Theorem quotcan 26273

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 26161	. . . . . . . . 9 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
2		simp2 1137	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ∈ (Poly‘𝑆))
3	1, 2	sselid 3931	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ∈ (Poly‘ℂ))
4		simp1 1136	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 ∈ (Poly‘𝑆))
5	1, 4	sselid 3931	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 ∈ (Poly‘ℂ))
6		quotcan.1	. . . . . . . . . . . 12 ⊢ 𝐻 = (𝐹 ∘f · 𝐺)
7		plymulcl 26182	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹 ∘f · 𝐺) ∈ (Poly‘ℂ))
8	6, 7	eqeltrid 2840	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐻 ∈ (Poly‘ℂ))
9	8	3adant3 1132	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐻 ∈ (Poly‘ℂ))
10		simp3 1138	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ≠ 0𝑝)
11		quotcl2 26266	. . . . . . . . . 10 ⊢ ((𝐻 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) ∈ (Poly‘ℂ))
12	9, 3, 10, 11	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) ∈ (Poly‘ℂ))
13		plysubcl 26183	. . . . . . . . 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 26244	. . . . . . . 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 11107	. . . . . . . . . . . . 13 ⊢ ℂ ∈ V
18	17	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ℂ ∈ V)
19		plyf 26159	. . . . . . . . . . . . 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 26159	. . . . . . . . . . . . 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 11112	. . . . . . . . . . . . 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 7659	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 ∘f · 𝐺) = (𝐺 ∘f · 𝐹))
26	6, 25	eqtrid 2783	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐻 = (𝐺 ∘f · 𝐹))
27	26	oveq1d 7373	. . . . . . . . 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 26159	. . . . . . . . . . 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 11570	. . . . . . . . . . 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 7664	. . . . . . . . 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 2774	. . . . . . . 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 2738	. . . . . . 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 2937	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ¬ 𝐺 = 0𝑝)
36		biorf 936	. . . . . . . 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 311	. . . . . 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 229	. . . . 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 2736	. . . . . . . . . . 11 ⊢ (deg‘𝐺) = (deg‘𝐺)
41		eqid 2736	. . . . . . . . . . 11 ⊢ (deg‘(𝐹 ∘f − (𝐻 quot 𝐺))) = (deg‘(𝐹 ∘f − (𝐻 quot 𝐺)))
42	40, 41	dgrmul 26232	. . . . . . . . . 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 456	. . . . . . . . 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 837	. . . . . . . 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 26194	. . . . . . . . . . . 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 12463	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘𝐺) ∈ ℝ)
48		dgrcl 26194	. . . . . . . . . . 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 12447	. . . . . . . . . 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 5102	. . . . . . . . 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 247	. . . . . . . 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 6838	. . . . . . . . 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 5110	. . . . . . . 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 26182	. . . . . . . . . . . . 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 26183	. . . . . . . . . . . 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 26194	. . . . . . . . . . 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 12463	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘(𝐻 ∘f − (𝐺 ∘f · (𝐻 quot 𝐺)))) ∈ ℝ)
64	47, 63	lenltd 11279	. . . . . . . 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 281	. . . . . . 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 239	. . . . . 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 2951	. . . . 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 2736	. . . . . . 7 ⊢ (𝐻 ∘f − (𝐺 ∘f · (𝐻 quot 𝐺))) = (𝐻 ∘f − (𝐺 ∘f · (𝐻 quot 𝐺)))
69	68	quotdgr 26267	. . . . . 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 1373	. . . . 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 860	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 ∘f − (𝐻 quot 𝐺)) = 0𝑝)
72		df-0p 25627	. . . 4 ⊢ 0𝑝 = (ℂ × {0})
73	71, 72	eqtrdi 2787	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 ∘f − (𝐻 quot 𝐺)) = (ℂ × {0}))
74		ofsubeq0 12142	. . . 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 1373	. . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐹 ∘f − (𝐻 quot 𝐺)) = (ℂ × {0}) ↔ 𝐹 = (𝐻 quot 𝐺)))
76	73, 75	mpbid 232	. 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 = (𝐻 quot 𝐺))
77	76	eqcomd 2742	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) = 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  Vcvv 3440  {csn 4580   class class class wbr 5098   × cxp 5622  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ∘_f cof 7620  ℂcc 11024  ℝcr 11025  0cc0 11026   + caddc 11029   · cmul 11031   < clt 11166   ≤ cle 11167   − cmin 11364  ℕ₀cn0 12401  0_𝑝c0p 25626  Polycply 26145  degcdgr 26148   quot cquot 26254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103  ax-pre-sup 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-map 8765  df-pm 8766  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9345  df-inf 9346  df-oi 9415  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-div 11795  df-nn 12146  df-2 12208  df-3 12209  df-n0 12402  df-z 12489  df-uz 12752  df-rp 12906  df-fz 13424  df-fzo 13571  df-fl 13712  df-seq 13925  df-exp 13985  df-hash 14254  df-cj 15022  df-re 15023  df-im 15024  df-sqrt 15158  df-abs 15159  df-clim 15411  df-rlim 15412  df-sum 15610  df-0p 25627  df-ply 26149  df-coe 26151  df-dgr 26152  df-quot 26255

This theorem is referenced by: (None)