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Metamath Proof Explorer

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 )