Theorem quotcan 26637

Description: Exact division with a multiple. (Contributed by Mario Carneiro, 26-Jul-2014.)

Hypothesis

Ref	Expression
quotcan.1	⊢ 𝐻 = (𝐹 ∘f · 𝐺)

Assertion

Ref	Expression
quotcan	⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) = 𝐹)

Proof of Theorem quotcan

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		plyssc 26523	. . . . . . . . 9 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
2		simp2 1155	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ∈ (Poly‘𝑆))
3	1, 2	sselid 3906	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ∈ (Poly‘ℂ))
4		simp1 1154	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 ∈ (Poly‘𝑆))
5	1, 4	sselid 3906	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 ∈ (Poly‘ℂ))
6		quotcan.1	. . . . . . . . . . . 12 ⊢ 𝐻 = (𝐹 ∘f · 𝐺)
7		plymulcl 26545	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘f · 𝐺) ∈ (Poly‘ℂ))
8	6, 7	eqeltrid 2864	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐻 ∈ (Poly‘ℂ))
9	8	3adant3 1150	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐻 ∈ (Poly‘ℂ))
10		simp3 1156	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ≠ 0𝑝)
11		quotcl2 26628	. . . . . . . . . 10 ⊢ ((𝐻 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) ∈ (Poly‘ℂ))
12	9, 3, 10, 11	syl3anc 1398	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) ∈ (Poly‘ℂ))
13		plysubcl 26546	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ (𝐻 quot 𝐺) ∈ (Poly‘ℂ)) → (𝐹 ∘f − (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
14	5, 12, 13	syl2anc 596	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 ∘f − (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
15		plymul0or 26604	. . . . . . . 8 ⊢ ((𝐺 ∈ (Poly‘ℂ) ∧ (𝐹 ∘f − (𝐻 quot 𝐺)) ∈ (Poly‘ℂ)) → ((𝐺 ∘f · (𝐹 ∘f − (𝐻 quot 𝐺))) = 0𝑝 ↔ (𝐺 = 0𝑝 ∨ (𝐹 ∘f − (𝐻 quot 𝐺)) = 0𝑝)))
16	3, 14, 15	syl2anc 596	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐺 ∘f · (𝐹 ∘f − (𝐻 quot 𝐺))) = 0𝑝 ↔ (𝐺 = 0𝑝 ∨ (𝐹 ∘f − (𝐻 quot 𝐺)) = 0𝑝)))
17		cnex 11284	. . . . . . . . . . . . 13 ⊢ ℂ ∈ V
18	17	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ℂ ∈ V)
19		plyf 26521	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
20	4, 19	syl 18	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹:ℂ⟶ℂ)
21		plyf 26521	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
22	2, 21	syl 18	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺:ℂ⟶ℂ)
23		mulcom 11289	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) = (𝑦 · 𝑥))
24	23	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) = (𝑦 · 𝑥))
25	18, 20, 22, 24	caofcom 7730	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 ∘f · 𝐺) = (𝐺 ∘f · 𝐹))
26	6, 25	eqtrid 2807	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐻 = (𝐺 ∘f · 𝐹))
27	26	oveq1d 7435	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 ∘f − (𝐺 ∘f · (𝐻 quot 𝐺))) = ((𝐺 ∘f · 𝐹) ∘f − (𝐺 ∘f · (𝐻 quot 𝐺))))
28		plyf 26521	. . . . . . . . . . 11 ⊢ ((𝐻 quot 𝐺) ∈ (Poly‘ℂ) → (𝐻 quot 𝐺):ℂ⟶ℂ)
29	12, 28	syl 18	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺):ℂ⟶ℂ)
30		subdi 11752	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 · (𝑦 − 𝑧)) = ((𝑥 · 𝑦) − (𝑥 · 𝑧)))
31	30	adantl 487	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → (𝑥 · (𝑦 − 𝑧)) = ((𝑥 · 𝑦) − (𝑥 · 𝑧)))
