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Metamath Proof Explorer

Theorem hlatmstcOLDN

Description: An atomic, complete, orthomodular lattice is atomistic i.e. every element is the join of the atoms under it. See remark before Proposition 1 in Kalmbach p. 140; also remark in BeltramettiCassinelli p. 98. ( hatomistici analog.) (Contributed by NM, 21-Oct-2011) (New usage is discouraged.)

Ref Expression
Hypotheses hlatmstc.b 
|- B = ( Base ` K )
hlatmstc.l 
|- .<_ = ( le ` K )
hlatmstc.u 
|- U = ( lub ` K )
hlatmstc.a 
|- A = ( Atoms ` K )
Assertion hlatmstcOLDN 
|- ( ( K e. HL /\ X e. B ) -> ( U ` { y e. A | y .<_ X } ) = X )

Proof

Step Hyp Ref Expression
1 hlatmstc.b 
 |-  B = ( Base ` K )
2 hlatmstc.l 
 |-  .<_ = ( le ` K )
3 hlatmstc.u 
 |-  U = ( lub ` K )
4 hlatmstc.a 
 |-  A = ( Atoms ` K )
5 hlomcmat 
 |-  ( K e. HL -> ( K e. OML /\ K e. CLat /\ K e. AtLat ) )
6 1 2 3 4 atlatmstc 
 |-  ( ( ( K e. OML /\ K e. CLat /\ K e. AtLat ) /\ X e. B ) -> ( U ` { y e. A | y .<_ X } ) = X )
7 5 6 sylan 
 |-  ( ( K e. HL /\ X e. B ) -> ( U ` { y e. A | y .<_ X } ) = X )