Theorem lclkrs

Description: The set of functionals having closed kernels and majorizing the orthocomplement of a given subspace R is a subspace of the dual space. TODO: This proof repeats large parts of the lclkr proof. Do we achieve overall shortening by breaking them out as subtheorems? Or make lclkr a special case of this? (Contributed by NM, 29-Jan-2015)

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

Proof

Step	Hyp	Ref	Expression
1		lclkrs.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		lclkrs.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
3		lclkrs.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4		lclkrs.s	⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
5		lclkrs.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
6		lclkrs.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
7		lclkrs.d	⊢ 𝐷 = ( LDual ‘ 𝑈 )
8		lclkrs.t	⊢ 𝑇 = ( LSubSp ‘ 𝐷 )
9		lclkrs.c	⊢ 𝐶 = { 𝑓 ∈ 𝐹 ∣ ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑅 ) }
10		lclkrs.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
11		lclkrs.r	⊢ ( 𝜑 → 𝑅 ∈ 𝑆 )
12		ssrab2	⊢ { 𝑓 ∈ 𝐹 ∣ ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑅 ) } ⊆ 𝐹
13	12	a1i	⊢ ( 𝜑 → { 𝑓 ∈ 𝐹 ∣ ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑅 ) } ⊆ 𝐹 )
14	9	a1i	⊢ ( 𝜑 → 𝐶 = { 𝑓 ∈ 𝐹 ∣ ( ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) ∧ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ⊆ 𝑅 ) } )
15		eqid	⊢ ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 )
16	1 3 10	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
17	5 7 15 16	ldualvbase	⊢ ( 𝜑 → ( Base ‘ 𝐷 ) = 𝐹 )
18	13 14 17	3sstr4d	⊢ ( 𝜑 → 𝐶 ⊆ ( Base ‘ 𝐷 ) )
19		eqid	⊢ ( Scalar ‘ 𝑈 ) = ( Scalar ‘ 𝑈 )
20		eqid	⊢ ( 0g ‘ ( Scalar ‘ 𝑈 ) ) = ( 0g ‘ ( Scalar ‘ 𝑈 ) )
21		eqid	⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 )
22	19 20 21 5	lfl0f	⊢ ( 𝑈 ∈ LMod → ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∈ 𝐹 )
23	16 22	syl	⊢ ( 𝜑 → ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∈ 𝐹 )
24	1 3 2 21 10	dochoc1	⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ ( Base ‘ 𝑈 ) ) ) = ( Base ‘ 𝑈 ) )
25		eqidd	⊢ ( 𝜑 → ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) = ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) )
26	19 20 21 5 6	lkr0f	⊢ ( ( 𝑈 ∈ LMod ∧ ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∈ 𝐹 ) → ( ( 𝐿 ‘ ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) = ( Base ‘ 𝑈 ) ↔ ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) = ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) )
27	16 23 26	syl2anc	⊢ ( 𝜑 → ( ( 𝐿 ‘ ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) = ( Base ‘ 𝑈 ) ↔ ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) = ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) )
28	25 27	mpbird	⊢ ( 𝜑 → ( 𝐿 ‘ ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) = ( Base ‘ 𝑈 ) )
29	28	fveq2d	⊢ ( 𝜑 → ( ⊥ ‘ ( 𝐿 ‘ ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ) = ( ⊥ ‘ ( Base ‘ 𝑈 ) ) )
