Theorem mapdsord

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

	Ref	Expression
Hypotheses	mapdcnvcl.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	mapdcnvcl.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	mapdcnvcl.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	mapdcnvcl.s	⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
	mapdcnvcl.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	mapdcl.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑆 )
	mapdsord.x	⊢ ( 𝜑 → 𝑌 ∈ 𝑆 )
Assertion	mapdsord	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑋 ) ⊊ ( 𝑀 ‘ 𝑌 ) ↔ 𝑋 ⊊ 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		mapdcnvcl.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		mapdcnvcl.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
3		mapdcnvcl.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4		mapdcnvcl.s	⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
5		mapdcnvcl.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
6		mapdcl.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑆 )
7		mapdsord.x	⊢ ( 𝜑 → 𝑌 ∈ 𝑆 )
8	1 3 4 2 5 6 7	mapdord	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑋 ) ⊆ ( 𝑀 ‘ 𝑌 ) ↔ 𝑋 ⊆ 𝑌 ) )
9	1 3 4 2 5 6 7	mapd11	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑋 ) = ( 𝑀 ‘ 𝑌 ) ↔ 𝑋 = 𝑌 ) )
10	9	necon3bid	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑋 ) ≠ ( 𝑀 ‘ 𝑌 ) ↔ 𝑋 ≠ 𝑌 ) )
11	8 10	anbi12d	⊢ ( 𝜑 → ( ( ( 𝑀 ‘ 𝑋 ) ⊆ ( 𝑀 ‘ 𝑌 ) ∧ ( 𝑀 ‘ 𝑋 ) ≠ ( 𝑀 ‘ 𝑌 ) ) ↔ ( 𝑋 ⊆ 𝑌 ∧ 𝑋 ≠ 𝑌 ) ) )
12		df-pss	⊢ ( ( 𝑀 ‘ 𝑋 ) ⊊ ( 𝑀 ‘ 𝑌 ) ↔ ( ( 𝑀 ‘ 𝑋 ) ⊆ ( 𝑀 ‘ 𝑌 ) ∧ ( 𝑀 ‘ 𝑋 ) ≠ ( 𝑀 ‘ 𝑌 ) ) )
13		df-pss	⊢ ( 𝑋 ⊊ 𝑌 ↔ ( 𝑋 ⊆ 𝑌 ∧ 𝑋 ≠ 𝑌 ) )
14	11 12 13	3bitr4g	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑋 ) ⊊ ( 𝑀 ‘ 𝑌 ) ↔ 𝑋 ⊊ 𝑌 ) )