32	18, 22, 20, 29, 31	caofdi 7735	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐺 ∘f · (𝐹 ∘f − (𝐻 quot 𝐺))) = ((𝐺 ∘f · 𝐹) ∘f − (𝐺 ∘f · (𝐻 quot 𝐺))))
33	27, 32	eqtr4d 2798	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 ∘f − (𝐺 ∘f · (𝐻 quot 𝐺))) = (𝐺 ∘f · (𝐹 ∘f − (𝐻 quot 𝐺))))
34	33	eqeq1d 2762	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐻 ∘f − (𝐺 ∘f · (𝐻 quot 𝐺))) = 0𝑝 ↔ (𝐺 ∘f · (𝐹 ∘f − (𝐻 quot 𝐺))) = 0𝑝))
35	10	neneqd 2960	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ¬ 𝐺 = 0𝑝)
36		biorf 950	. . . . . . . 8 ⊢ (¬ 𝐺 = 0𝑝 → ((𝐹 ∘f − (𝐻 quot 𝐺)) = 0𝑝 ↔ (𝐺 = 0𝑝 ∨ (𝐹 ∘f − (𝐻 quot 𝐺)) = 0𝑝)))
37	35, 36	syl 18	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐹 ∘f − (𝐻 quot 𝐺)) = 0𝑝 ↔ (𝐺 = 0𝑝 ∨ (𝐹 ∘f − (𝐻 quot 𝐺)) = 0𝑝)))
38	16, 34, 37	3bitr4d 314	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐻 ∘f − (𝐺 ∘f · (𝐻 quot 𝐺))) = 0𝑝 ↔ (𝐹 ∘f − (𝐻 quot 𝐺)) = 0𝑝))
39	38	biimpd 232	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐻 ∘f − (𝐺 ∘f · (𝐻 quot 𝐺))) = 0𝑝 → (𝐹 ∘f − (𝐻 quot 𝐺)) = 0𝑝))
40		eqid 2760	. . . . . . . . . . 11 ⊢ (deg‘𝐺) = (deg‘𝐺)
41		eqid 2760	. . . . . . . . . . 11 ⊢ (deg‘(𝐹 ∘f − (𝐻 quot 𝐺))) = (deg‘(𝐹 ∘f − (𝐻 quot 𝐺)))
42	40, 41	dgrmul 26594	. . . . . . . . . 10 ⊢ (((𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) ∧ ((𝐹 ∘f − (𝐻 quot 𝐺)) ∈ (Poly‘ℂ) ∧ (𝐹 ∘f − (𝐻 quot 𝐺)) ≠ 0𝑝)) → (deg‘(𝐺 ∘f · (𝐹 ∘f − (𝐻 quot 𝐺)))) = ((deg‘𝐺) + (deg‘(𝐹 ∘f − (𝐻 quot 𝐺)))))
43	42	expr 462	. . . . . . . . 9 ⊢ (((𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) ∧ (𝐹 ∘f − (𝐻 quot 𝐺)) ∈ (Poly‘ℂ)) → ((𝐹 ∘f − (𝐻 quot 𝐺)) ≠ 0𝑝 → (deg‘(𝐺 ∘f · (𝐹 ∘f − (𝐻 quot 𝐺)))) = ((deg‘𝐺) + (deg‘(𝐹 ∘f − (𝐻 quot 𝐺))))))
44	3, 10, 14, 43	syl21anc 851	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐹 ∘f − (𝐻 quot 𝐺)) ≠ 0𝑝 → (deg‘(𝐺 ∘f · (𝐹 ∘f − (𝐻 quot 𝐺)))) = ((deg‘𝐺) + (deg‘(𝐹 ∘f − (𝐻 quot 𝐺))))))
45		dgrcl 26557	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
46	2, 45	syl 18	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘𝐺) ∈ ℕ0)
47	46	nn0red 12671	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘𝐺) ∈ ℝ)
48		dgrcl 26557	. . . . . . . . . . 11 ⊢ ((𝐹 ∘f − (𝐻 quot 𝐺)) ∈ (Poly‘ℂ) → (deg‘(𝐹 ∘f − (𝐻 quot 𝐺))) ∈ ℕ0)
49	14, 48	syl 18	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘(𝐹 ∘f − (𝐻 quot 𝐺))) ∈ ℕ0)
50		nn0addge1 12655	. . . . . . . . . 10 ⊢ (((deg‘𝐺) ∈ ℝ ∧ (deg‘(𝐹 ∘f − (𝐻 quot 𝐺))) ∈ ℕ0) → (deg‘𝐺) ≤ ((deg‘𝐺) + (deg‘(𝐹 ∘f − (𝐻 quot 𝐺)))))
51	47, 49, 50	syl2anc 596	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘𝐺) ≤ ((deg‘𝐺) + (deg‘(𝐹 ∘f − (𝐻 quot 𝐺)))))
52		breq2 5106	. . . . . . . . 9 ⊢ ((deg‘(𝐺 ∘f · (𝐹 ∘f − (𝐻 quot 𝐺)))) = ((deg‘𝐺) + (deg‘(𝐹 ∘f − (𝐻 quot 𝐺)))) → ((deg‘𝐺) ≤ (deg‘(𝐺 ∘f · (𝐹 ∘f − (𝐻 quot 𝐺)))) ↔ (deg‘𝐺) ≤ ((deg‘𝐺) + (deg‘(𝐹 ∘f − (𝐻 quot 𝐺))))))
53	51, 52	syl5ibrcom 250	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((deg‘(𝐺 ∘f · (𝐹 ∘f − (𝐻 quot 𝐺)))) = ((deg‘𝐺) + (deg‘(𝐹 ∘f − (𝐻 quot 𝐺)))) → (deg‘𝐺) ≤ (deg‘(𝐺 ∘f · (𝐹 ∘f − (𝐻 quot 𝐺))))))
54	44, 53	syld 48	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐹 ∘f − (𝐻 quot 𝐺)) ≠ 0𝑝 → (deg‘𝐺) ≤ (deg‘(𝐺 ∘f · (𝐹 ∘f − (𝐻 quot 𝐺))))))
55	33	fveq2d 6889	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘(𝐻 ∘f − (𝐺 ∘f · (𝐻 quot 𝐺)))) = (deg‘(𝐺 ∘f · (𝐹 ∘f − (𝐻 quot 𝐺)))))
