Theorem eupth2lemsfi 16356

Description: Lemma for eupth2fi 16357 (induction step): The only vertices of odd degree in a graph with an Eulerian path are the endpoints, and then only if the endpoints are distinct, if the Eulerian path shortened by one edge has this property. (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
eupth2lemsfi	|- ( ( 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 ) ) } ) ) ) )

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

Allowed substitution hints:   ph( n)   P( x, n)   F( n)   G( x, n)   I( n)   V( n)

Proof of Theorem eupth2lemsfi

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0re 9414	. . . . . 6 |- ( n e. NN0 -> n e. RR )
2	1	adantl 277	. . . . 5 |- ( ( ph /\ n e. NN0 ) -> n e. RR )
3	2	lep1d 9114	. . . 4 |- ( ( ph /\ n e. NN0 ) -> n <_ ( n + 1 ) )
4		peano2re 8318	. . . . . 6 |- ( n e. RR -> ( n + 1 ) e. RR )
5	2, 4	syl 14	. . . . 5 |- ( ( ph /\ n e. NN0 ) -> ( n + 1 ) e. RR )
6		eupth2.p	. . . . . . . 8 |- ( ph -> F (EulerPaths ` G ) P )
7		eupthiswlk 16333	. . . . . . . 8 |- ( F (EulerPaths ` G ) P -> F (Walks ` G ) P )
8		wlkcl 16210	. . . . . . . 8 |- ( F (Walks ` G ) P -> (♯ ` F ) e. NN0 )
9	6, 7, 8	3syl 17	. . . . . . 7 |- ( ph -> (♯ ` F ) e. NN0 )
10	9	nn0red 9459	. . . . . 6 |- ( ph -> (♯ ` F ) e. RR )
11	10	adantr 276	. . . . 5 |- ( ( ph /\ n e. NN0 ) -> (♯ ` F ) e. RR )
12		letr 8265	. . . . 5 |- ( ( n e. RR /\ ( n + 1 ) e. RR /\ (♯ ` F ) e. RR ) -> ( ( n <_ ( n + 1 ) /\ ( n + 1 ) <_ (♯ ` F ) ) -> n <_ (♯ ` F ) ) )
13	2, 5, 11, 12	syl3anc 1273	. . . 4 |- ( ( ph /\ n e. NN0 ) -> ( ( n <_ ( n + 1 ) /\ ( n + 1 ) <_ (♯ ` F ) ) -> n <_ (♯ ` F ) ) )
14	3, 13	mpand 429	. . 3 |- ( ( ph /\ n e. NN0 ) -> ( ( n + 1 ) <_ (♯ ` F ) -> n <_ (♯ ` F ) ) )
15	14	imim1d 75	. 2 |- ( ( 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 ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) )
16		fveq2 5640	. . . . . . . . 9 |- ( x = y -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) = ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` y ) )
17	16	breq2d 4100	. . . . . . . 8 |- ( x = y -> ( 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) <-> 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` y ) ) )
18	17	notbid 673	. . . . . . 7 |- ( x = y -> ( -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) <-> -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` y ) ) )
19	18	elrab 2962	. . . . . 6 |- ( y e. { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) } <-> ( y e. V /\ -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` y ) ) )
20		eupth2.v	. . . . . . . . 9 |- V = (Vtx ` G )
21		eupth2.i	. . . . . . . . 9 |- I = (iEdg ` G )
22		eupth2fi.g	. . . . . . . . . 10 |- ( ph -> G e. UMGraph )
23	22	ad3antrrr 492	. . . . . . . . 9 |- ( ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) /\ y e. V ) -> G e. UMGraph )
24		eupth2.f	. . . . . . . . . 10 |- ( ph -> Fun I )
25	24	ad3antrrr 492	. . . . . . . . 9 |- ( ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) /\ y e. V ) -> Fun I )
26	6	ad3antrrr 492	. . . . . . . . 9 |- ( ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) /\ y e. V ) -> F (EulerPaths ` G ) P )
27		eupth2fi.fi	. . . . . . . . . 10 |- ( ph -> V e. Fin )
28	27	ad3antrrr 492	. . . . . . . . 9 |- ( ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) /\ y e. V ) -> V e. Fin )
29		eqid 2231	. . . . . . . . 9 |- <. V , ( I |` ( F " ( 0..^ n ) ) ) >. = <. V , ( I |` ( F " ( 0..^ n ) ) ) >.
30		eqid 2231	. . . . . . . . 9 |- <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. = <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >.
31		simpr 110	. . . . . . . . . 10 |- ( ( ph /\ n e. NN0 ) -> n e. NN0 )
32	31	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) /\ y e. V ) -> n e. NN0 )
