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Theorem quotcan 25813
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 25705 . . . . . . . . 9 (Polyβ€˜π‘†) βŠ† (Polyβ€˜β„‚)
2 simp2 1137 . . . . . . . . 9 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ 𝐺 ∈ (Polyβ€˜π‘†))
31, 2sselid 3979 . . . . . . . 8 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ 𝐺 ∈ (Polyβ€˜β„‚))
4 simp1 1136 . . . . . . . . . 10 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ 𝐹 ∈ (Polyβ€˜π‘†))
51, 4sselid 3979 . . . . . . . . 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 25726 . . . . . . . . . . . 12 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ (𝐹 ∘f Β· 𝐺) ∈ (Polyβ€˜β„‚))
86, 7eqeltrid 2837 . . . . . . . . . . 11 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†)) β†’ 𝐻 ∈ (Polyβ€˜β„‚))
983adant3 1132 . . . . . . . . . 10 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ 𝐻 ∈ (Polyβ€˜β„‚))
10 simp3 1138 . . . . . . . . . 10 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ 𝐺 β‰  0𝑝)
11 quotcl2 25806 . . . . . . . . . 10 ((𝐻 ∈ (Polyβ€˜β„‚) ∧ 𝐺 ∈ (Polyβ€˜β„‚) ∧ 𝐺 β‰  0𝑝) β†’ (𝐻 quot 𝐺) ∈ (Polyβ€˜β„‚))
129, 3, 10, 11syl3anc 1371 . . . . . . . . 9 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (𝐻 quot 𝐺) ∈ (Polyβ€˜β„‚))
13 plysubcl 25727 . . . . . . . . 9 ((𝐹 ∈ (Polyβ€˜β„‚) ∧ (𝐻 quot 𝐺) ∈ (Polyβ€˜β„‚)) β†’ (𝐹 ∘f βˆ’ (𝐻 quot 𝐺)) ∈ (Polyβ€˜β„‚))
145, 12, 13syl2anc 584 . . . . . . . 8 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (𝐹 ∘f βˆ’ (𝐻 quot 𝐺)) ∈ (Polyβ€˜β„‚))
15 plymul0or 25785 . . . . . . . 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 584 . . . . . . 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 11187 . . . . . . . . . . . . 13 β„‚ ∈ V
1817a1i 11 . . . . . . . . . . . 12 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ β„‚ ∈ V)
19 plyf 25703 . . . . . . . . . . . . 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 25703 . . . . . . . . . . . . 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 11192 . . . . . . . . . . . . 13 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚) β†’ (π‘₯ Β· 𝑦) = (𝑦 Β· π‘₯))
2423adantl 482 . . . . . . . . . . . 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 7701 . . . . . . . . . . 11 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (𝐹 ∘f Β· 𝐺) = (𝐺 ∘f Β· 𝐹))
266, 25eqtrid 2784 . . . . . . . . . 10 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ 𝐻 = (𝐺 ∘f Β· 𝐹))
2726oveq1d 7420 . . . . . . . . 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 25703 . . . . . . . . . . 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 11643 . . . . . . . . . . 11 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚ ∧ 𝑧 ∈ β„‚) β†’ (π‘₯ Β· (𝑦 βˆ’ 𝑧)) = ((π‘₯ Β· 𝑦) βˆ’ (π‘₯ Β· 𝑧)))
3130adantl 482 . . . . . . . . . 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 7705 . . . . . . . . 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 2775 . . . . . . . 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 2734 . . . . . . 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 2945 . . . . . . . 8 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ Β¬ 𝐺 = 0𝑝)
36 biorf 935 . . . . . . . 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 310 . . . . . 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 2732 . . . . . . . . . . 11 (degβ€˜πΊ) = (degβ€˜πΊ)
41 eqid 2732 . . . . . . . . . . 11 (degβ€˜(𝐹 ∘f βˆ’ (𝐻 quot 𝐺))) = (degβ€˜(𝐹 ∘f βˆ’ (𝐻 quot 𝐺)))
4240, 41dgrmul 25775 . . . . . . . . . 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 457 . . . . . . . . 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 836 . . . . . . . 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 25738 . . . . . . . . . . . 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 12529 . . . . . . . . . 10 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (degβ€˜πΊ) ∈ ℝ)
48 dgrcl 25738 . . . . . . . . . . 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 12514 . . . . . . . . . 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 584 . . . . . . . . 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 5151 . . . . . . . . 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 6892 . . . . . . . . 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 5159 . . . . . . . 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 25726 . . . . . . . . . . . . 13 ((𝐺 ∈ (Polyβ€˜β„‚) ∧ (𝐻 quot 𝐺) ∈ (Polyβ€˜β„‚)) β†’ (𝐺 ∘f Β· (𝐻 quot 𝐺)) ∈ (Polyβ€˜β„‚))
583, 12, 57syl2anc 584 . . . . . . . . . . . 12 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (𝐺 ∘f Β· (𝐻 quot 𝐺)) ∈ (Polyβ€˜β„‚))
59 plysubcl 25727 . . . . . . . . . . . 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 584 . . . . . . . . . . 11 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (𝐻 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺))) ∈ (Polyβ€˜β„‚))
61 dgrcl 25738 . . . . . . . . . . 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 12529 . . . . . . . . 9 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (degβ€˜(𝐻 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺)))) ∈ ℝ)
6447, 63lenltd 11356 . . . . . . . 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 280 . . . . . . 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 2959 . . . . 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 2732 . . . . . . 7 (𝐻 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺))) = (𝐻 ∘f βˆ’ (𝐺 ∘f Β· (𝐻 quot 𝐺)))
6968quotdgr 25807 . . . . . 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 1371 . . . . 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 858 . . . 4 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (𝐹 ∘f βˆ’ (𝐻 quot 𝐺)) = 0𝑝)
72 df-0p 25178 . . . 4 0𝑝 = (β„‚ Γ— {0})
7371, 72eqtrdi 2788 . . 3 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (𝐹 ∘f βˆ’ (𝐻 quot 𝐺)) = (β„‚ Γ— {0}))
74 ofsubeq0 12205 . . . 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 1371 . . 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 2738 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 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  Vcvv 3474  {csn 4627   class class class wbr 5147   Γ— cxp 5673  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405   ∘f cof 7664  β„‚cc 11104  β„cr 11105  0cc0 11106   + caddc 11109   Β· cmul 11111   < clt 11244   ≀ cle 11245   βˆ’ cmin 11440  β„•0cn0 12468  0𝑝c0p 25177  Polycply 25689  degcdgr 25692   quot cquot 25794
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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-inf2 9632  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-pm 8819  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-inf 9434  df-oi 9501  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-z 12555  df-uz 12819  df-rp 12971  df-fz 13481  df-fzo 13624  df-fl 13753  df-seq 13963  df-exp 14024  df-hash 14287  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-clim 15428  df-rlim 15429  df-sum 15629  df-0p 25178  df-ply 25693  df-coe 25695  df-dgr 25696  df-quot 25795
This theorem is referenced by: (None)
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