56	55	breq2d 5114	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((deg‘𝐺) ≤ (deg‘(𝐻 ∘f − (𝐺 ∘f · (𝐻 quot 𝐺)))) ↔ (deg‘𝐺) ≤ (deg‘(𝐺 ∘f · (𝐹 ∘f − (𝐻 quot 𝐺))))))
57		plymulcl 26545	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ (Poly‘ℂ) ∧ (𝐻 quot 𝐺) ∈ (Poly‘ℂ)) → (𝐺 ∘f · (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
58	3, 12, 57	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐺 ∘f · (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
59		plysubcl 26546	. . . . . . . . . . . 12 ⊢ ((𝐻 ∈ (Poly‘ℂ) ∧ (𝐺 ∘f · (𝐻 quot 𝐺)) ∈ (Poly‘ℂ)) → (𝐻 ∘f − (𝐺 ∘f · (𝐻 quot 𝐺))) ∈ (Poly‘ℂ))
60	9, 58, 59	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 ∘f − (𝐺 ∘f · (𝐻 quot 𝐺))) ∈ (Poly‘ℂ))
61		dgrcl 26557	. . . . . . . . . . 11 ⊢ ((𝐻 ∘f − (𝐺 ∘f · (𝐻 quot 𝐺))) ∈ (Poly‘ℂ) → (deg‘(𝐻 ∘f − (𝐺 ∘f · (𝐻 quot 𝐺)))) ∈ ℕ0)
62	60, 61	syl 18	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘(𝐻 ∘f − (𝐺 ∘f · (𝐻 quot 𝐺)))) ∈ ℕ0)
63	62	nn0red 12671	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘(𝐻 ∘f − (𝐺 ∘f · (𝐻 quot 𝐺)))) ∈ ℝ)
64	47, 63	lenltd 11459	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((deg‘𝐺) ≤ (deg‘(𝐻 ∘f − (𝐺 ∘f · (𝐻 quot 𝐺)))) ↔ ¬ (deg‘(𝐻 ∘f − (𝐺 ∘f · (𝐻 quot 𝐺)))) < (deg‘𝐺)))
65	56, 64	bitr3d 284	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((deg‘𝐺) ≤ (deg‘(𝐺 ∘f · (𝐹 ∘f − (𝐻 quot 𝐺)))) ↔ ¬ (deg‘(𝐻 ∘f − (𝐺 ∘f · (𝐻 quot 𝐺)))) < (deg‘𝐺)))
66	54, 65	sylibd 242	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐹 ∘f − (𝐻 quot 𝐺)) ≠ 0𝑝 → ¬ (deg‘(𝐻 ∘f − (𝐺 ∘f · (𝐻 quot 𝐺)))) < (deg‘𝐺)))
67	66	necon4ad 2974	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((deg‘(𝐻 ∘f − (𝐺 ∘f · (𝐻 quot 𝐺)))) < (deg‘𝐺) → (𝐹 ∘f − (𝐻 quot 𝐺)) = 0𝑝))
68		eqid 2760	. . . . . . 7 ⊢ (𝐻 ∘f − (𝐺 ∘f · (𝐻 quot 𝐺))) = (𝐻 ∘f − (𝐺 ∘f · (𝐻 quot 𝐺)))
69	68	quotdgr 26629	. . . . . 6 ⊢ ((𝐻 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → ((𝐻 ∘f − (𝐺 ∘f · (𝐻 quot 𝐺))) = 0𝑝 ∨ (deg‘(𝐻 ∘f − (𝐺 ∘f · (𝐻 quot 𝐺)))) < (deg‘𝐺)))
70	9, 3, 10, 69	syl3anc 1398	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐻 ∘f − (𝐺 ∘f · (𝐻 quot 𝐺))) = 0𝑝 ∨ (deg‘(𝐻 ∘f − (𝐺 ∘f · (𝐻 quot 𝐺)))) < (deg‘𝐺)))
71	39, 67, 70	mpjaod 874	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 ∘f − (𝐻 quot 𝐺)) = 0𝑝)
72		df-0p 25996	. . . 4 ⊢ 0𝑝 = (ℂ × {0})
73	71, 72	eqtrdi 2811	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 ∘f − (𝐻 quot 𝐺)) = (ℂ × {0}))
74		ofsubeq0 12320	. . . 4 ⊢ ((ℂ ∈ V ∧ 𝐹:ℂ⟶ℂ ∧ (𝐻 quot 𝐺):ℂ⟶ℂ) → ((𝐹 ∘f − (𝐻 quot 𝐺)) = (ℂ × {0}) ↔ 𝐹 = (𝐻 quot 𝐺)))
75	18, 20, 29, 74	syl3anc 1398	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐹 ∘f − (𝐻 quot 𝐺)) = (ℂ × {0}) ↔ 𝐹 = (𝐻 quot 𝐺)))
76	73, 75	mpbid 235	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 = (𝐻 quot 𝐺))
77	76	eqcomd 2766	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) = 𝐹)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  Vcvv 3450  {csn 4584   class class class wbr 5102   × cxp 5648  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ∘_f cof 7691  ℂcc 11201  ℝcr 11202  0cc0 11203   + caddc 11206   · cmul 11208   < clt 11346   ≤ cle 11347   − cmin 11544  ℕ₀cn0 12609  0_𝑝c0p 25995  Polycply 26507  degcdgr 26510   quot cquot 26616

