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Theorem quotcan 25575
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 25467 . . . . . . . . 9 (Poly‘𝑆) ⊆ (Poly‘ℂ)
2 simp2 1136 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ∈ (Poly‘𝑆))
31, 2sselid 3930 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ∈ (Poly‘ℂ))
4 simp1 1135 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 ∈ (Poly‘𝑆))
51, 4sselid 3930 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 ∈ (Poly‘ℂ))
6 quotcan.1 . . . . . . . . . . . 12 𝐻 = (𝐹f · 𝐺)
7 plymulcl 25488 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹f · 𝐺) ∈ (Poly‘ℂ))
86, 7eqeltrid 2841 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐻 ∈ (Poly‘ℂ))
983adant3 1131 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐻 ∈ (Poly‘ℂ))
10 simp3 1137 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ≠ 0𝑝)
11 quotcl2 25568 . . . . . . . . . 10 ((𝐻 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) ∈ (Poly‘ℂ))
129, 3, 10, 11syl3anc 1370 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) ∈ (Poly‘ℂ))
13 plysubcl 25489 . . . . . . . . 9 ((𝐹 ∈ (Poly‘ℂ) ∧ (𝐻 quot 𝐺) ∈ (Poly‘ℂ)) → (𝐹f − (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
145, 12, 13syl2anc 584 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹f − (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
15 plymul0or 25547 . . . . . . . 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 584 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐺f · (𝐹f − (𝐻 quot 𝐺))) = 0𝑝 ↔ (𝐺 = 0𝑝 ∨ (𝐹f − (𝐻 quot 𝐺)) = 0𝑝)))
17 cnex 11053 . . . . . . . . . . . . 13 ℂ ∈ V
1817a1i 11 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ℂ ∈ V)
19 plyf 25465 . . . . . . . . . . . . 13 (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
204, 19syl 17 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹:ℂ⟶ℂ)
21 plyf 25465 . . . . . . . . . . . . 13 (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
222, 21syl 17 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺:ℂ⟶ℂ)
23 mulcom 11058 . . . . . . . . . . . . 13 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) = (𝑦 · 𝑥))
2423adantl 482 . . . . . . . . . . . 12 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) = (𝑦 · 𝑥))
2518, 20, 22, 24caofcom 7630 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹f · 𝐺) = (𝐺f · 𝐹))
266, 25eqtrid 2788 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐻 = (𝐺f · 𝐹))
2726oveq1d 7352 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻f − (𝐺f · (𝐻 quot 𝐺))) = ((𝐺f · 𝐹) ∘f − (𝐺f · (𝐻 quot 𝐺))))
28 plyf 25465 . . . . . . . . . . 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 11509 . . . . . . . . . . 11 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 · (𝑦𝑧)) = ((𝑥 · 𝑦) − (𝑥 · 𝑧)))
3130adantl 482 . . . . . . . . . 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 7634 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐺f · (𝐹f − (𝐻 quot 𝐺))) = ((𝐺f · 𝐹) ∘f − (𝐺f · (𝐻 quot 𝐺))))
3327, 32eqtr4d 2779 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻f − (𝐺f · (𝐻 quot 𝐺))) = (𝐺f · (𝐹f − (𝐻 quot 𝐺))))
3433eqeq1d 2738 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐻f − (𝐺f · (𝐻 quot 𝐺))) = 0𝑝 ↔ (𝐺f · (𝐹f − (𝐻 quot 𝐺))) = 0𝑝))
3510neneqd 2945 . . . . . . . 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 2736 . . . . . . . . . . 11 (deg‘𝐺) = (deg‘𝐺)
41 eqid 2736 . . . . . . . . . . 11 (deg‘(𝐹f − (𝐻 quot 𝐺))) = (deg‘(𝐹f − (𝐻 quot 𝐺)))
4240, 41dgrmul 25537 . . . . . . . . . 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 457 . . . . . . . . 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 835 . . . . . . . 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 25500 . . . . . . . . . . . 12 (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
462, 45syl 17 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘𝐺) ∈ ℕ0)
