Theorem eupth2fi 16703

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 16699	. . . . . 6 |- ( ph -> (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) = (VtxDeg ` G ) )
8	7	fveq1d 5695	. . . . 5 |- ( ph -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) ` x ) = ( (VtxDeg ` G ) ` x ) )
9	8	breq2d 4140	. . . 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 16679	. . . 4 |- ( F (EulerPaths ` G ) P -> F (Walks ` G ) P )
13		wlkcl 16556	. . . 4 |- ( F (Walks ` G ) P -> (♯ ` F ) e. NN0 )
14	5, 12, 13	3syl 17	. . 3 |- ( ph -> (♯ ` F ) e. NN0 )
15		nn0re 9555	. . . . 5 |- ( (♯ ` F ) e. NN0 -> (♯ ` F ) e. RR )
16	15	leidd 8836	. . . 4 |- ( (♯ ` F ) e. NN0 -> (♯ ` F ) <_ (♯ ` F ) )
17		breq1 4131	. . . . . . 7 |- ( m = 0 -> ( m <_ (♯ ` F ) <-> 0 <_ (♯ ` F ) ) )
18		oveq2 6087	. . . . . . . . . . . . . . . 16 |- ( m = 0 -> ( 0..^ m ) = ( 0..^ 0 ) )
19	18	imaeq2d 5124	. . . . . . . . . . . . . . 15 |- ( m = 0 -> ( F " ( 0..^ m ) ) = ( F " ( 0..^ 0 ) ) )
20	19	reseq2d 5061	. . . . . . . . . . . . . 14 |- ( m = 0 -> ( I |` ( F " ( 0..^ m ) ) ) = ( I |` ( F " ( 0..^ 0 ) ) ) )
21	20	opeq2d 3909	. . . . . . . . . . . . 13 |- ( m = 0 -> <. V , ( I |` ( F " ( 0..^ m ) ) ) >. = <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. )
22	21	fveq2d 5697	. . . . . . . . . . . 12 |- ( m = 0 -> (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) = (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) )
23	22	fveq1d 5695	. . . . . . . . . . 11 |- ( m = 0 -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) = ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) ` x ) )
24	23	breq2d 4140	. . . . . . . . . 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 5693	. . . . . . . . . 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 3794	. . . . . . . . 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 4131	. . . . . . 7 |- ( m = n -> ( m <_ (♯ ` F ) <-> n <_ (♯ ` F ) ) )
35		oveq2 6087	. . . . . . . . . . . . . . . 16 |- ( m = n -> ( 0..^ m ) = ( 0..^ n ) )
36	35	imaeq2d 5124	. . . . . . . . . . . . . . 15 |- ( m = n -> ( F " ( 0..^ m ) ) = ( F " ( 0..^ n ) ) )
37	36	reseq2d 5061	. . . . . . . . . . . . . 14 |- ( m = n -> ( I |` ( F " ( 0..^ m ) ) ) = ( I |` ( F " ( 0..^ n ) ) ) )
38	37	opeq2d 3909	. . . . . . . . . . . . 13 |- ( m = n -> <. V , ( I |` ( F " ( 0..^ m ) ) ) >. = <. V , ( I |` ( F " ( 0..^ n ) ) ) >. )
39	38	fveq2d 5697	. . . . . . . . . . . 12 |- ( m = n -> (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) = (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) )
40	39	fveq1d 5695	. . . . . . . . . . 11 |- ( m = n -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) = ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) )
41	40	breq2d 4140	. . . . . . . . . 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 5693	. . . . . . . . . 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 3794	. . . . . . . . 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 4131	. . . . . . 7 |- ( m = ( n + 1 ) -> ( m <_ (♯ ` F ) <-> ( n + 1 ) <_ (♯ ` F ) ) )
52		oveq2 6087	. . . . . . . . . . . . . . . 16 |- ( m = ( n + 1 ) -> ( 0..^ m ) = ( 0..^ ( n + 1 ) ) )
53	52	imaeq2d 5124	. . . . . . . . . . . . . . 15 |- ( m = ( n + 1 ) -> ( F " ( 0..^ m ) ) = ( F " ( 0..^ ( n + 1 ) ) ) )
54	53	reseq2d 5061	. . . . . . . . . . . . . 14 |- ( m = ( n + 1 ) -> ( I |` ( F " ( 0..^ m ) ) ) = ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) )
55	54	opeq2d 3909	. . . . . . . . . . . . 13 |- ( m = ( n + 1 ) -> <. V , ( I |` ( F " ( 0..^ m ) ) ) >. = <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. )
