Theorem lswcl

Description: Closure of the last symbol: the last symbol of a not empty word belongs to the alphabet for the word. (Contributed by AV, 2-Aug-2018) (Proof shortened by AV, 29-Apr-2020)

	Ref	Expression
Assertion	lswcl	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅ ) → ( lastS ‘ 𝑊 ) ∈ 𝑉 )

Proof

Step	Hyp	Ref	Expression
1		lsw	⊢ ( 𝑊 ∈ Word 𝑉 → ( lastS ‘ 𝑊 ) = ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) )
2	1	adantr	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅ ) → ( lastS ‘ 𝑊 ) = ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) )
3		lennncl	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅ ) → ( ♯ ‘ 𝑊 ) ∈ ℕ )
4		fzo0end	⊢ ( ( ♯ ‘ 𝑊 ) ∈ ℕ → ( ( ♯ ‘ 𝑊 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) )
5	3 4	syl	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅ ) → ( ( ♯ ‘ 𝑊 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) )
6		wrdsymbcl	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ( ( ♯ ‘ 𝑊 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ∈ 𝑉 )
7	5 6	syldan	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅ ) → ( 𝑊 ‘ ( ( ♯ ‘ 𝑊 ) − 1 ) ) ∈ 𝑉 )
8	2 7	eqeltrd	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅ ) → ( lastS ‘ 𝑊 ) ∈ 𝑉 )