Theorem 3pthd

Description: A path of length 3 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017) (Revised by AV, 10-Feb-2021) (Revised by AV, 24-Mar-2021)

	Ref	Expression
Hypotheses	3wlkd.p	|- P = <" A B C D ">
	3wlkd.f	|- F = <" J K L ">
	3wlkd.s	|- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )
	3wlkd.n	|- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )
	3wlkd.e	|- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )
	3wlkd.v	|- V = ( Vtx ` G )
	3wlkd.i	|- I = ( iEdg ` G )
	3trld.n	|- ( ph -> ( J =/= K /\ J =/= L /\ K =/= L ) )
Assertion	3pthd	|- ( ph -> F ( Paths ` G ) P )

Proof

Step	Hyp	Ref	Expression
1		3wlkd.p	|- P = <" A B C D ">
2		3wlkd.f	|- F = <" J K L ">
3		3wlkd.s	|- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )
4		3wlkd.n	|- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )
5		3wlkd.e	|- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )
6		3wlkd.v	|- V = ( Vtx ` G )
7		3wlkd.i	|- I = ( iEdg ` G )
8		3trld.n	|- ( ph -> ( J =/= K /\ J =/= L /\ K =/= L ) )
9		s4cli	|- <" A B C D "> 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 L "> )
13		s3len	|- ( # ` <" J K L "> ) = 3
14	12 13	eqtri	|- ( # ` F ) = 3
15		4m1e3	|- ( 4 - 1 ) = 3
16	1	fveq2i	|- ( # ` P ) = ( # ` <" A B C D "> )
17		s4len	|- ( # ` <" A B C D "> ) = 4
18	16 17	eqtr2i	|- 4 = ( # ` P )
19	18	oveq1i	|- ( 4 - 1 ) = ( ( # ` P ) - 1 )
20	14 15 19	3eqtr2i	|- ( # ` F ) = ( ( # ` P ) - 1 )
21	1 2 3 4	3pthdlem1	|- ( 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	3trld	|- ( ph -> F ( Trails ` G ) P )
24	11 20 21 22 23	pthd	|- ( ph -> F ( Paths ` G ) P )