Metamath Proof Explorer


Theorem s4fv2

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

Ref Expression
Assertion s4fv2 ( 𝐶𝑉 → ( ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ ‘ 2 ) = 𝐶 )

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 s3fv2 ( 𝐶𝑉 → ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 2 ) = 𝐶 )
5 2nn0 2 ∈ ℕ0
6 2lt3 2 < 3
7 1 2 3 4 5 6 cats1fv ( 𝐶𝑉 → ( ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ ‘ 2 ) = 𝐶 )