Theorem irredcl

Description: An irreducible element is in the ring. (Contributed by Mario Carneiro, 4-Dec-2014)

	Ref	Expression
Hypotheses	irredn0.i	$$⊢I=\mathrm{Irred}\left(R\right)$$
	irredcl.b	$$⊢B={\mathrm{Base}}_R$$
Assertion	irredcl	$$⊢X\in I\to X\in B$$

Proof

Step	Hyp	Ref	Expression
1		irredn0.i	$$⊢I=\mathrm{Irred}\left(R\right)$$
2		irredcl.b	$$⊢B={\mathrm{Base}}_R$$
3		eqid	$$⊢\mathrm{Unit}\left(R\right)=\mathrm{Unit}\left(R\right)$$
4		eqid	$$⊢{\cdot }_R={\cdot }_R$$
5	2 3 1 4	isirred2	$$⊢X\in I\leftrightarrow \left(X\in B\wedge \neg X\in \mathrm{Unit}\left(R\right)\wedge \forall x\in B\forall y\in B\left(x{\cdot }_{R}y=X\to \left(x\in \mathrm{Unit}\left(R\right)\vee y\in \mathrm{Unit}\left(R\right)\right)\right)\right)$$
6	5	simp1bi	$$⊢X\in I\to X\in B$$