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

Theorem quotcan 26289
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 26179 . . . . . . . . 9 (Poly‘𝑆) ⊆ (Poly‘ℂ)
2 simp2 1134 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ∈ (Poly‘𝑆))
31, 2sselid 3974 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ∈ (Poly‘ℂ))
4 simp1 1133 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 ∈ (Poly‘𝑆))
51, 4sselid 3974 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 ∈ (Poly‘ℂ))
6 quotcan.1 . . . . . . . . . . . 12 𝐻 = (𝐹f · 𝐺)
7 plymulcl 26200 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹f · 𝐺) ∈ (Poly‘ℂ))
86, 7eqeltrid 2829 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐻 ∈ (Poly‘ℂ))
983adant3 1129 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐻 ∈ (Poly‘ℂ))
10 simp3 1135 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ≠ 0𝑝)
11 quotcl2 26282 . . . . . . . . . 10 ((𝐻 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) ∈ (Poly‘ℂ))
129, 3, 10, 11syl3anc 1368 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) ∈ (Poly‘ℂ))
13 plysubcl 26201 . . . . . . . . 9 ((𝐹 ∈ (Poly‘ℂ) ∧ (𝐻 quot 𝐺) ∈ (Poly‘ℂ)) → (𝐹f − (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
145, 12, 13syl2anc 582 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹f − (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
15 plymul0or 26260 . . . . . . . 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 582 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐺f · (𝐹f − (𝐻 quot 𝐺))) = 0𝑝 ↔ (𝐺 = 0𝑝 ∨ (𝐹f − (𝐻 quot 𝐺)) = 0𝑝)))
17 cnex 11221 . . . . . . . . . . . . 13 ℂ ∈ V
1817a1i 11 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ℂ ∈ V)
19 plyf 26177 . . . . . . . . . . . . 13 (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
204, 19syl 17 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹:ℂ⟶ℂ)
21 plyf 26177 . . . . . . . . . . . . 13 (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
222, 21syl 17 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺:ℂ⟶ℂ)
23 mulcom 11226 . . . . . . . . . . . . 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 7721 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹f · 𝐺) = (𝐺f · 𝐹))
266, 25eqtrid 2777 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐻 = (𝐺f · 𝐹))
2726oveq1d 7434 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻f − (𝐺f · (𝐻 quot 𝐺))) = ((𝐺f · 𝐹) ∘f − (𝐺f · (𝐻 quot 𝐺))))
28 plyf 26177 . . . . . . . . . . 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 11679 . . . . . . . . . . 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 7725 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐺f · (𝐹f − (𝐻 quot 𝐺))) = ((𝐺f · 𝐹) ∘f − (𝐺f · (𝐻 quot 𝐺))))
3327, 32eqtr4d 2768 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻f − (𝐺f · (𝐻 quot 𝐺))) = (𝐺f · (𝐹f − (𝐻 quot 𝐺))))
3433eqeq1d 2727 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐻f − (𝐺f · (𝐻 quot 𝐺))) = 0𝑝 ↔ (𝐺f · (𝐹f − (𝐻 quot 𝐺))) = 0𝑝))
3510neneqd 2934 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ¬ 𝐺 = 0𝑝)
36 biorf 934 . . . . . . . 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 310 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐻f − (𝐺f · (𝐻 quot 𝐺))) = 0𝑝 ↔ (𝐹f − (𝐻 quot 𝐺)) = 0𝑝))
3938biimpd 228 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐻f − (𝐺f · (𝐻 quot 𝐺))) = 0𝑝 → (𝐹f − (𝐻 quot 𝐺)) = 0𝑝))
40 eqid 2725 . . . . . . . . . . 11 (deg‘𝐺) = (deg‘𝐺)
41 eqid 2725 . . . . . . . . . . 11 (deg‘(𝐹f − (𝐻 quot 𝐺))) = (deg‘(𝐹f − (𝐻 quot 𝐺)))
4240, 41dgrmul 26250 . . . . . . . . . 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 455 . . . . . . . . 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 836 . . . . . . . 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 26212 . . . . . . . . . . . 12 (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
