Theorem eupth2lems 27098

Description: Lemma for eupth2 27099 (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. Formerly part of proof for eupth2 27099. (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 )
eupth2.g	|- ( ph -> G e. UPGraph )
eupth2.f	|- ( ph -> Fun I )
eupth2.p	|- ( ph -> F (EulerPaths ` G ) P )

Assertion

Ref	Expression
eupth2lems	|- ( ( 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 eupth2lems

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0re 11301	. . . . . 6 |- ( n e. NN0 -> n e. RR )
2	1	adantl 482	. . . . 5 |- ( ( ph /\ n e. NN0 ) -> n e. RR )
3	2	lep1d 10955	. . . 4 |- ( ( ph /\ n e. NN0 ) -> n <_ ( n + 1 ) )
4		peano2re 10209	. . . . . 6 |- ( n e. RR -> ( n + 1 ) e. RR )
5	2, 4	syl 17	. . . . 5 |- ( ( ph /\ n e. NN0 ) -> ( n + 1 ) e. RR )
6		eupth2.p	. . . . . . . 8 |- ( ph -> F (EulerPaths ` G ) P )
7		eupthiswlk 27072	. . . . . . . 8 |- ( F (EulerPaths ` G ) P -> F (Walks ` G ) P )
8		wlkcl 26511	. . . . . . . 8 |- ( F (Walks ` G ) P -> ( # ` F ) e. NN0 )
9	6, 7, 8	3syl 18	. . . . . . 7 |- ( ph -> ( # ` F ) e. NN0 )
10	9	nn0red 11352	. . . . . 6 |- ( ph -> ( # ` F ) e. RR )
11	10	adantr 481	. . . . 5 |- ( ( ph /\ n e. NN0 ) -> ( # ` F ) e. RR )
12		letr 10131	. . . . 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 1326	. . . 4 |- ( ( ph /\ n e. NN0 ) -> ( ( n <_ ( n + 1 ) /\ ( n + 1 ) <_ ( # ` F ) ) -> n <_ ( # ` F ) ) )
14	3, 13	mpand 711	. . 3 |- ( ( ph /\ n e. NN0 ) -> ( ( n + 1 ) <_ ( # ` F ) -> n <_ ( # ` F ) ) )
15	14	imim1d 82	. 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 6191	. . . . . . . . 9 |- ( x = y -> ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) = ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` y ) )
17	16	breq2d 4665	. . . . . . . 8 |- ( x = y -> ( 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) <-> 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` y ) ) )
18	17	notbid 308	. . . . . . 7 |- ( x = y -> ( -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` x ) <-> -. 2 || ( (VtxDeg ` <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. ) ` y ) ) )
19	18	elrab 3363	. . . . . 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		eupth2.g	. . . . . . . . . 10 |- ( ph -> G e. UPGraph )
23	22	ad3antrrr 766	. . . . . . . . 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. UPGraph )
24		eupth2.f	. . . . . . . . . 10 |- ( ph -> Fun I )
25	24	ad3antrrr 766	. . . . . . . . 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 766	. . . . . . . . 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		eqid 2622	. . . . . . . . 9 |- <. V , ( I |` ( F " ( 0..^ n ) ) ) >. = <. V , ( I |` ( F " ( 0..^ n ) ) ) >.
28		eqid 2622	. . . . . . . . 9 |- <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >. = <. V , ( I |` ( F " ( 0..^ ( n + 1 ) ) ) ) >.
29		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ n e. NN0 ) -> n e. NN0 )
30	29	ad2antrr 762	. . . . . . . . 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 )
31		simprl 794	. . . . . . . . . 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 ) )
32	31	adantr 481	. . . . . . . . 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 ) )
33		simpr 477	. . . . . . . . 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 )
34		simplrr 801	. . . . . . . . 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 ) } ) )
35	20, 21, 23, 25, 26, 27, 28, 30, 32, 33, 34	eupth2lem3 27096	. . . . . . . 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 ) ) } ) ) )
36	35	pm5.32da 673	. . . . . . 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 ) ) } ) ) ) )
37		0elpw 4834	. . . . . . . . . . 11 |- (/) e. ~P V
38	20	wlkepvtx 26556	. . . . . . . . . . . . . . . 16 |- ( F (Walks ` G ) P -> ( ( P ` 0 ) e. V /\ ( P ` ( # ` F ) ) e. V ) )
39	38	simpld 475	. . . . . . . . . . . . . . 15 |- ( F (Walks ` G ) P -> ( P ` 0 ) e. V )
40	6, 7, 39	3syl 18	. . . . . . . . . . . . . 14 |- ( ph -> ( P ` 0 ) e. V )
