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Metamath Proof Explorer

Theorem 3polN

Description: Triple polarity cancels to a single polarity. (Contributed by NM, 6-Mar-2012) (New usage is discouraged.)

Ref Expression
Hypotheses 2polss.a 
|- A = ( Atoms ` K )
2polss.p 
|- ._|_ = ( _|_P ` K )
Assertion 3polN 
|- ( ( K e. HL /\ S C_ A ) -> ( ._|_ ` ( ._|_ ` ( ._|_ ` S ) ) ) = ( ._|_ ` S ) )

Proof

Step Hyp Ref Expression
1 2polss.a 
 |-  A = ( Atoms ` K )
2 2polss.p 
 |-  ._|_ = ( _|_P ` K )
3 hlclat 
 |-  ( K e. HL -> K e. CLat )
4 eqid 
 |-  ( Base ` K ) = ( Base ` K )
5 4 1 atssbase 
 |-  A C_ ( Base ` K )
6 sstr 
 |-  ( ( S C_ A /\ A C_ ( Base ` K ) ) -> S C_ ( Base ` K ) )
7 5 6 mpan2 
 |-  ( S C_ A -> S C_ ( Base ` K ) )
8 eqid 
 |-  ( lub ` K ) = ( lub ` K )
9 4 8 clatlubcl 
 |-  ( ( K e. CLat /\ S C_ ( Base ` K ) ) -> ( ( lub ` K ) ` S ) e. ( Base ` K ) )
10 3 7 9 syl2an 
 |-  ( ( K e. HL /\ S C_ A ) -> ( ( lub ` K ) ` S ) e. ( Base ` K ) )
11 eqid 
 |-  ( oc ` K ) = ( oc ` K )
12 eqid 
 |-  ( pmap ` K ) = ( pmap ` K )
13 4 11 12 2 polpmapN 
 |-  ( ( K e. HL /\ ( ( lub ` K ) ` S ) e. ( Base ` K ) ) -> ( ._|_ ` ( ( pmap ` K ) ` ( ( lub ` K ) ` S ) ) ) = ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` S ) ) ) )
14 10 13 syldan 
 |-  ( ( K e. HL /\ S C_ A ) -> ( ._|_ ` ( ( pmap ` K ) ` ( ( lub ` K ) ` S ) ) ) = ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` S ) ) ) )
15 8 1 12 2 2polvalN 
 |-  ( ( K e. HL /\ S C_ A ) -> ( ._|_ ` ( ._|_ ` S ) ) = ( ( pmap ` K ) ` ( ( lub ` K ) ` S ) ) )
16 15 fveq2d 
 |-  ( ( K e. HL /\ S C_ A ) -> ( ._|_ ` ( ._|_ ` ( ._|_ ` S ) ) ) = ( ._|_ ` ( ( pmap ` K ) ` ( ( lub ` K ) ` S ) ) ) )
17 8 11 1 12 2 polval2N 
 |-  ( ( K e. HL /\ S C_ A ) -> ( ._|_ ` S ) = ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` S ) ) ) )
18 14 16 17 3eqtr4d 
 |-  ( ( K e. HL /\ S C_ A ) -> ( ._|_ ` ( ._|_ ` ( ._|_ ` S ) ) ) = ( ._|_ ` S ) )