MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  quotcan Structured version   Visualization version   GIF version

Theorem quotcan 23785
Description: Exact division with a multiple. (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypothesis
Ref Expression
quotcan.1 𝐻 = (𝐹𝑓 · 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.
StepHypRef Expression
1 plyssc 23677 . . . . . . . . 9 (Poly‘𝑆) ⊆ (Poly‘ℂ)
2 simp2 1054 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ∈ (Poly‘𝑆))
31, 2sseldi 3565 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ∈ (Poly‘ℂ))
4 simp1 1053 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 ∈ (Poly‘𝑆))
51, 4sseldi 3565 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 ∈ (Poly‘ℂ))
6 quotcan.1 . . . . . . . . . . . 12 𝐻 = (𝐹𝑓 · 𝐺)
7 plymulcl 23698 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹𝑓 · 𝐺) ∈ (Poly‘ℂ))
86, 7syl5eqel 2691 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐻 ∈ (Poly‘ℂ))
983adant3 1073 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐻 ∈ (Poly‘ℂ))
10 simp3 1055 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ≠ 0𝑝)
11 quotcl2 23778 . . . . . . . . . 10 ((𝐻 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) ∈ (Poly‘ℂ))
129, 3, 10, 11syl3anc 1317 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) ∈ (Poly‘ℂ))
13 plysubcl 23699 . . . . . . . . 9 ((𝐹 ∈ (Poly‘ℂ) ∧ (𝐻 quot 𝐺) ∈ (Poly‘ℂ)) → (𝐹𝑓 − (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
145, 12, 13syl2anc 690 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹𝑓 − (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
15 plymul0or 23757 . . . . . . . 8 ((𝐺 ∈ (Poly‘ℂ) ∧ (𝐹𝑓 − (𝐻 quot 𝐺)) ∈ (Poly‘ℂ)) → ((𝐺𝑓 · (𝐹𝑓 − (𝐻 quot 𝐺))) = 0𝑝 ↔ (𝐺 = 0𝑝 ∨ (𝐹𝑓 − (𝐻 quot 𝐺)) = 0𝑝)))
163, 14, 15syl2anc 690 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐺𝑓 · (𝐹𝑓 − (𝐻 quot 𝐺))) = 0𝑝 ↔ (𝐺 = 0𝑝 ∨ (𝐹𝑓 − (𝐻 quot 𝐺)) = 0𝑝)))
17 cnex 9873 . . . . . . . . . . . . 13 ℂ ∈ V
1817a1i 11 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ℂ ∈ V)
19 plyf 23675 . . . . . . . . . . . . 13 (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
204, 19syl 17 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹:ℂ⟶ℂ)
21 plyf 23675 . . . . . . . . . . . . 13 (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
222, 21syl 17 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺:ℂ⟶ℂ)
23 mulcom 9878 . . . . . . . . . . . . 13 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) = (𝑦 · 𝑥))
2423adantl 480 . . . . . . . . . . . 12 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) = (𝑦 · 𝑥))
2518, 20, 22, 24caofcom 6804 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹𝑓 · 𝐺) = (𝐺𝑓 · 𝐹))
266, 25syl5eq 2655 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐻 = (𝐺𝑓 · 𝐹))
2726oveq1d 6542 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺))) = ((𝐺𝑓 · 𝐹) ∘𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺))))
28 plyf 23675 . . . . . . . . . . 11 ((𝐻 quot 𝐺) ∈ (Poly‘ℂ) → (𝐻 quot 𝐺):ℂ⟶ℂ)
2912, 28syl 17 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺):ℂ⟶ℂ)
30 subdi 10314 . . . . . . . . . . 11 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 · (𝑦𝑧)) = ((𝑥 · 𝑦) − (𝑥 · 𝑧)))
3130adantl 480 . . . . . . . . . 10 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → (𝑥 · (𝑦𝑧)) = ((𝑥 · 𝑦) − (𝑥 · 𝑧)))
3218, 22, 20, 29, 31caofdi 6808 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐺𝑓 · (𝐹𝑓 − (𝐻 quot 𝐺))) = ((𝐺𝑓 · 𝐹) ∘𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺))))
3327, 32eqtr4d 2646 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺))) = (𝐺𝑓 · (𝐹𝑓 − (𝐻 quot 𝐺))))
