Theorem s7cld

Description: A length 7 string is a word. (Contributed by Mario Carneiro, 27-Feb-2016)

	Ref	Expression
Hypotheses	s2cld.1	|- ( ph -> A e. X )
	s2cld.2	|- ( ph -> B e. X )
	s3cld.3	|- ( ph -> C e. X )
	s4cld.4	|- ( ph -> D e. X )
	s5cld.5	|- ( ph -> E e. X )
	s6cld.6	|- ( ph -> F e. X )
	s7cld.7	|- ( ph -> G e. X )
Assertion	s7cld	|- ( ph -> <" A B C D E F G "> e. Word X )

Proof

Step	Hyp	Ref	Expression
1		s2cld.1	|- ( ph -> A e. X )
2		s2cld.2	|- ( ph -> B e. X )
3		s3cld.3	|- ( ph -> C e. X )
4		s4cld.4	|- ( ph -> D e. X )
5		s5cld.5	|- ( ph -> E e. X )
6		s6cld.6	|- ( ph -> F e. X )
7		s7cld.7	|- ( ph -> G e. X )
8		df-s7	|- <" A B C D E F G "> = ( <" A B C D E F "> ++ <" G "> )
9	1 2 3 4 5 6	s6cld	|- ( ph -> <" A B C D E F "> e. Word X )
10	8 9 7	cats1cld	|- ( ph -> <" A B C D E F G "> e. Word X )