Theorem stm1i

Description: State of component of unit meet. (Contributed by NM, 11-Nov-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	stle.1	|- A e. CH
	stle.2	|- B e. CH
Assertion	stm1i	|- ( S e. States -> ( ( S ` ( A i^i B ) ) = 1 -> ( S ` A ) = 1 ) )

Proof

Step	Hyp	Ref	Expression
1		stle.1	|- A e. CH
2		stle.2	|- B e. CH
3		inss1	|- ( A i^i B ) C_ A
4	1 2	chincli	|- ( A i^i B ) e. CH
5	4 1	stlei	|- ( S e. States -> ( ( A i^i B ) C_ A -> ( S ` ( A i^i B ) ) <_ ( S ` A ) ) )
6	3 5	mpi	|- ( S e. States -> ( S ` ( A i^i B ) ) <_ ( S ` A ) )
7		breq1	|- ( ( S ` ( A i^i B ) ) = 1 -> ( ( S ` ( A i^i B ) ) <_ ( S ` A ) <-> 1 <_ ( S ` A ) ) )
8	6 7	syl5ibcom	|- ( S e. States -> ( ( S ` ( A i^i B ) ) = 1 -> 1 <_ ( S ` A ) ) )
9	1	stge1i	|- ( S e. States -> ( 1 <_ ( S ` A ) <-> ( S ` A ) = 1 ) )
10	8 9	sylibd	|- ( S e. States -> ( ( S ` ( A i^i B ) ) = 1 -> ( S ` A ) = 1 ) )