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Theorem quotval 25216
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 25125 . . 3 (Poly‘𝑆) ⊆ (Poly‘ℂ)
21sseli 3913 . 2 (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
31sseli 3913 . . 3 (𝐺 ∈ (Poly‘𝑆) → 𝐺 ∈ (Poly‘ℂ))
4 eldifsn 4716 . . . . 5 (𝐺 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝))
5 oveq1 7241 . . . . . . . . . . 11 (𝑔 = 𝐺 → (𝑔f · 𝑞) = (𝐺f · 𝑞))
6 oveq12 7243 . . . . . . . . . . 11 ((𝑓 = 𝐹 ∧ (𝑔f · 𝑞) = (𝐺f · 𝑞)) → (𝑓f − (𝑔f · 𝑞)) = (𝐹f − (𝐺f · 𝑞)))
75, 6sylan2 596 . . . . . . . . . 10 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓f − (𝑔f · 𝑞)) = (𝐹f − (𝐺f · 𝑞)))
8 quotval.1 . . . . . . . . . 10 𝑅 = (𝐹f − (𝐺f · 𝑞))
97, 8eqtr4di 2798 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓f − (𝑔f · 𝑞)) = 𝑅)
109sbceq1d 3716 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → ([(𝑓f − (𝑔f · 𝑞)) / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔)) ↔ [𝑅 / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔))))
118ovexi 7268 . . . . . . . . . 10 𝑅 ∈ V
12 eqeq1 2743 . . . . . . . . . . 11 (𝑟 = 𝑅 → (𝑟 = 0𝑝𝑅 = 0𝑝))
13 fveq2 6738 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (deg‘𝑟) = (deg‘𝑅))
1413breq1d 5079 . . . . . . . . . . 11 (𝑟 = 𝑅 → ((deg‘𝑟) < (deg‘𝑔) ↔ (deg‘𝑅) < (deg‘𝑔)))
1512, 14orbi12d 919 . . . . . . . . . 10 (𝑟 = 𝑅 → ((𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔)) ↔ (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝑔))))
1611, 15sbcie 3754 . . . . . . . . 9 ([𝑅 / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔)) ↔ (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝑔)))
17 simpr 488 . . . . . . . . . . . 12 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑔 = 𝐺)
1817fveq2d 6742 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑔 = 𝐺) → (deg‘𝑔) = (deg‘𝐺))
1918breq2d 5081 . . . . . . . . . 10 ((𝑓 = 𝐹𝑔 = 𝐺) → ((deg‘𝑅) < (deg‘𝑔) ↔ (deg‘𝑅) < (deg‘𝐺)))
2019orbi2d 916 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝑔)) ↔ (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
2116, 20syl5bb 286 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → ([𝑅 / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔)) ↔ (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
2210, 21bitrd 282 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → ([(𝑓f − (𝑔f · 𝑞)) / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔)) ↔ (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
2322riotabidv 7193 . . . . . 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 25215 . . . . . 6 quot = (𝑓 ∈ (Poly‘ℂ), 𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↦ (𝑞 ∈ (Poly‘ℂ)[(𝑓f − (𝑔f · 𝑞)) / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔))))
25 riotaex 7195 . . . . . 6 (𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))) ∈ V
2623, 24, 25ovmpoa 7385 . . . . 5 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ ((Poly‘ℂ) ∖ {0𝑝})) → (𝐹 quot 𝐺) = (𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
274, 26sylan2br 598 . . . 4 ((𝐹 ∈ (Poly‘ℂ) ∧ (𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝)) → (𝐹 quot 𝐺) = (𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
28273impb 1117 . . 3 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) = (𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
293, 28syl3an2 1166 . 2 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) = (𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
302, 29syl3an1 1165 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 399  wo 847  w3a 1089   = wceq 1543  wcel 2112  wne 2943  [wsbc 3711  cdif 3880  {csn 4557   class class class wbr 5069  cfv 6400  crio 7190  (class class class)co 7234  f cof 7488  cc 10756   · cmul 10763   < clt 10896  cmin 11091  0𝑝c0p 24597  Polycply 25109  degcdgr 25112   quot cquot 25214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-rep 5195  ax-sep 5208  ax-nul 5215  ax-pow 5274  ax-pr 5338  ax-un 7544  ax-cnex 10814  ax-1cn 10816  ax-addcl 10818
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3425  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4456  df-pw 4531  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4836  df-iun 4922  df-br 5070  df-opab 5132  df-mpt 5152  df-tr 5178  df-id 5471  df-eprel 5477  df-po 5485  df-so 5486  df-fr 5526  df-we 5528  df-xp 5574  df-rel 5575  df-cnv 5576  df-co 5577  df-dm 5578  df-rn 5579  df-res 5580  df-ima 5581  df-pred 6178  df-ord 6236  df-on 6237  df-lim 6238  df-suc 6239  df-iota 6358  df-fun 6402  df-fn 6403  df-f 6404  df-f1 6405  df-fo 6406  df-f1o 6407  df-fv 6408  df-riota 7191  df-ov 7237  df-oprab 7238  df-mpo 7239  df-om 7666  df-1st 7782  df-2nd 7783  df-wrecs 8070  df-recs 8131  df-rdg 8169  df-map 8533  df-nn 11860  df-n0 12120  df-ply 25113  df-quot 25215
This theorem is referenced by:  quotlem  25224
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