Theorem quotcan 24355

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 24247	. . . . . . . . 9 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
2		simp2 1167	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ∈ (Poly‘𝑆))
3	1, 2	sseldi 3759	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ∈ (Poly‘ℂ))
4		simp1 1166	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 ∈ (Poly‘𝑆))
5	1, 4	sseldi 3759	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 ∈ (Poly‘ℂ))
6		quotcan.1	. . . . . . . . . . . 12 ⊢ 𝐻 = (𝐹 ∘𝑓 · 𝐺)
7		plymulcl 24268	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘𝑓 · 𝐺) ∈ (Poly‘ℂ))
8	6, 7	syl5eqel 2848	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐻 ∈ (Poly‘ℂ))
9	8	3adant3 1162	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐻 ∈ (Poly‘ℂ))
10		simp3 1168	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ≠ 0𝑝)
11		quotcl2 24348	. . . . . . . . . 10 ⊢ ((𝐻 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) ∈ (Poly‘ℂ))
12	9, 3, 10, 11	syl3anc 1490	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) ∈ (Poly‘ℂ))
13		plysubcl 24269	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ (𝐻 quot 𝐺) ∈ (Poly‘ℂ)) → (𝐹 ∘𝑓 − (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
14	5, 12, 13	syl2anc 579	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 ∘𝑓 − (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
15		plymul0or 24327	. . . . . . . 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 579	. . . . . . 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 10270	. . . . . . . . . . . . 13 ⊢ ℂ ∈ V
18	17	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ℂ ∈ V)
19		plyf 24245	. . . . . . . . . . . . 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 24245	. . . . . . . . . . . . 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 10275	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) = (𝑦 · 𝑥))
24	23	adantl 473	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) = (𝑦 · 𝑥))
25	18, 20, 22, 24	caofcom 7127	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 ∘𝑓 · 𝐺) = (𝐺 ∘𝑓 · 𝐹))
26	6, 25	syl5eq 2811	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐻 = (𝐺 ∘𝑓 · 𝐹))
27	26	oveq1d 6857	. . . . . . . . 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 24245	. . . . . . . . . . 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 10717	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 · (𝑦 − 𝑧)) = ((𝑥 · 𝑦) − (𝑥 · 𝑧)))
31	30	adantl 473	. . . . . . . . . 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 7131	. . . . . . . . 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 2802	. . . . . . . 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 2767	. . . . . . 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 2942	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ¬ 𝐺 = 0𝑝)
36		biorf 960	. . . . . . . 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 302	. . . . . 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 220	. . . . 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 2765	. . . . . . . . . . 11 ⊢ (deg‘𝐺) = (deg‘𝐺)
41		eqid 2765	. . . . . . . . . . 11 ⊢ (deg‘(𝐹 ∘𝑓 − (𝐻 quot 𝐺))) = (deg‘(𝐹 ∘𝑓 − (𝐻 quot 𝐺)))
42	40, 41	dgrmul 24317	. . . . . . . . . 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 448	. . . . . . . . 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 866	. . . . . . . 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 24280	. . . . . . . . . . . 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 11599	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘𝐺) ∈ ℝ)
48		dgrcl 24280	. . . . . . . . . . 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 11586	. . . . . . . . . 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 579	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘𝐺) ≤ ((deg‘𝐺) + (deg‘(𝐹 ∘𝑓 − (𝐻 quot 𝐺)))))
52		breq2 4813	. . . . . . . . 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 238	. . . . . . . 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 6379	. . . . . . . . 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 4821	. . . . . . . 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 24268	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ (Poly‘ℂ) ∧ (𝐻 quot 𝐺) ∈ (Poly‘ℂ)) → (𝐺 ∘𝑓 · (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
58	3, 12, 57	syl2anc 579	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐺 ∘𝑓 · (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
59		plysubcl 24269	. . . . . . . . . . . 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 579	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 ∘𝑓 − (𝐺 ∘𝑓 · (𝐻 quot 𝐺))) ∈ (Poly‘ℂ))
61		dgrcl 24280	. . . . . . . . . . 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 11599	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘(𝐻 ∘𝑓 − (𝐺 ∘𝑓 · (𝐻 quot 𝐺)))) ∈ ℝ)
64	47, 63	lenltd 10437	. . . . . . . 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 272	. . . . . . 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 230	. . . . . 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 2956	. . . . 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 2765	. . . . . . 7 ⊢ (𝐻 ∘𝑓 − (𝐺 ∘𝑓 · (𝐻 quot 𝐺))) = (𝐻 ∘𝑓 − (𝐺 ∘𝑓 · (𝐻 quot 𝐺)))
69	68	quotdgr 24349	. . . . . 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 1490	. . . . 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 886	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 ∘𝑓 − (𝐻 quot 𝐺)) = 0𝑝)
72		df-0p 23728	. . . 4 ⊢ 0𝑝 = (ℂ × {0})
73	71, 72	syl6eq 2815	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 ∘𝑓 − (𝐻 quot 𝐺)) = (ℂ × {0}))
74		ofsubeq0 11271	. . . 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 1490	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐹 ∘𝑓 − (𝐻 quot 𝐺)) = (ℂ × {0}) ↔ 𝐹 = (𝐻 quot 𝐺)))
76	73, 75	mpbid 223	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 = (𝐻 quot 𝐺))
77	76	eqcomd 2771	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) = 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 197   ∧ wa 384   ∨ wo 873   ∧ w3a 1107   = wceq 1652   ∈ wcel 2155   ≠ wne 2937  Vcvv 3350  {csn 4334   class class class wbr 4809   × cxp 5275  ⟶wf 6064  ‘cfv 6068  (class class class)co 6842   ∘_𝑓 cof 7093  ℂcc 10187  ℝcr 10188  0cc0 10189   + caddc 10192   · cmul 10194   < clt 10328   ≤ cle 10329   − cmin 10520  ℕ₀cn0 11538  0_𝑝c0p 23727  Polycply 24231  degcdgr 24234   quot cquot 24336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267  ax-addf 10268

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-sup 8555  df-inf 8556  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-z 11625  df-uz 11887  df-rp 12029  df-fz 12534  df-fzo 12674  df-fl 12801  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14126  df-re 14127  df-im 14128  df-sqrt 14262  df-abs 14263  df-clim 14506  df-rlim 14507  df-sum 14704  df-0p 23728  df-ply 24235  df-coe 24237  df-dgr 24238  df-quot 24337

This theorem is referenced by: (None)