Metamath Proof Explorer


Theorem ldualsca

Description: The ring of scalars of the dual of a vector space. (Contributed by NM, 18-Oct-2014)

Ref Expression
Hypotheses ldualsca.f F=ScalarW
ldualsca.o O=opprF
ldualsca.d D=LDualW
ldualsca.r R=ScalarD
ldualsca.w φWX
Assertion ldualsca φR=O

Proof

Step Hyp Ref Expression
1 ldualsca.f F=ScalarW
2 ldualsca.o O=opprF
3 ldualsca.d D=LDualW
4 ldualsca.r R=ScalarD
5 ldualsca.w φWX
6 eqid BaseW=BaseW
7 eqid +F=+F
8 eqid f+FLFnlW×LFnlW=f+FLFnlW×LFnlW
9 eqid LFnlW=LFnlW
10 eqid BaseF=BaseF
11 eqid F=F
12 eqid kBaseF,fLFnlWfFfBaseW×k=kBaseF,fLFnlWfFfBaseW×k
13 6 7 8 9 3 1 10 11 2 12 5 ldualset φD=BasendxLFnlW+ndxf+FLFnlW×LFnlWScalarndxOndxkBaseF,fLFnlWfFfBaseW×k
14 13 fveq2d φScalarD=ScalarBasendxLFnlW+ndxf+FLFnlW×LFnlWScalarndxOndxkBaseF,fLFnlWfFfBaseW×k
15 2 fvexi OV
16 eqid BasendxLFnlW+ndxf+FLFnlW×LFnlWScalarndxOndxkBaseF,fLFnlWfFfBaseW×k=BasendxLFnlW+ndxf+FLFnlW×LFnlWScalarndxOndxkBaseF,fLFnlWfFfBaseW×k
17 16 lmodsca OVO=ScalarBasendxLFnlW+ndxf+FLFnlW×LFnlWScalarndxOndxkBaseF,fLFnlWfFfBaseW×k
18 15 17 ax-mp O=ScalarBasendxLFnlW+ndxf+FLFnlW×LFnlWScalarndxOndxkBaseF,fLFnlWfFfBaseW×k
19 14 4 18 3eqtr4g φR=O