Theorem elpaddatriN

Description: Condition implying membership in a projective subspace sum with a point. (Contributed by NM, 1-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	elpaddatriN	|- ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ ( R e. X /\ S e. A /\ S .<_ ( R .\/ Q ) ) ) -> S e. ( X .+ { 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		simpl1	|- ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ ( R e. X /\ S e. A /\ S .<_ ( R .\/ Q ) ) ) -> K e. Lat )
6		simpl2	|- ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ ( R e. X /\ S e. A /\ S .<_ ( R .\/ Q ) ) ) -> X C_ A )
7		simpl3	|- ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ ( R e. X /\ S e. A /\ S .<_ ( R .\/ Q ) ) ) -> Q e. A )
8	7	snssd	|- ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ ( R e. X /\ S e. A /\ S .<_ ( R .\/ Q ) ) ) -> { Q } C_ A )
9		simpr1	|- ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ ( R e. X /\ S e. A /\ S .<_ ( R .\/ Q ) ) ) -> R e. X )
10		snidg	|- ( Q e. A -> Q e. { Q } )
11	7 10	syl	|- ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ ( R e. X /\ S e. A /\ S .<_ ( R .\/ Q ) ) ) -> Q e. { Q } )
12		simpr2	|- ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ ( R e. X /\ S e. A /\ S .<_ ( R .\/ Q ) ) ) -> S e. A )
13		simpr3	|- ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ ( R e. X /\ S e. A /\ S .<_ ( R .\/ Q ) ) ) -> S .<_ ( R .\/ Q ) )
14	1 2 3 4	elpaddri	|- ( ( ( K e. Lat /\ X C_ A /\ { Q } C_ A ) /\ ( R e. X /\ Q e. { Q } ) /\ ( S e. A /\ S .<_ ( R .\/ Q ) ) ) -> S e. ( X .+ { Q } ) )
15	5 6 8 9 11 12 13 14	syl322anc	|- ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ ( R e. X /\ S e. A /\ S .<_ ( R .\/ Q ) ) ) -> S e. ( X .+ { Q } ) )