41	40	ad2antrr 762	. . . . . . . . . . . . 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 )
42	20	wlkp 26512	. . . . . . . . . . . . . . . 16 |- ( F (Walks ` G ) P -> P : ( 0 ... ( # ` F ) ) --> V )
43	6, 7, 42	3syl 18	. . . . . . . . . . . . . . 15 |- ( ph -> P : ( 0 ... ( # ` F ) ) --> V )
44	43	ad2antrr 762	. . . . . . . . . . . . . 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 )
45		peano2nn0 11333	. . . . . . . . . . . . . . . . . . 19 |- ( n e. NN0 -> ( n + 1 ) e. NN0 )
46	45	adantl 482	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ n e. NN0 ) -> ( n + 1 ) e. NN0 )
47	46	adantr 481	. . . . . . . . . . . . . . . . 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 )
48		nn0uz 11722	. . . . . . . . . . . . . . . . 17 |- NN0 = ( ZZ>= ` 0 )
49	47, 48	syl6eleq 2711	. . . . . . . . . . . . . . . 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 ) )
50	9	ad2antrr 762	. . . . . . . . . . . . . . . . 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 )
51	50	nn0zd 11480	. . . . . . . . . . . . . . . 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 )
52		elfz5 12334	. . . . . . . . . . . . . . . 16 |- ( ( ( n + 1 ) e. ( ZZ>= ` 0 ) /\ ( # ` F ) e. ZZ ) -> ( ( n + 1 ) e. ( 0 ... ( # ` F ) ) <-> ( n + 1 ) <_ ( # ` F ) ) )
53	49, 51, 52	syl2anc 693	. . . . . . . . . . . . . . 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 ) ) )
54	31, 53	mpbird 247	. . . . . . . . . . . . . 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 ) ) )
55	44, 54	ffvelrnd 6360	. . . . . . . . . . . . 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 )
56	41, 55	prssd 4354	. . . . . . . . . . . 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 )
57		prex 4909	. . . . . . . . . . . . 13 |- { ( P ` 0 ) , ( P ` ( n + 1 ) ) } e. _V
58	57	elpw 4164	. . . . . . . . . . . 12 |- ( { ( P ` 0 ) , ( P ` ( n + 1 ) ) } e. ~P V <-> { ( P ` 0 ) , ( P ` ( n + 1 ) ) } C_ V )
59	56, 58	sylibr 224	. . . . . . . . . . 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 )
60		ifcl 4130	. . . . . . . . . . 11 |- ( ( (/) e. ~P V /\ { ( P ` 0 ) , ( P ` ( n + 1 ) ) } e. ~P V ) -> if ( ( P ` 0 ) = ( P ` ( n + 1 ) ) , (/) , { ( P ` 0 ) , ( P ` ( n + 1 ) ) } ) e. ~P V )
61	37, 59, 60	sylancr 695	. . . . . . . . . 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 )
62	61	elpwid 4170	. . . . . . . . 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 )
63	62	sseld 3602	. . . . . . . 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 ) )
64	63	pm4.71rd 667	. . . . . . 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 ) ) } ) ) ) )
65	36, 64	bitr4d 271	. . . . . 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 ) ) } ) ) )
66	19, 65	syl5bb 272	. . . . 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 ) ) } ) ) )
67	66	eqrdv 2620	. . . 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 ) ) } ) )
68	67	exp32 631	. . 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 ) ) } ) ) ) )
69	68	a2d 29	. 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 ) ) } ) ) ) )
70	15, 69	syld 47	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   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   (/)c0 3915   ifcif 4086   ~Pcpw 4158   {cpr 4179   <.cop 4183   class class class wbr 4653   |` cres 5116   "cima 5117   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   <_ cle 10075   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326  ..^cfzo 12465   #chash 13117   || cdvds 14983  Vtxcvtx 25874  iEdgciedg 25875   UPGraph cupgr 25975  VtxDegcvtxdg 26361  Walkscwlks 26492  EulerPathsceupth 27057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-xadd 11947  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-word 13299  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-vtx 25876  df-iedg 25877  df-edg 25940  df-uhgr 25953  df-ushgr 25954  df-upgr 25977  df-uspgr 26045  df-vtxdg 26362  df-wlks 26495  df-trls 26589  df-eupth 27058

This theorem is referenced by:  eupth2  27099