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Metamath Proof Explorer

Theorem deg1ge

Description: Conversely, a nonzero coefficient sets a lower bound on the degree. (Contributed by Stefan O'Rear, 23-Mar-2015)

Ref Expression
Hypotheses deg1leb.d ⊢ 𝐷 = ( deg1 ‘ 𝑅 )
deg1leb.p ⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
deg1leb.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
deg1leb.y ⊢ 0 = ( 0g ‘ 𝑅 )
deg1leb.a ⊢ 𝐴 = ( coe1 ‘ 𝐹 )
Assertion deg1ge ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ ( 𝐴 ‘ 𝐺 ) ≠ 0 ) → 𝐺 ≤ ( 𝐷 ‘ 𝐹 ) )

Proof

Step Hyp Ref Expression
1 deg1leb.d ⊢ 𝐷 = ( deg1 ‘ 𝑅 )
2 deg1leb.p ⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
3 deg1leb.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
4 deg1leb.y ⊢ 0 = ( 0g ‘ 𝑅 )
5 deg1leb.a ⊢ 𝐴 = ( coe1 ‘ 𝐹 )
6 1 2 3 deg1xrcl ⊢ ( 𝐹 ∈ 𝐵 → ( 𝐷 ‘ 𝐹 ) ∈ ℝ* )
7 nn0re ⊢ ( 𝐺 ∈ ℕ0 → 𝐺 ∈ ℝ )
8 7 rexrd ⊢ ( 𝐺 ∈ ℕ0 → 𝐺 ∈ ℝ* )
9 xrltnle ⊢ ( ( ( 𝐷 ‘ 𝐹 ) ∈ ℝ* ∧ 𝐺 ∈ ℝ* ) → ( ( 𝐷 ‘ 𝐹 ) < 𝐺 ↔ ¬ 𝐺 ≤ ( 𝐷 ‘ 𝐹 ) ) )
10 6 8 9 syl2an ⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ) → ( ( 𝐷 ‘ 𝐹 ) < 𝐺 ↔ ¬ 𝐺 ≤ ( 𝐷 ‘ 𝐹 ) ) )
11 1 2 3 4 5 deg1lt ⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ ( 𝐷 ‘ 𝐹 ) < 𝐺 ) → ( 𝐴 ‘ 𝐺 ) = 0 )
12 11 3expia ⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ) → ( ( 𝐷 ‘ 𝐹 ) < 𝐺 → ( 𝐴 ‘ 𝐺 ) = 0 ) )
13 10 12 sylbird ⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ) → ( ¬ 𝐺 ≤ ( 𝐷 ‘ 𝐹 ) → ( 𝐴 ‘ 𝐺 ) = 0 ) )
14 13 necon1ad ⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ) → ( ( 𝐴 ‘ 𝐺 ) ≠ 0 → 𝐺 ≤ ( 𝐷 ‘ 𝐹 ) ) )
15 14 3impia ⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ ( 𝐴 ‘ 𝐺 ) ≠ 0 ) → 𝐺 ≤ ( 𝐷 ‘ 𝐹 ) )