Theorem mapdrval

Description: Given a dual subspace R (of functionals with closed kernels), reconstruct the subspace Q that maps to it. (Contributed by NM, 12-Mar-2015)

	Ref	Expression
Hypotheses	mapdrval.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	mapdrval.o	⊢ 𝑂 = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
	mapdrval.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	mapdrval.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	mapdrval.s	⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
	mapdrval.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
	mapdrval.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
	mapdrval.d	⊢ 𝐷 = ( LDual ‘ 𝑈 )
	mapdrval.t	⊢ 𝑇 = ( LSubSp ‘ 𝐷 )
	mapdrval.c	⊢ 𝐶 = { 𝑔 ∈ 𝐹 ∣ ( 𝑂 ‘ ( 𝑂 ‘ ( 𝐿 ‘ 𝑔 ) ) ) = ( 𝐿 ‘ 𝑔 ) }
	mapdrval.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	mapdrval.r	⊢ ( 𝜑 → 𝑅 ∈ 𝑇 )
	mapdrval.e	⊢ ( 𝜑 → 𝑅 ⊆ 𝐶 )
	mapdrval.q	⊢ 𝑄 = ∪ ℎ ∈ 𝑅 ( 𝑂 ‘ ( 𝐿 ‘ ℎ ) )
Assertion	mapdrval	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑄 ) = 𝑅 )

Proof

Step	Hyp	Ref	Expression
1		mapdrval.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		mapdrval.o	⊢ 𝑂 = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
3		mapdrval.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
4		mapdrval.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
5		mapdrval.s	⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
6		mapdrval.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
7		mapdrval.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
8		mapdrval.d	⊢ 𝐷 = ( LDual ‘ 𝑈 )
9		mapdrval.t	⊢ 𝑇 = ( LSubSp ‘ 𝐷 )
10		mapdrval.c	⊢ 𝐶 = { 𝑔 ∈ 𝐹 ∣ ( 𝑂 ‘ ( 𝑂 ‘ ( 𝐿 ‘ 𝑔 ) ) ) = ( 𝐿 ‘ 𝑔 ) }
11		mapdrval.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
12		mapdrval.r	⊢ ( 𝜑 → 𝑅 ∈ 𝑇 )
13		mapdrval.e	⊢ ( 𝜑 → 𝑅 ⊆ 𝐶 )
14		mapdrval.q	⊢ 𝑄 = ∪ ℎ ∈ 𝑅 ( 𝑂 ‘ ( 𝐿 ‘ ℎ ) )
15	1 2 4 5 6 7 8 9 10 14 11 12 13	lcfr	⊢ ( 𝜑 → 𝑄 ∈ 𝑆 )
16	1 4 5 6 7 2 3 11 15 10	mapdvalc	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑄 ) = { 𝑓 ∈ 𝐶 ∣ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑄 } )
17		2fveq3	⊢ ( ℎ = 𝑖 → ( 𝑂 ‘ ( 𝐿 ‘ ℎ ) ) = ( 𝑂 ‘ ( 𝐿 ‘ 𝑖 ) ) )
18	17	cbviunv	⊢ ∪ ℎ ∈ 𝑅 ( 𝑂 ‘ ( 𝐿 ‘ ℎ ) ) = ∪ 𝑖 ∈ 𝑅 ( 𝑂 ‘ ( 𝐿 ‘ 𝑖 ) )
19	14 18	eqtri	⊢ 𝑄 = ∪ 𝑖 ∈ 𝑅 ( 𝑂 ‘ ( 𝐿 ‘ 𝑖 ) )
20		eqid	⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 )
21		eqid	⊢ ( LSAtoms ‘ 𝑈 ) = ( LSAtoms ‘ 𝑈 )
22		eqid	⊢ ( LSpan ‘ 𝑈 ) = ( LSpan ‘ 𝑈 )
23		eqid	⊢ ( 0g ‘ 𝑈 ) = ( 0g ‘ 𝑈 )
24		eqid	⊢ ( 0g ‘ 𝐷 ) = ( 0g ‘ 𝐷 )
25	1 2 3 4 5 6 7 8 9 10 11 12 13 19 20 21 22 23 24	mapdrvallem3	⊢ ( 𝜑 → { 𝑓 ∈ 𝐶 ∣ ( 𝑂 ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑄 } = 𝑅 )
26	16 25	eqtrd	⊢ ( 𝜑 → ( 𝑀 ‘ 𝑄 ) = 𝑅 )