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Theorem quotcan 25669
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 25561 . . . . . . . . 9 (Poly‘𝑆) ⊆ (Poly‘ℂ)
2 simp2 1137 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ∈ (Poly‘𝑆))
31, 2sselid 3942 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ∈ (Poly‘ℂ))
4 simp1 1136 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 ∈ (Poly‘𝑆))
51, 4sselid 3942 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹 ∈ (Poly‘ℂ))
6 quotcan.1 . . . . . . . . . . . 12 𝐻 = (𝐹f · 𝐺)
7 plymulcl 25582 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → (𝐹f · 𝐺) ∈ (Poly‘ℂ))
86, 7eqeltrid 2842 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆)) → 𝐻 ∈ (Poly‘ℂ))
983adant3 1132 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐻 ∈ (Poly‘ℂ))
10 simp3 1138 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺 ≠ 0𝑝)
11 quotcl2 25662 . . . . . . . . . 10 ((𝐻 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) ∈ (Poly‘ℂ))
129, 3, 10, 11syl3anc 1371 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻 quot 𝐺) ∈ (Poly‘ℂ))
13 plysubcl 25583 . . . . . . . . 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 25641 . . . . . . . 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 11132 . . . . . . . . . . . . 13 ℂ ∈ V
1817a1i 11 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ℂ ∈ V)
19 plyf 25559 . . . . . . . . . . . . 13 (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
204, 19syl 17 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐹:ℂ⟶ℂ)
21 plyf 25559 . . . . . . . . . . . . 13 (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
222, 21syl 17 . . . . . . . . . . . 12 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → 𝐺:ℂ⟶ℂ)
23 mulcom 11137 . . . . . . . . . . . . 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 7652 . . . . . . . . . . 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 7372 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐻f − (𝐺f · (𝐻 quot 𝐺))) = ((𝐺f · 𝐹) ∘f − (𝐺f · (𝐻 quot 𝐺))))
28 plyf 25559 . . . . . . . . . . 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 11588 . . . . . . . . . . 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 7656 . . . . . . . . 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 2948 . . . . . . . 8 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → ¬ 𝐺 = 0𝑝)
36 biorf 935 . . . . . . . 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 25631 . . . . . . . . . 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 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 25594 . . . . . . . . . . . 12 (𝐺 ∈ (Poly‘𝑆) → (deg‘𝐺) ∈ ℕ0)
462, 45syl 17 . . . . . . . . . . 11 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘𝐺) ∈ ℕ0)
4746nn0red 12474 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘𝐺) ∈ ℝ)
48 dgrcl 25594 . . . . . . . . . . 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 12459 . . . . . . . . . 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 5109 . . . . . . . . 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 6846 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘(𝐻f − (𝐺f · (𝐻 quot 𝐺)))) = (deg‘(𝐺f · (𝐹f − (𝐻 quot 𝐺)))))
5655breq2d 5117 . . . . . . . 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 25582 . . . . . . . . . . . . 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 25583 . . . . . . . . . . . 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 25594 . . . . . . . . . . 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 12474 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (deg‘(𝐻f − (𝐺f · (𝐻 quot 𝐺)))) ∈ ℝ)
6447, 63lenltd 11301 . . . . . . . 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 2962 . . . . 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 25663 . . . . . 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 1371 . . . . 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 25034 . . . 4 0𝑝 = (ℂ × {0})
7371, 72eqtrdi 2792 . . 3 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹f − (𝐻 quot 𝐺)) = (ℂ × {0}))
74 ofsubeq0 12150 . . . 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 1371 . . 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 845  w3a 1087   = wceq 1541  wcel 2106  wne 2943  Vcvv 3445  {csn 4586   class class class wbr 5105   × cxp 5631  wf 6492  cfv 6496  (class class class)co 7357  f cof 7615  cc 11049  cr 11050  0cc0 11051   + caddc 11054   · cmul 11056   < clt 11189  cle 11190  cmin 11385  0cn0 12413  0𝑝c0p 25033  Polycply 25545  degcdgr 25548   quot cquot 25650
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-sum 15571  df-0p 25034  df-ply 25549  df-coe 25551  df-dgr 25552  df-quot 25651
This theorem is referenced by: (None)
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