Theorem quotcan 23868

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 23760	. . . . . . . . 9 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
2		simp2 1055	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ∈ (Poly‘𝑆))
3	1, 2	sseldi 3566	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ∈ (Poly‘ℂ))
4		simp1 1054	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 ∈ (Poly‘𝑆))
5	1, 4	sseldi 3566	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 ∈ (Poly‘ℂ))
6		quotcan.1	. . . . . . . . . . . 12 ⊢ 𝐻 = (𝐹 ∘𝑓 · 𝐺)
7		plymulcl 23781	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘𝑓 · 𝐺) ∈ (Poly‘ℂ))
8	6, 7	syl5eqel 2692	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐻 ∈ (Poly‘ℂ))
9	8	3adant3 1074	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐻 ∈ (Poly‘ℂ))
10		simp3 1056	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ≠ 0𝑝)
11		quotcl2 23861	. . . . . . . . . 10 ⊢ ((𝐻 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) ∈ (Poly‘ℂ))
12	9, 3, 10, 11	syl3anc 1318	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) ∈ (Poly‘ℂ))
13		plysubcl 23782	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ (𝐻 quot 𝐺) ∈ (Poly‘ℂ)) → (𝐹 ∘𝑓 − (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
14	5, 12, 13	syl2anc 691	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 ∘𝑓 − (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
15		plymul0or 23840	. . . . . . . 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 691	. . . . . . 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 9896	. . . . . . . . . . . . 13 ⊢ ℂ ∈ V
18	17	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ℂ ∈ V)
19		plyf 23758	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
20	4, 19	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹:ℂ⟶ℂ)
21		plyf 23758	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
22	2, 21	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺:ℂ⟶ℂ)
23		mulcom 9901	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) = (𝑦 · 𝑥))
24	23	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) = (𝑦 · 𝑥))
25	18, 20, 22, 24	caofcom 6827	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 ∘𝑓 · 𝐺) = (𝐺 ∘𝑓 · 𝐹))
26	6, 25	syl5eq 2656	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐻 = (𝐺 ∘𝑓 · 𝐹))
27	26	oveq1d 6564	. . . . . . . . 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 23758	. . . . . . . . . . 11 ⊢ ((𝐻 quot 𝐺) ∈ (Poly‘ℂ) → (𝐻 quot 𝐺):ℂ⟶ℂ)
29	12, 28	syl 17	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺):ℂ⟶ℂ)
30		subdi 10342	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 · (𝑦 − 𝑧)) = ((𝑥 · 𝑦) − (𝑥 · 𝑧)))
31	30	adantl 481	. . . . . . . . . 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 6831	. . . . . . . . 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 2647	. . . . . . . 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 2612	. . . . . . 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 2787	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ¬ 𝐺 = 0𝑝)
36		biorf 419	. . . . . . . 8 ⊢ (¬ 𝐺 = 0𝑝 → ((𝐹 ∘𝑓 − (𝐻 quot 𝐺)) = 0𝑝 ↔ (𝐺 = 0𝑝 ∨ (𝐹 ∘𝑓 − (𝐻 quot 𝐺)) = 0𝑝)))
37	35, 36	syl 17	. . . . . . 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 299	. . . . . 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 218	. . . . 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 2610	. . . . . . . . . . 11 ⊢ (deg‘𝐺) = (deg‘𝐺)
41		eqid 2610	. . . . . . . . . . 11 ⊢ (deg‘(𝐹 ∘𝑓 − (𝐻 quot 𝐺))) = (deg‘(𝐹 ∘𝑓 − (𝐻 quot 𝐺)))
42	40, 41	dgrmul 23830	. . . . . . . . . 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 641	. . . . . . . . 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 1317	. . . . . . . 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 23793	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
46	2, 45	syl 17	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘𝐺) ∈ ℕ0)
47	46	nn0red 11229	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘𝐺) ∈ ℝ)
48		dgrcl 23793	. . . . . . . . . . 11 ⊢ ((𝐹 ∘𝑓 − (𝐻 quot 𝐺)) ∈ (Poly‘ℂ) → (deg‘(𝐹 ∘𝑓 − (𝐻 quot 𝐺))) ∈ ℕ0)
49	14, 48	syl 17	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘(𝐹 ∘𝑓 − (𝐻 quot 𝐺))) ∈ ℕ0)
50		nn0addge1 11216	. . . . . . . . . 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 691	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘𝐺) ≤ ((deg‘𝐺) + (deg‘(𝐹 ∘𝑓 − (𝐻 quot 𝐺)))))
52		breq2 4587	. . . . . . . . 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 236	. . . . . . . 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 46	. . . . . . 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 6107	. . . . . . . . 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 4595	. . . . . . . 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 23781	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ (Poly‘ℂ) ∧ (𝐻 quot 𝐺) ∈ (Poly‘ℂ)) → (𝐺 ∘𝑓 · (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
58	3, 12, 57	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐺 ∘𝑓 · (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
59		plysubcl 23782	. . . . . . . . . . . 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 691	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 ∘𝑓 − (𝐺 ∘𝑓 · (𝐻 quot 𝐺))) ∈ (Poly‘ℂ))
61		dgrcl 23793	. . . . . . . . . . 11 ⊢ ((𝐻 ∘𝑓 − (𝐺 ∘𝑓 · (𝐻 quot 𝐺))) ∈ (Poly‘ℂ) → (deg‘(𝐻 ∘𝑓 − (𝐺 ∘𝑓 · (𝐻 quot 𝐺)))) ∈ ℕ0)
62	60, 61	syl 17	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘(𝐻 ∘𝑓 − (𝐺 ∘𝑓 · (𝐻 quot 𝐺)))) ∈ ℕ0)
63	62	nn0red 11229	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘(𝐻 ∘𝑓 − (𝐺 ∘𝑓 · (𝐻 quot 𝐺)))) ∈ ℝ)
64	47, 63	lenltd 10062	. . . . . . . 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 269	. . . . . . 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 228	. . . . . 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 2801	. . . . 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 2610	. . . . . . 7 ⊢ (𝐻 ∘𝑓 − (𝐺 ∘𝑓 · (𝐻 quot 𝐺))) = (𝐻 ∘𝑓 − (𝐺 ∘𝑓 · (𝐻 quot 𝐺)))
69	68	quotdgr 23862	. . . . . 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 1318	. . . . 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 395	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 ∘𝑓 − (𝐻 quot 𝐺)) = 0𝑝)
72		df-0p 23243	. . . 4 ⊢ 0𝑝 = (ℂ × {0})
73	71, 72	syl6eq 2660	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 ∘𝑓 − (𝐻 quot 𝐺)) = (ℂ × {0}))
74		ofsubeq0 10894	. . . 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 1318	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐹 ∘𝑓 − (𝐻 quot 𝐺)) = (ℂ × {0}) ↔ 𝐹 = (𝐻 quot 𝐺)))
76	73, 75	mpbid 221	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 = (𝐻 quot 𝐺))
77	76	eqcomd 2616	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) = 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173  {csn 4125   class class class wbr 4583   × cxp 5036  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘_𝑓 cof 6793  ℂcc 9813  ℝcr 9814  0cc0 9815   + caddc 9818   · cmul 9820   < clt 9953   ≤ cle 9954   − cmin 10145  ℕ₀cn0 11169  0_𝑝c0p 23242  Polycply 23744  degcdgr 23747   quot cquot 23849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-0p 23243  df-ply 23748  df-coe 23750  df-dgr 23751  df-quot 23850

This theorem is referenced by: (None)