4746nn0red 12395 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘𝐺) ∈ ℝ)
48 dgrcl 25500 . . . . . . . . . . 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 12380 . . . . . . . . . 10 (((deg‘𝐺) ∈ ℝ ∧ (deg‘(𝐹f − (𝐻 quot 𝐺))) ∈ ℕ0) → (deg‘𝐺) ≤ ((deg‘𝐺) + (deg‘(𝐹f − (𝐻 quot 𝐺)))))
5147, 49, 50syl2anc 584 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘𝐺) ≤ ((deg‘𝐺) + (deg‘(𝐹f − (𝐻 quot 𝐺)))))
52 breq2 5096 . . . . . . . . 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 6829 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘(𝐻f − (𝐺f · (𝐻 quot 𝐺)))) = (deg‘(𝐺f · (𝐹f − (𝐻 quot 𝐺)))))
5655breq2d 5104 . . . . . . . 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 25488 . . . . . . . . . . . . 13 ((𝐺 ∈ (Poly‘ℂ) ∧ (𝐻 quot 𝐺) ∈ (Poly‘ℂ)) → (𝐺f · (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
583, 12, 57syl2anc 584 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐺f · (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
59 plysubcl 25489 . . . . . . . . . . . 12 ((𝐻 ∈ (Poly‘ℂ) ∧ (𝐺f · (𝐻 quot 𝐺)) ∈ (Poly‘ℂ)) → (𝐻f − (𝐺f · (𝐻 quot 𝐺))) ∈ (Poly‘ℂ))
609, 58, 59syl2anc 584 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻f − (𝐺f · (𝐻 quot 𝐺))) ∈ (Poly‘ℂ))
61 dgrcl 25500 . . . . . . . . . . 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 12395 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘(𝐻f − (𝐺f · (𝐻 quot 𝐺)))) ∈ ℝ)
6447, 63lenltd 11222 . . . . . . . 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 2959 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((deg‘(𝐻f − (𝐺f · (𝐻 quot 𝐺)))) < (deg‘𝐺) → (𝐹f − (𝐻 quot 𝐺)) = 0𝑝))
68 eqid 2736 . . . . . . 7 (𝐻f − (𝐺f · (𝐻 quot 𝐺))) = (𝐻f − (𝐺f · (𝐻 quot 𝐺)))
6968quotdgr 25569 . . . . . 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 1370 . . . . 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 857 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹f − (𝐻 quot 𝐺)) = 0𝑝)
72 df-0p 24940 . . . 4 0𝑝 = (ℂ × {0})
7371, 72eqtrdi 2792 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹f − (𝐻 quot 𝐺)) = (ℂ × {0}))
74 ofsubeq0 12071 . . . 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 1370 . . 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 2742 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 396  wo 844  w3a 1086   = wceq 1540  wcel 2105  wne 2940  Vcvv 3441  {csn 4573   class class class wbr 5092   × cxp 5618  wf 6475  cfv 6479  (class class class)co 7337  f cof 7593  cc 10970  cr 10971  0cc0 10972   + caddc 10975   · cmul 10977   < clt 11110  cle 11111  cmin 11306  0cn0 12334  0𝑝c0p 24939  Polycply 25451  degcdgr 25454   quot cquot 25556
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-inf2 9498  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-pre-mulgt0 11049  ax-pre-sup 11050
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-int 4895  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-se 5576  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-isom 6488  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-of 7595  df-om 7781  df-1st 7899  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-1o 8367  df-er 8569  df-map 8688  df-pm 8689  df-en 8805  df-dom 8806  df-sdom 8807  df-fin 8808  df-sup 9299  df-inf 9300  df-oi 9367  df-card 9796  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-sub 11308  df-neg 11309  df-div 11734  df-nn 12075  df-2 12137  df-3 12138  df-n0 12335  df-z 12421  df-uz 12684  df-rp 12832  df-fz 13341  df-fzo 13484  df-fl 13613  df-seq 13823  df-exp 13884  df-hash 14146  df-cj 14909  df-re 14910  df-im 14911  df-sqrt 15045  df-abs 15046  df-clim 15296  df-rlim 15297  df-sum 15497  df-0p 24940  df-ply 25455  df-coe 25457  df-dgr 25458  df-quot 25557
This theorem is referenced by: (None)
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