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Metamath Proof Explorer

Theorem abvf

Description: An absolute value is a function from the ring to the real numbers. (Contributed by Mario Carneiro, 8-Sep-2014)

Ref Expression
Hypotheses abvf.a ⊢ A = AbsVal ⁡ R
abvf.b ⊢ B = Base R
Assertion abvf ⊢ F ∈ A → F : B ⟶ ℝ

Proof

Step Hyp Ref Expression
1 abvf.a ⊢ A = AbsVal ⁡ R
2 abvf.b ⊢ B = Base R
3 1 2 abvfge0 ⊢ F ∈ A → F : B ⟶ 0 +∞
4 rge0ssre ⊢ 0 +∞ ⊆ ℝ
5 fss ⊢ F : B ⟶ 0 +∞ ∧ 0 +∞ ⊆ ℝ → F : B ⟶ ℝ
6 3 4 5 sylancl ⊢ F ∈ A → F : B ⟶ ℝ