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Theorem quotval 26042
Description: Value of the quotient function. (Contributed by Mario Carneiro, 23-Jul-2014.)
Hypothesis
Ref Expression
quotval.1 𝑅 = (𝐹 ∘f βˆ’ (𝐺 ∘f Β· π‘ž))
Assertion
Ref Expression
quotval ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (𝐹 quot 𝐺) = (β„©π‘ž ∈ (Polyβ€˜β„‚)(𝑅 = 0𝑝 ∨ (degβ€˜π‘…) < (degβ€˜πΊ))))
Distinct variable groups:   𝐹,π‘ž   𝐺,π‘ž
Allowed substitution hints:   𝑅(π‘ž)   𝑆(π‘ž)

Proof of Theorem quotval
Dummy variables 𝑓 𝑔 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plyssc 25950 . . 3 (Polyβ€˜π‘†) βŠ† (Polyβ€˜β„‚)
21sseli 3978 . 2 (𝐹 ∈ (Polyβ€˜π‘†) β†’ 𝐹 ∈ (Polyβ€˜β„‚))
31sseli 3978 . . 3 (𝐺 ∈ (Polyβ€˜π‘†) β†’ 𝐺 ∈ (Polyβ€˜β„‚))
4 eldifsn 4790 . . . . 5 (𝐺 ∈ ((Polyβ€˜β„‚) βˆ– {0𝑝}) ↔ (𝐺 ∈ (Polyβ€˜β„‚) ∧ 𝐺 β‰  0𝑝))
5 oveq1 7419 . . . . . . . . . . 11 (𝑔 = 𝐺 β†’ (𝑔 ∘f Β· π‘ž) = (𝐺 ∘f Β· π‘ž))
6 oveq12 7421 . . . . . . . . . . 11 ((𝑓 = 𝐹 ∧ (𝑔 ∘f Β· π‘ž) = (𝐺 ∘f Β· π‘ž)) β†’ (𝑓 ∘f βˆ’ (𝑔 ∘f Β· π‘ž)) = (𝐹 ∘f βˆ’ (𝐺 ∘f Β· π‘ž)))
75, 6sylan2 592 . . . . . . . . . 10 ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) β†’ (𝑓 ∘f βˆ’ (𝑔 ∘f Β· π‘ž)) = (𝐹 ∘f βˆ’ (𝐺 ∘f Β· π‘ž)))
8 quotval.1 . . . . . . . . . 10 𝑅 = (𝐹 ∘f βˆ’ (𝐺 ∘f Β· π‘ž))
97, 8eqtr4di 2789 . . . . . . . . 9 ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) β†’ (𝑓 ∘f βˆ’ (𝑔 ∘f Β· π‘ž)) = 𝑅)
109sbceq1d 3782 . . . . . . . 8 ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) β†’ ([(𝑓 ∘f βˆ’ (𝑔 ∘f Β· π‘ž)) / π‘Ÿ](π‘Ÿ = 0𝑝 ∨ (degβ€˜π‘Ÿ) < (degβ€˜π‘”)) ↔ [𝑅 / π‘Ÿ](π‘Ÿ = 0𝑝 ∨ (degβ€˜π‘Ÿ) < (degβ€˜π‘”))))
118ovexi 7446 . . . . . . . . . 10 𝑅 ∈ V
12 eqeq1 2735 . . . . . . . . . . 11 (π‘Ÿ = 𝑅 β†’ (π‘Ÿ = 0𝑝 ↔ 𝑅 = 0𝑝))
13 fveq2 6891 . . . . . . . . . . . 12 (π‘Ÿ = 𝑅 β†’ (degβ€˜π‘Ÿ) = (degβ€˜π‘…))
1413breq1d 5158 . . . . . . . . . . 11 (π‘Ÿ = 𝑅 β†’ ((degβ€˜π‘Ÿ) < (degβ€˜π‘”) ↔ (degβ€˜π‘…) < (degβ€˜π‘”)))
1512, 14orbi12d 916 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ ((π‘Ÿ = 0𝑝 ∨ (degβ€˜π‘Ÿ) < (degβ€˜π‘”)) ↔ (𝑅 = 0𝑝 ∨ (degβ€˜π‘…) < (degβ€˜π‘”))))
1611, 15sbcie 3820 . . . . . . . . 9 ([𝑅 / π‘Ÿ](π‘Ÿ = 0𝑝 ∨ (degβ€˜π‘Ÿ) < (degβ€˜π‘”)) ↔ (𝑅 = 0𝑝 ∨ (degβ€˜π‘…) < (degβ€˜π‘”)))
17 simpr 484 . . . . . . . . . . . 12 ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) β†’ 𝑔 = 𝐺)
1817fveq2d 6895 . . . . . . . . . . 11 ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) β†’ (degβ€˜π‘”) = (degβ€˜πΊ))
1918breq2d 5160 . . . . . . . . . 10 ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) β†’ ((degβ€˜π‘…) < (degβ€˜π‘”) ↔ (degβ€˜π‘…) < (degβ€˜πΊ)))
