MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  quotcan Structured version   Visualization version   GIF version
Theorem quotcan 25822
Description: Exact division with a multiple. (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypothesis
Ref Expression
quotcan.1 𝐻 = (𝐹 ∘f · 𝐺)
Assertion
Ref Expression
quotcan ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (𝐻 quot 𝐺) = 𝐹)
Proof of Theorem quotcan
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plyssc 25714 . . . . . . . . 9 (Polyβ€˜π‘†) βŠ† (Polyβ€˜β„‚)
2 simp2 1138 . . . . . . . . 9 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ 𝐺 ∈ (Polyβ€˜π‘†))
31, 2sselid 3981 . . . . . . . 8 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ 𝐺 ∈ (Polyβ€˜β„‚))
4 simp1 1137 . . . . . . . . . 10 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ 𝐹 ∈ (Polyβ€˜π‘†))
51, 4sselid 3981 . . . . . . . . 9 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ 𝐹 ∈ (Polyβ€˜β„‚))
6 quotcan.1 . . . . . . . . . . . 12 𝐻 = (𝐹 ∘f Β· 𝐺)
7 plymulcl 25735 . . . . . . . . . . . 12 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ (𝐹 ∘f Β· 𝐺) ∈ (Polyβ€˜β„‚))
86, 7eqeltrid 2838 . . . . . . . . . . 11 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ 𝐻 ∈ (Polyβ€˜β„‚))
983adant3 1133 . . . . . . . . . 10 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ 𝐻 ∈ (Polyβ€˜β„‚))
10 simp3 1139 . . . . . . . . . 10 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ 𝐺 β‰  0𝑝)
11 quotcl2 25815 . . . . . . . . . 10 ((𝐻 ∈ (Polyβ€˜β„‚) ∧ 𝐺 ∈ (Polyβ€˜β„‚) ∧ 𝐺 β‰  0𝑝) β†’ (𝐻 quot 𝐺) ∈ (Polyβ€˜β„‚))
129, 3, 10, 11syl3anc 1372 . . . . . . . . 9 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (𝐻 quot 𝐺) ∈ (Polyβ€˜β„‚))
13 plysubcl 25736 . . . . . . . . 9 ((𝐹 ∈ (Polyβ€˜β„‚) ∧ (𝐻 quot 𝐺) ∈ (Polyβ€˜β„‚)) β†’ (𝐹 ∘f βˆ’ (𝐻 quot 𝐺)) ∈ (Polyβ€˜β„‚))
145, 12, 13syl2anc 585 . . . . . . . 8 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (𝐹 ∘f βˆ’ (𝐻 quot 𝐺)) ∈ (Polyβ€˜β„‚))
15 plymul0or 25794 . . . . . . . 8 ((𝐺 ∈ (Polyβ€˜β„‚) ∧ (𝐹 ∘f βˆ’ (𝐻 quot 𝐺)) ∈ (Polyβ€˜β„‚)) β†’ ((𝐺 ∘f Β· (𝐹 ∘f βˆ’ (𝐻 quot 𝐺))) = 0𝑝 ↔ (𝐺 = 0𝑝 ∨ (𝐹 ∘f βˆ’ (𝐻 quot 𝐺)) = 0𝑝)))
163, 14, 15syl2anc 585 . . . . . . 7 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ ((𝐺 ∘f Β· (𝐹 ∘f βˆ’ (𝐻 quot 𝐺))) = 0𝑝 ↔ (𝐺 = 0𝑝 ∨ (𝐹 ∘f βˆ’ (𝐻 quot 𝐺)) = 0𝑝)))
17 cnex 11191 . . . . . . . . . . . . 13 β„‚ ∈ V
1817a1i 11 . . . . . . . . . . . 12 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ β„‚ ∈ V)
19 plyf 25712 . . . . . . . . . . . . 13 (𝐹 ∈ (Polyβ€˜π‘†) β†’ 𝐹:β„‚βŸΆβ„‚)
