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Theorem quotval 26041
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 25949 . . 3 (Polyβ€˜π‘†) βŠ† (Polyβ€˜β„‚)
21sseli 3977 . 2 (𝐹 ∈ (Polyβ€˜π‘†) β†’ 𝐹 ∈ (Polyβ€˜β„‚))
31sseli 3977 . . 3 (𝐺 ∈ (Polyβ€˜π‘†) β†’ 𝐺 ∈ (Polyβ€˜β„‚))
4 eldifsn 4789 . . . . 5 (𝐺 ∈ ((Polyβ€˜β„‚) βˆ– {0𝑝}) ↔ (𝐺 ∈ (Polyβ€˜β„‚) ∧ 𝐺 β‰  0𝑝))
5 oveq1 7418 . . . . . . . . . . 11 (𝑔 = 𝐺 β†’ (𝑔 ∘f Β· π‘ž) = (𝐺 ∘f Β· π‘ž))
6 oveq12 7420 . . . . . . . . . . 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 591 . . . . . . . . . 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 2788 . . . . . . . . 9 ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) β†’ (𝑓 ∘f βˆ’ (𝑔 ∘f Β· π‘ž)) = 𝑅)
109sbceq1d 3781 . . . . . . . 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 7445 . . . . . . . . . 10 𝑅 ∈ V
12 eqeq1 2734 . . . . . . . . . . 11 (π‘Ÿ = 𝑅 β†’ (π‘Ÿ = 0𝑝 ↔ 𝑅 = 0𝑝))
13 fveq2 6890 . . . . . . . . . . . 12 (π‘Ÿ = 𝑅 β†’ (degβ€˜π‘Ÿ) = (degβ€˜π‘…))
1413breq1d 5157 . . . . . . . . . . 11 (π‘Ÿ = 𝑅 β†’ ((degβ€˜π‘Ÿ) < (degβ€˜π‘”) ↔ (degβ€˜π‘…) < (degβ€˜π‘”)))
1512, 14orbi12d 915 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ ((π‘Ÿ = 0𝑝 ∨ (degβ€˜π‘Ÿ) < (degβ€˜π‘”)) ↔ (𝑅 = 0𝑝 ∨ (degβ€˜π‘…) < (degβ€˜π‘”))))
1611, 15sbcie 3819 . . . . . . . . 9 ([𝑅 / π‘Ÿ](π‘Ÿ = 0𝑝 ∨ (degβ€˜π‘Ÿ) < (degβ€˜π‘”)) ↔ (𝑅 = 0𝑝 ∨ (degβ€˜π‘…) < (degβ€˜π‘”)))
17 simpr 483 . . . . . . . . . . . 12 ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) β†’ 𝑔 = 𝐺)
1817fveq2d 6894 . . . . . . . . . . 11 ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) β†’ (degβ€˜π‘”) = (degβ€˜πΊ))
1918breq2d 5159 . . . . . . . . . 10 ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) β†’ ((degβ€˜π‘…) < (degβ€˜π‘”) ↔ (degβ€˜π‘…) < (degβ€˜πΊ)))
2019orbi2d 912 . . . . . . . . 9 ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) β†’ ((𝑅 = 0𝑝 ∨ (degβ€˜π‘…) < (degβ€˜π‘”)) ↔ (𝑅 = 0𝑝 ∨ (degβ€˜π‘…) < (degβ€˜πΊ))))
2116, 20bitrid 282 . . . . . . . 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 278 . . . . . . 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 7369 . . . . . 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 26040 . . . . . 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 7371 . . . . . 6 (β„©π‘ž ∈ (Polyβ€˜β„‚)(𝑅 = 0𝑝 ∨ (degβ€˜π‘…) < (degβ€˜πΊ))) ∈ V
2623, 24, 25ovmpoa 7565 . . . . 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 593 . . . 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 1113 . . 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 1162 . 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 1161 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 394   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  [wsbc 3776   βˆ– cdif 3944  {csn 4627   class class class wbr 5147  β€˜cfv 6542  β„©crio 7366  (class class class)co 7411   ∘f cof 7670  β„‚cc 11110   Β· cmul 11117   < clt 11252   βˆ’ cmin 11448  0𝑝c0p 25418  Polycply 25933  degcdgr 25936   quot cquot 26039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-1cn 11170  ax-addcl 11172
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-map 8824  df-nn 12217  df-n0 12477  df-ply 25937  df-quot 26040
This theorem is referenced by:  quotlem  26049
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