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Theorem quotval 26335
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 26240 . . 3 (Poly‘𝑆) ⊆ (Poly‘ℂ)
21sseli 3978 . 2 (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
31sseli 3978 . . 3 (𝐺 ∈ (Poly‘𝑆) → 𝐺 ∈ (Poly‘ℂ))
4 eldifsn 4785 . . . . 5 (𝐺 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝))
5 oveq1 7439 . . . . . . . . . . 11 (𝑔 = 𝐺 → (𝑔f · 𝑞) = (𝐺f · 𝑞))
6 oveq12 7441 . . . . . . . . . . 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 2794 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓f − (𝑔f · 𝑞)) = 𝑅)
109sbceq1d 3792 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → ([(𝑓f − (𝑔f · 𝑞)) / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔)) ↔ [𝑅 / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔))))
118ovexi 7466 . . . . . . . . . 10 𝑅 ∈ V
12 eqeq1 2740 . . . . . . . . . . 11 (𝑟 = 𝑅 → (𝑟 = 0𝑝𝑅 = 0𝑝))
13 fveq2 6905 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (deg‘𝑟) = (deg‘𝑅))
1413breq1d 5152 . . . . . . . . . . 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 3829 . . . . . . . . 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 6909 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑔 = 𝐺) → (deg‘𝑔) = (deg‘𝐺))
1918breq2d 5154 . . . . . . . . . 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 7391 . . . . . 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 26334 . . . . . 6 quot = (𝑓 ∈ (Poly‘ℂ), 𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↦ (𝑞 ∈ (Poly‘ℂ)[(𝑓f − (𝑔f · 𝑞)) / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔))))
25 riotaex 7393 . . . . . 6 (𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))) ∈ V
2623, 24, 25ovmpoa 7589 . . . . 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 1164 . 2 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) = (𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
302, 29syl3an1 1163 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 1539  wcel 2107  wne 2939  [wsbc 3787  cdif 3947  {csn 4625   class class class wbr 5142  cfv 6560  crio 7388  (class class class)co 7432  f cof 7696  cc 11154   · cmul 11161   < clt 11296  cmin 11493  0𝑝c0p 25705  Polycply 26224  degcdgr 26227   quot cquot 26333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-1cn 11214  ax-addcl 11216
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-map 8869  df-nn 12268  df-n0 12529  df-ply 26228  df-quot 26334
This theorem is referenced by:  quotlem  26343
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