Theorem psubspi2N

Description: Property of a projective subspace. (Contributed by NM, 13-Jan-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	psubspset.l	|- .<_ = ( le ` K )
	psubspset.j	|- .\/ = ( join ` K )
	psubspset.a	|- A = ( Atoms ` K )
	psubspset.s	|- S = ( PSubSp ` K )
Assertion	psubspi2N	|- ( ( ( K e. D /\ X e. S /\ P e. A ) /\ ( Q e. X /\ R e. X /\ P .<_ ( Q .\/ R ) ) ) -> P e. X )

Proof

Step	Hyp	Ref	Expression
1		psubspset.l	|- .<_ = ( le ` K )
2		psubspset.j	|- .\/ = ( join ` K )
3		psubspset.a	|- A = ( Atoms ` K )
4		psubspset.s	|- S = ( PSubSp ` K )
5		oveq1	|- ( q = Q -> ( q .\/ r ) = ( Q .\/ r ) )
6	5	breq2d	|- ( q = Q -> ( P .<_ ( q .\/ r ) <-> P .<_ ( Q .\/ r ) ) )
7		oveq2	|- ( r = R -> ( Q .\/ r ) = ( Q .\/ R ) )
8	7	breq2d	|- ( r = R -> ( P .<_ ( Q .\/ r ) <-> P .<_ ( Q .\/ R ) ) )
9	6 8	rspc2ev	|- ( ( Q e. X /\ R e. X /\ P .<_ ( Q .\/ R ) ) -> E. q e. X E. r e. X P .<_ ( q .\/ r ) )
10	1 2 3 4	psubspi	|- ( ( ( K e. D /\ X e. S /\ P e. A ) /\ E. q e. X E. r e. X P .<_ ( q .\/ r ) ) -> P e. X )
11	9 10	sylan2	|- ( ( ( K e. D /\ X e. S /\ P e. A ) /\ ( Q e. X /\ R e. X /\ P .<_ ( Q .\/ R ) ) ) -> P e. X )