56	55	fveq2d 5697	. . . . . . . . . . . 12 |- ( m = ( n + 1 ) -> (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) = (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) )
57	56	fveq1d 5695	. . . . . . . . . . 11 |- ( m = ( n + 1 ) -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) = ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) )
58	57	breq2d 4140	. . . . . . . . . 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 5693	. . . . . . . . . 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 3794	. . . . . . . . 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 4131	. . . . . . 7 |- ( m = (♯ ` F ) -> ( m <_ (♯ ` F ) <-> (♯ ` F ) <_ (♯ ` F ) ) )
69		oveq2 6087	. . . . . . . . . . . . . . . 16 |- ( m = (♯ ` F ) -> ( 0..^ m ) = ( 0..^ (♯ ` F ) ) )
70	69	imaeq2d 5124	. . . . . . . . . . . . . . 15 |- ( m = (♯ ` F ) -> ( F " ( 0..^ m ) ) = ( F " ( 0..^ (♯ ` F ) ) ) )
71	70	reseq2d 5061	. . . . . . . . . . . . . 14 |- ( m = (♯ ` F ) -> ( I |` ( F " ( 0..^ m ) ) ) = ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) )
72	71	opeq2d 3909	. . . . . . . . . . . . 13 |- ( m = (♯ ` F ) -> <. V , ( I |` ( F " ( 0..^ m ) ) ) >. = <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. )
73	72	fveq2d 5697	. . . . . . . . . . . 12 |- ( m = (♯ ` F ) -> (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) = (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) )
74	73	fveq1d 5695	. . . . . . . . . . 11 |- ( m = (♯ ` F ) -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) = ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) ` x ) )
75	74	breq2d 4140	. . . . . . . . . 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 5693	. . . . . . . . . 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 3794	. . . . . . . . 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 16701	. . . . . . 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 16702	. . . . . . 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 9743	. . . 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 ) ) } ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1402   e. wcel 2209   {crab 2532   (/)c0 3520   ifcif 3638   {cpr 3709   <.cop 3711   class class class wbr 4128   |` cres 4774   "cima 4775   Fun wfun 5369   ` cfv 5375  (class class class)co 6079   Fincfn 7016   0cc0 8173   1c1 8174   + caddc 8176   <_ cle 8355   2c2 9338   NN0cn0 9546  ..^cfzo 10532  ♯chash 11197   || cdvds 12537  Vtxcvtx 16236  iEdgciedg 16237  UMGraphcumgr 16316  VtxDegcvtxdg 16510  Walkscwlks 16541  EulerPathsceupth 16666

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-mulrcl 8272  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-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291  ax-arch 8292

This theorem 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 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-po 4439  df-iso 4440  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-irdg 6635  df-frec 6656  df-1o 6681  df-2o 6682  df-oadd 6685  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-reap 8897  df-ap 8904  df-div 8997  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-q 10003  df-rp 10038  df-xadd 10158  df-fz 10395  df-fzo 10533  df-fl 10688  df-mod 10743  df-seqfrec 10868  df-exp 10959  df-ihash 11198  df-word 11288  df-cj 11590  df-re 11591  df-im 11592  df-rsqrt 11747  df-abs 11748  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-umgren 16318  df-uspgren 16379  df-subgr 16478  df-vtxdg 16511  df-wlks 16542  df-trls 16605  df-eupth 16667

This theorem is referenced by:  eulerpathprum  16704