Theorem lclkr

Description: The set of functionals with closed kernels is a subspace. Part of proof of Theorem 3.6 of Holland95 p. 218, line 20, stating "The f_M that arise this way generate a subspace F of E'". Our proof was suggested by Mario Carneiro, 5-Jan-2015. (Contributed by NM, 18-Jan-2015)

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

Proof

Step	Hyp	Ref	Expression
1		lclkr.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		lclkr.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		lclkr.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
4		lclkr.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
5		lclkr.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
6		lclkr.d	⊢ 𝐷 = ( LDual ‘ 𝑈 )
7		lclkr.s	⊢ 𝑆 = ( LSubSp ‘ 𝐷 )
8		lclkr.c	⊢ 𝐶 = { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) }
9		lclkr.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
10		ssrab2	⊢ { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } ⊆ 𝐹
11	10	a1i	⊢ ( 𝜑 → { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } ⊆ 𝐹 )
12	8	a1i	⊢ ( 𝜑 → 𝐶 = { 𝑓 ∈ 𝐹 ∣ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ 𝑓 ) ) ) = ( 𝐿 ‘ 𝑓 ) } )
13		eqid	⊢ ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 )
14	1 2 9	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
15	4 6 13 14	ldualvbase	⊢ ( 𝜑 → ( Base ‘ 𝐷 ) = 𝐹 )
16	11 12 15	3sstr4d	⊢ ( 𝜑 → 𝐶 ⊆ ( Base ‘ 𝐷 ) )
17		eqid	⊢ ( Scalar ‘ 𝑈 ) = ( Scalar ‘ 𝑈 )
18		eqid	⊢ ( 0g ‘ ( Scalar ‘ 𝑈 ) ) = ( 0g ‘ ( Scalar ‘ 𝑈 ) )
19		eqid	⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 )
20	17 18 19 4	lfl0f	⊢ ( 𝑈 ∈ LMod → ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∈ 𝐹 )
21	14 20	syl	⊢ ( 𝜑 → ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∈ 𝐹 )
22	1 2 3 19 9	dochoc1	⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ ( Base ‘ 𝑈 ) ) ) = ( Base ‘ 𝑈 ) )
23		eqid	⊢ ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) = ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } )
24	17 18 19 4 5	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 ) ) } ) ) )
25	14 20 24	syl2anc2	⊢ ( 𝜑 → ( ( 𝐿 ‘ ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) = ( Base ‘ 𝑈 ) ↔ ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) = ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) )
26	23 25	mpbiri	⊢ ( 𝜑 → ( 𝐿 ‘ ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) = ( Base ‘ 𝑈 ) )
27	26	fveq2d	⊢ ( 𝜑 → ( ⊥ ‘ ( 𝐿 ‘ ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ) = ( ⊥ ‘ ( Base ‘ 𝑈 ) ) )
28	27	fveq2d	⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ) ) = ( ⊥ ‘ ( ⊥ ‘ ( Base ‘ 𝑈 ) ) ) )
29	22 28 26	3eqtr4d	⊢ ( 𝜑 → ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ) ) = ( 𝐿 ‘ ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) )
30	8	lcfl1lem	⊢ ( ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∈ 𝐶 ↔ ( ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∈ 𝐹 ∧ ( ⊥ ‘ ( ⊥ ‘ ( 𝐿 ‘ ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ) ) = ( 𝐿 ‘ ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ) )
31	21 29 30	sylanbrc	⊢ ( 𝜑 → ( ( Base ‘ 𝑈 ) × { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∈ 𝐶 )
32	31	ne0d	⊢ ( 𝜑 → 𝐶 ≠ ∅ )
33		eqid	⊢ ( +g ‘ 𝐷 ) = ( +g ‘ 𝐷 )
34	9	adantr	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
35		eqid	⊢ ( Base ‘ ( Scalar ‘ 𝑈 ) ) = ( Base ‘ ( Scalar ‘ 𝑈 ) )
36		eqid	⊢ ( ·𝑠 ‘ 𝐷 ) = ( ·𝑠 ‘ 𝐷 )
37		simpr1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) → 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) )
38		eqid	⊢ ( Scalar ‘ 𝐷 ) = ( Scalar ‘ 𝐷 )
39		eqid	⊢ ( Base ‘ ( Scalar ‘ 𝐷 ) ) = ( Base ‘ ( Scalar ‘ 𝐷 ) )
40	17 35 6 38 39 14	ldualsbase	⊢ ( 𝜑 → ( Base ‘ ( Scalar ‘ 𝐷 ) ) = ( Base ‘ ( Scalar ‘ 𝑈 ) ) )
41	40	adantr	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) → ( Base ‘ ( Scalar ‘ 𝐷 ) ) = ( Base ‘ ( Scalar ‘ 𝑈 ) ) )
42	37 41	eleqtrd	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) → 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) )
43		simpr2	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) → 𝑎 ∈ 𝐶 )
44	1 3 2 4 5 6 17 35 36 8 34 42 43	lclkrlem1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐷 ) 𝑎 ) ∈ 𝐶 )
45		simpr3	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) → 𝑏 ∈ 𝐶 )
46	1 3 2 4 5 6 33 8 34 44 45	lclkrlem2	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶 ) ) → ( ( 𝑥 ( ·𝑠 ‘ 𝐷 ) 𝑎 ) ( +g ‘ 𝐷 ) 𝑏 ) ∈ 𝐶 )
47	46	ralrimivvva	⊢ ( 𝜑 → ∀ 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) ∀ 𝑎 ∈ 𝐶 ∀ 𝑏 ∈ 𝐶 ( ( 𝑥 ( ·𝑠 ‘ 𝐷 ) 𝑎 ) ( +g ‘ 𝐷 ) 𝑏 ) ∈ 𝐶 )
48	38 39 13 33 36 7	islss	⊢ ( 𝐶 ∈ 𝑆 ↔ ( 𝐶 ⊆ ( Base ‘ 𝐷 ) ∧ 𝐶 ≠ ∅ ∧ ∀ 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐷 ) ) ∀ 𝑎 ∈ 𝐶 ∀ 𝑏 ∈ 𝐶 ( ( 𝑥 ( ·𝑠 ‘ 𝐷 ) 𝑎 ) ( +g ‘ 𝐷 ) 𝑏 ) ∈ 𝐶 ) )
49	16 32 47 48	syl3anbrc	⊢ ( 𝜑 → 𝐶 ∈ 𝑆 )