Theorem assasca

Description: An associative algebra's scalar field is a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015)

	Ref	Expression
Hypothesis	assasca.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
Assertion	assasca	$$⊢W\in \mathrm{AssAlg}\to F\in \mathrm{CRing}$$

Proof

Step	Hyp	Ref	Expression
1		assasca.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
2		eqid	$$⊢{\mathrm{Base}}_W={\mathrm{Base}}_W$$
3		eqid	$$⊢{\mathrm{Base}}_F={\mathrm{Base}}_F$$
4		eqid	$$⊢{\cdot }_W={\cdot }_W$$
5		eqid	$$⊢{\cdot }_W={\cdot }_W$$
6	2 1 3 4 5	isassa	$$⊢W\in \mathrm{AssAlg}\leftrightarrow \left(\left(W\in \mathrm{LMod}\wedge W\in \mathrm{Ring}\wedge F\in \mathrm{CRing}\right)\wedge \forall z\in {\mathrm{Base}}_F\forall x\in {\mathrm{Base}}_W\forall y\in {\mathrm{Base}}_W\left(\left(z{\cdot }_{W}x\right){\cdot }_{W}y=z{\cdot }_W\left(x{\cdot }_{W}y\right)\wedge x{\cdot }_W\left(z{\cdot }_{W}y\right)=z{\cdot }_W\left(x{\cdot }_{W}y\right)\right)\right)$$
7	6	simplbi	$$⊢W\in \mathrm{AssAlg}\to \left(W\in \mathrm{LMod}\wedge W\in \mathrm{Ring}\wedge F\in \mathrm{CRing}\right)$$
8	7	simp3d	$$⊢W\in \mathrm{AssAlg}\to F\in \mathrm{CRing}$$