Theorem xrmineq

Description: The minimum of two extended reals is equal to the second if the first is bigger. (Contributed by Mario Carneiro, 25-Mar-2015)

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

Proof

Step	Hyp	Ref	Expression
1		ifid	⊢ if ( 𝐴 ≤ 𝐵 , 𝐵 , 𝐵 ) = 𝐵
2		xrletri3	⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → ( 𝐵 = 𝐴 ↔ ( 𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) )
3	2	ancoms	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐵 = 𝐴 ↔ ( 𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) )
4	3	biimpar	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( 𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) → 𝐵 = 𝐴 )
5	4	anassrs	⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐵 ≤ 𝐴 ) ∧ 𝐴 ≤ 𝐵 ) → 𝐵 = 𝐴 )
6	5	ifeq1da	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐵 ≤ 𝐴 ) → if ( 𝐴 ≤ 𝐵 , 𝐵 , 𝐵 ) = if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) )
7	6	3impa	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐵 ≤ 𝐴 ) → if ( 𝐴 ≤ 𝐵 , 𝐵 , 𝐵 ) = if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) )
8	1 7	syl5reqr	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐵 ≤ 𝐴 ) → if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) = 𝐵 )