Theorem eupth2lemsfi 16490

Description: Lemma for eupth2fi 16491 (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 9507	. . . . . 6 |- ( n e. NN0 -> n e. RR )
2	1	adantl 277	. . . . 5 |- ( ( ph /\ n e. NN0 ) -> n e. RR )
3	2	lep1d 9207	. . . 4 |- ( ( ph /\ n e. NN0 ) -> n <_ ( n + 1 ) )
4		peano2re 8411	. . . . . 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 16467	. . . . . . . 8 |- ( F (EulerPaths ` G ) P -> F (Walks ` G ) P )
8		wlkcl 16344	. . . . . . . 8 |- ( F (Walks ` G ) P -> (♯ ` F ) e. NN0 )
9	6, 7, 8	3syl 17	. . . . . . 7 |- ( ph -> (♯ ` F ) e. NN0 )
10	9	nn0red 9556	. . . . . 6 |- ( ph -> (♯ ` F ) e. RR )
11	10	adantr 276	. . . . 5 |- ( ( ph /\ n e. NN0 ) -> (♯ ` F ) e. RR )
12		letr 8358	. . . . 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 5672	. . . . . . . . 9 |- ( x = y -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) = ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` y ) )
17	16	breq2d 4123	. . . . . . . 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 2975	. . . . . 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 2234	. . . . . . . . 9 |- <. V , ( I |` ( F " ( 0..^ n ) ) ) >. = <. V , ( I |` ( F " ( 0..^ n ) ) ) >.
30		eqid 2234	. . . . . . . . 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 16488	. . . . . . . 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 4279	. . . . . . . . . . . 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 16387	. . . . . . . . . . . . . . . 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 16346	. . . . . . . . . . . . . . . 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 9538	. . . . . . . . . . . . . . . . . . 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 9892	. . . . . . . . . . . . . . . . 17 |- NN0 = ( ZZ>= ` 0 )
52	50, 51	eleqtrdi 2327	. . . . . . . . . . . . . . . 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 9701	. . . . . . . . . . . . . . . 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 10354	. . . . . . . . . . . . . . . 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 5815	. . . . . . . . . . . . 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 3855	. . . . . . . . . . . 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 4327	. . . . . . . . . . . . . 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 3679	. . . . . . . . . . . . 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 7126	. . . . . . . . . . . 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 3662	. . . . . . . . . 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 3682	. . . . . . . . 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 3239	. . . . . . . 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 2232	. . . 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 2205   {crab 2526   _Vcvv 2815   C_ wss 3213   (/)c0 3510   ifcif 3622   ~Pcpw 3671   {cpr 3692   <.cop 3694   class class class wbr 4111   |` cres 4753   "cima 4754   Fun wfun 5348   -->wf 5350   ` cfv 5354  (class class class)co 6052   Fincfn 6977   RRcr 8128   0cc0 8129   1c1 8130   + caddc 8132   <_ cle 8311   2c2 9290   NN0cn0 9498   ZZcz 9579   ZZ>=cuz 9856   ...cfz 10345  ..^cfzo 10480  ♯chash 11142   || cdvds 12477  Vtxcvtx 16024  iEdgciedg 16025  UMGraphcumgr 16104  VtxDegcvtxdg 16298  Walkscwlks 16329  EulerPathsceupth 16454

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 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-mulrcl 8228  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-precex 8239  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246  ax-pre-mulext 8247  ax-arch 8248

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 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-frec 6624  df-1o 6649  df-2o 6650  df-oadd 6653  df-er 6769  df-map 6886  df-en 6978  df-dom 6979  df-fin 6980  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-reap 8851  df-ap 8858  df-div 8949  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-5 9301  df-6 9302  df-7 9303  df-8 9304  df-9 9305  df-n0 9499  df-z 9580  df-dec 9713  df-uz 9857  df-q 9955  df-rp 9990  df-xadd 10109  df-fz 10346  df-fzo 10481  df-fl 10634  df-mod 10689  df-seqfrec 10814  df-exp 10905  df-ihash 11143  df-word 11229  df-cj 11531  df-re 11532  df-im 11533  df-rsqrt 11687  df-abs 11688  df-dvds 12478  df-ndx 13232  df-slot 13233  df-base 13235  df-edgf 16017  df-vtx 16026  df-iedg 16027  df-edg 16070  df-uhgrm 16081  df-ushgrm 16082  df-upgren 16105  df-umgren 16106  df-uspgren 16167  df-subgr 16266  df-vtxdg 16299  df-wlks 16330  df-trls 16393  df-eupth 16455

This theorem is referenced by:  eupth2fi  16491