3433eqeq1d 2611 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺))) = 0𝑝 ↔ (𝐺𝑓 · (𝐹𝑓 − (𝐻 quot 𝐺))) = 0𝑝))
3510neneqd 2786 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ¬ 𝐺 = 0𝑝)
36 biorf 418 . . . . . . . 8 𝐺 = 0𝑝 → ((𝐹𝑓 − (𝐻 quot 𝐺)) = 0𝑝 ↔ (𝐺 = 0𝑝 ∨ (𝐹𝑓 − (𝐻 quot 𝐺)) = 0𝑝)))
3735, 36syl 17 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐹𝑓 − (𝐻 quot 𝐺)) = 0𝑝 ↔ (𝐺 = 0𝑝 ∨ (𝐹𝑓 − (𝐻 quot 𝐺)) = 0𝑝)))
3816, 34, 373bitr4d 298 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺))) = 0𝑝 ↔ (𝐹𝑓 − (𝐻 quot 𝐺)) = 0𝑝))
3938biimpd 217 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺))) = 0𝑝 → (𝐹𝑓 − (𝐻 quot 𝐺)) = 0𝑝))
40 eqid 2609 . . . . . . . . . . 11 (deg‘𝐺) = (deg‘𝐺)
41 eqid 2609 . . . . . . . . . . 11 (deg‘(𝐹𝑓 − (𝐻 quot 𝐺))) = (deg‘(𝐹𝑓 − (𝐻 quot 𝐺)))
4240, 41dgrmul 23747 . . . . . . . . . 10 (((𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) ∧ ((𝐹𝑓 − (𝐻 quot 𝐺)) ∈ (Poly‘ℂ) ∧ (𝐹𝑓 − (𝐻 quot 𝐺)) ≠ 0𝑝)) → (deg‘(𝐺𝑓 · (𝐹𝑓 − (𝐻 quot 𝐺)))) = ((deg‘𝐺) + (deg‘(𝐹𝑓 − (𝐻 quot 𝐺)))))
4342expr 640 . . . . . . . . 9 (((𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) ∧ (𝐹𝑓 − (𝐻 quot 𝐺)) ∈ (Poly‘ℂ)) → ((𝐹𝑓 − (𝐻 quot 𝐺)) ≠ 0𝑝 → (deg‘(𝐺𝑓 · (𝐹𝑓 − (𝐻 quot 𝐺)))) = ((deg‘𝐺) + (deg‘(𝐹𝑓 − (𝐻 quot 𝐺))))))
443, 10, 14, 43syl21anc 1316 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐹𝑓 − (𝐻 quot 𝐺)) ≠ 0𝑝 → (deg‘(𝐺𝑓 · (𝐹𝑓 − (𝐻 quot 𝐺)))) = ((deg‘𝐺) + (deg‘(𝐹𝑓 − (𝐻 quot 𝐺))))))
45 dgrcl 23710 . . . . . . . . . . . 12 (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
462, 45syl 17 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘𝐺) ∈ ℕ0)
4746nn0red 11199 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘𝐺) ∈ ℝ)
48 dgrcl 23710 . . . . . . . . . . 11 ((𝐹𝑓 − (𝐻 quot 𝐺)) ∈ (Poly‘ℂ) → (deg‘(𝐹𝑓 − (𝐻 quot 𝐺))) ∈ ℕ0)
4914, 48syl 17 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘(𝐹𝑓 − (𝐻 quot 𝐺))) ∈ ℕ0)
50 nn0addge1 11186 . . . . . . . . . 10 (((deg‘𝐺) ∈ ℝ ∧ (deg‘(𝐹𝑓 − (𝐻 quot 𝐺))) ∈ ℕ0) → (deg‘𝐺) ≤ ((deg‘𝐺) + (deg‘(𝐹𝑓 − (𝐻 quot 𝐺)))))
5147, 49, 50syl2anc 690 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘𝐺) ≤ ((deg‘𝐺) + (deg‘(𝐹𝑓 − (𝐻 quot 𝐺)))))
52 breq2 4581 . . . . . . . . 9 ((deg‘(𝐺𝑓 · (𝐹𝑓 − (𝐻 quot 𝐺)))) = ((deg‘𝐺) + (deg‘(𝐹𝑓 − (𝐻 quot 𝐺)))) → ((deg‘𝐺) ≤ (deg‘(𝐺𝑓 · (𝐹𝑓 − (𝐻 quot 𝐺)))) ↔ (deg‘𝐺) ≤ ((deg‘𝐺) + (deg‘(𝐹𝑓 − (𝐻 quot 𝐺))))))
5351, 52syl5ibrcom 235 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((deg‘(𝐺𝑓 · (𝐹𝑓 − (𝐻 quot 𝐺)))) = ((deg‘𝐺) + (deg‘(𝐹𝑓 − (𝐻 quot 𝐺)))) → (deg‘𝐺) ≤ (deg‘(𝐺𝑓 · (𝐹𝑓 − (𝐻 quot 𝐺))))))
5444, 53syld 45 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐹𝑓 − (𝐻 quot 𝐺)) ≠ 0𝑝 → (deg‘𝐺) ≤ (deg‘(𝐺𝑓 · (𝐹𝑓 − (𝐻 quot 𝐺))))))
5533fveq2d 6092 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘(𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺)))) = (deg‘(𝐺𝑓 · (𝐹𝑓 − (𝐻 quot 𝐺)))))
5655breq2d 4589 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((deg‘𝐺) ≤ (deg‘(𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺)))) ↔ (deg‘𝐺) ≤ (deg‘(𝐺𝑓 · (𝐹𝑓 − (𝐻 quot 𝐺))))))
57 plymulcl 23698 . . . . . . . . . . . . 13 ((𝐺 ∈ (Poly‘ℂ) ∧ (𝐻 quot 𝐺) ∈ (Poly‘ℂ)) → (𝐺𝑓 · (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
583, 12, 57syl2anc 690 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐺𝑓 · (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
59 plysubcl 23699 . . . . . . . . . . . 12 ((𝐻 ∈ (Poly‘ℂ) ∧ (𝐺𝑓 · (𝐻 quot 𝐺)) ∈ (Poly‘ℂ)) → (𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺))) ∈ (Poly‘ℂ))
609, 58, 59syl2anc 690 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺))) ∈ (Poly‘ℂ))
