Theorem lcdlmod

Description: The dual vector space of functionals with closed kernels is a left module. (Contributed by NM, 13-Mar-2015)

	Ref	Expression
Hypotheses	lcdlmod.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	lcdlmod.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	lcdlmod.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
Assertion	lcdlmod	⊢ ( 𝜑 → 𝐶 ∈ LMod )

Proof

Step	Hyp	Ref	Expression
1		lcdlmod.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		lcdlmod.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
3		lcdlmod.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
4	1 2 3	lcdlvec	⊢ ( 𝜑 → 𝐶 ∈ LVec )
5		lveclmod	⊢ ( 𝐶 ∈ LVec → 𝐶 ∈ LMod )
6	4 5	syl	⊢ ( 𝜑 → 𝐶 ∈ LMod )