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Theorem quotval 26349
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 26254 . . 3 (Poly‘𝑆) ⊆ (Poly‘ℂ)
21sseli 3991 . 2 (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
31sseli 3991 . . 3 (𝐺 ∈ (Poly‘𝑆) → 𝐺 ∈ (Poly‘ℂ))
4 eldifsn 4791 . . . . 5 (𝐺 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝))
5 oveq1 7438 . . . . . . . . . . 11 (𝑔 = 𝐺 → (𝑔f · 𝑞) = (𝐺f · 𝑞))
6 oveq12 7440 . . . . . . . . . . 11 ((𝑓 = 𝐹 ∧ (𝑔f · 𝑞) = (𝐺f · 𝑞)) → (𝑓f − (𝑔f · 𝑞)) = (𝐹f − (𝐺f · 𝑞)))
75, 6sylan2 593 . . . . . . . . . 10 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓f − (𝑔f · 𝑞)) = (𝐹f − (𝐺f · 𝑞)))
8 quotval.1 . . . . . . . . . 10 𝑅 = (𝐹f − (𝐺f · 𝑞))
97, 8eqtr4di 2793 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓f − (𝑔f · 𝑞)) = 𝑅)
109sbceq1d 3796 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → ([(𝑓f − (𝑔f · 𝑞)) / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔)) ↔ [𝑅 / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔))))
118ovexi 7465 . . . . . . . . . 10 𝑅 ∈ V
12 eqeq1 2739 . . . . . . . . . . 11 (𝑟 = 𝑅 → (𝑟 = 0𝑝𝑅 = 0𝑝))
13 fveq2 6907 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (deg‘𝑟) = (deg‘𝑅))
1413breq1d 5158 . . . . . . . . . . 11 (𝑟 = 𝑅 → ((deg‘𝑟) < (deg‘𝑔) ↔ (deg‘𝑅) < (deg‘𝑔)))
1512, 14orbi12d 918 . . . . . . . . . 10 (𝑟 = 𝑅 → ((𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔)) ↔ (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝑔))))
1611, 15sbcie 3835 . . . . . . . . 9 ([𝑅 / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔)) ↔ (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝑔)))
17 simpr 484 . . . . . . . . . . . 12 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑔 = 𝐺)
1817fveq2d 6911 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑔 = 𝐺) → (deg‘𝑔) = (deg‘𝐺))
1918breq2d 5160 . . . . . . . . . 10 ((𝑓 = 𝐹𝑔 = 𝐺) → ((deg‘𝑅) < (deg‘𝑔) ↔ (deg‘𝑅) < (deg‘𝐺)))
2019orbi2d 915 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝑔)) ↔ (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
2116, 20bitrid 283 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → ([𝑅 / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔)) ↔ (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
2210, 21bitrd 279 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → ([(𝑓f − (𝑔f · 𝑞)) / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔)) ↔ (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
2322riotabidv 7390 . . . . . 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 26348 . . . . . 6 quot = (𝑓 ∈ (Poly‘ℂ), 𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↦ (𝑞 ∈ (Poly‘ℂ)[(𝑓f − (𝑔f · 𝑞)) / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔))))
25 riotaex 7392 . . . . . 6 (𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))) ∈ V
2623, 24, 25ovmpoa 7588 . . . . 5 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ ((Poly‘ℂ) ∖ {0𝑝})) → (𝐹 quot 𝐺) = (𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
274, 26sylan2br 595 . . . 4 ((𝐹 ∈ (Poly‘ℂ) ∧ (𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝)) → (𝐹 quot 𝐺) = (𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
28273impb 1114 . . 3 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) = (𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
293, 28syl3an2 1163 . 2 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) = (𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
302, 29syl3an1 1162 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 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wne 2938  [wsbc 3791  cdif 3960  {csn 4631   class class class wbr 5148  cfv 6563  crio 7387  (class class class)co 7431  f cof 7695  cc 11151   · cmul 11158   < clt 11293  cmin 11490  0𝑝c0p 25718  Polycply 26238  degcdgr 26241   quot cquot 26347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-1cn 11211  ax-addcl 11213
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-map 8867  df-nn 12265  df-n0 12525  df-ply 26242  df-quot 26348
This theorem is referenced by:  quotlem  26357
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