Theorem s4fv1

Description: Extract the second symbol from a length 4 string. (Contributed by Thierry Arnoux, 8-Oct-2020)

	Ref	Expression
Assertion	s4fv1	⊢ ( 𝐵 ∈ 𝑉 → ( ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ ‘ 1 ) = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		df-s4	⊢ ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ = ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ++ ⟨“ 𝐷 ”⟩ )
2		s3cli	⊢ ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∈ Word V
3		s3len	⊢ ( ♯ ‘ ⟨“ 𝐴 𝐵 𝐶 ”⟩ ) = 3
4		s3fv1	⊢ ( 𝐵 ∈ 𝑉 → ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 1 ) = 𝐵 )
5		1nn0	⊢ 1 ∈ ℕ0
6		1lt3	⊢ 1 < 3
7	1 2 3 4 5 6	cats1fv	⊢ ( 𝐵 ∈ 𝑉 → ( ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ ‘ 1 ) = 𝐵 )