Theorem maxle

Description: Two ways of saying the maximum of two numbers is less than or equal to a third. (Contributed by NM, 29-Sep-2005)

	Ref	Expression
Assertion	maxle	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( if ( 𝐴 ≤ 𝐵 , 𝐵 , 𝐴 ) ≤ 𝐶 ↔ ( 𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		rexr	⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℝ* )
2		rexr	⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℝ* )
3		rexr	⊢ ( 𝐶 ∈ ℝ → 𝐶 ∈ ℝ* )
4		xrmaxle	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( if ( 𝐴 ≤ 𝐵 , 𝐵 , 𝐴 ) ≤ 𝐶 ↔ ( 𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐶 ) ) )
5	1 2 3 4	syl3an	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( if ( 𝐴 ≤ 𝐵 , 𝐵 , 𝐴 ) ≤ 𝐶 ↔ ( 𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐶 ) ) )