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Metamath Proof Explorer

Theorem 2resupmax

Description: The supremum of two real numbers is the maximum of these two numbers. (Contributed by AV, 8-Jun-2021)

Ref Expression
Assertion 2resupmax ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → sup ( { 𝐴 , 𝐵 } , ℝ , < ) = if ( 𝐴 ≤ 𝐵 , 𝐵 , 𝐴 ) )

Proof

Step Hyp Ref Expression
1 ltso ⊢ < Or ℝ
2 suppr ⊢ ( ( < Or ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → sup ( { 𝐴 , 𝐵 } , ℝ , < ) = if ( 𝐵 < 𝐴 , 𝐴 , 𝐵 ) )
3 1 2 mp3an1 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → sup ( { 𝐴 , 𝐵 } , ℝ , < ) = if ( 𝐵 < 𝐴 , 𝐴 , 𝐵 ) )
4 ifnot ⊢ if ( ¬ 𝐵 < 𝐴 , 𝐵 , 𝐴 ) = if ( 𝐵 < 𝐴 , 𝐴 , 𝐵 )
5 lenlt ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴 ) )
6 5 bicomd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ¬ 𝐵 < 𝐴 ↔ 𝐴 ≤ 𝐵 ) )
7 6 ifbid ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → if ( ¬ 𝐵 < 𝐴 , 𝐵 , 𝐴 ) = if ( 𝐴 ≤ 𝐵 , 𝐵 , 𝐴 ) )
8 4 7 eqtr3id ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → if ( 𝐵 < 𝐴 , 𝐴 , 𝐵 ) = if ( 𝐴 ≤ 𝐵 , 𝐵 , 𝐴 ) )
9 3 8 eqtrd ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → sup ( { 𝐴 , 𝐵 } , ℝ , < ) = if ( 𝐴 ≤ 𝐵 , 𝐵 , 𝐴 ) )