	Expression
eupth2	|- ( 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 eupth2

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		eupth2.g	. . . . . . 7 |- ( ph -> G e. UPGraph )
4		eupth2.f	. . . . . . 7 |- ( ph -> Fun I )
5		eupth2.p	. . . . . . 7 |- ( ph -> F (EulerPaths ` G ) P )
6		eqid 2622	. . . . . . 7 |- <. V , ( I |` ( F " ( 0..^ ( # ` F ) ) ) ) >. = <. V , ( I |` ( F " ( 0..^ ( # ` F ) ) ) ) >.
7	1, 2, 3, 4, 5, 6	eupthvdres 27095	. . . . . 6 |- ( ph -> (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( # ` F ) ) ) ) >. ) = (VtxDeg ` G ) )
8	7	fveq1d 6193	. . . . 5 |- ( ph -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( # ` F ) ) ) ) >. ) ` x ) = ( (VtxDeg ` G ) ` x ) )
9	8	breq2d 4665	. . . 4 |- ( ph -> ( 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( # ` F ) ) ) ) >. ) ` x ) <-> 2 || ( (VtxDeg ` G ) ` x ) ) )
10	9	notbid 308	. . 3 |- ( ph -> ( -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( # ` F ) ) ) ) >. ) ` x ) <-> -. 2 || ( (VtxDeg ` G ) ` x ) ) )
11	10	rabbidv 3189	. 2 |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( # ` F ) ) ) ) >. ) ` x ) } = { x e. V | -. 2 || ( (VtxDeg ` G ) ` x ) } )
12		eupthiswlk 27072	. . . 4 |- ( F (EulerPaths ` G ) P -> F (Walks ` G ) P )
13		wlkcl 26511	. . . 4 |- ( F (Walks ` G ) P -> ( # ` F ) e. NN0 )
14	5, 12, 13	3syl 18	. . 3 |- ( ph -> ( # ` F ) e. NN0 )
15		nn0re 11301	. . . . 5 |- ( ( # ` F ) e. NN0 -> ( # ` F ) e. RR )
16	15	leidd 10594	. . . 4 |- ( ( # ` F ) e. NN0 -> ( # ` F ) <_ ( # ` F ) )
17		breq1 4656	. . . . . . 7 |- ( m = 0 -> ( m <_ ( # ` F ) <-> 0 <_ ( # ` F ) ) )
18		oveq2 6658	. . . . . . . . . . . . . . . 16 |- ( m = 0 -> ( 0..^ m ) = ( 0..^ 0 ) )
19	18	imaeq2d 5466	. . . . . . . . . . . . . . 15 |- ( m = 0 -> ( F " ( 0..^ m ) ) = ( F " ( 0..^ 0 ) ) )
20	19	reseq2d 5396	. . . . . . . . . . . . . 14 |- ( m = 0 -> ( I |` ( F " ( 0..^ m ) ) ) = ( I |` ( F " ( 0..^ 0 ) ) ) )
21	20	opeq2d 4409	. . . . . . . . . . . . 13 |- ( m = 0 -> <. V , ( I |` ( F " ( 0..^ m ) ) ) >. = <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. )
22	21	fveq2d 6195	. . . . . . . . . . . 12 |- ( m = 0 -> (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) = (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) )
23	22	fveq1d 6193	. . . . . . . . . . 11 |- ( m = 0 -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) = ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) ` x ) )
24	23	breq2d 4665	. . . . . . . . . 10 |- ( m = 0 -> ( 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) <-> 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) ` x ) ) )
25	24	notbid 308	. . . . . . . . 9 |- ( m = 0 -> ( -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) <-> -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) ` x ) ) )
26	25	rabbidv 3189	. . . . . . . 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 6191	. . . . . . . . . 10 |- ( m = 0 -> ( P ` m ) = ( P ` 0 ) )
28	27	eqeq2d 2632	. . . . . . . . 9 |- ( m = 0 -> ( ( P ` 0 ) = ( P ` m ) <-> ( P ` 0 ) = ( P ` 0 ) ) )
29	27	preq2d 4275	. . . . . . . . 9 |- ( m = 0 -> { ( P ` 0 ) , ( P ` m ) } = { ( P ` 0 ) , ( P ` 0 ) } )
30	28, 29	ifbieq2d 4111	. . . . . . . 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 2637	. . . . . . 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 334	. . . . . 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 330	. . . . 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 4656	. . . . . . 7 |- ( m = n -> ( m <_ ( # ` F ) <-> n <_ ( # ` F ) ) )
35		oveq2 6658	. . . . . . . . . . . . . . . 16 |- ( m = n -> ( 0..^ m ) = ( 0..^ n ) )
36	35	imaeq2d 5466	. . . . . . . . . . . . . . 15 |- ( m = n -> ( F " ( 0..^ m ) ) = ( F " ( 0..^ n ) ) )
37	36	reseq2d 5396	. . . . . . . . . . . . . 14 |- ( m = n -> ( I |` ( F " ( 0..^ m ) ) ) = ( I |` ( F " ( 0..^ n ) ) ) )
