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Metamath Proof Explorer

Theorem atpsubN

Description: The set of all atoms is a projective subspace. Remark below Definition 15.1 of MaedaMaeda p. 61. (Contributed by NM, 13-Oct-2011) (New usage is discouraged.)

Ref Expression
Hypotheses atpsub.a 
|- A = ( Atoms ` K )
atpsub.s 
|- S = ( PSubSp ` K )
Assertion atpsubN 
|- ( K e. V -> A e. S )

Proof

Step Hyp Ref Expression
1 atpsub.a 
 |-  A = ( Atoms ` K )
2 atpsub.s 
 |-  S = ( PSubSp ` K )
3 ssid 
 |-  A C_ A
4 ax-1 
 |-  ( r e. A -> ( r ( le ` K ) ( p ( join ` K ) q ) -> r e. A ) )
5 4 rgen 
 |-  A. r e. A ( r ( le ` K ) ( p ( join ` K ) q ) -> r e. A )
6 5 rgen2w 
 |-  A. p e. A A. q e. A A. r e. A ( r ( le ` K ) ( p ( join ` K ) q ) -> r e. A )
7 3 6 pm3.2i 
 |-  ( A C_ A /\ A. p e. A A. q e. A A. r e. A ( r ( le ` K ) ( p ( join ` K ) q ) -> r e. A ) )
8 eqid 
 |-  ( le ` K ) = ( le ` K )
9 eqid 
 |-  ( join ` K ) = ( join ` K )
10 8 9 1 2 ispsubsp 
 |-  ( K e. V -> ( A e. S <-> ( A C_ A /\ A. p e. A A. q e. A A. r e. A ( r ( le ` K ) ( p ( join ` K ) q ) -> r e. A ) ) ) )
11 7 10 mpbiri 
 |-  ( K e. V -> A e. S )