Theorem eupth2fi 16600

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 2234	. . . . . . 7 |- <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. = <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >.
7	1, 2, 3, 4, 5, 6	eupthvdres 16596	. . . . . 6 |- ( ph -> (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) = (VtxDeg ` G ) )
8	7	fveq1d 5677	. . . . 5 |- ( ph -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) ` x ) = ( (VtxDeg ` G ) ` x ) )
9	8	breq2d 4126	. . . 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 2804	. 2 |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) ` x ) } = { x e. V | -. 2 || ( (VtxDeg ` G ) ` x ) } )
12		eupthiswlk 16576	. . . 4 |- ( F (EulerPaths ` G ) P -> F (Walks ` G ) P )
13		wlkcl 16453	. . . 4 |- ( F (Walks ` G ) P -> (♯ ` F ) e. NN0 )
14	5, 12, 13	3syl 17	. . 3 |- ( ph -> (♯ ` F ) e. NN0 )
15		nn0re 9522	. . . . 5 |- ( (♯ ` F ) e. NN0 -> (♯ ` F ) e. RR )
16	15	leidd 8805	. . . 4 |- ( (♯ ` F ) e. NN0 -> (♯ ` F ) <_ (♯ ` F ) )
17		breq1 4117	. . . . . . 7 |- ( m = 0 -> ( m <_ (♯ ` F ) <-> 0 <_ (♯ ` F ) ) )
18		oveq2 6066	. . . . . . . . . . . . . . . 16 |- ( m = 0 -> ( 0..^ m ) = ( 0..^ 0 ) )
19	18	imaeq2d 5106	. . . . . . . . . . . . . . 15 |- ( m = 0 -> ( F " ( 0..^ m ) ) = ( F " ( 0..^ 0 ) ) )
20	19	reseq2d 5043	. . . . . . . . . . . . . 14 |- ( m = 0 -> ( I |` ( F " ( 0..^ m ) ) ) = ( I |` ( F " ( 0..^ 0 ) ) ) )
21	20	opeq2d 3895	. . . . . . . . . . . . 13 |- ( m = 0 -> <. V , ( I |` ( F " ( 0..^ m ) ) ) >. = <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. )
22	21	fveq2d 5679	. . . . . . . . . . . 12 |- ( m = 0 -> (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) = (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) )
23	22	fveq1d 5677	. . . . . . . . . . 11 |- ( m = 0 -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) = ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) ` x ) )
24	23	breq2d 4126	. . . . . . . . . 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 2804	. . . . . . . 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 5675	. . . . . . . . . 10 |- ( m = 0 -> ( P ` m ) = ( P ` 0 ) )
28	27	eqeq2d 2246	. . . . . . . . 9 |- ( m = 0 -> ( ( P ` 0 ) = ( P ` m ) <-> ( P ` 0 ) = ( P ` 0 ) ) )
29	27	preq2d 3780	. . . . . . . . 9 |- ( m = 0 -> { ( P ` 0 ) , ( P ` m ) } = { ( P ` 0 ) , ( P ` 0 ) } )
30	28, 29	ifbieq2d 3651	. . . . . . . 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 2249	. . . . . . 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 4117	. . . . . . 7 |- ( m = n -> ( m <_ (♯ ` F ) <-> n <_ (♯ ` F ) ) )
35		oveq2 6066	. . . . . . . . . . . . . . . 16 |- ( m = n -> ( 0..^ m ) = ( 0..^ n ) )
36	35	imaeq2d 5106	. . . . . . . . . . . . . . 15 |- ( m = n -> ( F " ( 0..^ m ) ) = ( F " ( 0..^ n ) ) )
37	36	reseq2d 5043	. . . . . . . . . . . . . 14 |- ( m = n -> ( I |` ( F " ( 0..^ m ) ) ) = ( I |` ( F " ( 0..^ n ) ) ) )
38	37	opeq2d 3895	. . . . . . . . . . . . 13 |- ( m = n -> <. V , ( I |` ( F " ( 0..^ m ) ) ) >. = <. V , ( I |` ( F " ( 0..^ n ) ) ) >. )
39	38	fveq2d 5679	. . . . . . . . . . . 12 |- ( m = n -> (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) = (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) )
40	39	fveq1d 5677	. . . . . . . . . . 11 |- ( m = n -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) = ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) )
41	40	breq2d 4126	. . . . . . . . . 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 2804	. . . . . . . 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 5675	. . . . . . . . . 10 |- ( m = n -> ( P ` m ) = ( P ` n ) )
45	44	eqeq2d 2246	. . . . . . . . 9 |- ( m = n -> ( ( P ` 0 ) = ( P ` m ) <-> ( P ` 0 ) = ( P ` n ) ) )
46	44	preq2d 3780	. . . . . . . . 9 |- ( m = n -> { ( P ` 0 ) , ( P ` m ) } = { ( P ` 0 ) , ( P ` n ) } )
47	45, 46	ifbieq2d 3651	. . . . . . . 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 2249	. . . . . . 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 4117	. . . . . . 7 |- ( m = ( n + 1 ) -> ( m <_ (♯ ` F ) <-> ( n + 1 ) <_ (♯ ` F ) ) )
52		oveq2 6066	. . . . . . . . . . . . . . . 16 |- ( m = ( n + 1 ) -> ( 0..^ m ) = ( 0..^ ( n + 1 ) ) )
53	52	imaeq2d 5106	. . . . . . . . . . . . . . 15 |- ( m = ( n + 1 ) -> ( F " ( 0..^ m ) ) = ( F " ( 0..^ ( n + 1 ) ) ) )
54	53	reseq2d 5043	. . . . . . . . . . . . . 14 |- ( m = ( n + 1 ) -> ( I |` ( F " ( 0..^ m ) ) ) = ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) )
55	54	opeq2d 3895	. . . . . . . . . . . . 13 |- ( m = ( n + 1 ) -> <. V , ( I |` ( F " ( 0..^ m ) ) ) >. = <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. )