61 dgrcl 23710 . . . . . . . . . . 11 ((𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺))) ∈ (Poly‘ℂ) → (deg‘(𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺)))) ∈ ℕ0)
6260, 61syl 17 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘(𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺)))) ∈ ℕ0)
6362nn0red 11199 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘(𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺)))) ∈ ℝ)
6447, 63lenltd 10034 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((deg‘𝐺) ≤ (deg‘(𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺)))) ↔ ¬ (deg‘(𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺)))) < (deg‘𝐺)))
6556, 64bitr3d 268 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((deg‘𝐺) ≤ (deg‘(𝐺𝑓 · (𝐹𝑓 − (𝐻 quot 𝐺)))) ↔ ¬ (deg‘(𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺)))) < (deg‘𝐺)))
6654, 65sylibd 227 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐹𝑓 − (𝐻 quot 𝐺)) ≠ 0𝑝 → ¬ (deg‘(𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺)))) < (deg‘𝐺)))
6766necon4ad 2800 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((deg‘(𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺)))) < (deg‘𝐺) → (𝐹𝑓 − (𝐻 quot 𝐺)) = 0𝑝))
68 eqid 2609 . . . . . . 7 (𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺))) = (𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺)))
6968quotdgr 23779 . . . . . 6 ((𝐻 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → ((𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺))) = 0𝑝 ∨ (deg‘(𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺)))) < (deg‘𝐺)))
709, 3, 10, 69syl3anc 1317 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺))) = 0𝑝 ∨ (deg‘(𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺)))) < (deg‘𝐺)))
7139, 67, 70mpjaod 394 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹𝑓 − (𝐻 quot 𝐺)) = 0𝑝)
72 df-0p 23160 . . . 4 0𝑝 = (ℂ × {0})
7371, 72syl6eq 2659 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹𝑓 − (𝐻 quot 𝐺)) = (ℂ × {0}))
74 ofsubeq0 10864 . . . 4 ((ℂ ∈ V ∧ 𝐹:ℂ⟶ℂ ∧ (𝐻 quot 𝐺):ℂ⟶ℂ) → ((𝐹𝑓 − (𝐻 quot 𝐺)) = (ℂ × {0}) ↔ 𝐹 = (𝐻 quot 𝐺)))
7518, 20, 29, 74syl3anc 1317 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐹𝑓 − (𝐻 quot 𝐺)) = (ℂ × {0}) ↔ 𝐹 = (𝐻 quot 𝐺)))
7673, 75mpbid 220 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 = (𝐻 quot 𝐺))
7776eqcomd 2615 1 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  Vcvv 3172  {csn 4124   class class class wbr 4577   × cxp 5026  wf 5786  cfv 5790  (class class class)co 6527  𝑓 cof 6770  cc 9790  cr 9791  0cc0 9792   + caddc 9795   · cmul 9797   < clt 9930  cle 9931  cmin 10117  0cn0 11139  0𝑝c0p 23159  Polycply 23661  degcdgr 23664   quot cquot 23766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870  ax-addf 9871
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-of 6772  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-map 7723  df-pm 7724  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-sup 8208  df-inf 8209  df-oi 8275  df-card 8625  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534  df-nn 10868  df-2 10926  df-3 10927  df-n0 11140  df-z 11211  df-uz 11520  df-rp 11665  df-fz 12153  df-fzo 12290  df-fl 12410  df-seq 12619  df-exp 12678  df-hash 12935  df-cj 13633  df-re 13634  df-im 13635  df-sqrt 13769  df-abs 13770  df-clim 14013  df-rlim 14014  df-sum 14211  df-0p 23160  df-ply 23665  df-coe 23667  df-dgr 23668  df-quot 23767
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator