Metamath Proof Explorer


Theorem lcd0vcl

Description: Closure of the zero functional in the set of functionals with closed kernels. (Contributed by NM, 15-Mar-2015)

Ref Expression
Hypotheses lcdv0cl.h 𝐻 = ( LHyp ‘ 𝐾 )
lcdv0cl.c 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
lcdv0cl.v 𝑉 = ( Base ‘ 𝐶 )
lcdv0cl.o 𝑂 = ( 0g𝐶 )
lcdv0cl.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
Assertion lcd0vcl ( 𝜑𝑂𝑉 )

Proof

Step Hyp Ref Expression
1 lcdv0cl.h 𝐻 = ( LHyp ‘ 𝐾 )
2 lcdv0cl.c 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
3 lcdv0cl.v 𝑉 = ( Base ‘ 𝐶 )
4 lcdv0cl.o 𝑂 = ( 0g𝐶 )
5 lcdv0cl.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
6 1 2 5 lcdlmod ( 𝜑𝐶 ∈ LMod )
7 3 4 lmod0vcl ( 𝐶 ∈ LMod → 𝑂𝑉 )
8 6 7 syl ( 𝜑𝑂𝑉 )