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Metamath Proof Explorer

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 xrletri3 ⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → ( 𝐵 = 𝐴 ↔ ( 𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) )
2 1 ancoms ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐵 = 𝐴 ↔ ( 𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) )
3 2 biimpar ⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( 𝐵 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) → 𝐵 = 𝐴 )
4 3 anassrs ⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐵 ≤ 𝐴 ) ∧ 𝐴 ≤ 𝐵 ) → 𝐵 = 𝐴 )
5 4 ifeq1da ⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐵 ≤ 𝐴 ) → if ( 𝐴 ≤ 𝐵 , 𝐵 , 𝐵 ) = if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) )
6 5 3impa ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐵 ≤ 𝐴 ) → if ( 𝐴 ≤ 𝐵 , 𝐵 , 𝐵 ) = if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) )
7 ifid ⊢ if ( 𝐴 ≤ 𝐵 , 𝐵 , 𝐵 ) = 𝐵
8 6 7 eqtr3di ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐵 ≤ 𝐴 ) → if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) = 𝐵 )