Theorem 0wlkon

Description: A walk of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017) (Revised by AV, 3-Jan-2021) (Revised by AV, 23-Mar-2021) (Proof shortened by AV, 30-Oct-2021)

	Ref	Expression
Hypothesis	0wlk.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion	0wlkon	⊢ ( ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ∧ ( 𝑃 ‘ 0 ) = 𝑁 ) → ∅ ( 𝑁 ( WalksOn ‘ 𝐺 ) 𝑁 ) 𝑃 )

Proof

Step	Hyp	Ref	Expression
1		0wlk.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		simpl	⊢ ( ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ∧ ( 𝑃 ‘ 0 ) = 𝑁 ) → 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 )
3	1	0wlkonlem1	⊢ ( ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ∧ ( 𝑃 ‘ 0 ) = 𝑁 ) → ( 𝑁 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ) )
4	1	1vgrex	⊢ ( 𝑁 ∈ 𝑉 → 𝐺 ∈ V )
5	4	adantr	⊢ ( ( 𝑁 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ) → 𝐺 ∈ V )
6	1	0wlk	⊢ ( 𝐺 ∈ V → ( ∅ ( Walks ‘ 𝐺 ) 𝑃 ↔ 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ) )
7	3 5 6	3syl	⊢ ( ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ∧ ( 𝑃 ‘ 0 ) = 𝑁 ) → ( ∅ ( Walks ‘ 𝐺 ) 𝑃 ↔ 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ) )
8	2 7	mpbird	⊢ ( ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ∧ ( 𝑃 ‘ 0 ) = 𝑁 ) → ∅ ( Walks ‘ 𝐺 ) 𝑃 )
9		simpr	⊢ ( ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ∧ ( 𝑃 ‘ 0 ) = 𝑁 ) → ( 𝑃 ‘ 0 ) = 𝑁 )
10		hash0	⊢ ( ♯ ‘ ∅ ) = 0
11	10	fveq2i	⊢ ( 𝑃 ‘ ( ♯ ‘ ∅ ) ) = ( 𝑃 ‘ 0 )
12	11 9	eqtrid	⊢ ( ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ∧ ( 𝑃 ‘ 0 ) = 𝑁 ) → ( 𝑃 ‘ ( ♯ ‘ ∅ ) ) = 𝑁 )
13		0ex	⊢ ∅ ∈ V
14	13	a1i	⊢ ( ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ∧ ( 𝑃 ‘ 0 ) = 𝑁 ) → ∅ ∈ V )
15	1	0wlkonlem2	⊢ ( ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ∧ ( 𝑃 ‘ 0 ) = 𝑁 ) → 𝑃 ∈ ( 𝑉 ↑pm ( 0 ... 0 ) ) )
16	1	iswlkon	⊢ ( ( ( 𝑁 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ) ∧ ( ∅ ∈ V ∧ 𝑃 ∈ ( 𝑉 ↑pm ( 0 ... 0 ) ) ) ) → ( ∅ ( 𝑁 ( WalksOn ‘ 𝐺 ) 𝑁 ) 𝑃 ↔ ( ∅ ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝑁 ∧ ( 𝑃 ‘ ( ♯ ‘ ∅ ) ) = 𝑁 ) ) )
17	3 14 15 16	syl12anc	⊢ ( ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ∧ ( 𝑃 ‘ 0 ) = 𝑁 ) → ( ∅ ( 𝑁 ( WalksOn ‘ 𝐺 ) 𝑁 ) 𝑃 ↔ ( ∅ ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝑁 ∧ ( 𝑃 ‘ ( ♯ ‘ ∅ ) ) = 𝑁 ) ) )
18	8 9 12 17	mpbir3and	⊢ ( ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ∧ ( 𝑃 ‘ 0 ) = 𝑁 ) → ∅ ( 𝑁 ( WalksOn ‘ 𝐺 ) 𝑁 ) 𝑃 )