Theorem lcdsbase

Description: Base set of scalar ring for the closed kernel dual of a vector space. (Contributed by NM, 18-Mar-2015)

	Ref	Expression
Hypotheses	lcdsbase.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	lcdsbase.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
	lcdsbase.f	$$⊢F=\mathrm{Scalar}\left(U\right)$$
	lcdsbase.l	$$⊢L={\mathrm{Base}}_F$$
	lcdsbase.c	$$⊢C=\mathrm{LCDual}\left(K\right)\left(W\right)$$
	lcdsbase.s	$$⊢S=\mathrm{Scalar}\left(C\right)$$
	lcdsbase.r	$$⊢R={\mathrm{Base}}_S$$
	lcdsbase.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
Assertion	lcdsbase	$$⊢\phi \to R=L$$

Proof

Step	Hyp	Ref	Expression
1		lcdsbase.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
2		lcdsbase.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
3		lcdsbase.f	$$⊢F=\mathrm{Scalar}\left(U\right)$$
4		lcdsbase.l	$$⊢L={\mathrm{Base}}_F$$
5		lcdsbase.c	$$⊢C=\mathrm{LCDual}\left(K\right)\left(W\right)$$
6		lcdsbase.s	$$⊢S=\mathrm{Scalar}\left(C\right)$$
7		lcdsbase.r	$$⊢R={\mathrm{Base}}_S$$
8		lcdsbase.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	$$⊢\phi \to {\mathrm{Base}}_S={\mathrm{Base}}_{{opp}_r\left(F\right)}$$
12	9 4	opprbas	$$⊢L={\mathrm{Base}}_{{opp}_r\left(F\right)}$$
13	11 7 12	3eqtr4g	$$⊢\phi \to R=L$$