MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-mdeg Structured version   Visualization version   GIF version

Definition df-mdeg 25569
Description: Define the degree of a polynomial. Note (SO): as an experiment I am using a definition which makes the degree of the zero polynomial -∞, contrary to the convention used in df-dgr 25704. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by AV, 25-Jun-2019.)
Assertion
Ref Expression
df-mdeg mDeg = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ sup(ran ( ∈ (𝑓 supp (0g𝑟)) ↦ (ℂfld Σg )), ℝ*, < )))
Distinct variable group:   𝑖,𝑟,,𝑓

Detailed syntax breakdown of Definition df-mdeg
StepHypRef Expression
1 cmdg 25567 . 2 class mDeg
2 vi . . 3 setvar 𝑖
3 vr . . 3 setvar 𝑟
4 cvv 3474 . . 3 class V
5 vf . . . 4 setvar 𝑓
62cv 1540 . . . . . 6 class 𝑖
73cv 1540 . . . . . 6 class 𝑟
8 cmpl 21458 . . . . . 6 class mPoly
96, 7, 8co 7408 . . . . 5 class (𝑖 mPoly 𝑟)
10 cbs 17143 . . . . 5 class Base
119, 10cfv 6543 . . . 4 class (Base‘(𝑖 mPoly 𝑟))
12 vh . . . . . . 7 setvar
135cv 1540 . . . . . . . 8 class 𝑓
14 c0g 17384 . . . . . . . . 9 class 0g
157, 14cfv 6543 . . . . . . . 8 class (0g𝑟)
16 csupp 8145 . . . . . . . 8 class supp
1713, 15, 16co 7408 . . . . . . 7 class (𝑓 supp (0g𝑟))
18 ccnfld 20943 . . . . . . . 8 class fld
1912cv 1540 . . . . . . . 8 class
20 cgsu 17385 . . . . . . . 8 class Σg
2118, 19, 20co 7408 . . . . . . 7 class (ℂfld Σg )
2212, 17, 21cmpt 5231 . . . . . 6 class ( ∈ (𝑓 supp (0g𝑟)) ↦ (ℂfld Σg ))
2322crn 5677 . . . . 5 class ran ( ∈ (𝑓 supp (0g𝑟)) ↦ (ℂfld Σg ))
24 cxr 11246 . . . . 5 class *
25 clt 11247 . . . . 5 class <
2623, 24, 25csup 9434 . . . 4 class sup(ran ( ∈ (𝑓 supp (0g𝑟)) ↦ (ℂfld Σg )), ℝ*, < )
275, 11, 26cmpt 5231 . . 3 class (𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ sup(ran ( ∈ (𝑓 supp (0g𝑟)) ↦ (ℂfld Σg )), ℝ*, < ))
282, 3, 4, 4, 27cmpo 7410 . 2 class (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ sup(ran ( ∈ (𝑓 supp (0g𝑟)) ↦ (ℂfld Σg )), ℝ*, < )))
291, 28wceq 1541 1 wff mDeg = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ sup(ran ( ∈ (𝑓 supp (0g𝑟)) ↦ (ℂfld Σg )), ℝ*, < )))
Colors of variables: wff setvar class
This definition is referenced by:  reldmmdeg  25571  mdegfval  25579
  Copyright terms: Public domain W3C validator