Theorem prstr

Description: "Less than or equal to" is transitive in a proset. (Contributed by Stefan O'Rear, 1-Feb-2015)

	Ref	Expression
Hypotheses	isprs.b	$$⊢B={\mathrm{Base}}_K$$
	isprs.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
Assertion	prstr	$$⊢\left(K\in \mathrm{Proset}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\wedge \left(X\underset{\dot{}}{\le }Y\wedge Y\underset{\dot{}}{\le }Z\right)\right)\to X\underset{\dot{}}{\le }Z$$

Proof

Step	Hyp	Ref	Expression
1		isprs.b	$$⊢B={\mathrm{Base}}_K$$
2		isprs.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3	1 2	prslem	$$⊢\left(K\in \mathrm{Proset}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(X\underset{\dot{}}{\le }X\wedge \left(\left(X\underset{\dot{}}{\le }Y\wedge Y\underset{\dot{}}{\le }Z\right)\to X\underset{\dot{}}{\le }Z\right)\right)$$
4	3	simprd	$$⊢\left(K\in \mathrm{Proset}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(\left(X\underset{\dot{}}{\le }Y\wedge Y\underset{\dot{}}{\le }Z\right)\to X\underset{\dot{}}{\le }Z\right)$$
5	4	3impia	$$⊢\left(K\in \mathrm{Proset}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\wedge \left(X\underset{\dot{}}{\le }Y\wedge Y\underset{\dot{}}{\le }Z\right)\right)\to X\underset{\dot{}}{\le }Z$$