204, 19syl 17 . . . . . . . . . . . 12 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ 𝐹:β„‚βŸΆβ„‚)
21 plyf 25712 . . . . . . . . . . . . 13 (𝐺 ∈ (Polyβ€˜π‘†) β†’ 𝐺:β„‚βŸΆβ„‚)
222, 21syl 17 . . . . . . . . . . . 12 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ 𝐺:β„‚βŸΆβ„‚)
23 mulcom 11196 . . . . . . . . . . . . 13 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚) β†’ (π‘₯ Β· 𝑦) = (𝑦 Β· π‘₯))
2423adantl 483 . . . . . . . . . . . 12 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) ∧ (π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚)) β†’ (π‘₯ Β· 𝑦) = (𝑦 Β· π‘₯))
2518, 20, 22, 24caofcom 7705 . . . . . . . . . . 11 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (𝐹 ∘f Β· 𝐺) = (𝐺 ∘f Β· 𝐹))
266, 25eqtrid 2785 . . . . . . . . . 10 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ 𝐻 = (𝐺 ∘f Β· 𝐹))
2726oveq1d 7424 . . . . . . . . 9 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (𝐻 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺))) = ((𝐺 ∘f Β· 𝐹) ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺))))
28 plyf 25712 . . . . . . . . . . 11 ((𝐻 quot 𝐺) ∈ (Polyβ€˜β„‚) β†’ (𝐻 quot 𝐺):β„‚βŸΆβ„‚)
2912, 28syl 17 . . . . . . . . . 10 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (𝐻 quot 𝐺):β„‚βŸΆβ„‚)
30 subdi 11647 . . . . . . . . . . 11 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚ ∧ 𝑧 ∈ β„‚) β†’ (π‘₯ Β· (𝑦 βˆ’ 𝑧)) = ((π‘₯ Β· 𝑦) βˆ’ (π‘₯ Β· 𝑧)))
3130adantl 483 . . . . . . . . . 10 (((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) ∧ (π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚ ∧ 𝑧 ∈ β„‚)) β†’ (π‘₯ Β· (𝑦 βˆ’ 𝑧)) = ((π‘₯ Β· 𝑦) βˆ’ (π‘₯ Β· 𝑧)))
3218, 22, 20, 29, 31caofdi 7709 . . . . . . . . 9 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (𝐺 ∘f Β· (𝐹 ∘f βˆ’ (𝐻 quot 𝐺))) = ((𝐺 ∘f Β· 𝐹) ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺))))
3327, 32eqtr4d 2776 . . . . . . . 8 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (𝐻 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺))) = (𝐺 ∘f Β· (𝐹 ∘f βˆ’ (𝐻 quot 𝐺))))
3433eqeq1d 2735 . . . . . . 7 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ ((𝐻 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺))) = 0𝑝 ↔ (𝐺 ∘f Β· (𝐹 ∘f βˆ’ (𝐻 quot 𝐺))) = 0𝑝))
3510neneqd 2946 . . . . . . . 8 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ Β¬ 𝐺 = 0𝑝)
36 biorf 936 . . . . . . . 8 (Β¬ 𝐺 = 0𝑝 β†’ ((𝐹 ∘f βˆ’ (𝐻 quot 𝐺)) = 0𝑝 ↔ (𝐺 = 0𝑝 ∨ (𝐹 ∘f βˆ’ (𝐻 quot 𝐺)) = 0𝑝)))
