Theorem mapdcnvid2

Description: Value of the converse of the map defined by df-mapd . (Contributed by NM, 13-Mar-2015)

	Ref	Expression
Hypotheses	mapdcnvid2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	mapdcnvid2.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	mapdcnvid2.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	mapdcnvid2.x	⊢ ( 𝜑 → 𝑋 ∈ ran 𝑀 )
Assertion	mapdcnvid2	⊢ ( 𝜑 → ( 𝑀 ‘ ( ◡ 𝑀 ‘ 𝑋 ) ) = 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		mapdcnvid2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		mapdcnvid2.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
3		mapdcnvid2.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
4		mapdcnvid2.x	⊢ ( 𝜑 → 𝑋 ∈ ran 𝑀 )
5		eqid	⊢ ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
6		eqid	⊢ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
7		eqid	⊢ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) )
8		eqid	⊢ ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) )
9		eqid	⊢ ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) )
10		eqid	⊢ ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) )
11		eqid	⊢ ( LSubSp ‘ ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) = ( LSubSp ‘ ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) )
12		eqid	⊢ { 𝑔 ∈ ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ‘ 𝑔 ) ) ) = ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ‘ 𝑔 ) } = { 𝑔 ∈ ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ‘ 𝑔 ) ) ) = ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ‘ 𝑔 ) }
13	1 5 2 6 7 8 9 10 11 12 3	mapd1o	⊢ ( 𝜑 → 𝑀 : ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) –1-1-onto→ ( ( LSubSp ‘ ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) ∩ 𝒫 { 𝑔 ∈ ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ‘ 𝑔 ) ) ) = ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ‘ 𝑔 ) } ) )
14		f1of1	⊢ ( 𝑀 : ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) –1-1-onto→ ( ( LSubSp ‘ ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) ∩ 𝒫 { 𝑔 ∈ ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ‘ 𝑔 ) ) ) = ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ‘ 𝑔 ) } ) → 𝑀 : ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) –1-1→ ( ( LSubSp ‘ ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) ∩ 𝒫 { 𝑔 ∈ ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ‘ 𝑔 ) ) ) = ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ‘ 𝑔 ) } ) )
15		f1f1orn	⊢ ( 𝑀 : ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) –1-1→ ( ( LSubSp ‘ ( LDual ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) ∩ 𝒫 { 𝑔 ∈ ( LFnl ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ∣ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ‘ 𝑔 ) ) ) = ( ( LKer ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ‘ 𝑔 ) } ) → 𝑀 : ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) –1-1-onto→ ran 𝑀 )
16	13 14 15	3syl	⊢ ( 𝜑 → 𝑀 : ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) –1-1-onto→ ran 𝑀 )
17		f1ocnvfv2	⊢ ( ( 𝑀 : ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) –1-1-onto→ ran 𝑀 ∧ 𝑋 ∈ ran 𝑀 ) → ( 𝑀 ‘ ( ◡ 𝑀 ‘ 𝑋 ) ) = 𝑋 )
18	16 4 17	syl2anc	⊢ ( 𝜑 → ( 𝑀 ‘ ( ◡ 𝑀 ‘ 𝑋 ) ) = 𝑋 )