Theorem mapdcnvcl

Description: Closure of the converse of the map 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 ) )
	mapdcnvcl.x	⊢ ( 𝜑 → 𝑋 ∈ ran 𝑀 )
Assertion	mapdcnvcl	⊢ ( 𝜑 → ( ◡ 𝑀 ‘ 𝑋 ) ∈ 𝑆 )

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		mapdcnvcl.x	⊢ ( 𝜑 → 𝑋 ∈ ran 𝑀 )
7		eqid	⊢ ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
8		eqid	⊢ ( LFnl ‘ 𝑈 ) = ( LFnl ‘ 𝑈 )
9		eqid	⊢ ( LKer ‘ 𝑈 ) = ( LKer ‘ 𝑈 )
10		eqid	⊢ ( LDual ‘ 𝑈 ) = ( LDual ‘ 𝑈 )
11		eqid	⊢ ( LSubSp ‘ ( LDual ‘ 𝑈 ) ) = ( LSubSp ‘ ( LDual ‘ 𝑈 ) )
12		eqid	⊢ { 𝑔 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LKer ‘ 𝑈 ) ‘ 𝑔 ) ) ) = ( ( LKer ‘ 𝑈 ) ‘ 𝑔 ) } = { 𝑔 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LKer ‘ 𝑈 ) ‘ 𝑔 ) ) ) = ( ( LKer ‘ 𝑈 ) ‘ 𝑔 ) }
13	1 7 2 3 4 8 9 10 11 12 5	mapd1o	⊢ ( 𝜑 → 𝑀 : 𝑆 –1-1-onto→ ( ( LSubSp ‘ ( LDual ‘ 𝑈 ) ) ∩ 𝒫 { 𝑔 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LKer ‘ 𝑈 ) ‘ 𝑔 ) ) ) = ( ( LKer ‘ 𝑈 ) ‘ 𝑔 ) } ) )
14		f1of1	⊢ ( 𝑀 : 𝑆 –1-1-onto→ ( ( LSubSp ‘ ( LDual ‘ 𝑈 ) ) ∩ 𝒫 { 𝑔 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LKer ‘ 𝑈 ) ‘ 𝑔 ) ) ) = ( ( LKer ‘ 𝑈 ) ‘ 𝑔 ) } ) → 𝑀 : 𝑆 –1-1→ ( ( LSubSp ‘ ( LDual ‘ 𝑈 ) ) ∩ 𝒫 { 𝑔 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LKer ‘ 𝑈 ) ‘ 𝑔 ) ) ) = ( ( LKer ‘ 𝑈 ) ‘ 𝑔 ) } ) )
15		f1f1orn	⊢ ( 𝑀 : 𝑆 –1-1→ ( ( LSubSp ‘ ( LDual ‘ 𝑈 ) ) ∩ 𝒫 { 𝑔 ∈ ( LFnl ‘ 𝑈 ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LKer ‘ 𝑈 ) ‘ 𝑔 ) ) ) = ( ( LKer ‘ 𝑈 ) ‘ 𝑔 ) } ) → 𝑀 : 𝑆 –1-1-onto→ ran 𝑀 )
16	13 14 15	3syl	⊢ ( 𝜑 → 𝑀 : 𝑆 –1-1-onto→ ran 𝑀 )
17		f1ocnvdm	⊢ ( ( 𝑀 : 𝑆 –1-1-onto→ ran 𝑀 ∧ 𝑋 ∈ ran 𝑀 ) → ( ◡ 𝑀 ‘ 𝑋 ) ∈ 𝑆 )
18	16 6 17	syl2anc	⊢ ( 𝜑 → ( ◡ 𝑀 ‘ 𝑋 ) ∈ 𝑆 )