3735, 36syl 17 . . . . . . 7 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ ((𝐹 ∘f βˆ’ (𝐻 quot 𝐺)) = 0𝑝 ↔ (𝐺 = 0𝑝 ∨ (𝐹 ∘f βˆ’ (𝐻 quot 𝐺)) = 0𝑝)))
3816, 34, 373bitr4d 311 . . . . . 6 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ ((𝐻 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺))) = 0𝑝 ↔ (𝐹 ∘f βˆ’ (𝐻 quot 𝐺)) = 0𝑝))
3938biimpd 228 . . . . 5 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ ((𝐻 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺))) = 0𝑝 β†’ (𝐹 ∘f βˆ’ (𝐻 quot 𝐺)) = 0𝑝))
40 eqid 2733 . . . . . . . . . . 11 (degβ€˜πΊ) = (degβ€˜πΊ)
41 eqid 2733 . . . . . . . . . . 11 (degβ€˜(𝐹 ∘f βˆ’ (𝐻 quot 𝐺))) = (degβ€˜(𝐹 ∘f βˆ’ (𝐻 quot 𝐺)))
4240, 41dgrmul 25784 . . . . . . . . . 10 (((𝐺 ∈ (Polyβ€˜β„‚) ∧ 𝐺 β‰  0𝑝) ∧ ((𝐹 ∘f βˆ’ (𝐻 quot 𝐺)) ∈ (Polyβ€˜β„‚) ∧ (𝐹 ∘f βˆ’ (𝐻 quot 𝐺)) β‰  0𝑝)) β†’ (degβ€˜(𝐺 ∘f Β· (𝐹 ∘f βˆ’ (𝐻 quot 𝐺)))) = ((degβ€˜πΊ) + (degβ€˜(𝐹 ∘f βˆ’ (𝐻 quot 𝐺)))))
4342expr 458 . . . . . . . . 9 (((𝐺 ∈ (Polyβ€˜β„‚) ∧ 𝐺 β‰  0𝑝) ∧ (𝐹 ∘f βˆ’ (𝐻 quot 𝐺)) ∈ (Polyβ€˜β„‚)) β†’ ((𝐹 ∘f βˆ’ (𝐻 quot 𝐺)) β‰  0𝑝 β†’ (degβ€˜(𝐺 ∘f Β· (𝐹 ∘f βˆ’ (𝐻 quot 𝐺)))) = ((degβ€˜πΊ) + (degβ€˜(𝐹 ∘f βˆ’ (𝐻 quot 𝐺))))))
443, 10, 14, 43syl21anc 837 . . . . . . . 8 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ ((𝐹 ∘f βˆ’ (𝐻 quot 𝐺)) β‰  0𝑝 β†’ (degβ€˜(𝐺 ∘f Β· (𝐹 ∘f βˆ’ (𝐻 quot 𝐺)))) = ((degβ€˜πΊ) + (degβ€˜(𝐹 ∘f βˆ’ (𝐻 quot 𝐺))))))
45 dgrcl 25747 . . . . . . . . . . . 12 (𝐺 ∈ (Polyβ€˜π‘†) β†’ (degβ€˜πΊ) ∈ β„•0)
462, 45syl 17 . . . . . . . . . . 11 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (degβ€˜πΊ) ∈ β„•0)
4746nn0red 12533 . . . . . . . . . 10 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (degβ€˜πΊ) ∈ ℝ)
48 dgrcl 25747 . . . . . . . . . . 11 ((𝐹 ∘f βˆ’ (𝐻 quot 𝐺)) ∈ (Polyβ€˜β„‚) β†’ (degβ€˜(𝐹 ∘f βˆ’ (𝐻 quot 𝐺))) ∈ β„•0)
4914, 48syl 17 . . . . . . . . . 10 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (degβ€˜(𝐹 ∘f βˆ’ (𝐻 quot 𝐺))) ∈ β„•0)
50 nn0addge1 12518 . . . . . . . . . 10 (((degβ€˜πΊ) ∈ ℝ ∧ (degβ€˜(𝐹 ∘f βˆ’ (𝐻 quot 𝐺))) ∈ β„•0) β†’ (degβ€˜πΊ) ≀ ((degβ€˜πΊ) + (degβ€˜(𝐹 ∘f βˆ’ (𝐻 quot 𝐺)))))
5147, 49, 50syl2anc 585 . . . . . . . . 9 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (degβ€˜πΊ) ≀ ((degβ€˜πΊ) + (degβ€˜(𝐹 ∘f βˆ’ (𝐻 quot 𝐺)))))
52 breq2 5153 . . . . . . . . 9 ((degβ€˜(𝐺 ∘f Β· (𝐹 ∘f βˆ’ (𝐻 quot 𝐺)))) = ((degβ€˜πΊ) + (degβ€˜(𝐹 ∘f βˆ’ (𝐻 quot 𝐺)))) β†’ ((degβ€˜πΊ) ≀ (degβ€˜(𝐺 ∘f Β· (𝐹 ∘f βˆ’ (𝐻 quot 𝐺)))) ↔ (degβ€˜πΊ) ≀ ((degβ€˜πΊ) + (degβ€˜(𝐹 ∘f βˆ’ (𝐻 quot 𝐺))))))