30	29	fveq2d	⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ) ) = ( ⊥ ‘ ( ⊥ ‘ ( Base ‘ 𝑈 ) ) ) )
31	24 30 28	3eqtr4d	⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ) ) = ( 𝐿 ‘ ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) )
32		eqid	⊢ ( 0g ‘ 𝑈 ) = ( 0g ‘ 𝑈 )
33	1 3 2 21 32	doch1	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ⊥ ‘ ( Base ‘ 𝑈 ) ) = { ( 0g ‘ 𝑈 ) } )
34	10 33	syl	⊢ ( 𝜑 → ( ⊥ ‘ ( Base ‘ 𝑈 ) ) = { ( 0g ‘ 𝑈 ) } )
35	29 34	eqtrd	⊢ ( 𝜑 → ( ⊥ ‘ ( 𝐿 ‘ ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ) = { ( 0g ‘ 𝑈 ) } )
36	32 4	lss0ss	⊢ ( ( 𝑈 ∈ LMod ∧ 𝑅 ∈ 𝑆 ) → { ( 0g ‘ 𝑈 ) } ⊆ 𝑅 )
37	16 11 36	syl2anc	⊢ ( 𝜑 → { ( 0g ‘ 𝑈 ) } ⊆ 𝑅 )
38	35 37	eqsstrd	⊢ ( 𝜑 → ( ⊥ ‘ ( 𝐿 ‘ ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ) ⊆ 𝑅 )
39	9	lcfls1lem	⊢ ( ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∈ 𝐶 ↔ ( ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∈ 𝐹 ∧ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ) ) = ( 𝐿 ‘ ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ∧ ( ⊥ ‘ ( 𝐿 ‘ ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ) ⊆ 𝑅 ) )
40	23 31 38 39	syl3anbrc	⊢ ( 𝜑 → ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∈ 𝐶 )
41	40	ne0d	⊢ ( 𝜑 → 𝐶 ≠ ∅ )
42		eqid	⊢ ( Base ‘ ( Scalar ‘ 𝑈 ) ) = ( Base ‘ ( Scalar ‘ 𝑈 ) )
43		eqid	⊢ ( ·𝑠 ‘ 𝐷 ) = ( ·𝑠 ‘ 𝐷 )
44	10	adantr	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
45	11	adantr	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) → 𝑅 ∈ 𝑆 )
46		simpr3	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) → 𝑏 ∈ 𝐶 )
47		eqid	⊢ ( +g ‘ 𝐷 ) = ( +g ‘ 𝐷 )
48		simpr2	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) → 𝑎 ∈ 𝐶 )
49		simpr1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) → 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) )
50		eqid	⊢ ( Scalar ‘ 𝐷 ) = ( Scalar ‘ 𝐷 )
51		eqid	⊢ ( Base ‘ ( Scalar ‘ 𝐷 ) ) = ( Base ‘ ( Scalar ‘ 𝐷 ) )
52	19 42 7 50 51 16	ldualsbase	⊢ ( 𝜑 → ( Base ‘ ( Scalar ‘ 𝐷 ) ) = ( Base ‘ ( Scalar ‘ 𝑈 ) ) )
53	52	adantr	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) → ( Base ‘ ( Scalar ‘ 𝐷 ) ) = ( Base ‘ ( Scalar ‘ 𝑈 ) ) )
54	49 53	eleqtrd	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) → 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) )
55	1 2 3 4 5 6 7 19 42 43 9 44 45 48 54	lclkrslem1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐷 ) 𝑎 ) ∈ 𝐶 )
56	1 2 3 4 5 6 7 19 42 43 9 44 45 46 47 55	lclkrslem2	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) → ( ( 𝑥 ( ·𝑠 ‘ 𝐷 ) 𝑎 ) ( +g ‘ 𝐷 ) 𝑏 ) ∈ 𝐶 )
57	56	ralrimivvva	⊢ ( 𝜑 → ∀ 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) ∀ 𝑎 ∈ 𝐶 ∀ 𝑏 ∈ 𝐶 ( ( 𝑥 ( ·𝑠 ‘ 𝐷 ) 𝑎 ) ( +g ‘ 𝐷 ) 𝑏 ) ∈ 𝐶 )
58	50 51 15 47 43 8	islss	⊢ ( 𝐶 ∈ 𝑇 ↔ ( 𝐶 ⊆ ( Base ‘ 𝐷 ) ∧ 𝐶 ≠ ∅ ∧ ∀ 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) ∀ 𝑎 ∈ 𝐶 ∀ 𝑏 ∈ 𝐶 ( ( 𝑥 ( ·𝑠 ‘ 𝐷 ) 𝑎 ) ( +g ‘ 𝐷 ) 𝑏 ) ∈ 𝐶 ) )
59	18 41 57 58	syl3anbrc	⊢ ( 𝜑 → 𝐶 ∈ 𝑇 )