Theorem mon1pn0

Description: Monic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015)

	Ref	Expression
Hypotheses	uc1pn0.p	$$⊢P={\mathrm{Poly}}_1\left(R\right)$$
	uc1pn0.z	$$⊢\underset{\dot{}}{0}=0_P$$
	mon1pn0.m	$$⊢M={Monic}_{\mathrm{1p}}\left(R\right)$$
Assertion	mon1pn0	$$⊢F\in M\to F\ne \underset{\dot{}}{0}$$

Proof

Step	Hyp	Ref	Expression
1		uc1pn0.p	$$⊢P={\mathrm{Poly}}_1\left(R\right)$$
2		uc1pn0.z	$$⊢\underset{\dot{}}{0}=0_P$$
3		mon1pn0.m	$$⊢M={Monic}_{\mathrm{1p}}\left(R\right)$$
4		eqid	$$⊢{\mathrm{Base}}_P={\mathrm{Base}}_P$$
5		eqid	$$⊢{\mathrm{deg}}_1\left(R\right)={\mathrm{deg}}_1\left(R\right)$$
6		eqid	$$⊢1_R=1_R$$
7	1 4 2 5 3 6	ismon1p	$$⊢F\in M\leftrightarrow \left(F\in {\mathrm{Base}}_P\wedge F\ne \underset{\dot{}}{0}\wedge {\mathrm{coe}}_1\left(F\right)\left({\mathrm{deg}}_1\left(R\right)\left(F\right)\right)=1_R\right)$$
8	7	simp2bi	$$⊢F\in M\to F\ne \underset{\dot{}}{0}$$