Theorem eupth2lemsfi 16941

Description: Lemma for eupth2fi 16942 (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 9583	. . . . . 6 |- ( n e. NN0 -> n e. RR )
2	1	adantl 277	. . . . 5 |- ( ( ph /\ n e. NN0 ) -> n e. RR )
3	2	lep1d 9269	. . . 4 |- ( ( ph /\ n e. NN0 ) -> n <_ ( n + 1 ) )
4		peano2re 8469	. . . . . 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 16918	. . . . . . . 8 |- ( F (EulerPaths ` G ) P -> F (Walks ` G ) P )
8		wlkcl 16795	. . . . . . . 8 |- ( F (Walks ` G ) P -> (♯ ` F ) e. NN0 )
9	6, 7, 8	3syl 17	. . . . . . 7 |- ( ph -> (♯ ` F ) e. NN0 )
10	9	nn0red 9630	. . . . . 6 |- ( ph -> (♯ ` F ) e. RR )
11	10	adantr 276	. . . . 5 |- ( ( ph /\ n e. NN0 ) -> (♯ ` F ) e. RR )
12		letr 8414	. . . . 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 1278	. . . 4 |- ( ( ph /\ n e. NN0 ) -> ( ( n <_ ( n + 1 ) /\ ( n + 1 ) <_ (♯ ` F ) ) -> n <_ (♯ ` F ) ) )
14	3, 13	mpand 433	. . 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 5698	. . . . . . . . 9 |- ( x = y -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) = ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` y ) )
17	16	breq2d 4143	. . . . . . . 8 |- ( x = y -> ( 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) <-> 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` y ) ) )
18	17	notbid 677	. . . . . . 7 |- ( x = y -> ( -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) <-> -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` y ) ) )
19	18	elrab 2982	. . . . . 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 496	. . . . . . . . 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 496	. . . . . . . . 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 496	. . . . . . . . 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 496	. . . . . . . . 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 2238	. . . . . . . . 9 |- <. V , ( I |` ( F " ( 0..^ n ) ) ) >. = <. V , ( I |` ( F " ( 0..^ n ) ) ) >.
30		eqid 2238	. . . . . . . . 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 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 ) -> n e. NN0 )
33		simprl 535	. . . . . . . . . 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 542	. . . . . . . . 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 16939	. . . . . . . 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 456	. . . . . . 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 4303	. . . . . . . . . . . 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 16838	. . . . . . . . . . . . . . . 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 492	. . . . . . . . . . . . 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 16797	. . . . . . . . . . . . . . . 16 |- ( F (Walks ` G ) P -> P : ( 0 ... (♯ ` F ) ) --> V )
46	6, 7, 45	3syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> P : ( 0 ... (♯ ` F ) ) --> V )
47	46	ad2antrr 492	. . . . . . . . . . . . . 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 9614	. . . . . . . . . . . . . . . . . . 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 9972	. . . . . . . . . . . . . . . . 17 |- NN0 = ( ZZ>= ` 0 )
52	50, 51	eleqtrdi 2331	. . . . . . . . . . . . . . . 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 492	. . . . . . . . . . . . . . . . 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 9776	. . . . . . . . . . . . . . . 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 10436	. . . . . . . . . . . . . . . 16 |- ( ( ( n + 1 ) e. ( ZZ>= ` 0 ) /\ (♯ ` F ) e. ZZ ) -> ( ( n + 1 ) e. ( 0 ... (♯ ` F ) ) <-> ( n + 1 ) <_ (♯ ` F ) ) )
56	52, 54, 55	syl2anc 415	. . . . . . . . . . . . . . 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 5847	. . . . . . . . . . . . 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 3874	. . . . . . . . . . . 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 4351	. . . . . . . . . . . . . 14 |- ( ( ( P ` 0 ) e. V /\ ( P ` ( n + 1 ) ) e. V ) -> { ( P ` 0 ) , ( P ` ( n + 1 ) ) } e. _V )
61	44, 58, 60	syl2anc 415	. . . . . . . . . . . . 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 3696	. . . . . . . . . . . . 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 492	. . . . . . . . . . . 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 7177	. . . . . . . . . . . 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 1278	. . . . . . . . . . 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 3678	. . . . . . . . . 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 3700	. . . . . . . . 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 3247	. . . . . . . 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 398	. . . . . . 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 2236	. . . 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 ) ) } ) ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 846   = wceq 1402   e. wcel 2209   {crab 2532   _Vcvv 2821   C_ wss 3220   (/)c0 3520   ifcif 3638   ~Pcpw 3688   {cpr 3710   <.cop 3712   class class class wbr 4131   |` cres 4778   "cima 4779   Fun wfun 5374   -->wf 5376   ` cfv 5380  (class class class)co 6088   Fincfn 7028   RRcr 8184   0cc0 8185   1c1 8186   + caddc 8188   <_ cle 8367   2c2 9364   NN0cn0 9574   ZZcz 9653   ZZ>=cuz 9936   ...cfz 10427  ..^cfzo 10565  ♯chash 11235   || cdvds 12578  Vtxcvtx 16475  iEdgciedg 16476  UMGraphcumgr 16555  VtxDegcvtxdg 16749  Walkscwlks 16780  EulerPathsceupth 16905

This proof depends on 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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6581  df-irdg 6646  df-frec 6667  df-1o 6692  df-2o 6693  df-oadd 6696  df-er 6812  df-map 6929  df-en 7029  df-dom 7030  df-fin 7031  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-7 9377  df-8 9378  df-9 9379  df-n0 9575  df-z 9654  df-dec 9788  df-uz 9937  df-q 10035  df-rp 10071  df-xadd 10191  df-fz 10428  df-fzo 10566  df-fl 10721  df-mod 10780  df-seqfrec 10905  df-exp 10996  df-ihash 11236  df-word 11326  df-cj 11628  df-re 11629  df-im 11630  df-rsqrt 11785  df-abs 11786  df-dvds 12579  df-ndx 13412  df-slot 13413  df-base 13415  df-edgf 16468  df-vtx 16477  df-iedg 16478  df-edg 16521  df-uhgrm 16532  df-ushgrm 16533  df-upgren 16556  df-umgren 16557  df-uspgren 16618  df-subgr 16717  df-vtxdg 16750  df-wlks 16781  df-trls 16844  df-eupth 16906

This theorem is used by:  eupth2fi  16942