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Theorem quotval 24808
Description: Value of the quotient function. (Contributed by Mario Carneiro, 23-Jul-2014.)
Hypothesis
Ref Expression
quotval.1 𝑅 = (𝐹f − (𝐺f · 𝑞))
Assertion
Ref Expression
quotval ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) = (𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
Distinct variable groups:   𝐹,𝑞   𝐺,𝑞
Allowed substitution hints:   𝑅(𝑞)   𝑆(𝑞)

Proof of Theorem quotval
Dummy variables 𝑓 𝑔 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plyssc 24717 . . 3 (Poly‘𝑆) ⊆ (Poly‘ℂ)
21sseli 3960 . 2 (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
31sseli 3960 . . 3 (𝐺 ∈ (Poly‘𝑆) → 𝐺 ∈ (Poly‘ℂ))
4 eldifsn 4711 . . . . 5 (𝐺 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝))
5 oveq1 7152 . . . . . . . . . . 11 (𝑔 = 𝐺 → (𝑔f · 𝑞) = (𝐺f · 𝑞))
6 oveq12 7154 . . . . . . . . . . 11 ((𝑓 = 𝐹 ∧ (𝑔f · 𝑞) = (𝐺f · 𝑞)) → (𝑓f − (𝑔f · 𝑞)) = (𝐹f − (𝐺f · 𝑞)))
75, 6sylan2 592 . . . . . . . . . 10 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓f − (𝑔f · 𝑞)) = (𝐹f − (𝐺f · 𝑞)))
8 quotval.1 . . . . . . . . . 10 𝑅 = (𝐹f − (𝐺f · 𝑞))
97, 8syl6eqr 2871 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓f − (𝑔f · 𝑞)) = 𝑅)
109sbceq1d 3774 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → ([(𝑓f − (𝑔f · 𝑞)) / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔)) ↔ [𝑅 / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔))))
118ovexi 7179 . . . . . . . . . 10 𝑅 ∈ V
12 eqeq1 2822 . . . . . . . . . . 11 (𝑟 = 𝑅 → (𝑟 = 0𝑝𝑅 = 0𝑝))
13 fveq2 6663 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (deg‘𝑟) = (deg‘𝑅))
1413breq1d 5067 . . . . . . . . . . 11 (𝑟 = 𝑅 → ((deg‘𝑟) < (deg‘𝑔) ↔ (deg‘𝑅) < (deg‘𝑔)))
1512, 14orbi12d 912 . . . . . . . . . 10 (𝑟 = 𝑅 → ((𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔)) ↔ (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝑔))))
1611, 15sbcie 3809 . . . . . . . . 9 ([𝑅 / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔)) ↔ (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝑔)))
17 simpr 485 . . . . . . . . . . . 12 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑔 = 𝐺)
1817fveq2d 6667 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑔 = 𝐺) → (deg‘𝑔) = (deg‘𝐺))
1918breq2d 5069 . . . . . . . . . 10 ((𝑓 = 𝐹𝑔 = 𝐺) → ((deg‘𝑅) < (deg‘𝑔) ↔ (deg‘𝑅) < (deg‘𝐺)))
2019orbi2d 909 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝑔)) ↔ (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
2116, 20syl5bb 284 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → ([𝑅 / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔)) ↔ (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
2210, 21bitrd 280 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → ([(𝑓f − (𝑔f · 𝑞)) / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔)) ↔ (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
2322riotabidv 7105 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑞 ∈ (Poly‘ℂ)[(𝑓f − (𝑔f · 𝑞)) / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔))) = (𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
24 df-quot 24807 . . . . . 6 quot = (𝑓 ∈ (Poly‘ℂ), 𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↦ (𝑞 ∈ (Poly‘ℂ)[(𝑓f − (𝑔f · 𝑞)) / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔))))
25 riotaex 7107 . . . . . 6 (𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))) ∈ V
2623, 24, 25ovmpoa 7294 . . . . 5 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ ((Poly‘ℂ) ∖ {0𝑝})) → (𝐹 quot 𝐺) = (𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
274, 26sylan2br 594 . . . 4 ((𝐹 ∈ (Poly‘ℂ) ∧ (𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝)) → (𝐹 quot 𝐺) = (𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
28273impb 1107 . . 3 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) = (𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
293, 28syl3an2 1156 . 2 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) = (𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
302, 29syl3an1 1155 1 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) = (𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 841  w3a 1079   = wceq 1528  wcel 2105  wne 3013  [wsbc 3769  cdif 3930  {csn 4557   class class class wbr 5057  cfv 6348  crio 7102  (class class class)co 7145  f cof 7396  cc 10523   · cmul 10530   < clt 10663  cmin 10858  0𝑝c0p 24197  Polycply 24701  degcdgr 24704   quot cquot 24806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-1cn 10583  ax-addcl 10585
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-map 8397  df-nn 11627  df-n0 11886  df-ply 24705  df-quot 24807
This theorem is referenced by:  quotlem  24816
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