Theorem pclun2N

Description: The projective subspace closure of the union of two subspaces equals their projective sum. (Contributed by NM, 12-Sep-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pclun2.s	|- S = ( PSubSp ` K )
	pclun2.p	|- .+ = ( +P ` K )
	pclun2.c	|- U = ( PCl ` K )
Assertion	pclun2N	|- ( ( K e. HL /\ X e. S /\ Y e. S ) -> ( U ` ( X u. Y ) ) = ( X .+ Y ) )

Proof

Step	Hyp	Ref	Expression
1		pclun2.s	|- S = ( PSubSp ` K )
2		pclun2.p	|- .+ = ( +P ` K )
3		pclun2.c	|- U = ( PCl ` K )
4		simp1	|- ( ( K e. HL /\ X e. S /\ Y e. S ) -> K e. HL )
5		eqid	|- ( Atoms ` K ) = ( Atoms ` K )
6	5 1	psubssat	|- ( ( K e. HL /\ X e. S ) -> X C_ ( Atoms ` K ) )
7	6	3adant3	|- ( ( K e. HL /\ X e. S /\ Y e. S ) -> X C_ ( Atoms ` K ) )
8	5 1	psubssat	|- ( ( K e. HL /\ Y e. S ) -> Y C_ ( Atoms ` K ) )
9	8	3adant2	|- ( ( K e. HL /\ X e. S /\ Y e. S ) -> Y C_ ( Atoms ` K ) )
10	5 2 3	pclunN	|- ( ( K e. HL /\ X C_ ( Atoms ` K ) /\ Y C_ ( Atoms ` K ) ) -> ( U ` ( X u. Y ) ) = ( U ` ( X .+ Y ) ) )
11	4 7 9 10	syl3anc	|- ( ( K e. HL /\ X e. S /\ Y e. S ) -> ( U ` ( X u. Y ) ) = ( U ` ( X .+ Y ) ) )
12	1 2	paddclN	|- ( ( K e. HL /\ X e. S /\ Y e. S ) -> ( X .+ Y ) e. S )
13	1 3	pclidN	|- ( ( K e. HL /\ ( X .+ Y ) e. S ) -> ( U ` ( X .+ Y ) ) = ( X .+ Y ) )
14	4 12 13	syl2anc	|- ( ( K e. HL /\ X e. S /\ Y e. S ) -> ( U ` ( X .+ Y ) ) = ( X .+ Y ) )
15	11 14	eqtrd	|- ( ( K e. HL /\ X e. S /\ Y e. S ) -> ( U ` ( X u. Y ) ) = ( X .+ Y ) )