Theorem eupth2lemsfi 16460

Description: Lemma for eupth2fi 16461 (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 9501	. . . . . 6 |- ( n e. NN0 -> n e. RR )
2	1	adantl 277	. . . . 5 |- ( ( ph /\ n e. NN0 ) -> n e. RR )
3	2	lep1d 9201	. . . 4 |- ( ( ph /\ n e. NN0 ) -> n <_ ( n + 1 ) )
4		peano2re 8405	. . . . . 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 16437	. . . . . . . 8 |- ( F (EulerPaths ` G ) P -> F (Walks ` G ) P )
8		wlkcl 16314	. . . . . . . 8 |- ( F (Walks ` G ) P -> (♯ ` F ) e. NN0 )
9	6, 7, 8	3syl 17	. . . . . . 7 |- ( ph -> (♯ ` F ) e. NN0 )
10	9	nn0red 9550	. . . . . 6 |- ( ph -> (♯ ` F ) e. RR )
11	10	adantr 276	. . . . 5 |- ( ( ph /\ n e. NN0 ) -> (♯ ` F ) e. RR )
12		letr 8352	. . . . 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 1274	. . . 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 5669	. . . . . . . . 9 |- ( x = y -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) = ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` y ) )
17	16	breq2d 4120	. . . . . . . 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 2972	. . . . . 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 2232	. . . . . . . . 9 |- <. V , ( I |` ( F " ( 0..^ n ) ) ) >. = <. V , ( I |` ( F " ( 0..^ n ) ) ) >.
30		eqid 2232	. . . . . . . . 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 16458	. . . . . . . 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 4276	. . . . . . . . . . . 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 16357	. . . . . . . . . . . . . . . 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 16316	. . . . . . . . . . . . . . . 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 9532	. . . . . . . . . . . . . . . . . . 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 9885	. . . . . . . . . . . . . . . . 17 |- NN0 = ( ZZ>= ` 0 )
52	50, 51	eleqtrdi 2325	. . . . . . . . . . . . . . . 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 9694	. . . . . . . . . . . . . . . 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 10347	. . . . . . . . . . . . . . . 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 5812	. . . . . . . . . . . . 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 3852	. . . . . . . . . . . 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 4324	. . . . . . . . . . . . . 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 3676	. . . . . . . . . . . . 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 7123	. . . . . . . . . . . 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 1274	. . . . . . . . . . 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 3659	. . . . . . . . . 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 3679	. . . . . . . . 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 3236	. . . . . . . 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 2230	. . . 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 842   = wceq 1398   e. wcel 2203   {crab 2524   _Vcvv 2812   C_ wss 3210   (/)c0 3507   ifcif 3619   ~Pcpw 3668   {cpr 3689   <.cop 3691   class class class wbr 4108   |` cres 4750   "cima 4751   Fun wfun 5345   -->wf 5347   ` cfv 5351  (class class class)co 6049   Fincfn 6974   RRcr 8122   0cc0 8123   1c1 8124   + caddc 8126   <_ cle 8305   2c2 9284   NN0cn0 9492   ZZcz 9573   ZZ>=cuz 9849   ...cfz 10338  ..^cfzo 10472  ♯chash 11133   || cdvds 12466  Vtxcvtx 15994  iEdgciedg 15995  UMGraphcumgr 16074  VtxDegcvtxdg 16268  Walkscwlks 16299  EulerPathsceupth 16424

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-mulrcl 8222  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-precex 8233  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-apti 8238  ax-pre-ltadd 8239  ax-pre-mulgt0 8240  ax-pre-mulext 8241  ax-arch 8242

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-frec 6621  df-1o 6646  df-2o 6647  df-oadd 6650  df-er 6766  df-map 6883  df-en 6975  df-dom 6976  df-fin 6977  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-reap 8845  df-ap 8852  df-div 8943  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-5 9295  df-6 9296  df-7 9297  df-8 9298  df-9 9299  df-n0 9493  df-z 9574  df-dec 9706  df-uz 9850  df-q 9948  df-rp 9983  df-xadd 10102  df-fz 10339  df-fzo 10473  df-fl 10626  df-mod 10681  df-seqfrec 10806  df-exp 10897  df-ihash 11134  df-word 11218  df-cj 11520  df-re 11521  df-im 11522  df-rsqrt 11676  df-abs 11677  df-dvds 12467  df-ndx 13204  df-slot 13205  df-base 13207  df-edgf 15987  df-vtx 15996  df-iedg 15997  df-edg 16040  df-uhgrm 16051  df-ushgrm 16052  df-upgren 16075  df-umgren 16076  df-uspgren 16137  df-subgr 16236  df-vtxdg 16269  df-wlks 16300  df-trls 16363  df-eupth 16425

This theorem is referenced by:  eupth2fi  16461