Theorem lsws1

Description: The last symbol of a singleton word is its symbol. (Contributed by AV, 22-Oct-2018)

	Ref	Expression
Assertion	lsws1	⊢ ( 𝐴 ∈ 𝑉 → ( lastS ‘ ⟨“ 𝐴 ”⟩ ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		s1cl	⊢ ( 𝐴 ∈ 𝑉 → ⟨“ 𝐴 ”⟩ ∈ Word 𝑉 )
2		s1len	⊢ ( ♯ ‘ ⟨“ 𝐴 ”⟩ ) = 1
3		lsw1	⊢ ( ( ⟨“ 𝐴 ”⟩ ∈ Word 𝑉 ∧ ( ♯ ‘ ⟨“ 𝐴 ”⟩ ) = 1 ) → ( lastS ‘ ⟨“ 𝐴 ”⟩ ) = ( ⟨“ 𝐴 ”⟩ ‘ 0 ) )
4	1 2 3	sylancl	⊢ ( 𝐴 ∈ 𝑉 → ( lastS ‘ ⟨“ 𝐴 ”⟩ ) = ( ⟨“ 𝐴 ”⟩ ‘ 0 ) )
5		s1fv	⊢ ( 𝐴 ∈ 𝑉 → ( ⟨“ 𝐴 ”⟩ ‘ 0 ) = 𝐴 )
6	4 5	eqtrd	⊢ ( 𝐴 ∈ 𝑉 → ( lastS ‘ ⟨“ 𝐴 ”⟩ ) = 𝐴 )