5351, 52syl5ibrcom 246 . . . . . . . 8 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ ((degβ€˜(𝐺 ∘f Β· (𝐹 ∘f βˆ’ (𝐻 quot 𝐺)))) = ((degβ€˜πΊ) + (degβ€˜(𝐹 ∘f βˆ’ (𝐻 quot 𝐺)))) β†’ (degβ€˜πΊ) ≀ (degβ€˜(𝐺 ∘f Β· (𝐹 ∘f βˆ’ (𝐻 quot 𝐺))))))
5444, 53syld 47 . . . . . . 7 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ ((𝐹 ∘f βˆ’ (𝐻 quot 𝐺)) β‰  0𝑝 β†’ (degβ€˜πΊ) ≀ (degβ€˜(𝐺 ∘f Β· (𝐹 ∘f βˆ’ (𝐻 quot 𝐺))))))
5533fveq2d 6896 . . . . . . . . 9 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (degβ€˜(𝐻 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺)))) = (degβ€˜(𝐺 ∘f Β· (𝐹 ∘f βˆ’ (𝐻 quot 𝐺)))))
5655breq2d 5161 . . . . . . . 8 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ ((degβ€˜πΊ) ≀ (degβ€˜(𝐻 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺)))) ↔ (degβ€˜πΊ) ≀ (degβ€˜(𝐺 ∘f Β· (𝐹 ∘f βˆ’ (𝐻 quot 𝐺))))))
57 plymulcl 25735 . . . . . . . . . . . . 13 ((𝐺 ∈ (Polyβ€˜β„‚) ∧ (𝐻 quot 𝐺) ∈ (Polyβ€˜β„‚)) β†’ (𝐺 ∘f Β· (𝐻 quot 𝐺)) ∈ (Polyβ€˜β„‚))
583, 12, 57syl2anc 585 . . . . . . . . . . . 12 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (𝐺 ∘f Β· (𝐻 quot 𝐺)) ∈ (Polyβ€˜β„‚))
59 plysubcl 25736 . . . . . . . . . . . 12 ((𝐻 ∈ (Polyβ€˜β„‚) ∧ (𝐺 ∘f Β· (𝐻 quot 𝐺)) ∈ (Polyβ€˜β„‚)) β†’ (𝐻 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺))) ∈ (Polyβ€˜β„‚))
609, 58, 59syl2anc 585 . . . . . . . . . . 11 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (𝐻 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺))) ∈ (Polyβ€˜β„‚))
61 dgrcl 25747 . . . . . . . . . . 11 ((𝐻 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺))) ∈ (Polyβ€˜β„‚) β†’ (degβ€˜(𝐻 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺)))) ∈ β„•0)
6260, 61syl 17 . . . . . . . . . 10 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (degβ€˜(𝐻 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺)))) ∈ β„•0)
6362nn0red 12533 . . . . . . . . 9 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (degβ€˜(𝐻 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺)))) ∈ ℝ)
6447, 63lenltd 11360 . . . . . . . 8 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ ((degβ€˜πΊ) ≀ (degβ€˜(𝐻 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺)))) ↔ Β¬ (degβ€˜(𝐻 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺)))) < (degβ€˜πΊ)))
6556, 64bitr3d 281 . . . . . . 7 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ ((degβ€˜πΊ) ≀ (degβ€˜(𝐺 ∘f Β· (𝐹 ∘f βˆ’ (𝐻 quot 𝐺)))) ↔ Β¬ (degβ€˜(𝐻 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺)))) < (degβ€˜πΊ)))
6654, 65sylibd 238 . . . . . 6 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ ((𝐹 ∘f βˆ’ (𝐻 quot 𝐺)) β‰  0𝑝 β†’ Β¬ (degβ€˜(𝐻 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺)))) < (degβ€˜πΊ)))