33		simprl 531	. . . . . . . . . 10 |- ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) -> ( n + 1 ) <_ (♯ ` F ) )
34	33	adantr 276	. . . . . . . . 9 |- ( ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) /\ y e. V ) -> ( n + 1 ) <_ (♯ ` F ) )
35		simpr 110	. . . . . . . . 9 |- ( ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) /\ y e. V ) -> y e. V )
36		simplrr 538	. . . . . . . . 9 |- ( ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) /\ y e. V ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) )
37	20, 21, 23, 25, 26, 28, 29, 30, 32, 34, 35, 36	eupth2lem3fi 16354	. . . . . . . 8 |- ( ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) /\ y e. V ) -> ( -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` y ) <-> y e. if ( ( P ` 0 ) = ( P ` ( n + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( n + 1 ) ) } ) ) )
38	37	pm5.32da 452	. . . . . . 7 |- ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) -> ( ( y e. V /\ -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` y ) ) <-> ( y e. V /\ y e. if ( ( P ` 0 ) = ( P ` ( n + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( n + 1 ) ) } ) ) ) )
39		0elpw 4254	. . . . . . . . . . . 12 |- (/) e. ~P V
40	39	a1i 9	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) -> (/) e. ~P V )
41	20	wlkepvtx 16253	. . . . . . . . . . . . . . . 16 |- ( F (Walks ` G ) P -> ( ( P ` 0 ) e. V /\ ( P ` (♯ ` F ) ) e. V ) )
42	41	simpld 112	. . . . . . . . . . . . . . 15 |- ( F (Walks ` G ) P -> ( P ` 0 ) e. V )
43	6, 7, 42	3syl 17	. . . . . . . . . . . . . 14 |- ( ph -> ( P ` 0 ) e. V )
44	43	ad2antrr 488	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) -> ( P ` 0 ) e. V )
45	20	wlkp 16212	. . . . . . . . . . . . . . . 16 |- ( F (Walks ` G ) P -> P : ( 0 ... (♯ ` F ) ) --> V )
46	6, 7, 45	3syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> P : ( 0 ... (♯ ` F ) ) --> V )
47	46	ad2antrr 488	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) -> P : ( 0 ... (♯ ` F ) ) --> V )
48		peano2nn0 9445	. . . . . . . . . . . . . . . . . . 19 |- ( n e. NN0 -> ( n + 1 ) e. NN0 )
49	48	adantl 277	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ n e. NN0 ) -> ( n + 1 ) e. NN0 )
50	49	adantr 276	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) -> ( n + 1 ) e. NN0 )
51		nn0uz 9794	. . . . . . . . . . . . . . . . 17 |- NN0 = ( ZZ>= ` 0 )
52	50, 51	eleqtrdi 2324	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) -> ( n + 1 ) e. ( ZZ>= ` 0 ) )
53	9	ad2antrr 488	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) -> (♯ ` F ) e. NN0 )
54	53	nn0zd 9603	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) -> (♯ ` F ) e. ZZ )
55		elfz5 10255	. . . . . . . . . . . . . . . 16 |- ( ( ( n + 1 ) e. ( ZZ>= ` 0 ) /\ (♯ ` F ) e. ZZ ) -> ( ( n + 1 ) e. ( 0 ... (♯ ` F ) ) <-> ( n + 1 ) <_ (♯ ` F ) ) )
56	52, 54, 55	syl2anc 411	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) -> ( ( n + 1 ) e. ( 0 ... (♯ ` F ) ) <-> ( n + 1 ) <_ (♯ ` F ) ) )
57	33, 56	mpbird 167	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) -> ( n + 1 ) e. ( 0 ... (♯ ` F ) ) )
58	47, 57	ffvelcdmd 5784	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) -> ( P ` ( n + 1 ) ) e. V )
59	44, 58	prssd 3832	. . . . . . . . . . . 12 |- ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) -> { ( P ` 0 ) , ( P ` ( n + 1 ) ) } C_ V )
60		prexg 4301	. . . . . . . . . . . . . 14 |- ( ( ( P ` 0 ) e. V /\ ( P ` ( n + 1 ) ) e. V ) -> { ( P ` 0 ) , ( P ` ( n + 1 ) ) } e. _V )
61	44, 58, 60	syl2anc 411	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) -> { ( P ` 0 ) , ( P ` ( n + 1 ) ) } e. _V )
62		elpwg 3660	. . . . . . . . . . . . 13 |- ( { ( P ` 0 ) , ( P ` ( n + 1 ) ) } e. _V -> ( { ( P ` 0 ) , ( P ` ( n + 1 ) ) } e. ~P V <-> { ( P ` 0 ) , ( P ` ( n + 1 ) ) } C_ V ) )
