Theorem rprmasso3

Description: In an integral domain, if a prime element divides another, they are associates. (Contributed by Thierry Arnoux, 27-May-2025)

	Ref	Expression
Hypotheses	rprmasso.b	$$⊢B={\mathrm{Base}}_R$$
	rprmasso.p	$$⊢P=\mathrm{RPrime}\left(R\right)$$
	rprmasso.d	$$⊢\underset{\dot{}}{\parallel }={\parallel }_r\left(R\right)$$
	rprmasso.r	$$⊢\phi \to R\in \mathrm{IDomn}$$
	rprmasso.x	$$⊢\phi \to X\in P$$
	rprmasso.1	$$⊢\phi \to X\underset{\dot{}}{\parallel }Y$$
	rprmasso2.y	$$⊢\phi \to Y\in P$$
	rprmasso3.1	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
	rprmasso3.u	$$⊢U=\mathrm{Unit}\left(R\right)$$
Assertion	rprmasso3	$$⊢\phi \to \exists t\in Ut\underset{\dot{}}{·}X=Y$$

Proof

Step	Hyp	Ref	Expression
1		rprmasso.b	$$⊢B={\mathrm{Base}}_R$$
2		rprmasso.p	$$⊢P=\mathrm{RPrime}\left(R\right)$$
3		rprmasso.d	$$⊢\underset{\dot{}}{\parallel }={\parallel }_r\left(R\right)$$
4		rprmasso.r	$$⊢\phi \to R\in \mathrm{IDomn}$$
5		rprmasso.x	$$⊢\phi \to X\in P$$
6		rprmasso.1	$$⊢\phi \to X\underset{\dot{}}{\parallel }Y$$
7		rprmasso2.y	$$⊢\phi \to Y\in P$$
8		rprmasso3.1	$$⊢\underset{\dot{}}{·}={\cdot }_R$$
9		rprmasso3.u	$$⊢U=\mathrm{Unit}\left(R\right)$$
10	1 2 3 4 5 6 7	rprmasso2	$$⊢\phi \to Y\underset{\dot{}}{\parallel }X$$
11		eqid	$$⊢\mathrm{RSpan}\left(R\right)=\mathrm{RSpan}\left(R\right)$$
12	1 2 4 5	rprmcl	$$⊢\phi \to X\in B$$
13	1 2 4 7	rprmcl	$$⊢\phi \to Y\in B$$
14	1 11 3 12 13 9 8 4	dvdsruasso	$$⊢\phi \to \left(\left(X\underset{\dot{}}{\parallel }Y\wedge Y\underset{\dot{}}{\parallel }X\right)\leftrightarrow \exists t\in Ut\underset{\dot{}}{·}X=Y\right)$$
15	6 10 14	mpbi2and	$$⊢\phi \to \exists t\in Ut\underset{\dot{}}{·}X=Y$$