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Theorem quotcan 24263
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 24155 . . . . . . . . 9 (Poly‘𝑆) ⊆ (Poly‘ℂ)
2 simp2 1132 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ∈ (Poly‘𝑆))
31, 2sseldi 3742 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ∈ (Poly‘ℂ))
4 simp1 1131 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 ∈ (Poly‘𝑆))
51, 4sseldi 3742 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 ∈ (Poly‘ℂ))
6 quotcan.1 . . . . . . . . . . . 12 𝐻 = (𝐹𝑓 · 𝐺)
7 plymulcl 24176 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹𝑓 · 𝐺) ∈ (Poly‘ℂ))
86, 7syl5eqel 2843 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐻 ∈ (Poly‘ℂ))
983adant3 1127 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐻 ∈ (Poly‘ℂ))
10 simp3 1133 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ≠ 0𝑝)
11 quotcl2 24256 . . . . . . . . . 10 ((𝐻 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) ∈ (Poly‘ℂ))
129, 3, 10, 11syl3anc 1477 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) ∈ (Poly‘ℂ))
13 plysubcl 24177 . . . . . . . . 9 ((𝐹 ∈ (Poly‘ℂ) ∧ (𝐻 quot 𝐺) ∈ (Poly‘ℂ)) → (𝐹𝑓 − (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
145, 12, 13syl2anc 696 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹𝑓 − (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
15 plymul0or 24235 . . . . . . . 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 696 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐺𝑓 · (𝐹𝑓 − (𝐻 quot 𝐺))) = 0𝑝 ↔ (𝐺 = 0𝑝 ∨ (𝐹𝑓 − (𝐻 quot 𝐺)) = 0𝑝)))
17 cnex 10209 . . . . . . . . . . . . 13 ℂ ∈ V
1817a1i 11 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ℂ ∈ V)
19 plyf 24153 . . . . . . . . . . . . 13 (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
204, 19syl 17 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹:ℂ⟶ℂ)
21 plyf 24153 . . . . . . . . . . . . 13 (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
222, 21syl 17 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺:ℂ⟶ℂ)
23 mulcom 10214 . . . . . . . . . . . . 13 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) = (𝑦 · 𝑥))
2423adantl 473 . . . . . . . . . . . 12 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) = (𝑦 · 𝑥))
2518, 20, 22, 24caofcom 7094 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹𝑓 · 𝐺) = (𝐺𝑓 · 𝐹))
266, 25syl5eq 2806 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐻 = (𝐺𝑓 · 𝐹))
2726oveq1d 6828 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺))) = ((𝐺𝑓 · 𝐹) ∘𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺))))
28 plyf 24153 . . . . . . . . . . 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 10655 . . . . . . . . . . 11 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 · (𝑦𝑧)) = ((𝑥 · 𝑦) − (𝑥 · 𝑧)))
3130adantl 473 . . . . . . . . . 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 7098 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐺𝑓 · (𝐹𝑓 − (𝐻 quot 𝐺))) = ((𝐺𝑓 · 𝐹) ∘𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺))))
3327, 32eqtr4d 2797 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺))) = (𝐺𝑓 · (𝐹𝑓 − (𝐻 quot 𝐺))))
3433eqeq1d 2762 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺))) = 0𝑝 ↔ (𝐺𝑓 · (𝐹𝑓 − (𝐻 quot 𝐺))) = 0𝑝))
3510neneqd 2937 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ¬ 𝐺 = 0𝑝)
36 biorf 419 . . . . . . . 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 300 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺))) = 0𝑝 ↔ (𝐹𝑓 − (𝐻 quot 𝐺)) = 0𝑝))
3938biimpd 219 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺))) = 0𝑝 → (𝐹𝑓 − (𝐻 quot 𝐺)) = 0𝑝))
40 eqid 2760 . . . . . . . . . . 11 (deg‘𝐺) = (deg‘𝐺)
41 eqid 2760 . . . . . . . . . . 11 (deg‘(𝐹𝑓 − (𝐻 quot 𝐺))) = (deg‘(𝐹𝑓 − (𝐻 quot 𝐺)))
4240, 41dgrmul 24225 . . . . . . . . . 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 644 . . . . . . . . 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 1476 . . . . . . . 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 24188 . . . . . . . . . . . 12 (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
