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Definition df-quot 26028
Description: Define the quotient function on polynomials. This is the π‘ž of the expression 𝑓 = 𝑔 Β· π‘ž + π‘Ÿ in the division algorithm. (Contributed by Mario Carneiro, 23-Jul-2014.)
Assertion
Ref Expression
df-quot quot = (𝑓 ∈ (Polyβ€˜β„‚), 𝑔 ∈ ((Polyβ€˜β„‚) βˆ– {0𝑝}) ↦ (β„©π‘ž ∈ (Polyβ€˜β„‚)[(𝑓 ∘f βˆ’ (𝑔 ∘f Β· π‘ž)) / π‘Ÿ](π‘Ÿ = 0𝑝 ∨ (degβ€˜π‘Ÿ) < (degβ€˜π‘”))))
Distinct variable group:   𝑓,𝑔,π‘ž,π‘Ÿ

Detailed syntax breakdown of Definition df-quot
StepHypRef Expression
1 cquot 26027 . 2 class quot
2 vf . . 3 setvar 𝑓
3 vg . . 3 setvar 𝑔
4 cc 11110 . . . 4 class β„‚
5 cply 25922 . . . 4 class Poly
64, 5cfv 6543 . . 3 class (Polyβ€˜β„‚)
7 c0p 25410 . . . . 5 class 0𝑝
87csn 4628 . . . 4 class {0𝑝}
96, 8cdif 3945 . . 3 class ((Polyβ€˜β„‚) βˆ– {0𝑝})
10 vr . . . . . . . 8 setvar π‘Ÿ
1110cv 1540 . . . . . . 7 class π‘Ÿ
1211, 7wceq 1541 . . . . . 6 wff π‘Ÿ = 0𝑝
13 cdgr 25925 . . . . . . . 8 class deg
1411, 13cfv 6543 . . . . . . 7 class (degβ€˜π‘Ÿ)
153cv 1540 . . . . . . . 8 class 𝑔
1615, 13cfv 6543 . . . . . . 7 class (degβ€˜π‘”)
17 clt 11252 . . . . . . 7 class <
1814, 16, 17wbr 5148 . . . . . 6 wff (degβ€˜π‘Ÿ) < (degβ€˜π‘”)
1912, 18wo 845 . . . . 5 wff (π‘Ÿ = 0𝑝 ∨ (degβ€˜π‘Ÿ) < (degβ€˜π‘”))
202cv 1540 . . . . . 6 class 𝑓
21 vq . . . . . . . 8 setvar π‘ž
2221cv 1540 . . . . . . 7 class π‘ž
23 cmul 11117 . . . . . . . 8 class Β·
2423cof 7670 . . . . . . 7 class ∘f Β·
2515, 22, 24co 7411 . . . . . 6 class (𝑔 ∘f Β· π‘ž)
26 cmin 11448 . . . . . . 7 class βˆ’
2726cof 7670 . . . . . 6 class ∘f βˆ’
2820, 25, 27co 7411 . . . . 5 class (𝑓 ∘f βˆ’ (𝑔 ∘f Β· π‘ž))
2919, 10, 28wsbc 3777 . . . 4 wff [(𝑓 ∘f βˆ’ (𝑔 ∘f Β· π‘ž)) / π‘Ÿ](π‘Ÿ = 0𝑝 ∨ (degβ€˜π‘Ÿ) < (degβ€˜π‘”))
3029, 21, 6crio 7366 . . 3 class (β„©π‘ž ∈ (Polyβ€˜β„‚)[(𝑓 ∘f βˆ’ (𝑔 ∘f Β· π‘ž)) / π‘Ÿ](π‘Ÿ = 0𝑝 ∨ (degβ€˜π‘Ÿ) < (degβ€˜π‘”)))
312, 3, 6, 9, 30cmpo 7413 . 2 class (𝑓 ∈ (Polyβ€˜β„‚), 𝑔 ∈ ((Polyβ€˜β„‚) βˆ– {0𝑝}) ↦ (β„©π‘ž ∈ (Polyβ€˜β„‚)[(𝑓 ∘f βˆ’ (𝑔 ∘f Β· π‘ž)) / π‘Ÿ](π‘Ÿ = 0𝑝 ∨ (degβ€˜π‘Ÿ) < (degβ€˜π‘”))))
321, 31wceq 1541 1 wff quot = (𝑓 ∈ (Polyβ€˜β„‚), 𝑔 ∈ ((Polyβ€˜β„‚) βˆ– {0𝑝}) ↦ (β„©π‘ž ∈ (Polyβ€˜β„‚)[(𝑓 ∘f βˆ’ (𝑔 ∘f Β· π‘ž)) / π‘Ÿ](π‘Ÿ = 0𝑝 ∨ (degβ€˜π‘Ÿ) < (degβ€˜π‘”))))
Colors of variables: wff setvar class
This definition is referenced by:  quotval  26029
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