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 ) )