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Metamath Proof Explorer

Theorem deg1xrcl

Description: Closure of univariate polynomial degree in extended reals. (Contributed by Stefan O'Rear, 23-Mar-2015)

Ref Expression
Hypotheses deg1xrf.d ⊢ 𝐷 = ( deg1 ‘ 𝑅 )
deg1xrf.p ⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
deg1xrf.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
Assertion deg1xrcl ( 𝐹 ∈ 𝐵 → ( 𝐷 ‘ 𝐹 ) ∈ ℝ* )

Proof

Step Hyp Ref Expression
1 deg1xrf.d ⊢ 𝐷 = ( deg1 ‘ 𝑅 )
2 deg1xrf.p ⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
3 deg1xrf.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
4 1 2 3 deg1xrf ⊢ 𝐷 : 𝐵 ⟶ ℝ*
5 4 ffvelcdmi ⊢ ( 𝐹 ∈ 𝐵 → ( 𝐷 ‘ 𝐹 ) ∈ ℝ* )