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	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → 𝐴 ≤ if ( 𝐴 ≤ 𝐵 , 𝐵 , 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		xrleid	⊢ ( 𝐴 ∈ ℝ* → 𝐴 ≤ 𝐴 )
2		iffalse	⊢ ( ¬ 𝐴 ≤ 𝐵 → if ( 𝐴 ≤ 𝐵 , 𝐵 , 𝐴 ) = 𝐴 )
3	2	breq2d	⊢ ( ¬ 𝐴 ≤ 𝐵 → ( 𝐴 ≤ if ( 𝐴 ≤ 𝐵 , 𝐵 , 𝐴 ) ↔ 𝐴 ≤ 𝐴 ) )
4	1 3	syl5ibrcom	⊢ ( 𝐴 ∈ ℝ* → ( ¬ 𝐴 ≤ 𝐵 → 𝐴 ≤ if ( 𝐴 ≤ 𝐵 , 𝐵 , 𝐴 ) ) )
5		id	⊢ ( 𝐴 ≤ 𝐵 → 𝐴 ≤ 𝐵 )
6		iftrue	⊢ ( 𝐴 ≤ 𝐵 → if ( 𝐴 ≤ 𝐵 , 𝐵 , 𝐴 ) = 𝐵 )
7	5 6	breqtrrd	⊢ ( 𝐴 ≤ 𝐵 → 𝐴 ≤ if ( 𝐴 ≤ 𝐵 , 𝐵 , 𝐴 ) )
8	4 7	pm2.61d2	⊢ ( 𝐴 ∈ ℝ* → 𝐴 ≤ if ( 𝐴 ≤ 𝐵 , 𝐵 , 𝐴 ) )
9	8	adantr	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → 𝐴 ≤ if ( 𝐴 ≤ 𝐵 , 𝐵 , 𝐴 ) )