Theorem s4cld

Description: A length 4 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016)

	Ref	Expression
Hypotheses	s2cld.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑋 )
	s2cld.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑋 )
	s3cld.3	⊢ ( 𝜑 → 𝐶 ∈ 𝑋 )
	s4cld.4	⊢ ( 𝜑 → 𝐷 ∈ 𝑋 )
Assertion	s4cld	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ ∈ Word 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		s2cld.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑋 )
2		s2cld.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑋 )
3		s3cld.3	⊢ ( 𝜑 → 𝐶 ∈ 𝑋 )
4		s4cld.4	⊢ ( 𝜑 → 𝐷 ∈ 𝑋 )
5		df-s4	⊢ ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ = ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ++ ⟨“ 𝐷 ”⟩ )
6	1 2 3	s3cld	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∈ Word 𝑋 )
7	5 6 4	cats1cld	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ ∈ Word 𝑋 )