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Metamath Proof Explorer

Theorem quotcl

Description: The quotient of two polynomials in a field S is also in the field. (Contributed by Mario Carneiro, 26-Jul-2014)

Ref Expression
Hypotheses plydiv.pl ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝑆 )
plydiv.tm ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 · 𝑦 ) ∈ 𝑆 )
plydiv.rc ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑥 ≠ 0 ) ) → ( 1 / 𝑥 ) ∈ 𝑆 )
plydiv.m1 ⊢ ( 𝜑 → - 1 ∈ 𝑆 )
plydiv.f ⊢ ( 𝜑 → 𝐹 ∈ ( Poly ‘ 𝑆 ) )
plydiv.g ⊢ ( 𝜑 → 𝐺 ∈ ( Poly ‘ 𝑆 ) )
plydiv.z ⊢ ( 𝜑 → 𝐺 ≠ 0𝑝 )
Assertion quotcl ( 𝜑 → ( 𝐹 quot 𝐺 ) ∈ ( Poly ‘ 𝑆 ) )

Proof

Step Hyp Ref Expression
1 plydiv.pl ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝑆 )
2 plydiv.tm ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 · 𝑦 ) ∈ 𝑆 )
3 plydiv.rc ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑥 ≠ 0 ) ) → ( 1 / 𝑥 ) ∈ 𝑆 )
4 plydiv.m1 ⊢ ( 𝜑 → - 1 ∈ 𝑆 )
5 plydiv.f ⊢ ( 𝜑 → 𝐹 ∈ ( Poly ‘ 𝑆 ) )
6 plydiv.g ⊢ ( 𝜑 → 𝐺 ∈ ( Poly ‘ 𝑆 ) )
7 plydiv.z ⊢ ( 𝜑 → 𝐺 ≠ 0𝑝 )
8 eqid ⊢ ( 𝐹 ∘f − ( 𝐺 ∘f · ( 𝐹 quot 𝐺 ) ) ) = ( 𝐹 ∘f − ( 𝐺 ∘f · ( 𝐹 quot 𝐺 ) ) )
9 1 2 3 4 5 6 7 8 quotlem ⊢ ( 𝜑 → ( ( 𝐹 quot 𝐺 ) ∈ ( Poly ‘ 𝑆 ) ∧ ( ( 𝐹 ∘f − ( 𝐺 ∘f · ( 𝐹 quot 𝐺 ) ) ) = 0𝑝 ∨ ( deg ‘ ( 𝐹 ∘f − ( 𝐺 ∘f · ( 𝐹 quot 𝐺 ) ) ) ) < ( deg ‘ 𝐺 ) ) ) )
10 9 simpld ⊢ ( 𝜑 → ( 𝐹 quot 𝐺 ) ∈ ( Poly ‘ 𝑆 ) )