Theorem stm1addi

Description: Sum of states whose meet is 1. (Contributed by NM, 11-Nov-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	stle.1	|- A e. CH
	stle.2	|- B e. CH
Assertion	stm1addi	|- ( S e. States -> ( ( S ` ( A i^i B ) ) = 1 -> ( ( S ` A ) + ( S ` B ) ) = 2 ) )

Proof

Step	Hyp	Ref	Expression
1		stle.1	|- A e. CH
2		stle.2	|- B e. CH
3	1 2	stm1i	|- ( S e. States -> ( ( S ` ( A i^i B ) ) = 1 -> ( S ` A ) = 1 ) )
4	1 2	stm1ri	|- ( S e. States -> ( ( S ` ( A i^i B ) ) = 1 -> ( S ` B ) = 1 ) )
5	3 4	jcad	|- ( S e. States -> ( ( S ` ( A i^i B ) ) = 1 -> ( ( S ` A ) = 1 /\ ( S ` B ) = 1 ) ) )
6		oveq12	|- ( ( ( S ` A ) = 1 /\ ( S ` B ) = 1 ) -> ( ( S ` A ) + ( S ` B ) ) = ( 1 + 1 ) )
7		df-2	|- 2 = ( 1 + 1 )
8	6 7	eqtr4di	|- ( ( ( S ` A ) = 1 /\ ( S ` B ) = 1 ) -> ( ( S ` A ) + ( S ` B ) ) = 2 )
9	5 8	syl6	|- ( S e. States -> ( ( S ` ( A i^i B ) ) = 1 -> ( ( S ` A ) + ( S ` B ) ) = 2 ) )