Definition df-dgr 24781

Description: Define the degree of a polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)

Assertion

Ref	Expression
df-dgr	⊢ deg = (𝑓 ∈ (Poly‘ℂ) ↦ sup((◡(coeff‘𝑓) “ (ℂ ∖ {0})), ℕ0, < ))

Detailed syntax breakdown of Definition df-dgr

Step	Hyp	Ref	Expression
1		cdgr 24777	. 2 class deg
2		vf	. . 3 setvar 𝑓
3		cc 10535	. . . 4 class ℂ
4		cply 24774	. . . 4 class Poly
5	3, 4	cfv 6355	. . 3 class (Poly‘ℂ)
6	2	cv 1536	. . . . . . 7 class 𝑓
7		ccoe 24776	. . . . . . 7 class coeff
8	6, 7	cfv 6355	. . . . . 6 class (coeff‘𝑓)
9	8	ccnv 5554	. . . . 5 class ◡(coeff‘𝑓)
10		cc0 10537	. . . . . . 7 class 0
11	10	csn 4567	. . . . . 6 class {0}
12	3, 11	cdif 3933	. . . . 5 class (ℂ ∖ {0})
13	9, 12	cima 5558	. . . 4 class (◡(coeff‘𝑓) “ (ℂ ∖ {0}))
14		cn0 11898	. . . 4 class ℕ0
15		clt 10675	. . . 4 class <
16	13, 14, 15	csup 8904	. . 3 class sup((◡(coeff‘𝑓) “ (ℂ ∖ {0})), ℕ0, < )
17	2, 5, 16	cmpt 5146	. 2 class (𝑓 ∈ (Poly‘ℂ) ↦ sup((◡(coeff‘𝑓) “ (ℂ ∖ {0})), ℕ0, < ))
18	1, 17	wceq 1537	1 wff deg = (𝑓 ∈ (Poly‘ℂ) ↦ sup((◡(coeff‘𝑓) “ (ℂ ∖ {0})), ℕ0, < ))

This definition is referenced by:  dgrval  24818