2019orbi2d 913 . . . . . . . . 9 ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) β†’ ((𝑅 = 0𝑝 ∨ (degβ€˜π‘…) < (degβ€˜π‘”)) ↔ (𝑅 = 0𝑝 ∨ (degβ€˜π‘…) < (degβ€˜πΊ))))
2116, 20bitrid 283 . . . . . . . 8 ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) β†’ ([𝑅 / π‘Ÿ](π‘Ÿ = 0𝑝 ∨ (degβ€˜π‘Ÿ) < (degβ€˜π‘”)) ↔ (𝑅 = 0𝑝 ∨ (degβ€˜π‘…) < (degβ€˜πΊ))))
2210, 21bitrd 279 . . . . . . 7 ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) β†’ ([(𝑓 ∘f βˆ’ (𝑔 ∘f Β· π‘ž)) / π‘Ÿ](π‘Ÿ = 0𝑝 ∨ (degβ€˜π‘Ÿ) < (degβ€˜π‘”)) ↔ (𝑅 = 0𝑝 ∨ (degβ€˜π‘…) < (degβ€˜πΊ))))
2322riotabidv 7370 . . . . . 6 ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) β†’ (β„©π‘ž ∈ (Polyβ€˜β„‚)[(𝑓 ∘f βˆ’ (𝑔 ∘f Β· π‘ž)) / π‘Ÿ](π‘Ÿ = 0𝑝 ∨ (degβ€˜π‘Ÿ) < (degβ€˜π‘”))) = (β„©π‘ž ∈ (Polyβ€˜β„‚)(𝑅 = 0𝑝 ∨ (degβ€˜π‘…) < (degβ€˜πΊ))))
24 df-quot 26041 . . . . . 6 quot = (𝑓 ∈ (Polyβ€˜β„‚), 𝑔 ∈ ((Polyβ€˜β„‚) βˆ– {0𝑝}) ↦ (β„©π‘ž ∈ (Polyβ€˜β„‚)[(𝑓 ∘f βˆ’ (𝑔 ∘f Β· π‘ž)) / π‘Ÿ](π‘Ÿ = 0𝑝 ∨ (degβ€˜π‘Ÿ) < (degβ€˜π‘”))))
25 riotaex 7372 . . . . . 6 (β„©π‘ž ∈ (Polyβ€˜β„‚)(𝑅 = 0𝑝 ∨ (degβ€˜π‘…) < (degβ€˜πΊ))) ∈ V
2623, 24, 25ovmpoa 7566 . . . . 5 ((𝐹 ∈ (Polyβ€˜β„‚) ∧ 𝐺 ∈ ((Polyβ€˜β„‚) βˆ– {0𝑝})) β†’ (𝐹 quot 𝐺) = (β„©π‘ž ∈ (Polyβ€˜β„‚)(𝑅 = 0𝑝 ∨ (degβ€˜π‘…) < (degβ€˜πΊ))))
274, 26sylan2br 594 . . . 4 ((𝐹 ∈ (Polyβ€˜β„‚) ∧ (𝐺 ∈ (Polyβ€˜β„‚) ∧ 𝐺 β‰  0𝑝)) β†’ (𝐹 quot 𝐺) = (β„©π‘ž ∈ (Polyβ€˜β„‚)(𝑅 = 0𝑝 ∨ (degβ€˜π‘…) < (degβ€˜πΊ))))
28273impb 1114 . . 3 ((𝐹 ∈ (Polyβ€˜β„‚) ∧ 𝐺 ∈ (Polyβ€˜β„‚) ∧ 𝐺 β‰  0𝑝) β†’ (𝐹 quot 𝐺) = (β„©π‘ž ∈ (Polyβ€˜β„‚)(𝑅 = 0𝑝 ∨ (degβ€˜π‘…) < (degβ€˜πΊ))))
293, 28syl3an2 1163 . 2 ((𝐹 ∈ (Polyβ€˜β„‚) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (𝐹 quot 𝐺) = (β„©π‘ž ∈ (Polyβ€˜β„‚)(𝑅 = 0𝑝 ∨ (degβ€˜π‘…) < (degβ€˜πΊ))))
302, 29syl3an1 1162 1 ((𝐹 ∈ (Polyβ€˜π‘†) ∧ 𝐺 ∈ (Polyβ€˜π‘†) ∧ 𝐺 β‰  0𝑝) β†’ (𝐹 quot 𝐺) = (β„©π‘ž ∈ (Polyβ€˜β„‚)(𝑅 = 0𝑝 ∨ (degβ€˜π‘…) < (degβ€˜πΊ))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∨ wo 844   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   β‰  wne 2939  [wsbc 3777   βˆ– cdif 3945  {csn 4628   class class class wbr 5148  β€˜cfv 6543  β„©crio 7367  (class class class)co 7412   ∘f cof 7671  β„‚cc 11111   Β· cmul 11118   < clt 11253   βˆ’ cmin 11449  0𝑝c0p 25419  Polycply 25934  degcdgr 25937   quot cquot 26040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728  ax-cnex 11169  ax-1cn 11171  ax-addcl 11173
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  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 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-map 8825  df-nn 12218  df-n0 12478  df-ply 25938  df-quot 26041
This theorem is referenced by:  quotlem  26050
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