Theorem 2pthd

Description: A path of length 2 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017) (Revised by AV, 24-Jan-2021) (Revised by AV, 24-Mar-2021)

	Ref	Expression
Hypotheses	2wlkd.p	|- P = <" A B C ">
	2wlkd.f	|- F = <" J K ">
	2wlkd.s	|- ( ph -> ( A e. V /\ B e. V /\ C e. V ) )
	2wlkd.n	|- ( ph -> ( A =/= B /\ B =/= C ) )
	2wlkd.e	|- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) ) )
	2wlkd.v	|- V = ( Vtx ` G )
	2wlkd.i	|- I = ( iEdg ` G )
	2trld.n	|- ( ph -> J =/= K )
Assertion	2pthd	|- ( ph -> F ( Paths ` G ) P )

Proof

Step	Hyp	Ref	Expression
1		2wlkd.p	|- P = <" A B C ">
2		2wlkd.f	|- F = <" J K ">
3		2wlkd.s	|- ( ph -> ( A e. V /\ B e. V /\ C e. V ) )
4		2wlkd.n	|- ( ph -> ( A =/= B /\ B =/= C ) )
5		2wlkd.e	|- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) ) )
6		2wlkd.v	|- V = ( Vtx ` G )
7		2wlkd.i	|- I = ( iEdg ` G )
8		2trld.n	|- ( ph -> J =/= K )
9		s3cli	|- <" A B C "> e. Word _V
10	1 9	eqeltri	|- P e. Word _V
11	10	a1i	|- ( ph -> P e. Word _V )
12	2	fveq2i	|- ( # ` F ) = ( # ` <" J K "> )
13		s2len	|- ( # ` <" J K "> ) = 2
14	12 13	eqtri	|- ( # ` F ) = 2
15		3m1e2	|- ( 3 - 1 ) = 2
16	1	fveq2i	|- ( # ` P ) = ( # ` <" A B C "> )
17		s3len	|- ( # ` <" A B C "> ) = 3
18	16 17	eqtr2i	|- 3 = ( # ` P )
19	18	oveq1i	|- ( 3 - 1 ) = ( ( # ` P ) - 1 )
20	14 15 19	3eqtr2i	|- ( # ` F ) = ( ( # ` P ) - 1 )
21	1 2 3 4	2pthdlem1	|- ( ph -> A. k e. ( 0 ..^ ( # ` P ) ) A. j e. ( 1 ..^ ( # ` F ) ) ( k =/= j -> ( P ` k ) =/= ( P ` j ) ) )
22		eqid	|- ( # ` F ) = ( # ` F )
23	1 2 3 4 5 6 7 8	2trld	|- ( ph -> F ( Trails ` G ) P )
24	11 20 21 22 23	pthd	|- ( ph -> F ( Paths ` G ) P )