462, 45syl 17 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘𝐺) ∈ ℕ0)
4746nn0red 11544 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘𝐺) ∈ ℝ)
48 dgrcl 24188 . . . . . . . . . . 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 11531 . . . . . . . . . 10 (((deg‘𝐺) ∈ ℝ ∧ (deg‘(𝐹𝑓 − (𝐻 quot 𝐺))) ∈ ℕ0) → (deg‘𝐺) ≤ ((deg‘𝐺) + (deg‘(𝐹𝑓 − (𝐻 quot 𝐺)))))
5147, 49, 50syl2anc 696 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘𝐺) ≤ ((deg‘𝐺) + (deg‘(𝐹𝑓 − (𝐻 quot 𝐺)))))
52 breq2 4808 . . . . . . . . 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 237 . . . . . . . 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 47 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐹𝑓 − (𝐻 quot 𝐺)) ≠ 0𝑝 → (deg‘𝐺) ≤ (deg‘(𝐺𝑓 · (𝐹𝑓 − (𝐻 quot 𝐺))))))
5533fveq2d 6356 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘(𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺)))) = (deg‘(𝐺𝑓 · (𝐹𝑓 − (𝐻 quot 𝐺)))))
5655breq2d 4816 . . . . . . . 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 24176 . . . . . . . . . . . . 13 ((𝐺 ∈ (Poly‘ℂ) ∧ (𝐻 quot 𝐺) ∈ (Poly‘ℂ)) → (𝐺𝑓 · (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
583, 12, 57syl2anc 696 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐺𝑓 · (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
59 plysubcl 24177 . . . . . . . . . . . 12 ((𝐻 ∈ (Poly‘ℂ) ∧ (𝐺𝑓 · (𝐻 quot 𝐺)) ∈ (Poly‘ℂ)) → (𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺))) ∈ (Poly‘ℂ))
609, 58, 59syl2anc 696 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺))) ∈ (Poly‘ℂ))
61 dgrcl 24188 . . . . . . . . . . 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 11544 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘(𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺)))) ∈ ℝ)
6447, 63lenltd 10375 . . . . . . . 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 270 . . . . . . 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 229 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐹𝑓 − (𝐻 quot 𝐺)) ≠ 0𝑝 → ¬ (deg‘(𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺)))) < (deg‘𝐺)))
6766necon4ad 2951 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((deg‘(𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺)))) < (deg‘𝐺) → (𝐹𝑓 − (𝐻 quot 𝐺)) = 0𝑝))
68 eqid 2760 . . . . . . 7 (𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺))) = (𝐻𝑓 − (𝐺𝑓 · (𝐻 quot 𝐺)))
6968quotdgr 24257 . . . . . 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 1477 . . . . 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 395 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹𝑓 − (𝐻 quot 𝐺)) = 0𝑝)
72 df-0p 23636 . . . 4 0𝑝 = (ℂ × {0})
7371, 72syl6eq 2810 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹𝑓 − (𝐻 quot 𝐺)) = (ℂ × {0}))
74 ofsubeq0 11209 . . . 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 1477 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐹𝑓 − (𝐻 quot 𝐺)) = (ℂ × {0}) ↔ 𝐹 = (𝐻 quot 𝐺)))
7673, 75mpbid 222 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 = (𝐻 quot 𝐺))
7776eqcomd 2766 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 196  wo 382  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  Vcvv 3340  {csn 4321   class class class wbr 4804   × cxp 5264  wf 6045  cfv 6049  (class class class)co 6813  𝑓 cof 7060  cc 10126  cr 10127  0cc0 10128   + caddc 10131   · cmul 10133   < clt 10266  cle 10267  cmin 10458  0cn0 11484  0𝑝c0p 23635  Polycply 24139  degcdgr 24142   quot cquot 24244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206  ax-addf 10207
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-sup 8513  df-inf 8514  df-oi 8580  df-card 8955  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-n0 11485  df-z 11570  df-uz 11880  df-rp 12026  df-fz 12520  df-fzo 12660  df-fl 12787  df-seq 12996  df-exp 13055  df-hash 13312  df-cj 14038  df-re 14039  df-im 14040  df-sqrt 14174  df-abs 14175  df-clim 14418  df-rlim 14419  df-sum 14616  df-0p 23636  df-ply 24143  df-coe 24145  df-dgr 24146  df-quot 24245
This theorem is referenced by: (None)
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