Theorem quotcan 24581

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 24473	. . . . . . . . 9 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
2		simp2 1130	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ∈ (Poly‘𝑆))
3	1, 2	sseldi 3887	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ∈ (Poly‘ℂ))
4		simp1 1129	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 ∈ (Poly‘𝑆))
5	1, 4	sseldi 3887	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 ∈ (Poly‘ℂ))
6		quotcan.1	. . . . . . . . . . . 12 ⊢ 𝐻 = (𝐹 ∘𝑓 · 𝐺)
7		plymulcl 24494	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘𝑓 · 𝐺) ∈ (Poly‘ℂ))
8	6, 7	syl5eqel 2887	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐻 ∈ (Poly‘ℂ))
9	8	3adant3 1125	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐻 ∈ (Poly‘ℂ))
10		simp3 1131	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ≠ 0𝑝)
11		quotcl2 24574	. . . . . . . . . 10 ⊢ ((𝐻 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) ∈ (Poly‘ℂ))
12	9, 3, 10, 11	syl3anc 1364	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) ∈ (Poly‘ℂ))
13		plysubcl 24495	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ (𝐻 quot 𝐺) ∈ (Poly‘ℂ)) → (𝐹 ∘𝑓 − (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
14	5, 12, 13	syl2anc 584	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 ∘𝑓 − (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
15		plymul0or 24553	. . . . . . . 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 584	. . . . . . 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 10464	. . . . . . . . . . . . 13 ⊢ ℂ ∈ V
18	17	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ℂ ∈ V)
19		plyf 24471	. . . . . . . . . . . . 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 24471	. . . . . . . . . . . . 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 10469	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) = (𝑦 · 𝑥))
24	23	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) = (𝑦 · 𝑥))
25	18, 20, 22, 24	caofcom 7299	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 ∘𝑓 · 𝐺) = (𝐺 ∘𝑓 · 𝐹))
26	6, 25	syl5eq 2843	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐻 = (𝐺 ∘𝑓 · 𝐹))
27	26	oveq1d 7031	. . . . . . . . 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 24471	. . . . . . . . . . 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 10921	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 · (𝑦 − 𝑧)) = ((𝑥 · 𝑦) − (𝑥 · 𝑧)))
31	30	adantl 482	. . . . . . . . . 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 7303	. . . . . . . . 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 2834	. . . . . . . 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 2797	. . . . . . 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 2989	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ¬ 𝐺 = 0𝑝)
36		biorf 931	. . . . . . . 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 312	. . . . . 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 230	. . . . 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 2795	. . . . . . . . . . 11 ⊢ (deg‘𝐺) = (deg‘𝐺)
41		eqid 2795	. . . . . . . . . . 11 ⊢ (deg‘(𝐹 ∘𝑓 − (𝐻 quot 𝐺))) = (deg‘(𝐹 ∘𝑓 − (𝐻 quot 𝐺)))
42	40, 41	dgrmul 24543	. . . . . . . . . 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 457	. . . . . . . . 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 834	. . . . . . . 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 24506	. . . . . . . . . . . 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 11804	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘𝐺) ∈ ℝ)
48		dgrcl 24506	. . . . . . . . . . 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 11791	. . . . . . . . . 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 584	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘𝐺) ≤ ((deg‘𝐺) + (deg‘(𝐹 ∘𝑓 − (𝐻 quot 𝐺)))))
52		breq2 4966	. . . . . . . . 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 248	. . . . . . . 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 47	. . . . . . 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 6542	. . . . . . . . 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 4974	. . . . . . . 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 24494	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ (Poly‘ℂ) ∧ (𝐻 quot 𝐺) ∈ (Poly‘ℂ)) → (𝐺 ∘𝑓 · (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
58	3, 12, 57	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐺 ∘𝑓 · (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
59		plysubcl 24495	. . . . . . . . . . . 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 584	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 ∘𝑓 − (𝐺 ∘𝑓 · (𝐻 quot 𝐺))) ∈ (Poly‘ℂ))
61		dgrcl 24506	. . . . . . . . . . 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 11804	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘(𝐻 ∘𝑓 − (𝐺 ∘𝑓 · (𝐻 quot 𝐺)))) ∈ ℝ)
64	47, 63	lenltd 10633	. . . . . . . 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 282	. . . . . . 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 240	. . . . . 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 3003	. . . . 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 2795	. . . . . . 7 ⊢ (𝐻 ∘𝑓 − (𝐺 ∘𝑓 · (𝐻 quot 𝐺))) = (𝐻 ∘𝑓 − (𝐺 ∘𝑓 · (𝐻 quot 𝐺)))
69	68	quotdgr 24575	. . . . . 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 1364	. . . . 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 855	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 ∘𝑓 − (𝐻 quot 𝐺)) = 0𝑝)
72		df-0p 23954	. . . 4 ⊢ 0𝑝 = (ℂ × {0})
73	71, 72	syl6eq 2847	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 ∘𝑓 − (𝐻 quot 𝐺)) = (ℂ × {0}))
74		ofsubeq0 11483	. . . 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 1364	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐹 ∘𝑓 − (𝐻 quot 𝐺)) = (ℂ × {0}) ↔ 𝐹 = (𝐻 quot 𝐺)))
76	73, 75	mpbid 233	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 = (𝐻 quot 𝐺))
77	76	eqcomd 2801	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) = 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 842   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081   ≠ wne 2984  Vcvv 3437  {csn 4472   class class class wbr 4962   × cxp 5441  ⟶wf 6221  ‘cfv 6225  (class class class)co 7016   ∘_𝑓 cof 7265  ℂcc 10381  ℝcr 10382  0cc0 10383   + caddc 10386   · cmul 10388   < clt 10521   ≤ cle 10522   − cmin 10717  ℕ₀cn0 11745  0_𝑝c0p 23953  Polycply 24457  degcdgr 24460   quot cquot 24562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-inf2 8950  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460  ax-pre-sup 10461  ax-addf 10462

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-of 7267  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-oadd 7957  df-er 8139  df-map 8258  df-pm 8259  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-sup 8752  df-inf 8753  df-oi 8820  df-card 9214  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-div 11146  df-nn 11487  df-2 11548  df-3 11549  df-n0 11746  df-z 11830  df-uz 12094  df-rp 12240  df-fz 12743  df-fzo 12884  df-fl 13012  df-seq 13220  df-exp 13280  df-hash 13541  df-cj 14292  df-re 14293  df-im 14294  df-sqrt 14428  df-abs 14429  df-clim 14679  df-rlim 14680  df-sum 14877  df-0p 23954  df-ply 24461  df-coe 24463  df-dgr 24464  df-quot 24563

This theorem is referenced by: (None)