Theorem mapdord

Description: Ordering property of the map defined by df-mapd . Property (b) of Baer p. 40. (Contributed by NM, 27-Jan-2015)

	Ref	Expression
Hypotheses	mapdord.h	|- H = ( LHyp ` K )
	mapdord.u	|- U = ( ( DVecH ` K ) ` W )
	mapdord.s	|- S = ( LSubSp ` U )
	mapdord.m	|- M = ( ( mapd ` K ) ` W )
	mapdord.k	|- ( ph -> ( K e. HL /\ W e. H ) )
	mapdord.x	|- ( ph -> X e. S )
	mapdord.y	|- ( ph -> Y e. S )
Assertion	mapdord	|- ( ph -> ( ( M ` X ) C_ ( M ` Y ) <-> X C_ Y ) )

Proof

Step	Hyp	Ref	Expression
1		mapdord.h	|- H = ( LHyp ` K )
2		mapdord.u	|- U = ( ( DVecH ` K ) ` W )
3		mapdord.s	|- S = ( LSubSp ` U )
4		mapdord.m	|- M = ( ( mapd ` K ) ` W )
5		mapdord.k	|- ( ph -> ( K e. HL /\ W e. H ) )
6		mapdord.x	|- ( ph -> X e. S )
7		mapdord.y	|- ( ph -> Y e. S )
8		eqid	|- ( ( ocH ` K ) ` W ) = ( ( ocH ` K ) ` W )
9		eqid	|- ( LSAtoms ` U ) = ( LSAtoms ` U )
10		eqid	|- ( LFnl ` U ) = ( LFnl ` U )
11		eqid	|- ( LSHyp ` U ) = ( LSHyp ` U )
12		eqid	|- ( LKer ` U ) = ( LKer ` U )
13		eqid	|- { g e. ( LFnl ` U ) | ( ( ( ocH ` K ) ` W ) ` ( ( ( ocH ` K ) ` W ) ` ( ( LKer ` U ) ` g ) ) ) e. ( LSHyp ` U ) } = { g e. ( LFnl ` U ) | ( ( ( ocH ` K ) ` W ) ` ( ( ( ocH ` K ) ` W ) ` ( ( LKer ` U ) ` g ) ) ) e. ( LSHyp ` U ) }
14		eqid	|- { g e. ( LFnl ` U ) | ( ( ( ocH ` K ) ` W ) ` ( ( ( ocH ` K ) ` W ) ` ( ( LKer ` U ) ` g ) ) ) = ( ( LKer ` U ) ` g ) } = { g e. ( LFnl ` U ) | ( ( ( ocH ` K ) ` W ) ` ( ( ( ocH ` K ) ` W ) ` ( ( LKer ` U ) ` g ) ) ) = ( ( LKer ` U ) ` g ) }
15	1 2 3 4 5 6 7 8 9 10 11 12 13 14	mapdordlem2	|- ( ph -> ( ( M ` X ) C_ ( M ` Y ) <-> X C_ Y ) )