Theorem mapdsord

Description: Strong ordering property of themap defined by df-mapd . (Contributed by NM, 13-Mar-2015)

	Ref	Expression
Hypotheses	mapdcnvcl.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	mapdcnvcl.m	$$⊢M=\mathrm{mapd}\left(K\right)\left(W\right)$$
	mapdcnvcl.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
	mapdcnvcl.s	$$⊢S=\mathrm{LSubSp}\left(U\right)$$
	mapdcnvcl.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
	mapdcl.x	$$⊢\phi \to X\in S$$
	mapdsord.x	$$⊢\phi \to Y\in S$$
Assertion	mapdsord	$$⊢\phi \to \left(M\left(X\right)\subset M\left(Y\right)\leftrightarrow X\subset Y\right)$$

Proof

Step	Hyp	Ref	Expression
1		mapdcnvcl.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
2		mapdcnvcl.m	$$⊢M=\mathrm{mapd}\left(K\right)\left(W\right)$$
3		mapdcnvcl.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
4		mapdcnvcl.s	$$⊢S=\mathrm{LSubSp}\left(U\right)$$
5		mapdcnvcl.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
6		mapdcl.x	$$⊢\phi \to X\in S$$
7		mapdsord.x	$$⊢\phi \to Y\in S$$
8	1 3 4 2 5 6 7	mapdord	$$⊢\phi \to \left(M\left(X\right)\subseteq M\left(Y\right)\leftrightarrow X\subseteq Y\right)$$
9	1 3 4 2 5 6 7	mapd11	$$⊢\phi \to \left(M\left(X\right)=M\left(Y\right)\leftrightarrow X=Y\right)$$
10	9	necon3bid	$$⊢\phi \to \left(M\left(X\right)\ne M\left(Y\right)\leftrightarrow X\ne Y\right)$$
11	8 10	anbi12d	$$⊢\phi \to \left(\left(M\left(X\right)\subseteq M\left(Y\right)\wedge M\left(X\right)\ne M\left(Y\right)\right)\leftrightarrow \left(X\subseteq Y\wedge X\ne Y\right)\right)$$
12		df-pss	$$⊢M\left(X\right)\subset M\left(Y\right)\leftrightarrow \left(M\left(X\right)\subseteq M\left(Y\right)\wedge M\left(X\right)\ne M\left(Y\right)\right)$$
13		df-pss	$$⊢X\subset Y\leftrightarrow \left(X\subseteq Y\wedge X\ne Y\right)$$
14	11 12 13	3bitr4g	$$⊢\phi \to \left(M\left(X\right)\subset M\left(Y\right)\leftrightarrow X\subset Y\right)$$