462, 45syl 17 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘𝐺) ∈ ℕ0)
4746nn0red 12566 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘𝐺) ∈ ℝ)
48 dgrcl 26212 . . . . . . . . . . 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 12551 . . . . . . . . . 10 (((deg‘𝐺) ∈ ℝ ∧ (deg‘(𝐹f − (𝐻 quot 𝐺))) ∈ ℕ0) → (deg‘𝐺) ≤ ((deg‘𝐺) + (deg‘(𝐹f − (𝐻 quot 𝐺)))))
5147, 49, 50syl2anc 582 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘𝐺) ≤ ((deg‘𝐺) + (deg‘(𝐹f − (𝐻 quot 𝐺)))))
52 breq2 5153 . . . . . . . . 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 246 . . . . . . . 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 6900 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘(𝐻f − (𝐺f · (𝐻 quot 𝐺)))) = (deg‘(𝐺f · (𝐹f − (𝐻 quot 𝐺)))))
5655breq2d 5161 . . . . . . . 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 26200 . . . . . . . . . . . . 13 ((𝐺 ∈ (Poly‘ℂ) ∧ (𝐻 quot 𝐺) ∈ (Poly‘ℂ)) → (𝐺f · (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
583, 12, 57syl2anc 582 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐺f · (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
59 plysubcl 26201 . . . . . . . . . . . 12 ((𝐻 ∈ (Poly‘ℂ) ∧ (𝐺f · (𝐻 quot 𝐺)) ∈ (Poly‘ℂ)) → (𝐻f − (𝐺f · (𝐻 quot 𝐺))) ∈ (Poly‘ℂ))
609, 58, 59syl2anc 582 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻f − (𝐺f · (𝐻 quot 𝐺))) ∈ (Poly‘ℂ))
61 dgrcl 26212 . . . . . . . . . . 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 12566 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘(𝐻f − (𝐺f · (𝐻 quot 𝐺)))) ∈ ℝ)
6447, 63lenltd 11392 . . . . . . . 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 280 . . . . . . 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 238 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐹f − (𝐻 quot 𝐺)) ≠ 0𝑝 → ¬ (deg‘(𝐻f − (𝐺f · (𝐻 quot 𝐺)))) < (deg‘𝐺)))
6766necon4ad 2948 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((deg‘(𝐻f − (𝐺f · (𝐻 quot 𝐺)))) < (deg‘𝐺) → (𝐹f − (𝐻 quot 𝐺)) = 0𝑝))
68 eqid 2725 . . . . . . 7 (𝐻f − (𝐺f · (𝐻 quot 𝐺))) = (𝐻f − (𝐺f · (𝐻 quot 𝐺)))
6968quotdgr 26283 . . . . . 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 1368 . . . . 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 858 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹f − (𝐻 quot 𝐺)) = 0𝑝)
72 df-0p 25643 . . . 4 0𝑝 = (ℂ × {0})
7371, 72eqtrdi 2781 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹f − (𝐻 quot 𝐺)) = (ℂ × {0}))
74 ofsubeq0 12242 . . . 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 1368 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐹f − (𝐻 quot 𝐺)) = (ℂ × {0}) ↔ 𝐹 = (𝐻 quot 𝐺)))
7673, 75mpbid 231 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 = (𝐻 quot 𝐺))
7776eqcomd 2731 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 205  wa 394  wo 845  w3a 1084   = wceq 1533  wcel 2098  wne 2929  Vcvv 3461  {csn 4630   class class class wbr 5149   × cxp 5676  wf 6545  cfv 6549  (class class class)co 7419  f cof 7683  cc 11138  cr 11139  0cc0 11140   + caddc 11143   · cmul 11145   < clt 11280  cle 11281  cmin 11476  0cn0 12505  0𝑝c0p 25642  Polycply 26163  degcdgr 26166   quot cquot 26270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7685  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-map 8847  df-pm 8848  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9467  df-inf 9468  df-oi 9535  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-3 12309  df-n0 12506  df-z 12592  df-uz 12856  df-rp 13010  df-fz 13520  df-fzo 13663  df-fl 13793  df-seq 14003  df-exp 14063  df-hash 14326  df-cj 15082  df-re 15083  df-im 15084  df-sqrt 15218  df-abs 15219  df-clim 15468  df-rlim 15469  df-sum 15669  df-0p 25643  df-ply 26167  df-coe 26169  df-dgr 26170  df-quot 26271
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator