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 )