Theorem xrmax1

Description: An extended real is less than or equal to the maximum of it and another. (Contributed by NM, 7-Feb-2007)

	Ref	Expression
Assertion	xrmax1	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\right)\to A\le if\left(A\le B,B,A\right)$$

Proof

Step	Hyp	Ref	Expression
1		xrleid	$$⊢A\in {ℝ}^{*}\to A\le A$$
2		iffalse	$$⊢\neg A\le B\to if\left(A\le B,B,A\right)=A$$
3	2	breq2d	$$⊢\neg A\le B\to \left(A\le if\left(A\le B,B,A\right)\leftrightarrow A\le A\right)$$
4	1 3	syl5ibrcom	$$⊢A\in {ℝ}^{*}\to \left(\neg A\le B\to A\le if\left(A\le B,B,A\right)\right)$$
5		id	$$⊢A\le B\to A\le B$$
6		iftrue	$$⊢A\le B\to if\left(A\le B,B,A\right)=B$$
7	5 6	breqtrrd	$$⊢A\le B\to A\le if\left(A\le B,B,A\right)$$
8	4 7	pm2.61d2	$$⊢A\in {ℝ}^{*}\to A\le if\left(A\le B,B,A\right)$$
9	8	adantr	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\right)\to A\le if\left(A\le B,B,A\right)$$