Theorem ply1sclf

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

	Ref	Expression
Hypotheses	ply1scl.p	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
	ply1scl.a	⊢ 𝐴 = ( algSc ‘ 𝑃 )
	coe1scl.k	⊢ 𝐾 = ( Base ‘ 𝑅 )
	ply1sclf.b	⊢ 𝐵 = ( Base ‘ 𝑃 )
Assertion	ply1sclf	⊢ ( 𝑅 ∈ Ring → 𝐴 : 𝐾 ⟶ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ply1scl.p	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
2		ply1scl.a	⊢ 𝐴 = ( algSc ‘ 𝑃 )
3		coe1scl.k	⊢ 𝐾 = ( Base ‘ 𝑅 )
4		ply1sclf.b	⊢ 𝐵 = ( Base ‘ 𝑃 )
5	1	ply1sca2	⊢ ( I ‘ 𝑅 ) = ( Scalar ‘ 𝑃 )
6	1	ply1ring	⊢ ( 𝑅 ∈ Ring → 𝑃 ∈ Ring )
7	1	ply1lmod	⊢ ( 𝑅 ∈ Ring → 𝑃 ∈ LMod )
8		df-base	⊢ Base = Slot 1
9	8 3	strfvi	⊢ 𝐾 = ( Base ‘ ( I ‘ 𝑅 ) )
10	2 5 6 7 9 4	asclf	⊢ ( 𝑅 ∈ Ring → 𝐴 : 𝐾 ⟶ 𝐵 )