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	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	lcdlmod.c	$$⊢C=\mathrm{LCDual}\left(K\right)\left(W\right)$$
	lcdlmod.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
Assertion	lcdlmod	$$⊢\phi \to C\in \mathrm{LMod}$$

Proof

Step	Hyp	Ref	Expression
1		lcdlmod.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
2		lcdlmod.c	$$⊢C=\mathrm{LCDual}\left(K\right)\left(W\right)$$
3		lcdlmod.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
4	1 2 3	lcdlvec	$$⊢\phi \to C\in \mathrm{LVec}$$
5		lveclmod	$$⊢C\in \mathrm{LVec}\to C\in \mathrm{LMod}$$
6	4 5	syl	$$⊢\phi \to C\in \mathrm{LMod}$$