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-inf2 9645  ax-cnex 11259  ax-resscn 11260  ax-1cn 11261  ax-icn 11262  ax-addcl 11263  ax-addrcl 11264  ax-mulcl 11265  ax-mulrcl 11266  ax-mulcom 11267  ax-addass 11268  ax-mulass 11269  ax-distr 11270  ax-i2m1 11271  ax-1ne0 11272  ax-1rid 11273  ax-rnegex 11274  ax-rrecex 11275  ax-cnre 11276  ax-pre-lttri 11277  ax-pre-lttrn 11278  ax-pre-ltadd 11279  ax-pre-mulgt0 11280  ax-pre-sup 11281

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7693  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-er 8720  df-map 8852  df-pm 8853  df-en 8977  df-dom 8978  df-sdom 8979  df-fin 8980  df-sup 9437  df-inf 9438  df-oi 9507  df-card 10023  df-pnf 11348  df-mnf 11349  df-xr 11350  df-ltxr 11351  df-le 11352  df-sub 11546  df-neg 11547  df-div 11977  df-nn 12339  df-2 12408  df-3 12409  df-n0 12610  df-z 12697  df-uz 12969  df-rp 13124  df-fz 13643  df-fzo 13792  df-fl 13935  df-seq 14148  df-exp 14208  df-hash 14478  df-cj 15271  df-re 15272  df-im 15273  df-sqrt 15407  df-abs 15408  df-clim 15660  df-rlim 15661  df-sum 15859  df-0p 25996  df-ply 26511  df-coe 26513  df-dgr 26514  df-quot 26617

This theorem is used by: (None)