Metamath Proof Explorer

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 )