38	37	opeq2d 4409	. . . . . . . . . . . . 13 |- ( m = n -> <. V , ( I |` ( F " ( 0..^ m ) ) ) >. = <. V , ( I |` ( F " ( 0..^ n ) ) ) >. )
39	38	fveq2d 6195	. . . . . . . . . . . 12 |- ( m = n -> (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) = (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) )
40	39	fveq1d 6193	. . . . . . . . . . 11 |- ( m = n -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) = ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) )
41	40	breq2d 4665	. . . . . . . . . 10 |- ( m = n -> ( 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) <-> 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) ) )
42	41	notbid 308	. . . . . . . . 9 |- ( m = n -> ( -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) <-> -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) ) )
43	42	rabbidv 3189	. . . . . . . 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 6191	. . . . . . . . . 10 |- ( m = n -> ( P ` m ) = ( P ` n ) )
45	44	eqeq2d 2632	. . . . . . . . 9 |- ( m = n -> ( ( P ` 0 ) = ( P ` m ) <-> ( P ` 0 ) = ( P ` n ) ) )
46	44	preq2d 4275	. . . . . . . . 9 |- ( m = n -> { ( P ` 0 ) , ( P ` m ) } = { ( P ` 0 ) , ( P ` n ) } )
47	45, 46	ifbieq2d 4111	. . . . . . . 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 2637	. . . . . . 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 334	. . . . . 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 330	. . . . 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 4656	. . . . . . 7 |- ( m = ( n + 1 ) -> ( m <_ ( # ` F ) <-> ( n + 1 ) <_ ( # ` F ) ) )
52		oveq2 6658	. . . . . . . . . . . . . . . 16 |- ( m = ( n + 1 ) -> ( 0..^ m ) = ( 0..^ ( n + 1 ) ) )
53	52	imaeq2d 5466	. . . . . . . . . . . . . . 15 |- ( m = ( n + 1 ) -> ( F " ( 0..^ m ) ) = ( F " ( 0..^ ( n + 1 ) ) ) )
54	53	reseq2d 5396	. . . . . . . . . . . . . 14 |- ( m = ( n + 1 ) -> ( I |` ( F " ( 0..^ m ) ) ) = ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) )
55	54	opeq2d 4409	. . . . . . . . . . . . 13 |- ( m = ( n + 1 ) -> <. V , ( I |` ( F " ( 0..^ m ) ) ) >. = <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. )
56	55	fveq2d 6195	. . . . . . . . . . . 12 |- ( m = ( n + 1 ) -> (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) = (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) )
57	56	fveq1d 6193	. . . . . . . . . . 11 |- ( m = ( n + 1 ) -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) = ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) )
58	57	breq2d 4665	. . . . . . . . . 10 |- ( m = ( n + 1 ) -> ( 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) <-> 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) ) )
59	58	notbid 308	. . . . . . . . 9 |- ( m = ( n + 1 ) -> ( -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) <-> -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) ) )
60	59	rabbidv 3189	. . . . . . . 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 6191	. . . . . . . . . 10 |- ( m = ( n + 1 ) -> ( P ` m ) = ( P ` ( n + 1 ) ) )
62	61	eqeq2d 2632	. . . . . . . . 9 |- ( m = ( n + 1 ) -> ( ( P ` 0 ) = ( P ` m ) <-> ( P ` 0 ) = ( P ` ( n + 1 ) ) ) )
63	61	preq2d 4275	. . . . . . . . 9 |- ( m = ( n + 1 ) -> { ( P ` 0 ) , ( P ` m ) } = { ( P ` 0 ) , ( P ` ( n + 1 ) ) } )
64	62, 63	ifbieq2d 4111	. . . . . . . 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 2637	. . . . . . 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 334	. . . . . 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 330	. . . . 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 4656	. . . . . . 7 |- ( m = ( # ` F ) -> ( m <_ ( # ` F ) <-> ( # ` F ) <_ ( # ` F ) ) )
69		oveq2 6658	. . . . . . . . . . . . . . . 16 |- ( m = ( # ` F ) -> ( 0..^ m ) = ( 0..^ ( # ` F ) ) )
70	69	imaeq2d 5466	. . . . . . . . . . . . . . 15 |- ( m = ( # ` F ) -> ( F " ( 0..^ m ) ) = ( F " ( 0..^ ( # ` F ) ) ) )
