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Metamath Proof Explorer

Theorem 2polvalN

Description: Value of double polarity. (Contributed by NM, 25-Jan-2012) (New usage is discouraged.)

Ref Expression
Hypotheses 2polval.u 
|- U = ( lub ` K )
2polval.a 
|- A = ( Atoms ` K )
2polval.m 
|- M = ( pmap ` K )
2polval.p 
|- ._|_ = ( _|_P ` K )
Assertion 2polvalN 
|- ( ( K e. HL /\ X C_ A ) -> ( ._|_ ` ( ._|_ ` X ) ) = ( M ` ( U ` X ) ) )

Proof

Step Hyp Ref Expression
1 2polval.u 
 |-  U = ( lub ` K )
2 2polval.a 
 |-  A = ( Atoms ` K )
3 2polval.m 
 |-  M = ( pmap ` K )
4 2polval.p 
 |-  ._|_ = ( _|_P ` K )
5 eqid 
 |-  ( oc ` K ) = ( oc ` K )
6 1 5 2 3 4 polval2N 
 |-  ( ( K e. HL /\ X C_ A ) -> ( ._|_ ` X ) = ( M ` ( ( oc ` K ) ` ( U ` X ) ) ) )
7 6 fveq2d 
 |-  ( ( K e. HL /\ X C_ A ) -> ( ._|_ ` ( ._|_ ` X ) ) = ( ._|_ ` ( M ` ( ( oc ` K ) ` ( U ` X ) ) ) ) )
8 hlop 
 |-  ( K e. HL -> K e. OP )
9 8 adantr 
 |-  ( ( K e. HL /\ X C_ A ) -> K e. OP )
10 hlclat 
 |-  ( K e. HL -> K e. CLat )
11 eqid 
 |-  ( Base ` K ) = ( Base ` K )
12 11 2 atssbase 
 |-  A C_ ( Base ` K )
13 sstr 
 |-  ( ( X C_ A /\ A C_ ( Base ` K ) ) -> X C_ ( Base ` K ) )
14 12 13 mpan2 
 |-  ( X C_ A -> X C_ ( Base ` K ) )
15 11 1 clatlubcl 
 |-  ( ( K e. CLat /\ X C_ ( Base ` K ) ) -> ( U ` X ) e. ( Base ` K ) )
16 10 14 15 syl2an 
 |-  ( ( K e. HL /\ X C_ A ) -> ( U ` X ) e. ( Base ` K ) )
17 11 5 opoccl 
 |-  ( ( K e. OP /\ ( U ` X ) e. ( Base ` K ) ) -> ( ( oc ` K ) ` ( U ` X ) ) e. ( Base ` K ) )
18 9 16 17 syl2anc 
 |-  ( ( K e. HL /\ X C_ A ) -> ( ( oc ` K ) ` ( U ` X ) ) e. ( Base ` K ) )
19 11 5 3 4 polpmapN 
 |-  ( ( K e. HL /\ ( ( oc ` K ) ` ( U ` X ) ) e. ( Base ` K ) ) -> ( ._|_ ` ( M ` ( ( oc ` K ) ` ( U ` X ) ) ) ) = ( M ` ( ( oc ` K ) ` ( ( oc ` K ) ` ( U ` X ) ) ) ) )
20 18 19 syldan 
 |-  ( ( K e. HL /\ X C_ A ) -> ( ._|_ ` ( M ` ( ( oc ` K ) ` ( U ` X ) ) ) ) = ( M ` ( ( oc ` K ) ` ( ( oc ` K ) ` ( U ` X ) ) ) ) )
21 11 5 opococ 
 |-  ( ( K e. OP /\ ( U ` X ) e. ( Base ` K ) ) -> ( ( oc ` K ) ` ( ( oc ` K ) ` ( U ` X ) ) ) = ( U ` X ) )
22 9 16 21 syl2anc 
 |-  ( ( K e. HL /\ X C_ A ) -> ( ( oc ` K ) ` ( ( oc ` K ) ` ( U ` X ) ) ) = ( U ` X ) )
23 22 fveq2d 
 |-  ( ( K e. HL /\ X C_ A ) -> ( M ` ( ( oc ` K ) ` ( ( oc ` K ) ` ( U ` X ) ) ) ) = ( M ` ( U ` X ) ) )
24 7 20 23 3eqtrd 
 |-  ( ( K e. HL /\ X C_ A ) -> ( ._|_ ` ( ._|_ ` X ) ) = ( M ` ( U ` X ) ) )