Theorem elpadd2at2

Description: Membership in a projective subspace sum of two points. (Contributed by NM, 8-Mar-2012)

	Ref	Expression
Hypotheses	paddfval.l	|- .<_ = ( le ` K )
	paddfval.j	|- .\/ = ( join ` K )
	paddfval.a	|- A = ( Atoms ` K )
	paddfval.p	|- .+ = ( +P ` K )
Assertion	elpadd2at2	|- ( ( K e. Lat /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( S e. ( { Q } .+ { R } ) <-> S .<_ ( Q .\/ R ) ) )

Proof

Step	Hyp	Ref	Expression
1		paddfval.l	|- .<_ = ( le ` K )
2		paddfval.j	|- .\/ = ( join ` K )
3		paddfval.a	|- A = ( Atoms ` K )
4		paddfval.p	|- .+ = ( +P ` K )
5	1 2 3 4	elpadd2at	|- ( ( K e. Lat /\ Q e. A /\ R e. A ) -> ( S e. ( { Q } .+ { R } ) <-> ( S e. A /\ S .<_ ( Q .\/ R ) ) ) )
6	5	3adant3r3	|- ( ( K e. Lat /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( S e. ( { Q } .+ { R } ) <-> ( S e. A /\ S .<_ ( Q .\/ R ) ) ) )
7		simpr3	|- ( ( K e. Lat /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> S e. A )
8	7	biantrurd	|- ( ( K e. Lat /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( S .<_ ( Q .\/ R ) <-> ( S e. A /\ S .<_ ( Q .\/ R ) ) ) )
9	6 8	bitr4d	|- ( ( K e. Lat /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( S e. ( { Q } .+ { R } ) <-> S .<_ ( Q .\/ R ) ) )