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Theorem quotcan 24908
 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 24800 . . . . . . . . 9 (Poly‘𝑆) ⊆ (Poly‘ℂ)
2 simp2 1134 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ∈ (Poly‘𝑆))
31, 2sseldi 3916 . . . . . . . 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, 4sseldi 3916 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 ∈ (Poly‘ℂ))
6 quotcan.1 . . . . . . . . . . . 12 𝐻 = (𝐹f · 𝐺)
7 plymulcl 24821 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹f · 𝐺) ∈ (Poly‘ℂ))
86, 7eqeltrid 2897 . . . . . . . . . . 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 24901 . . . . . . . . . 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 24822 . . . . . . . . 9 ((𝐹 ∈ (Poly‘ℂ) ∧ (𝐻 quot 𝐺) ∈ (Poly‘ℂ)) → (𝐹f − (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
145, 12, 13syl2anc 587 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹f − (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
15 plymul0or 24880 . . . . . . . 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 587 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐺f · (𝐹f − (𝐻 quot 𝐺))) = 0𝑝 ↔ (𝐺 = 0𝑝 ∨ (𝐹f − (𝐻 quot 𝐺)) = 0𝑝)))
17 cnex 10611 . . . . . . . . . . . . 13 ℂ ∈ V
1817a1i 11 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ℂ ∈ V)
19 plyf 24798 . . . . . . . . . . . . 13 (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
204, 19syl 17 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹:ℂ⟶ℂ)
21 plyf 24798 . . . . . . . . . . . . 13 (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
222, 21syl 17 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺:ℂ⟶ℂ)
23 mulcom 10616 . . . . . . . . . . . . 13 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) = (𝑦 · 𝑥))
2423adantl 485 . . . . . . . . . . . 12 (((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) = (𝑦 · 𝑥))
2518, 20, 22, 24caofcom 7425 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹f · 𝐺) = (𝐺f · 𝐹))
266, 25syl5eq 2848 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐻 = (𝐺f · 𝐹))
2726oveq1d 7154 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻f − (𝐺f · (𝐻 quot 𝐺))) = ((𝐺f · 𝐹) ∘f − (𝐺f · (𝐻 quot 𝐺))))
28 plyf 24798 . . . . . . . . . . 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 11066 . . . . . . . . . . 11 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 · (𝑦𝑧)) = ((𝑥 · 𝑦) − (𝑥 · 𝑧)))
3130adantl 485 . . . . . . . . . 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 7429 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐺f · (𝐹f − (𝐻 quot 𝐺))) = ((𝐺f · 𝐹) ∘f − (𝐺f · (𝐻 quot 𝐺))))
3327, 32eqtr4d 2839 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻f − (𝐺f · (𝐻 quot 𝐺))) = (𝐺f · (𝐹f − (𝐻 quot 𝐺))))
3433eqeq1d 2803 . . . . . . 7 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐻f − (𝐺f · (𝐻 quot 𝐺))) = 0𝑝 ↔ (𝐺f · (𝐹f − (𝐻 quot 𝐺))) = 0𝑝))
3510neneqd 2995 . . . . . . . 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 314 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐻f − (𝐺f · (𝐻 quot 𝐺))) = 0𝑝 ↔ (𝐹f − (𝐻 quot 𝐺)) = 0𝑝))
3938biimpd 232 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐻f − (𝐺f · (𝐻 quot 𝐺))) = 0𝑝 → (𝐹f − (𝐻 quot 𝐺)) = 0𝑝))
40 eqid 2801 . . . . . . . . . . 11 (deg‘𝐺) = (deg‘𝐺)
41 eqid 2801 . . . . . . . . . . 11 (deg‘(𝐹f − (𝐻 quot 𝐺))) = (deg‘(𝐹f − (𝐻 quot 𝐺)))
4240, 41dgrmul 24870 . . . . . . . . . 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 460 . . . . . . . . 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 24833 . . . . . . . . . . . 12 (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
