Theorem s7cld

Description: A length 7 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	$$⊢\phi \to D\in X$$
	s5cld.5	$$⊢\phi \to E\in X$$
	s6cld.6	$$⊢\phi \to F\in X$$
	s7cld.7	$$⊢\phi \to G\in X$$
Assertion	s7cld	$$⊢\phi \to ⟨“ABCDEFG”⟩\in \mathrm{Word}X$$

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		s5cld.5	$$⊢\phi \to E\in X$$
6		s6cld.6	$$⊢\phi \to F\in X$$
7		s7cld.7	$$⊢\phi \to G\in X$$
8		df-s7	$$⊢⟨“ABCDEFG”⟩=⟨“ABCDEF”⟩\mathrm{++}⟨“G”⟩$$
9	1 2 3 4 5 6	s6cld	$$⊢\phi \to ⟨“ABCDEF”⟩\in \mathrm{Word}X$$
10	8 9 7	cats1cld	$$⊢\phi \to ⟨“ABCDEFG”⟩\in \mathrm{Word}X$$