Theorem min1d

Description: The minimum of two numbers is less than or equal to the first. (Contributed by Glauco Siliprandi, 5-Feb-2022)

	Ref	Expression
Hypotheses	min1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	min1d.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
Assertion	min1d	⊢ ( 𝜑 → if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) ≤ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		min1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		min1d.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		min1	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) ≤ 𝐴 )
4	1 2 3	syl2anc	⊢ ( 𝜑 → if ( 𝐴 ≤ 𝐵 , 𝐴 , 𝐵 ) ≤ 𝐴 )