Theorem lattrd

Description: A lattice ordering is transitive. Deduction version of lattr . (Contributed by NM, 3-Sep-2012)

	Ref	Expression
Hypotheses	lattrd.b	$$⊢B={\mathrm{Base}}_K$$
	lattrd.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	lattrd.1	$$⊢\phi \to K\in \mathrm{Lat}$$
	lattrd.2	$$⊢\phi \to X\in B$$
	lattrd.3	$$⊢\phi \to Y\in B$$
	lattrd.4	$$⊢\phi \to Z\in B$$
	lattrd.5	$$⊢\phi \to X\underset{\dot{}}{\le }Y$$
	lattrd.6	$$⊢\phi \to Y\underset{\dot{}}{\le }Z$$
Assertion	lattrd	$$⊢\phi \to X\underset{\dot{}}{\le }Z$$

Proof

Step	Hyp	Ref	Expression
1		lattrd.b	$$⊢B={\mathrm{Base}}_K$$
2		lattrd.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		lattrd.1	$$⊢\phi \to K\in \mathrm{Lat}$$
4		lattrd.2	$$⊢\phi \to X\in B$$
5		lattrd.3	$$⊢\phi \to Y\in B$$
6		lattrd.4	$$⊢\phi \to Z\in B$$
7		lattrd.5	$$⊢\phi \to X\underset{\dot{}}{\le }Y$$
8		lattrd.6	$$⊢\phi \to Y\underset{\dot{}}{\le }Z$$
9	1 2	lattr	$$⊢\left(K\in \mathrm{Lat}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(\left(X\underset{\dot{}}{\le }Y\wedge Y\underset{\dot{}}{\le }Z\right)\to X\underset{\dot{}}{\le }Z\right)$$
10	3 4 5 6 9	syl13anc	$$⊢\phi \to \left(\left(X\underset{\dot{}}{\le }Y\wedge Y\underset{\dot{}}{\le }Z\right)\to X\underset{\dot{}}{\le }Z\right)$$
11	7 8 10	mp2and	$$⊢\phi \to X\underset{\dot{}}{\le }Z$$