Metamath Proof Explorer

Theorem 2clwwlk

Description: Value of operation C , mapping a vertex v and an integer n greater than 1 to the "closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) with v(n-2) = v" according to definition 6 in Huneke p. 2. Such closed walks are "double loops" consisting of a closed (n-2)-walk v = v(0) ... v(n-2) = v and a closed 2-walk v = v(n-2) v(n-1) v(n) = v, see 2clwwlk2clwwlk . ( X C N ) is called the "set of double loops of length N on vertex X " in the following. (Contributed by Alexander van der Vekens, 14-Sep-2018) (Revised by AV, 29-May-2021) (Revised by AV, 20-Apr-2022)

Ref Expression
Hypothesis 2clwwlk.c ⊢C=v∈V,n∈ℤ≥2⟼w∈vClWWalksNOn⁡Gn|w⁡n−2=v
Assertion 2clwwlk ⊢X∈V∧N∈ℤ≥2→XCN=w∈XClWWalksNOn⁡GN|w⁡N−2=X

Proof

Step Hyp Ref Expression
1 2clwwlk.c ⊢C=v∈V,n∈ℤ≥2⟼w∈vClWWalksNOn⁡Gn|w⁡n−2=v
2 oveq12 ⊢v=X∧n=N→vClWWalksNOn⁡Gn=XClWWalksNOn⁡GN
3 fvoveq1 ⊢n=N→w⁡n−2=w⁡N−2
4 3 adantl ⊢v=X∧n=N→w⁡n−2=w⁡N−2
5 simpl ⊢v=X∧n=N→v=X
6 4 5 eqeq12d ⊢v=X∧n=N→w⁡n−2=v↔w⁡N−2=X
7 2 6 rabeqbidv ⊢v=X∧n=N→w∈vClWWalksNOn⁡Gn|w⁡n−2=v=w∈XClWWalksNOn⁡GN|w⁡N−2=X
8 ovex ⊢XClWWalksNOn⁡GN∈V
9 8 rabex ⊢w∈XClWWalksNOn⁡GN|w⁡N−2=X∈V
10 7 1 9 ovmpoa ⊢X∈V∧N∈ℤ≥2→XCN=w∈XClWWalksNOn⁡GN|w⁡N−2=X