Theorem ply1sclf

Description: A scalar polynomial is a polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015)

	Ref	Expression
Hypotheses	ply1scl.p	$$⊢P={\mathrm{Poly}}_1\left(R\right)$$
	ply1scl.a	$$⊢A=\mathrm{algSc}\left(P\right)$$
	coe1scl.k	$$⊢K={\mathrm{Base}}_R$$
	ply1sclf.b	$$⊢B={\mathrm{Base}}_P$$
Assertion	ply1sclf	$$⊢R\in \mathrm{Ring}\to A:K⟶B$$

Proof

Step	Hyp	Ref	Expression
1		ply1scl.p	$$⊢P={\mathrm{Poly}}_1\left(R\right)$$
2		ply1scl.a	$$⊢A=\mathrm{algSc}\left(P\right)$$
3		coe1scl.k	$$⊢K={\mathrm{Base}}_R$$
4		ply1sclf.b	$$⊢B={\mathrm{Base}}_P$$
5	1	ply1sca2	$$⊢\mathrm{I}\left(R\right)=\mathrm{Scalar}\left(P\right)$$
6	1	ply1ring	$$⊢R\in \mathrm{Ring}\to P\in \mathrm{Ring}$$
7	1	ply1lmod	$$⊢R\in \mathrm{Ring}\to P\in \mathrm{LMod}$$
8		df-base	$$⊢\mathrm{Base}=\mathrm{Slot}1$$
9	8 3	strfvi	$$⊢K={\mathrm{Base}}_{\mathrm{I}\left(R\right)}$$
10	2 5 6 7 9 4	asclf	$$⊢R\in \mathrm{Ring}\to A:K⟶B$$