Theorem ccat2s1fst

Description: The first symbol of the concatenation of a word with two single symbols. (Contributed by Alexander van der Vekens, 22-Sep-2018) (Revised by AV, 28-Jan-2024)

	Ref	Expression
Assertion	ccat2s1fst	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 0 < ( ♯ ‘ 𝑊 ) ) → ( ( ( 𝑊 ++ ⟨“ 𝑋 ”⟩ ) ++ ⟨“ 𝑌 ”⟩ ) ‘ 0 ) = ( 𝑊 ‘ 0 ) )

Proof

Step	Hyp	Ref	Expression
1		0nn0	⊢ 0 ∈ ℕ0
2		ccat2s1fvw	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 0 ∈ ℕ0 ∧ 0 < ( ♯ ‘ 𝑊 ) ) → ( ( ( 𝑊 ++ ⟨“ 𝑋 ”⟩ ) ++ ⟨“ 𝑌 ”⟩ ) ‘ 0 ) = ( 𝑊 ‘ 0 ) )
3	1 2	mp3an2	⊢ ( ( 𝑊 ∈ Word 𝑉 ∧ 0 < ( ♯ ‘ 𝑊 ) ) → ( ( ( 𝑊 ++ ⟨“ 𝑋 ”⟩ ) ++ ⟨“ 𝑌 ”⟩ ) ‘ 0 ) = ( 𝑊 ‘ 0 ) )