Theorem s3fv1

Description: Extract the second symbol from a length 3 string. (Contributed by Mario Carneiro, 13-Jan-2017)

	Ref	Expression
Assertion	s3fv1	⊢ ( 𝐵 ∈ 𝑉 → ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 1 ) = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		df-s3	⊢ ⟨“ 𝐴 𝐵 𝐶 ”⟩ = ( ⟨“ 𝐴 𝐵 ”⟩ ++ ⟨“ 𝐶 ”⟩ )
2		s2cli	⊢ ⟨“ 𝐴 𝐵 ”⟩ ∈ Word V
3		s2len	⊢ ( ♯ ‘ ⟨“ 𝐴 𝐵 ”⟩ ) = 2
4		s2fv1	⊢ ( 𝐵 ∈ 𝑉 → ( ⟨“ 𝐴 𝐵 ”⟩ ‘ 1 ) = 𝐵 )
5		1nn0	⊢ 1 ∈ ℕ0
6		1lt2	⊢ 1 < 2
7	1 2 3 4 5 6	cats1fv	⊢ ( 𝐵 ∈ 𝑉 → ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 1 ) = 𝐵 )