56	55	fveq2d 5679	. . . . . . . . . . . 12 |- ( m = ( n + 1 ) -> (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) = (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) )
57	56	fveq1d 5677	. . . . . . . . . . 11 |- ( m = ( n + 1 ) -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) = ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) )
58	57	breq2d 4126	. . . . . . . . . 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 2804	. . . . . . . 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 5675	. . . . . . . . . 10 |- ( m = ( n + 1 ) -> ( P ` m ) = ( P ` ( n + 1 ) ) )
62	61	eqeq2d 2246	. . . . . . . . 9 |- ( m = ( n + 1 ) -> ( ( P ` 0 ) = ( P ` m ) <-> ( P ` 0 ) = ( P ` ( n + 1 ) ) ) )
63	61	preq2d 3780	. . . . . . . . 9 |- ( m = ( n + 1 ) -> { ( P ` 0 ) , ( P ` m ) } = { ( P ` 0 ) , ( P ` ( n + 1 ) ) } )
64	62, 63	ifbieq2d 3651	. . . . . . . 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 2249	. . . . . . 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 4117	. . . . . . 7 |- ( m = (♯ ` F ) -> ( m <_ (♯ ` F ) <-> (♯ ` F ) <_ (♯ ` F ) ) )
69		oveq2 6066	. . . . . . . . . . . . . . . 16 |- ( m = (♯ ` F ) -> ( 0..^ m ) = ( 0..^ (♯ ` F ) ) )
70	69	imaeq2d 5106	. . . . . . . . . . . . . . 15 |- ( m = (♯ ` F ) -> ( F " ( 0..^ m ) ) = ( F " ( 0..^ (♯ ` F ) ) ) )
71	70	reseq2d 5043	. . . . . . . . . . . . . 14 |- ( m = (♯ ` F ) -> ( I |` ( F " ( 0..^ m ) ) ) = ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) )
72	71	opeq2d 3895	. . . . . . . . . . . . 13 |- ( m = (♯ ` F ) -> <. V , ( I |` ( F " ( 0..^ m ) ) ) >. = <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. )
73	72	fveq2d 5679	. . . . . . . . . . . 12 |- ( m = (♯ ` F ) -> (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) = (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) )
74	73	fveq1d 5677	. . . . . . . . . . 11 |- ( m = (♯ ` F ) -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ m ) ) ) >. ) ` x ) = ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ (♯ ` F ) ) ) ) >. ) ` x ) )
75	74	breq2d 4126	. . . . . . . . . 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 2804	. . . . . . . 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 5675	. . . . . . . . . 10 |- ( m = (♯ ` F ) -> ( P ` m ) = ( P ` (♯ ` F ) ) )
79	78	eqeq2d 2246	. . . . . . . . 9 |- ( m = (♯ ` F ) -> ( ( P ` 0 ) = ( P ` m ) <-> ( P ` 0 ) = ( P ` (♯ ` F ) ) ) )
80	78	preq2d 3780	. . . . . . . . 9 |- ( m = (♯ ` F ) -> { ( P ` 0 ) , ( P ` m ) } = { ( P ` 0 ) , ( P ` (♯ ` F ) ) } )
81	79, 80	ifbieq2d 3651	. . . . . . . 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 2249	. . . . . . 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 16598	. . . . . . 7 |- ( ph -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ 0 ) ) ) >. ) ` x ) } = (/) )
87		eqid 2234	. . . . . . . 8 |- ( P ` 0 ) = ( P ` 0 )
88	87	iftruei 3632	. . . . . . 7 |- if ( ( P ` 0 ) = ( P ` 0 ) , (/) , { ( P ` 0 ) , ( P ` 0 ) } ) = (/)
89	86, 88	eqtr4di 2285	. . . . . 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 16599	. . . . . . 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 9710	. . . 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 2269	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 1398   e. wcel 2205   {crab 2526   (/)c0 3512   ifcif 3624   {cpr 3695   <.cop 3697   class class class wbr 4114   |` cres 4756   "cima 4757   Fun wfun 5351   ` cfv 5357  (class class class)co 6058   Fincfn 6988   0cc0 8143   1c1 8144   + caddc 8146   <_ cle 8325   2c2 9305   NN0cn0 9513  ..^cfzo 10498  ♯chash 11163   || cdvds 12498  Vtxcvtx 16133  iEdgciedg 16134  UMGraphcumgr 16213  VtxDegcvtxdg 16407  Walkscwlks 16438  EulerPathsceupth 16563

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-ifp 987  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-xor 1421  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6780  df-map 6897  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-z 9595  df-dec 9728  df-uz 9872  df-q 9970  df-rp 10005  df-xadd 10125  df-fz 10362  df-fzo 10499  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925  df-ihash 11164  df-word 11250  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-dvds 12499  df-ndx 13299  df-slot 13300  df-base 13302  df-edgf 16126  df-vtx 16135  df-iedg 16136  df-edg 16179  df-uhgrm 16190  df-ushgrm 16191  df-upgren 16214  df-umgren 16215  df-uspgren 16276  df-subgr 16375  df-vtxdg 16408  df-wlks 16439  df-trls 16502  df-eupth 16564

This theorem is referenced by:  eulerpathprum  16601