462, 45syl 17 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘𝐺) ∈ ℕ0)
4746nn0red 11948 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘𝐺) ∈ ℝ)
48 dgrcl 24833 . . . . . . . . . . 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 11935 . . . . . . . . . 10 (((deg‘𝐺) ∈ ℝ ∧ (deg‘(𝐹f − (𝐻 quot 𝐺))) ∈ ℕ0) → (deg‘𝐺) ≤ ((deg‘𝐺) + (deg‘(𝐹f − (𝐻 quot 𝐺)))))
5147, 49, 50syl2anc 587 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘𝐺) ≤ ((deg‘𝐺) + (deg‘(𝐹f − (𝐻 quot 𝐺)))))
52 breq2 5037 . . . . . . . . 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 250 . . . . . . . 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 6653 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘(𝐻f − (𝐺f · (𝐻 quot 𝐺)))) = (deg‘(𝐺f · (𝐹f − (𝐻 quot 𝐺)))))
5655breq2d 5045 . . . . . . . 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 24821 . . . . . . . . . . . . 13 ((𝐺 ∈ (Poly‘ℂ) ∧ (𝐻 quot 𝐺) ∈ (Poly‘ℂ)) → (𝐺f · (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
583, 12, 57syl2anc 587 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐺f · (𝐻 quot 𝐺)) ∈ (Poly‘ℂ))
59 plysubcl 24822 . . . . . . . . . . . 12 ((𝐻 ∈ (Poly‘ℂ) ∧ (𝐺f · (𝐻 quot 𝐺)) ∈ (Poly‘ℂ)) → (𝐻f − (𝐺f · (𝐻 quot 𝐺))) ∈ (Poly‘ℂ))
609, 58, 59syl2anc 587 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻f − (𝐺f · (𝐻 quot 𝐺))) ∈ (Poly‘ℂ))
61 dgrcl 24833 . . . . . . . . . . 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 11948 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘(𝐻f − (𝐺f · (𝐻 quot 𝐺)))) ∈ ℝ)
6447, 63lenltd 10779 . . . . . . . 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 284 . . . . . . 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 242 . . . . . 6 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((𝐹f − (𝐻 quot 𝐺)) ≠ 0𝑝 → ¬ (deg‘(𝐻f − (𝐺f · (𝐻 quot 𝐺)))) < (deg‘𝐺)))
6766necon4ad 3009 . . . . 5 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ((deg‘(𝐻f − (𝐺f · (𝐻 quot 𝐺)))) < (deg‘𝐺) → (𝐹f − (𝐻 quot 𝐺)) = 0𝑝))
68 eqid 2801 . . . . . . 7 (𝐻f − (𝐺f · (𝐻 quot 𝐺))) = (𝐻f − (𝐺f · (𝐻 quot 𝐺)))
6968quotdgr 24902 . . . . . 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 857 . . . 4 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹f − (𝐻 quot 𝐺)) = 0𝑝)
72 df-0p 24277 . . . 4 0𝑝 = (ℂ × {0})
7371, 72eqtrdi 2852 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹f − (𝐻 quot 𝐺)) = (ℂ × {0}))
74 ofsubeq0 11626 . . . 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 235 . 2 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 = (𝐻 quot 𝐺))
7776eqcomd 2807 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 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112   ≠ wne 2990  Vcvv 3444  {csn 4528   class class class wbr 5033   × cxp 5521  ⟶wf 6324  ‘cfv 6328  (class class class)co 7139   ∘f cof 7391  ℂcc 10528  ℝcr 10529  0cc0 10530   + caddc 10533   · cmul 10535   < clt 10668   ≤ cle 10669   − cmin 10863  ℕ0cn0 11889  0𝑝c0p 24276  Polycply 24784  degcdgr 24787   quot cquot 24889 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608  ax-addf 10609 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-rp 12382  df-fz 12890  df-fzo 13033  df-fl 13161  df-seq 13369  df-exp 13430  df-hash 13691  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-clim 14840  df-rlim 14841  df-sum 15038  df-0p 24277  df-ply 24788  df-coe 24790  df-dgr 24791  df-quot 24890 This theorem is referenced by: (None)
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