63	61, 62	syl 14	. . . . . . . . . . . 12 |- ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) -> ( { ( P ` 0 ) , ( P ` ( n + 1 ) ) } e. ~P V <-> { ( P ` 0 ) , ( P ` ( n + 1 ) ) } C_ V ) )
64	59, 63	mpbird 167	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) -> { ( P ` 0 ) , ( P ` ( n + 1 ) ) } e. ~P V )
65	27	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) -> V e. Fin )
66		fidceq 7059	. . . . . . . . . . . 12 |- ( ( V e. Fin /\ ( P ` 0 ) e. V /\ ( P ` ( n + 1 ) ) e. V ) -> DECID ( P ` 0 ) = ( P ` ( n + 1 ) ) )
67	65, 44, 58, 66	syl3anc 1273	. . . . . . . . . . 11 |- ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) -> DECID ( P ` 0 ) = ( P ` ( n + 1 ) ) )
68	40, 64, 67	ifcldcd 3643	. . . . . . . . . 10 |- ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) -> if ( ( P ` 0 ) = ( P ` ( n + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( n + 1 ) ) } ) e. ~P V )
69	68	elpwid 3663	. . . . . . . . 9 |- ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) -> if ( ( P ` 0 ) = ( P ` ( n + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( n + 1 ) ) } ) C_ V )
70	69	sseld 3226	. . . . . . . 8 |- ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) -> ( y e. if ( ( P ` 0 ) = ( P ` ( n + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( n + 1 ) ) } ) -> y e. V ) )
71	70	pm4.71rd 394	. . . . . . 7 |- ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) -> ( y e. if ( ( P ` 0 ) = ( P ` ( n + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( n + 1 ) ) } ) <-> ( y e. V /\ y e. if ( ( P ` 0 ) = ( P ` ( n + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( n + 1 ) ) } ) ) ) )
72	38, 71	bitr4d 191	. . . . . 6 |- ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) -> ( ( y e. V /\ -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` y ) ) <-> y e. if ( ( P ` 0 ) = ( P ` ( n + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( n + 1 ) ) } ) ) )
73	19, 72	bitrid 192	. . . . 5 |- ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) -> ( y e. { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) } <-> y e. if ( ( P ` 0 ) = ( P ` ( n + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( n + 1 ) ) } ) ) )
74	73	eqrdv 2229	. . . 4 |- ( ( ( ph /\ n e. NN0 ) /\ ( ( n + 1 ) <_ (♯ ` F ) /\ { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) ) ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` ( n + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( n + 1 ) ) } ) )
75	74	exp32 365	. . 3 |- ( ( ph /\ n e. NN0 ) -> ( ( n + 1 ) <_ (♯ ` F ) -> ( { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ n ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` n ) , (/) , { ( P ` 0 ) , ( P ` n ) } ) -> { x e. V | -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) } = if ( ( P ` 0 ) = ( P ` ( n + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( n + 1 ) ) } ) ) ) )
76	75	a2d 26	. 2 |- ( ( ph /\ n e. NN0 ) -> ( ( ( n + 1 ) <_ (♯ ` 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 ) ) } ) ) ) )
77	15, 76	syld 45	1 |- ( ( 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 ) ) } ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 841   = wceq 1397   e. wcel 2202   {crab 2514   _Vcvv 2802   C_ wss 3200   (/)c0 3494   ifcif 3605   ~Pcpw 3652   {cpr 3670   <.cop 3672   class class class wbr 4088   |` cres 4727   "cima 4728   Fun wfun 5320   -->wf 5322   ` cfv 5326  (class class class)co 6021   Fincfn 6912   RRcr 8034   0cc0 8035   1c1 8036   + caddc 8038   <_ cle 8218   2c2 9197   NN0cn0 9405   ZZcz 9482   ZZ>=cuz 9758   ...cfz 10246  ..^cfzo 10380  ♯chash 11041   || cdvds 12369  Vtxcvtx 15890  iEdgciedg 15891  UMGraphcumgr 15970  VtxDegcvtxdg 16164  Walkscwlks 16195  EulerPathsceupth 16320