6766necon4ad 2960 . . . . 5 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ ((degβ€˜(𝐻 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺)))) < (degβ€˜πΊ) β†’ (𝐹 ∘f βˆ’ (𝐻 quot 𝐺)) = 0𝑝))
68 eqid 2733 . . . . . . 7 (𝐻 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺))) = (𝐻 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺)))
6968quotdgr 25816 . . . . . 6 ((𝐻 ∈ (Polyβ€˜β„‚) ∧ 𝐺 ∈ (Polyβ€˜β„‚) ∧ 𝐺 β‰  0𝑝) β†’ ((𝐻 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺))) = 0𝑝 ∨ (degβ€˜(𝐻 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺)))) < (degβ€˜πΊ)))
709, 3, 10, 69syl3anc 1372 . . . . 5 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ ((𝐻 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺))) = 0𝑝 ∨ (degβ€˜(𝐻 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺)))) < (degβ€˜πΊ)))
7139, 67, 70mpjaod 859 . . . 4 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (𝐹 ∘f βˆ’ (𝐻 quot 𝐺)) = 0𝑝)
72 df-0p 25187 . . . 4 0𝑝 = (β„‚ Γ— {0})
7371, 72eqtrdi 2789 . . 3 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (𝐹 ∘f βˆ’ (𝐻 quot 𝐺)) = (β„‚ Γ— {0}))
74 ofsubeq0 12209 . . . 4 ((β„‚ ∈ V ∧ 𝐹:β„‚βŸΆβ„‚ ∧ (𝐻 quot 𝐺):β„‚βŸΆβ„‚) β†’ ((𝐹 ∘f βˆ’ (𝐻 quot 𝐺)) = (β„‚ Γ— {0}) ↔ 𝐹 = (𝐻 quot 𝐺)))
7518, 20, 29, 74syl3anc 1372 . . 3 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ ((𝐹 ∘f βˆ’ (𝐻 quot 𝐺)) = (β„‚ Γ— {0}) ↔ 𝐹 = (𝐻 quot 𝐺)))
7673, 75mpbid 231 . 2 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ 𝐹 = (𝐻 quot 𝐺))
7776eqcomd 2739 1 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (𝐻 quot 𝐺) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  Vcvv 3475  {csn 4629   class class class wbr 5149   Γ— cxp 5675  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ∘f cof 7668  β„‚cc 11108  β„cr 11109  0cc0 11110   + caddc 11113   Β· cmul 11115   < clt 11248   ≀ cle 11249   βˆ’ cmin 11444  β„•0cn0 12472  0𝑝c0p 25186  Polycply 25698  degcdgr 25701   quot cquot 25803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9437  df-inf 9438  df-oi 9505  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-n0 12473  df-z 12559  df-uz 12823  df-rp 12975  df-fz 13485  df-fzo 13628  df-fl 13757  df-seq 13967  df-exp 14028  df-hash 14291  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-clim 15432  df-rlim 15433  df-sum 15633  df-0p 25187  df-ply 25702  df-coe 25704  df-dgr 25705  df-quot 25804
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator