Theorem mapdvalc

Description: Value of projectivity from vector space H to dual space. (Contributed by NM, 27-Jan-2015)

	Ref	Expression
Hypotheses	mapdval.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	mapdval.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	mapdval.s	⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
	mapdval.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
	mapdval.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
	mapdval.o	⊢ 𝑂 = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
	mapdval.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	mapdval.k	⊢ ( 𝜑 → ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) )
	mapdval.t	⊢ ( 𝜑 → 𝑇 ∈ 𝑆 )
	mapdvalc.c	⊢ 𝐶 = { 𝑔 ∈ 𝐹 ∣ ( 𝑂 ‘ ( 𝑂 ‘ ( 𝐿 ‘ 𝑔 ) ) ) = ( 𝐿 ‘ 𝑔 ) }
Assertion	mapdvalc	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑇 ) = { 𝑓 ∈ 𝐶 ∣ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑇 } )

Proof

Step	Hyp	Ref	Expression
1		mapdval.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		mapdval.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		mapdval.s	⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
4		mapdval.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
5		mapdval.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
6		mapdval.o	⊢ 𝑂 = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
7		mapdval.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
8		mapdval.k	⊢ ( 𝜑 → ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) )
9		mapdval.t	⊢ ( 𝜑 → 𝑇 ∈ 𝑆 )
10		mapdvalc.c	⊢ 𝐶 = { 𝑔 ∈ 𝐹 ∣ ( 𝑂 ‘ ( 𝑂 ‘ ( 𝐿 ‘ 𝑔 ) ) ) = ( 𝐿 ‘ 𝑔 ) }
11	1 2 3 4 5 6 7 8 9	mapdval	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑇 ) = { 𝑓 ∈ 𝐹 ∣ ( ( 𝑂 ‘ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) ∧ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑇 ) } )
12		anass	⊢ ( ( ( 𝑓 ∈ 𝐹 ∧ ( 𝑂 ‘ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) ) ∧ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑇 ) ↔ ( 𝑓 ∈ 𝐹 ∧ ( ( 𝑂 ‘ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) ∧ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑇 ) ) )
13	10	lcfl1lem	⊢ ( 𝑓 ∈ 𝐶 ↔ ( 𝑓 ∈ 𝐹 ∧ ( 𝑂 ‘ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) ) )
14	13	anbi1i	⊢ ( ( 𝑓 ∈ 𝐶 ∧ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑇 ) ↔ ( ( 𝑓 ∈ 𝐹 ∧ ( 𝑂 ‘ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) ) ∧ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑇 ) )
15	14	bicomi	⊢ ( ( ( 𝑓 ∈ 𝐹 ∧ ( 𝑂 ‘ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) ) ∧ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑇 ) ↔ ( 𝑓 ∈ 𝐶 ∧ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑇 ) )
16	15	a1i	⊢ ( 𝜑 → ( ( ( 𝑓 ∈ 𝐹 ∧ ( 𝑂 ‘ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) ) ∧ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑇 ) ↔ ( 𝑓 ∈ 𝐶 ∧ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑇 ) ) )
17	12 16	bitr3id	⊢ ( 𝜑 → ( ( 𝑓 ∈ 𝐹 ∧ ( ( 𝑂 ‘ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) ∧ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑇 ) ) ↔ ( 𝑓 ∈ 𝐶 ∧ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑇 ) ) )
18	17	rabbidva2	⊢ ( 𝜑 → { 𝑓 ∈ 𝐹 ∣ ( ( 𝑂 ‘ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) ∧ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑇 ) } = { 𝑓 ∈ 𝐶 ∣ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑇 } )
19	11 18	eqtrd	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑇 ) = { 𝑓 ∈ 𝐶 ∣ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑇 } )