Theorem lswccats1

Description: The last symbol of a word concatenated with a singleton word is the symbol of the singleton word. (Contributed by AV, 6-Aug-2018) (Proof shortened by AV, 22-Oct-2018)

	Ref	Expression
Assertion	lswccats1	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ) → ( lastS ‘ ( 𝑊 ++ ⟨“ 𝑆 ”⟩ ) ) = 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		simpl	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ) → 𝑊 ∈ Word 𝑉 )
2		s1cl	⊢ ( 𝑆 ∈ 𝑉 → ⟨“ 𝑆 ”⟩ ∈ Word 𝑉 )
3	2	adantl	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ) → ⟨“ 𝑆 ”⟩ ∈ Word 𝑉 )
4		s1nz	⊢ ⟨“ 𝑆 ”⟩ ≠ ∅
5	4	a1i	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ) → ⟨“ 𝑆 ”⟩ ≠ ∅ )
6		lswccatn0lsw	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ ⟨“ 𝑆 ”⟩ ∈ Word 𝑉 ∧ ⟨“ 𝑆 ”⟩ ≠ ∅ ) → ( lastS ‘ ( 𝑊 ++ ⟨“ 𝑆 ”⟩ ) ) = ( lastS ‘ ⟨“ 𝑆 ”⟩ ) )
7	1 3 5 6	syl3anc	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ) → ( lastS ‘ ( 𝑊 ++ ⟨“ 𝑆 ”⟩ ) ) = ( lastS ‘ ⟨“ 𝑆 ”⟩ ) )
8		lsws1	⊢ ( 𝑆 ∈ 𝑉 → ( lastS ‘ ⟨“ 𝑆 ”⟩ ) = 𝑆 )
9	8	adantl	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ) → ( lastS ‘ ⟨“ 𝑆 ”⟩ ) = 𝑆 )
10	7 9	eqtrd	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 𝑆 ∈ 𝑉 ) → ( lastS ‘ ( 𝑊 ++ ⟨“ 𝑆 ”⟩ ) ) = 𝑆 )