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Metamath Proof Explorer

Theorem polcon3N

Description: Contraposition law for polarity. Remark in Holland95 p. 223. (Contributed by NM, 23-Mar-2012) (New usage is discouraged.)

Ref Expression
Hypotheses 2polss.a 
|- A = ( Atoms ` K )
2polss.p 
|- ._|_ = ( _|_P ` K )
Assertion polcon3N 
|- ( ( K e. HL /\ Y C_ A /\ X C_ Y ) -> ( ._|_ ` Y ) C_ ( ._|_ ` X ) )

Proof

Step Hyp Ref Expression
1 2polss.a 
 |-  A = ( Atoms ` K )
2 2polss.p 
 |-  ._|_ = ( _|_P ` K )
3 simp3 
 |-  ( ( K e. HL /\ Y C_ A /\ X C_ Y ) -> X C_ Y )
4 iinss1 
 |-  ( X C_ Y -> |^|_ p e. Y ( ( pmap ` K ) ` ( ( oc ` K ) ` p ) ) C_ |^|_ p e. X ( ( pmap ` K ) ` ( ( oc ` K ) ` p ) ) )
5 sslin 
 |-  ( |^|_ p e. Y ( ( pmap ` K ) ` ( ( oc ` K ) ` p ) ) C_ |^|_ p e. X ( ( pmap ` K ) ` ( ( oc ` K ) ` p ) ) -> ( A i^i |^|_ p e. Y ( ( pmap ` K ) ` ( ( oc ` K ) ` p ) ) ) C_ ( A i^i |^|_ p e. X ( ( pmap ` K ) ` ( ( oc ` K ) ` p ) ) ) )
6 3 4 5 3syl 
 |-  ( ( K e. HL /\ Y C_ A /\ X C_ Y ) -> ( A i^i |^|_ p e. Y ( ( pmap ` K ) ` ( ( oc ` K ) ` p ) ) ) C_ ( A i^i |^|_ p e. X ( ( pmap ` K ) ` ( ( oc ` K ) ` p ) ) ) )
7 eqid 
 |-  ( oc ` K ) = ( oc ` K )
8 eqid 
 |-  ( pmap ` K ) = ( pmap ` K )
9 7 1 8 2 polvalN 
 |-  ( ( K e. HL /\ Y C_ A ) -> ( ._|_ ` Y ) = ( A i^i |^|_ p e. Y ( ( pmap ` K ) ` ( ( oc ` K ) ` p ) ) ) )
10 9 3adant3 
 |-  ( ( K e. HL /\ Y C_ A /\ X C_ Y ) -> ( ._|_ ` Y ) = ( A i^i |^|_ p e. Y ( ( pmap ` K ) ` ( ( oc ` K ) ` p ) ) ) )
11 simp1 
 |-  ( ( K e. HL /\ Y C_ A /\ X C_ Y ) -> K e. HL )
12 simp2 
 |-  ( ( K e. HL /\ Y C_ A /\ X C_ Y ) -> Y C_ A )
13 3 12 sstrd 
 |-  ( ( K e. HL /\ Y C_ A /\ X C_ Y ) -> X C_ A )
14 7 1 8 2 polvalN 
 |-  ( ( K e. HL /\ X C_ A ) -> ( ._|_ ` X ) = ( A i^i |^|_ p e. X ( ( pmap ` K ) ` ( ( oc ` K ) ` p ) ) ) )
15 11 13 14 syl2anc 
 |-  ( ( K e. HL /\ Y C_ A /\ X C_ Y ) -> ( ._|_ ` X ) = ( A i^i |^|_ p e. X ( ( pmap ` K ) ` ( ( oc ` K ) ` p ) ) ) )
16 6 10 15 3sstr4d 
 |-  ( ( K e. HL /\ Y C_ A /\ X C_ Y ) -> ( ._|_ ` Y ) C_ ( ._|_ ` X ) )