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Theorem quotval 26221
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 26128 . . 3 (Poly‘𝑆) ⊆ (Poly‘ℂ)
21sseli 3975 . 2 (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
31sseli 3975 . . 3 (𝐺 ∈ (Poly‘𝑆) → 𝐺 ∈ (Poly‘ℂ))
4 eldifsn 4787 . . . . 5 (𝐺 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝))
5 oveq1 7422 . . . . . . . . . . 11 (𝑔 = 𝐺 → (𝑔f · 𝑞) = (𝐺f · 𝑞))
6 oveq12 7424 . . . . . . . . . . 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, 8eqtr4di 2786 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓f − (𝑔f · 𝑞)) = 𝑅)
109sbceq1d 3780 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → ([(𝑓f − (𝑔f · 𝑞)) / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔)) ↔ [𝑅 / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔))))
118ovexi 7449 . . . . . . . . . 10 𝑅 ∈ V
12 eqeq1 2732 . . . . . . . . . . 11 (𝑟 = 𝑅 → (𝑟 = 0𝑝𝑅 = 0𝑝))
13 fveq2 6892 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (deg‘𝑟) = (deg‘𝑅))
1413breq1d 5153 . . . . . . . . . . 11 (𝑟 = 𝑅 → ((deg‘𝑟) < (deg‘𝑔) ↔ (deg‘𝑅) < (deg‘𝑔)))
1512, 14orbi12d 917 . . . . . . . . . 10 (𝑟 = 𝑅 → ((𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔)) ↔ (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝑔))))
1611, 15sbcie 3818 . . . . . . . . 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 6896 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑔 = 𝐺) → (deg‘𝑔) = (deg‘𝐺))
1918breq2d 5155 . . . . . . . . . 10 ((𝑓 = 𝐹𝑔 = 𝐺) → ((deg‘𝑅) < (deg‘𝑔) ↔ (deg‘𝑅) < (deg‘𝐺)))
2019orbi2d 914 . . . . . . . . 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 7373 . . . . . 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 26220 . . . . . 6 quot = (𝑓 ∈ (Poly‘ℂ), 𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↦ (𝑞 ∈ (Poly‘ℂ)[(𝑓f − (𝑔f · 𝑞)) / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔))))
25 riotaex 7375 . . . . . 6 (𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))) ∈ V
2623, 24, 25ovmpoa 7571 . . . . 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 1113 . . 3 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) = (𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
293, 28syl3an2 1162 . 2 ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) = (𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
302, 29syl3an1 1161 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 846  w3a 1085   = wceq 1534  wcel 2099  wne 2936  [wsbc 3775  cdif 3942  {csn 4625   class class class wbr 5143  cfv 6543  crio 7370  (class class class)co 7415  f cof 7678  cc 11131   · cmul 11138   < clt 11273  cmin 11469  0𝑝c0p 25592  Polycply 26112  degcdgr 26115   quot cquot 26219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-1cn 11191  ax-addcl 11193
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7866  df-1st 7988  df-2nd 7989  df-frecs 8281  df-wrecs 8312  df-recs 8386  df-rdg 8425  df-map 8841  df-nn 12238  df-n0 12498  df-ply 26116  df-quot 26220
This theorem is referenced by:  quotlem  26229
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