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Metamath Proof Explorer

Theorem deg1lt

Description: If the degree of a univariate polynomial is less than some index, then that coefficient must be zero. (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 deg1lt ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ ( 𝐷 ‘ 𝐹 ) < 𝐺 ) → ( 𝐴 ‘ 𝐺 ) = 0 )

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 simp3 ⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ ( 𝐷 ‘ 𝐹 ) < 𝐺 ) → ( 𝐷 ‘ 𝐹 ) < 𝐺 )
7 breq2 ⊢ ( 𝑥 = 𝐺 → ( ( 𝐷 ‘ 𝐹 ) < 𝑥 ↔ ( 𝐷 ‘ 𝐹 ) < 𝐺 ) )
8 fveqeq2 ⊢ ( 𝑥 = 𝐺 → ( ( 𝐴 ‘ 𝑥 ) = 0 ↔ ( 𝐴 ‘ 𝐺 ) = 0 ) )
9 7 8 imbi12d ⊢ ( 𝑥 = 𝐺 → ( ( ( 𝐷 ‘ 𝐹 ) < 𝑥 → ( 𝐴 ‘ 𝑥 ) = 0 ) ↔ ( ( 𝐷 ‘ 𝐹 ) < 𝐺 → ( 𝐴 ‘ 𝐺 ) = 0 ) ) )
10 1 2 3 deg1xrcl ⊢ ( 𝐹 ∈ 𝐵 → ( 𝐷 ‘ 𝐹 ) ∈ ℝ* )
11 10 3ad2ant1 ⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ ( 𝐷 ‘ 𝐹 ) < 𝐺 ) → ( 𝐷 ‘ 𝐹 ) ∈ ℝ* )
12 11 xrleidd ⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ ( 𝐷 ‘ 𝐹 ) < 𝐺 ) → ( 𝐷 ‘ 𝐹 ) ≤ ( 𝐷 ‘ 𝐹 ) )
13 simp1 ⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ ( 𝐷 ‘ 𝐹 ) < 𝐺 ) → 𝐹 ∈ 𝐵 )
14 1 2 3 4 5 deg1leb ⊢ ( ( 𝐹 ∈ 𝐵 ∧ ( 𝐷 ‘ 𝐹 ) ∈ ℝ* ) → ( ( 𝐷 ‘ 𝐹 ) ≤ ( 𝐷 ‘ 𝐹 ) ↔ ∀ 𝑥 ∈ ℕ0 ( ( 𝐷 ‘ 𝐹 ) < 𝑥 → ( 𝐴 ‘ 𝑥 ) = 0 ) ) )
15 13 10 14 syl2anc2 ⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ ( 𝐷 ‘ 𝐹 ) < 𝐺 ) → ( ( 𝐷 ‘ 𝐹 ) ≤ ( 𝐷 ‘ 𝐹 ) ↔ ∀ 𝑥 ∈ ℕ0 ( ( 𝐷 ‘ 𝐹 ) < 𝑥 → ( 𝐴 ‘ 𝑥 ) = 0 ) ) )
16 12 15 mpbid ⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ ( 𝐷 ‘ 𝐹 ) < 𝐺 ) → ∀ 𝑥 ∈ ℕ0 ( ( 𝐷 ‘ 𝐹 ) < 𝑥 → ( 𝐴 ‘ 𝑥 ) = 0 ) )
17 simp2 ⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ ( 𝐷 ‘ 𝐹 ) < 𝐺 ) → 𝐺 ∈ ℕ0 )
18 9 16 17 rspcdva ⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ ( 𝐷 ‘ 𝐹 ) < 𝐺 ) → ( ( 𝐷 ‘ 𝐹 ) < 𝐺 → ( 𝐴 ‘ 𝐺 ) = 0 ) )
19 6 18 mpd ⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℕ0 ∧ ( 𝐷 ‘ 𝐹 ) < 𝐺 ) → ( 𝐴 ‘ 𝐺 ) = 0 )