71	70	reseq2d 5396	. . . . . . . . . . . . . 14 |- ( m = ( # ` F ) -> ( I |` ( F " ( 0..^ m ) ) ) = ( I |` ( F " ( 0..^ ( # ` F ) ) ) ) )
72	71	opeq2d 4409	. . . . . . . . . . . . 13 |- ( m = ( # ` F ) -> <. V , ( I |` ( F " ( 0..^ m ) ) ) >. = <. V , ( I |` ( F " ( 0..^ ( # ` F ) ) ) ) >. )
73	72	fveq2d 6195	. . . . . . . . . . . 12 |- ( m = ( # ` F ) -> (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) = (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( # ` F ) ) ) ) >. ) )
74	73	fveq1d 6193	. . . . . . . . . . 11 |- ( m = ( # ` F ) -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) = ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( # ` F ) ) ) ) >. ) ` x ) )
75	74	breq2d 4665	. . . . . . . . . 10 |- ( m = ( # ` F ) -> ( 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) <-> 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( # ` F ) ) ) ) >. ) ` x ) ) )
76	75	notbid 308	. . . . . . . . 9 |- ( m = ( # ` F ) -> ( -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) <-> -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( # ` F ) ) ) ) >. ) ` x ) ) )
77	76	rabbidv 3189	. . . . . . . 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 6191	. . . . . . . . . 10 |- ( m = ( # ` F ) -> ( P ` m ) = ( P ` ( # ` F ) ) )
79	78	eqeq2d 2632	. . . . . . . . 9 |- ( m = ( # ` F ) -> ( ( P ` 0 ) = ( P ` m ) <-> ( P ` 0 ) = ( P ` ( # ` F ) ) ) )
80	78	preq2d 4275	. . . . . . . . 9 |- ( m = ( # ` F ) -> { ( P ` 0 ) , ( P ` m ) } = { ( P ` 0 ) , ( P ` ( # ` F ) ) } )
81	79, 80	ifbieq2d 4111	. . . . . . . 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 2637	. . . . . . 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 334	. . . . . 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 330	. . . . 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	1, 2, 3, 4, 5	eupth2lemb 27097	. . . . . . 7 |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) ` x ) } = (/) )
86		eqid 2622	. . . . . . . 8 |- ( P ` 0 ) = ( P ` 0 )
87	86	iftruei 4093	. . . . . . 7 |- if ( ( P ` 0 ) = ( P ` 0 ) , (/) , { ( P ` 0 ) , ( P ` 0 ) } ) = (/)
88	85, 87	syl6eqr 2674	. . . . . 6 |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` 0 ) , (/) , { ( P ` 0 ) , ( P ` 0 ) } ) )
89	88	a1d 25	. . . . 5 |- ( ph -> ( 0 <_ ( # ` F ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` 0 ) , (/) , { ( P ` 0 ) , ( P ` 0 ) } ) ) )
90	1, 2, 3, 4, 5	eupth2lems 27098	. . . . . . 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 ) ) } ) ) ) )
91	90	expcom 451	. . . . . 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 ) ) } ) ) ) ) )
92	91	a2d 29	. . . . 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 ) ) } ) ) ) ) )
93	33, 50, 67, 84, 89, 92	nn0ind 11472	. . . 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 ) ) } ) ) ) )
94	16, 93	mpid 44	. . 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 ) ) } ) ) )
95	14, 94	mpcom 38	. 2 |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( # ` F ) ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` ( # ` F ) ) , (/) , { ( P ` 0 ) , ( P ` ( # ` F ) ) } ) )
96	11, 95	eqtr3d 2658	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 1483   e. wcel 1990   {crab 2916   (/)c0 3915   ifcif 4086   {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   <_ cle 10075   2c2 11070   NN0cn0 11292  ..^cfzo 12465   #chash 13117   || cdvds 14983  Vtxcvtx 25874  iEdgciedg 25875   UPGraph cupgr 25975  VtxDegcvtxdg 26361  Walkscwlks 26492  EulerPathsceupth 27057

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-2o 7561  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-vtx 25876  df-iedg 25877  df-edg 25940  df-uhgr 25953  df-ushgr 25954  df-upgr 25977  df-uspgr 26045  df-vtxdg 26362  df-wlks 26495  df-trls 26589  df-eupth 27058

This theorem is referenced by:  eulerpathpr  27100  eulercrct  27102