Theorem mapdval3N

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

	Ref	Expression
Hypotheses	mapdval2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	mapdval2.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	mapdval2.s	⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
	mapdval2.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	mapdval2.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
	mapdval2.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
	mapdval2.o	⊢ 𝑂 = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
	mapdval2.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	mapdval2.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	mapdval2.t	⊢ ( 𝜑 → 𝑇 ∈ 𝑆 )
	mapdval2.c	⊢ 𝐶 = { 𝑔 ∈ 𝐹 ∣ ( 𝑂 ‘ ( 𝑂 ‘ ( 𝐿 ‘ 𝑔 ) ) ) = ( 𝐿 ‘ 𝑔 ) }
Assertion	mapdval3N	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑇 ) = ∪ 𝑣 ∈ 𝑇 { 𝑓 ∈ 𝐶 ∣ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) = ( 𝑁 ‘ { 𝑣 } ) } )

Proof

Step	Hyp	Ref	Expression
1		mapdval2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		mapdval2.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		mapdval2.s	⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
4		mapdval2.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
5		mapdval2.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
6		mapdval2.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
7		mapdval2.o	⊢ 𝑂 = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
8		mapdval2.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
9		mapdval2.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
10		mapdval2.t	⊢ ( 𝜑 → 𝑇 ∈ 𝑆 )
11		mapdval2.c	⊢ 𝐶 = { 𝑔 ∈ 𝐹 ∣ ( 𝑂 ‘ ( 𝑂 ‘ ( 𝐿 ‘ 𝑔 ) ) ) = ( 𝐿 ‘ 𝑔 ) }
12	1 2 3 4 5 6 7 8 9 10 11	mapdval2N	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑇 ) = { 𝑓 ∈ 𝐶 ∣ ∃ 𝑣 ∈ 𝑇 ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) = ( 𝑁 ‘ { 𝑣 } ) } )
13		iunrab	⊢ ∪ 𝑣 ∈ 𝑇 { 𝑓 ∈ 𝐶 ∣ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) = ( 𝑁 ‘ { 𝑣 } ) } = { 𝑓 ∈ 𝐶 ∣ ∃ 𝑣 ∈ 𝑇 ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) = ( 𝑁 ‘ { 𝑣 } ) }
14	12 13	eqtr4di	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑇 ) = ∪ 𝑣 ∈ 𝑇 { 𝑓 ∈ 𝐶 ∣ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) = ( 𝑁 ‘ { 𝑣 } ) } )