Theorem xrmin2

Description: The minimum of two extended reals is less than or equal to one of them. (Contributed by NM, 7-Feb-2007)

	Ref	Expression
Assertion	xrmin2	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) ≤ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		xrleid	⊢ ( 𝐵 ∈ ℝ* → 𝐵 ≤ 𝐵 )
2		iffalse	⊢ ( ¬ 𝐴 ≤ 𝐵 → if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) = 𝐵 )
3	2	breq1d	⊢ ( ¬ 𝐴 ≤ 𝐵 → ( if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) ≤ 𝐵 ↔ 𝐵 ≤ 𝐵 ) )
4	1 3	syl5ibrcom	⊢ ( 𝐵 ∈ ℝ* → ( ¬ 𝐴 ≤ 𝐵 → if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) ≤ 𝐵 ) )
5		iftrue	⊢ ( 𝐴 ≤ 𝐵 → if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) = 𝐴 )
6		id	⊢ ( 𝐴 ≤ 𝐵 → 𝐴 ≤ 𝐵 )
7	5 6	eqbrtrd	⊢ ( 𝐴 ≤ 𝐵 → if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) ≤ 𝐵 )
8	4 7	pm2.61d2	⊢ ( 𝐵 ∈ ℝ* → if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) ≤ 𝐵 )
9	8	adantl	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) ≤ 𝐵 )