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Metamath Proof Explorer

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 )