Theorem xrletrd

Description: Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015)

	Ref	Expression
Hypotheses	xrlttrd.1	$$⊢\phi \to A\in {ℝ}^{*}$$
	xrlttrd.2	$$⊢\phi \to B\in {ℝ}^{*}$$
	xrlttrd.3	$$⊢\phi \to C\in {ℝ}^{*}$$
	xrletrd.4	$$⊢\phi \to A\le B$$
	xrletrd.5	$$⊢\phi \to B\le C$$
Assertion	xrletrd	$$⊢\phi \to A\le C$$

Proof

Step	Hyp	Ref	Expression
1		xrlttrd.1	$$⊢\phi \to A\in {ℝ}^{*}$$
2		xrlttrd.2	$$⊢\phi \to B\in {ℝ}^{*}$$
3		xrlttrd.3	$$⊢\phi \to C\in {ℝ}^{*}$$
4		xrletrd.4	$$⊢\phi \to A\le B$$
5		xrletrd.5	$$⊢\phi \to B\le C$$
6		xrletr	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\wedge C\in {ℝ}^{*}\right)\to \left(\left(A\le B\wedge B\le C\right)\to A\le C\right)$$
7	1 2 3 6	syl3anc	$$⊢\phi \to \left(\left(A\le B\wedge B\le C\right)\to A\le C\right)$$
8	4 5 7	mp2and	$$⊢\phi \to A\le C$$