Theorem mapdval5N

Description: Value of projectivity from vector space H to dual space. (Contributed by NM, 31-Jan-2015) (New usage is discouraged.)

	Ref	Expression
Hypotheses	mapdval4.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	mapdval4.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	mapdval4.s	⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
	mapdval4.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
	mapdval4.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
	mapdval4.o	⊢ 𝑂 = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
	mapdval4.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	mapdval4.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	mapdval4.t	⊢ ( 𝜑 → 𝑇 ∈ 𝑆 )
Assertion	mapdval5N	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑇 ) = ∪ 𝑣 ∈ 𝑇 { 𝑓 ∈ 𝐹 ∣ ( 𝑂 ‘ { 𝑣 } ) = ( 𝐿 ‘ 𝑓 ) } )

Proof

Step	Hyp	Ref	Expression
1		mapdval4.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		mapdval4.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		mapdval4.s	⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
4		mapdval4.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
5		mapdval4.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
6		mapdval4.o	⊢ 𝑂 = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
7		mapdval4.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
8		mapdval4.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9		mapdval4.t	⊢ ( 𝜑 → 𝑇 ∈ 𝑆 )
10	1 2 3 4 5 6 7 8 9	mapdval4N	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑇 ) = { 𝑓 ∈ 𝐹 ∣ ∃ 𝑣 ∈ 𝑇 ( 𝑂 ‘ { 𝑣 } ) = ( 𝐿 ‘ 𝑓 ) } )
11		iunrab	⊢ ∪ 𝑣 ∈ 𝑇 { 𝑓 ∈ 𝐹 ∣ ( 𝑂 ‘ { 𝑣 } ) = ( 𝐿 ‘ 𝑓 ) } = { 𝑓 ∈ 𝐹 ∣ ∃ 𝑣 ∈ 𝑇 ( 𝑂 ‘ { 𝑣 } ) = ( 𝐿 ‘ 𝑓 ) }
12	10 11	eqtr4di	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑇 ) = ∪ 𝑣 ∈ 𝑇 { 𝑓 ∈ 𝐹 ∣ ( 𝑂 ‘ { 𝑣 } ) = ( 𝐿 ‘ 𝑓 ) } )