Theorem clwwlk1loop

Description: A closed walk of length 1 is a loop. See also clwlkl1loop . (Contributed by AV, 24-Apr-2021)

	Ref	Expression
Assertion	clwwlk1loop	⊢ ( ( 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = 1 ) → { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
2		eqid	⊢ ( Edg ‘ 𝐺 ) = ( Edg ‘ 𝐺 )
3	1 2	isclwwlk	⊢ ( 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) ↔ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) )
4		lsw1	⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = 1 ) → ( lastS ‘ 𝑊 ) = ( 𝑊 ‘ 0 ) )
5	4	preq1d	⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = 1 ) → { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } = { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 0 ) } )
6	5	eleq1d	⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = 1 ) → ( { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ↔ { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) )
7	6	biimpd	⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = 1 ) → ( { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) → { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) )
8	7	ex	⊢ ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) → ( ( ♯ ‘ 𝑊 ) = 1 → ( { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) → { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ) )
9	8	com23	⊢ ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) → ( { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) → ( ( ♯ ‘ 𝑊 ) = 1 → { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ) )
10	9	adantr	⊢ ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) → ( { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) → ( ( ♯ ‘ 𝑊 ) = 1 → { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ) )
11	10	imp	⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) → ( ( ♯ ‘ 𝑊 ) = 1 → { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) )
12	11	3adant2	⊢ ( ( ( 𝑊 ∈ Word ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ ∅ ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑊 ) − 1 ) ) { ( 𝑊 ‘ 𝑖 ) , ( 𝑊 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ∧ { ( lastS ‘ 𝑊 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) → ( ( ♯ ‘ 𝑊 ) = 1 → { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) )
13	3 12	sylbi	⊢ ( 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) → ( ( ♯ ‘ 𝑊 ) = 1 → { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) )
14	13	imp	⊢ ( ( 𝑊 ∈ ( ClWWalks ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = 1 ) → { ( 𝑊 ‘ 0 ) , ( 𝑊 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) )