Theorem paddunssN

Description: Projective subspace sum includes the set union of its arguments. (Contributed by NM, 12-Jan-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	padd0.a	|- A = ( Atoms ` K )
	padd0.p	|- .+ = ( +P ` K )
Assertion	paddunssN	|- ( ( K e. B /\ X C_ A /\ Y C_ A ) -> ( X u. Y ) C_ ( X .+ Y ) )

Proof

Step	Hyp	Ref	Expression
1		padd0.a	|- A = ( Atoms ` K )
2		padd0.p	|- .+ = ( +P ` K )
3		ssun1	|- ( X u. Y ) C_ ( ( X u. Y ) u. { p e. A | E. q e. X E. r e. Y p ( le ` K ) ( q ( join ` K ) r ) } )
4		eqid	|- ( le ` K ) = ( le ` K )
5		eqid	|- ( join ` K ) = ( join ` K )
6	4 5 1 2	paddval	|- ( ( K e. B /\ X C_ A /\ Y C_ A ) -> ( X .+ Y ) = ( ( X u. Y ) u. { p e. A | E. q e. X E. r e. Y p ( le ` K ) ( q ( join ` K ) r ) } ) )
7	3 6	sseqtrrid	|- ( ( K e. B /\ X C_ A /\ Y C_ A ) -> ( X u. Y ) C_ ( X .+ Y ) )