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 = ( deg1𝑅 )
deg1leb.p 𝑃 = ( Poly1𝑅 )
deg1leb.b 𝐵 = ( Base ‘ 𝑃 )
deg1leb.y 0 = ( 0g𝑅 )
deg1leb.a 𝐴 = ( coe1𝐹 )
Assertion deg1lt ( ( 𝐹𝐵𝐺 ∈ ℕ0 ∧ ( 𝐷𝐹 ) < 𝐺 ) → ( 𝐴𝐺 ) = 0 )

Proof

Step Hyp Ref Expression
1 deg1leb.d 𝐷 = ( deg1𝑅 )
2 deg1leb.p 𝑃 = ( Poly1𝑅 )
3 deg1leb.b 𝐵 = ( Base ‘ 𝑃 )
4 deg1leb.y 0 = ( 0g𝑅 )
5 deg1leb.a 𝐴 = ( coe1𝐹 )
6 simp3 ( ( 𝐹𝐵𝐺 ∈ ℕ0 ∧ ( 𝐷𝐹 ) < 𝐺 ) → ( 𝐷𝐹 ) < 𝐺 )
7 breq2 ( 𝑥 = 𝐺 → ( ( 𝐷𝐹 ) < 𝑥 ↔ ( 𝐷𝐹 ) < 𝐺 ) )
8 fveqeq2 ( 𝑥 = 𝐺 → ( ( 𝐴𝑥 ) = 0 ↔ ( 𝐴𝐺 ) = 0 ) )
9 7 8 imbi12d ( 𝑥 = 𝐺 → ( ( ( 𝐷𝐹 ) < 𝑥 → ( 𝐴𝑥 ) = 0 ) ↔ ( ( 𝐷𝐹 ) < 𝐺 → ( 𝐴𝐺 ) = 0 ) ) )
10 1 2 3 deg1xrcl ( 𝐹𝐵 → ( 𝐷𝐹 ) ∈ ℝ* )
11 10 3ad2ant1 ( ( 𝐹𝐵𝐺 ∈ ℕ0 ∧ ( 𝐷𝐹 ) < 𝐺 ) → ( 𝐷𝐹 ) ∈ ℝ* )
12 11 xrleidd ( ( 𝐹𝐵𝐺 ∈ ℕ0 ∧ ( 𝐷𝐹 ) < 𝐺 ) → ( 𝐷𝐹 ) ≤ ( 𝐷𝐹 ) )
13 simp1 ( ( 𝐹𝐵𝐺 ∈ ℕ0 ∧ ( 𝐷𝐹 ) < 𝐺 ) → 𝐹𝐵 )
14 1 2 3 4 5 deg1leb ( ( 𝐹𝐵 ∧ ( 𝐷𝐹 ) ∈ ℝ* ) → ( ( 𝐷𝐹 ) ≤ ( 𝐷𝐹 ) ↔ ∀ 𝑥 ∈ ℕ0 ( ( 𝐷𝐹 ) < 𝑥 → ( 𝐴𝑥 ) = 0 ) ) )
15 13 10 14 syl2anc2 ( ( 𝐹𝐵𝐺 ∈ ℕ0 ∧ ( 𝐷𝐹 ) < 𝐺 ) → ( ( 𝐷𝐹 ) ≤ ( 𝐷𝐹 ) ↔ ∀ 𝑥 ∈ ℕ0 ( ( 𝐷𝐹 ) < 𝑥 → ( 𝐴𝑥 ) = 0 ) ) )
16 12 15 mpbid ( ( 𝐹𝐵𝐺 ∈ ℕ0 ∧ ( 𝐷𝐹 ) < 𝐺 ) → ∀ 𝑥 ∈ ℕ0 ( ( 𝐷𝐹 ) < 𝑥 → ( 𝐴𝑥 ) = 0 ) )
17 simp2 ( ( 𝐹𝐵𝐺 ∈ ℕ0 ∧ ( 𝐷𝐹 ) < 𝐺 ) → 𝐺 ∈ ℕ0 )
18 9 16 17 rspcdva ( ( 𝐹𝐵𝐺 ∈ ℕ0 ∧ ( 𝐷𝐹 ) < 𝐺 ) → ( ( 𝐷𝐹 ) < 𝐺 → ( 𝐴𝐺 ) = 0 ) )
19 6 18 mpd ( ( 𝐹𝐵𝐺 ∈ ℕ0 ∧ ( 𝐷𝐹 ) < 𝐺 ) → ( 𝐴𝐺 ) = 0 )