Theorem uc1pldg

Description: Unitic polynomials have unit leading coefficients. (Contributed by Stefan O'Rear, 28-Mar-2015)

	Ref	Expression
Hypotheses	uc1pldg.d	$$⊢D={\mathrm{deg}}_1\left(R\right)$$
	uc1pldg.u	$$⊢U=\mathrm{Unit}\left(R\right)$$
	uc1pldg.c	$$⊢C={Unic}_{\mathrm{1p}}\left(R\right)$$
Assertion	uc1pldg	$$⊢F\in C\to {\mathrm{coe}}_1\left(F\right)\left(D\left(F\right)\right)\in U$$

Proof

Step	Hyp	Ref	Expression
1		uc1pldg.d	$$⊢D={\mathrm{deg}}_1\left(R\right)$$
2		uc1pldg.u	$$⊢U=\mathrm{Unit}\left(R\right)$$
3		uc1pldg.c	$$⊢C={Unic}_{\mathrm{1p}}\left(R\right)$$
4		eqid	$$⊢{\mathrm{Poly}}_1\left(R\right)={\mathrm{Poly}}_1\left(R\right)$$
5		eqid	$$⊢{\mathrm{Base}}_{{\mathrm{Poly}}_1\left(R\right)}={\mathrm{Base}}_{{\mathrm{Poly}}_1\left(R\right)}$$
6		eqid	$$⊢0_{{\mathrm{Poly}}_1\left(R\right)}=0_{{\mathrm{Poly}}_1\left(R\right)}$$
7	4 5 6 1 3 2	isuc1p	$$⊢F\in C\leftrightarrow \left(F\in {\mathrm{Base}}_{{\mathrm{Poly}}_1\left(R\right)}\wedge F\ne 0_{{\mathrm{Poly}}_1\left(R\right)}\wedge {\mathrm{coe}}_1\left(F\right)\left(D\left(F\right)\right)\in U\right)$$
8	7	simp3bi	$$⊢F\in C\to {\mathrm{coe}}_1\left(F\right)\left(D\left(F\right)\right)\in U$$