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Metamath Proof Explorer

Theorem rpgecld

Description: A number greater than or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses rpgecld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
rpgecld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
rpgecld.3 ⊢ ( 𝜑 → 𝐵 ≤ 𝐴 )
Assertion rpgecld ( 𝜑 → 𝐴 ∈ ℝ+ )

Proof

Step Hyp Ref Expression
1 rpgecld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 rpgecld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
3 rpgecld.3 ⊢ ( 𝜑 → 𝐵 ≤ 𝐴 )
4 rpgecl ⊢ ( ( 𝐵 ∈ ℝ+ ∧ 𝐴 ∈ ℝ ∧ 𝐵 ≤ 𝐴 ) → 𝐴 ∈ ℝ+ )
5 2 1 3 4 syl3anc ⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )