Theorem maxle

Description: Two ways of saying the maximum of two numbers is less than or equal to a third. (Contributed by NM, 29-Sep-2005)

	Ref	Expression
Assertion	maxle	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge C\in ℝ\right)\to \left(if\left(A\le B,B,A\right)\le C\leftrightarrow \left(A\le C\wedge B\le C\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		rexr	$$⊢A\in ℝ\to A\in {ℝ}^{*}$$
2		rexr	$$⊢B\in ℝ\to B\in {ℝ}^{*}$$
3		rexr	$$⊢C\in ℝ\to C\in {ℝ}^{*}$$
4		xrmaxle	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\wedge C\in {ℝ}^{*}\right)\to \left(if\left(A\le B,B,A\right)\le C\leftrightarrow \left(A\le C\wedge B\le C\right)\right)$$
5	1 2 3 4	syl3an	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge C\in ℝ\right)\to \left(if\left(A\le B,B,A\right)\le C\leftrightarrow \left(A\le C\wedge B\le C\right)\right)$$