Theorem lcd0vvalN

Description: Value of the zero functional at any vector. (Contributed by NM, 28-Mar-2015) (New usage is discouraged.)

	Ref	Expression
Hypotheses	lcd0vval.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	lcd0vval.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
	lcd0vval.v	$$⊢V={\mathrm{Base}}_U$$
	lcd0vval.s	$$⊢S=\mathrm{Scalar}\left(U\right)$$
	lcd0vval.z	$$⊢\underset{\dot{}}{0}=0_S$$
	lcd0vval.c	$$⊢C=\mathrm{LCDual}\left(K\right)\left(W\right)$$
	lcd0vval.o	$$⊢O=0_C$$
	lcd0vval.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
	lcd0vval.x	$$⊢\phi \to X\in V$$
Assertion	lcd0vvalN	$$⊢\phi \to O\left(X\right)=\underset{\dot{}}{0}$$

Proof

Step	Hyp	Ref	Expression
1		lcd0vval.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
2		lcd0vval.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
3		lcd0vval.v	$$⊢V={\mathrm{Base}}_U$$
4		lcd0vval.s	$$⊢S=\mathrm{Scalar}\left(U\right)$$
5		lcd0vval.z	$$⊢\underset{\dot{}}{0}=0_S$$
6		lcd0vval.c	$$⊢C=\mathrm{LCDual}\left(K\right)\left(W\right)$$
7		lcd0vval.o	$$⊢O=0_C$$
8		lcd0vval.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
9		lcd0vval.x	$$⊢\phi \to X\in V$$
10	1 2 3 4 5 6 7 8	lcd0v	$$⊢\phi \to O=V\times \left\{\underset{\dot{}}{0}\right\}$$
11	10	fveq1d	$$⊢\phi \to O\left(X\right)=\left(V\times \left\{\underset{\dot{}}{0}\right\}\right)\left(X\right)$$
12	5	fvexi	$$⊢\underset{\dot{}}{0}\in \mathrm{V}$$
13	12	fvconst2	$$⊢X\in V\to \left(V\times \left\{\underset{\dot{}}{0}\right\}\right)\left(X\right)=\underset{\dot{}}{0}$$
14	9 13	syl	$$⊢\phi \to \left(V\times \left\{\underset{\dot{}}{0}\right\}\right)\left(X\right)=\underset{\dot{}}{0}$$
15	11 14	eqtrd	$$⊢\phi \to O\left(X\right)=\underset{\dot{}}{0}$$