Theorem s4cld

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

	Ref	Expression
Hypotheses	s2cld.1	$$⊢\phi \to A\in X$$
	s2cld.2	$$⊢\phi \to B\in X$$
	s3cld.3	$$⊢\phi \to C\in X$$
	s4cld.4	$$⊢\phi \to D\in X$$
Assertion	s4cld	$$⊢\phi \to ⟨“ABCD”⟩\in \mathrm{Word}X$$

Proof

Step	Hyp	Ref	Expression
1		s2cld.1	$$⊢\phi \to A\in X$$
2		s2cld.2	$$⊢\phi \to B\in X$$
3		s3cld.3	$$⊢\phi \to C\in X$$
4		s4cld.4	$$⊢\phi \to D\in X$$
5		df-s4	$$⊢⟨“ABCD”⟩=⟨“ABC”⟩\mathrm{++}⟨“D”⟩$$
6	1 2 3	s3cld	$$⊢\phi \to ⟨“ABC”⟩\in \mathrm{Word}X$$
7	5 6 4	cats1cld	$$⊢\phi \to ⟨“ABCD”⟩\in \mathrm{Word}X$$