Theorem 0trlon

Description: A trail of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017) (Revised by AV, 8-Jan-2021) (Revised by AV, 23-Mar-2021) (Proof shortened by AV, 30-Oct-2021)

	Ref	Expression
Hypothesis	0wlk.v	|- V = ( Vtx ` G )
Assertion	0trlon	|- ( ( P : ( 0 ... 0 ) --> V /\ ( P ` 0 ) = N ) -> (/) ( N ( TrailsOn ` G ) N ) P )

Proof

Step	Hyp	Ref	Expression
1		0wlk.v	|- V = ( Vtx ` G )
2	1	0wlkon	|- ( ( P : ( 0 ... 0 ) --> V /\ ( P ` 0 ) = N ) -> (/) ( N ( WalksOn ` G ) N ) P )
3		simpl	|- ( ( P : ( 0 ... 0 ) --> V /\ ( P ` 0 ) = N ) -> P : ( 0 ... 0 ) --> V )
4	1	0wlkonlem1	|- ( ( P : ( 0 ... 0 ) --> V /\ ( P ` 0 ) = N ) -> ( N e. V /\ N e. V ) )
5	1	1vgrex	|- ( N e. V -> G e. _V )
6	5	adantr	|- ( ( N e. V /\ N e. V ) -> G e. _V )
7	1	0trl	|- ( G e. _V -> ( (/) ( Trails ` G ) P <-> P : ( 0 ... 0 ) --> V ) )
8	4 6 7	3syl	|- ( ( P : ( 0 ... 0 ) --> V /\ ( P ` 0 ) = N ) -> ( (/) ( Trails ` G ) P <-> P : ( 0 ... 0 ) --> V ) )
9	3 8	mpbird	|- ( ( P : ( 0 ... 0 ) --> V /\ ( P ` 0 ) = N ) -> (/) ( Trails ` G ) P )
10		0ex	|- (/) e. _V
11	10	a1i	|- ( ( P : ( 0 ... 0 ) --> V /\ ( P ` 0 ) = N ) -> (/) e. _V )
12	1	0wlkonlem2	|- ( ( P : ( 0 ... 0 ) --> V /\ ( P ` 0 ) = N ) -> P e. ( V ^pm ( 0 ... 0 ) ) )
13	1	istrlson	|- ( ( ( N e. V /\ N e. V ) /\ ( (/) e. _V /\ P e. ( V ^pm ( 0 ... 0 ) ) ) ) -> ( (/) ( N ( TrailsOn ` G ) N ) P <-> ( (/) ( N ( WalksOn ` G ) N ) P /\ (/) ( Trails ` G ) P ) ) )
14	4 11 12 13	syl12anc	|- ( ( P : ( 0 ... 0 ) --> V /\ ( P ` 0 ) = N ) -> ( (/) ( N ( TrailsOn ` G ) N ) P <-> ( (/) ( N ( WalksOn ` G ) N ) P /\ (/) ( Trails ` G ) P ) ) )
15	2 9 14	mpbir2and	|- ( ( P : ( 0 ... 0 ) --> V /\ ( P ` 0 ) = N ) -> (/) ( N ( TrailsOn ` G ) N ) P )