Metamath Proof Explorer

Theorem clwwnonrepclwwnon

Description: If the initial vertex of a closed walk occurs another time in the walk, the walk starts with a closed walk on this vertex. See also the remarks in clwwnrepclwwn . (Contributed by AV, 24-Apr-2022) (Revised by AV, 10-May-2022) (Revised by AV, 30-Oct-2022)

Ref Expression
Assertion clwwnonrepclwwnon 
|- ( ( N e. ( ZZ>= ` 3 ) /\ W e. ( X ( ClWWalksNOn ` G ) N ) /\ ( W ` ( N - 2 ) ) = X ) -> ( W prefix ( N - 2 ) ) e. ( X ( ClWWalksNOn ` G ) ( N - 2 ) ) )

Proof

Step Hyp Ref Expression
1 simp1 
 |-  ( ( N e. ( ZZ>= ` 3 ) /\ W e. ( X ( ClWWalksNOn ` G ) N ) /\ ( W ` ( N - 2 ) ) = X ) -> N e. ( ZZ>= ` 3 ) )
2 isclwwlknon 
 |-  ( W e. ( X ( ClWWalksNOn ` G ) N ) <-> ( W e. ( N ClWWalksN G ) /\ ( W ` 0 ) = X ) )
3 2 simplbi 
 |-  ( W e. ( X ( ClWWalksNOn ` G ) N ) -> W e. ( N ClWWalksN G ) )
4 3 3ad2ant2 
 |-  ( ( N e. ( ZZ>= ` 3 ) /\ W e. ( X ( ClWWalksNOn ` G ) N ) /\ ( W ` ( N - 2 ) ) = X ) -> W e. ( N ClWWalksN G ) )
5 simpr 
 |-  ( ( W e. ( N ClWWalksN G ) /\ ( W ` 0 ) = X ) -> ( W ` 0 ) = X )
6 5 eqcomd 
 |-  ( ( W e. ( N ClWWalksN G ) /\ ( W ` 0 ) = X ) -> X = ( W ` 0 ) )
7 2 6 sylbi 
 |-  ( W e. ( X ( ClWWalksNOn ` G ) N ) -> X = ( W ` 0 ) )
8 7 eqeq2d 
 |-  ( W e. ( X ( ClWWalksNOn ` G ) N ) -> ( ( W ` ( N - 2 ) ) = X <-> ( W ` ( N - 2 ) ) = ( W ` 0 ) ) )
9 8 biimpa 
 |-  ( ( W e. ( X ( ClWWalksNOn ` G ) N ) /\ ( W ` ( N - 2 ) ) = X ) -> ( W ` ( N - 2 ) ) = ( W ` 0 ) )
10 9 3adant1 
 |-  ( ( N e. ( ZZ>= ` 3 ) /\ W e. ( X ( ClWWalksNOn ` G ) N ) /\ ( W ` ( N - 2 ) ) = X ) -> ( W ` ( N - 2 ) ) = ( W ` 0 ) )
11 clwwnrepclwwn 
 |-  ( ( N e. ( ZZ>= ` 3 ) /\ W e. ( N ClWWalksN G ) /\ ( W ` ( N - 2 ) ) = ( W ` 0 ) ) -> ( W prefix ( N - 2 ) ) e. ( ( N - 2 ) ClWWalksN G ) )
12 1 4 10 11 syl3anc 
 |-  ( ( N e. ( ZZ>= ` 3 ) /\ W e. ( X ( ClWWalksNOn ` G ) N ) /\ ( W ` ( N - 2 ) ) = X ) -> ( W prefix ( N - 2 ) ) e. ( ( N - 2 ) ClWWalksN G ) )
13 2clwwlklem 
 |-  ( ( W e. ( N ClWWalksN G ) /\ N e. ( ZZ>= ` 3 ) ) -> ( ( W prefix ( N - 2 ) ) ` 0 ) = ( W ` 0 ) )
14 3 13 sylan 
 |-  ( ( W e. ( X ( ClWWalksNOn ` G ) N ) /\ N e. ( ZZ>= ` 3 ) ) -> ( ( W prefix ( N - 2 ) ) ` 0 ) = ( W ` 0 ) )
15 14 ancoms 
 |-  ( ( N e. ( ZZ>= ` 3 ) /\ W e. ( X ( ClWWalksNOn ` G ) N ) ) -> ( ( W prefix ( N - 2 ) ) ` 0 ) = ( W ` 0 ) )
16 15 3adant3 
 |-  ( ( N e. ( ZZ>= ` 3 ) /\ W e. ( X ( ClWWalksNOn ` G ) N ) /\ ( W ` ( N - 2 ) ) = X ) -> ( ( W prefix ( N - 2 ) ) ` 0 ) = ( W ` 0 ) )
17 2 simprbi 
 |-  ( W e. ( X ( ClWWalksNOn ` G ) N ) -> ( W ` 0 ) = X )
18 17 3ad2ant2 
 |-  ( ( N e. ( ZZ>= ` 3 ) /\ W e. ( X ( ClWWalksNOn ` G ) N ) /\ ( W ` ( N - 2 ) ) = X ) -> ( W ` 0 ) = X )
19 16 18 eqtrd 
 |-  ( ( N e. ( ZZ>= ` 3 ) /\ W e. ( X ( ClWWalksNOn ` G ) N ) /\ ( W ` ( N - 2 ) ) = X ) -> ( ( W prefix ( N - 2 ) ) ` 0 ) = X )
20 isclwwlknon 
 |-  ( ( W prefix ( N - 2 ) ) e. ( X ( ClWWalksNOn ` G ) ( N - 2 ) ) <-> ( ( W prefix ( N - 2 ) ) e. ( ( N - 2 ) ClWWalksN G ) /\ ( ( W prefix ( N - 2 ) ) ` 0 ) = X ) )
21 12 19 20 sylanbrc 
 |-  ( ( N e. ( ZZ>= ` 3 ) /\ W e. ( X ( ClWWalksNOn ` G ) N ) /\ ( W ` ( N - 2 ) ) = X ) -> ( W prefix ( N - 2 ) ) e. ( X ( ClWWalksNOn ` G ) ( N - 2 ) ) )