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 8126  ax-resscn 8127  ax-1cn 8128  ax-1re 8129  ax-icn 8130  ax-addcl 8131  ax-addrcl 8132  ax-mulcl 8133  ax-mulrcl 8134  ax-addcom 8135  ax-mulcom 8136  ax-addass 8137  ax-mulass 8138  ax-distr 8139  ax-i2m1 8140  ax-0lt1 8141  ax-1rid 8142  ax-0id 8143  ax-rnegex 8144  ax-precex 8145  ax-cnre 8146  ax-pre-ltirr 8147  ax-pre-ltwlin 8148  ax-pre-lttrn 8149  ax-pre-apti 8150  ax-pre-ltadd 8151  ax-pre-mulgt0 8152  ax-pre-mulext 8153  ax-arch 8154

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 5974  df-ov 6024  df-oprab 6025  df-mpo 6026  df-1st 6306  df-2nd 6307  df-recs 6474  df-irdg 6539  df-frec 6560  df-1o 6585  df-2o 6586  df-oadd 6589  df-er 6705  df-map 6822  df-en 6913  df-dom 6914  df-fin 6915  df-pnf 8219  df-mnf 8220  df-xr 8221  df-ltxr 8222  df-le 8223  df-sub 8355  df-neg 8356  df-reap 8758  df-ap 8765  df-div 8856  df-inn 9147  df-2 9205  df-3 9206  df-4 9207  df-5 9208  df-6 9209  df-7 9210  df-8 9211  df-9 9212  df-n0 9406  df-z 9483  df-dec 9615  df-uz 9759  df-q 9857  df-rp 9892  df-xadd 10011  df-fz 10247  df-fzo 10381  df-fl 10534  df-mod 10589  df-seqfrec 10714  df-exp 10805  df-ihash 11042  df-word 11121  df-cj 11423  df-re 11424  df-im 11425  df-rsqrt 11579  df-abs 11580  df-dvds 12370  df-ndx 13106  df-slot 13107  df-base 13109  df-edgf 15883  df-vtx 15892  df-iedg 15893  df-edg 15936  df-uhgrm 15947  df-ushgrm 15948  df-upgren 15971  df-umgren 15972  df-uspgren 16033  df-subgr 16132  df-vtxdg 16165  df-wlks 16196  df-trls 16259  df-eupth 16321

This theorem is referenced by:  eupth2fi  16357