Theorem elpaddatiN

Description: Consequence of membership in a projective subspace sum with a point. (Contributed by NM, 2-Feb-2012) (New usage is discouraged.)

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

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	elpaddat	|- ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ X =/= (/) ) -> ( R e. ( X .+ { Q } ) <-> ( R e. A /\ E. p e. X R .<_ ( p .\/ Q ) ) ) )
6		simpr	|- ( ( R e. A /\ E. p e. X R .<_ ( p .\/ Q ) ) -> E. p e. X R .<_ ( p .\/ Q ) )
7	5 6	syl6bi	|- ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ X =/= (/) ) -> ( R e. ( X .+ { Q } ) -> E. p e. X R .<_ ( p .\/ Q ) ) )
8	7	impr	|- ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ ( X =/= (/) /\ R e. ( X .+ { Q } ) ) ) -> E. p e. X R .<_ ( p .\/ Q ) )