Theorem lcd0

Description: The zero scalar of the closed kernel dual of a vector space. (Contributed by NM, 20-Mar-2015)

	Ref	Expression
Hypotheses	lcd0.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	lcd0.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
	lcd0.f	$$⊢F=\mathrm{Scalar}\left(U\right)$$
	lcd0.z	$$⊢\underset{\dot{}}{0}=0_F$$
	lcd0.c	$$⊢C=\mathrm{LCDual}\left(K\right)\left(W\right)$$
	lcd0.s	$$⊢S=\mathrm{Scalar}\left(C\right)$$
	lcd0.o	$$⊢O=0_S$$
	lcd0.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
Assertion	lcd0	$$⊢\phi \to O=\underset{\dot{}}{0}$$

Proof

Step	Hyp	Ref	Expression
1		lcd0.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
2		lcd0.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
3		lcd0.f	$$⊢F=\mathrm{Scalar}\left(U\right)$$
4		lcd0.z	$$⊢\underset{\dot{}}{0}=0_F$$
5		lcd0.c	$$⊢C=\mathrm{LCDual}\left(K\right)\left(W\right)$$
6		lcd0.s	$$⊢S=\mathrm{Scalar}\left(C\right)$$
7		lcd0.o	$$⊢O=0_S$$
8		lcd0.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
9		eqid	$$⊢{opp}_r\left(F\right)={opp}_r\left(F\right)$$
10	1 2 3 9 5 6 8	lcdsca	$$⊢\phi \to S={opp}_r\left(F\right)$$
11	10	fveq2d	$$⊢\phi \to 0_S=0_{{opp}_r\left(F\right)}$$
12	9 4	oppr0	$$⊢\underset{\dot{}}{0}=0_{{opp}_r\left(F\right)}$$
13	11 7 12	3eqtr4g	$$⊢\phi \to O=\underset{\dot{}}{0}$$