Theorem quotval 26258

Description: Value of the quotient function. (Contributed by Mario Carneiro, 23-Jul-2014.)

Hypothesis

Ref	Expression
quotval.1	⊢ 𝑅 = (𝐹 ∘f − (𝐺 ∘f · 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.

Step	Hyp	Ref	Expression
1		plyssc 26165	. . 3 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
2	1	sseli 3917	. 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
3	1	sseli 3917	. . 3 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺 ∈ (Poly‘ℂ))
4		eldifsn 4731	. . . . 5 ⊢ (𝐺 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝))
5		oveq1 7374	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (𝑔 ∘f · 𝑞) = (𝐺 ∘f · 𝑞))
6		oveq12 7376	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ (𝑔 ∘f · 𝑞) = (𝐺 ∘f · 𝑞)) → (𝑓 ∘f − (𝑔 ∘f · 𝑞)) = (𝐹 ∘f − (𝐺 ∘f · 𝑞)))
7	5, 6	sylan2 594	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓 ∘f − (𝑔 ∘f · 𝑞)) = (𝐹 ∘f − (𝐺 ∘f · 𝑞)))
8		quotval.1	. . . . . . . . . 10 ⊢ 𝑅 = (𝐹 ∘f − (𝐺 ∘f · 𝑞))
9	7, 8	eqtr4di 2789	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓 ∘f − (𝑔 ∘f · 𝑞)) = 𝑅)
10	9	sbceq1d 3733	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ([(𝑓 ∘f − (𝑔 ∘f · 𝑞)) / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔)) ↔ [𝑅 / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔))))
11	8	ovexi 7401	. . . . . . . . . 10 ⊢ 𝑅 ∈ V
12		eqeq1 2740	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (𝑟 = 0𝑝 ↔ 𝑅 = 0𝑝))
13		fveq2 6840	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (deg‘𝑟) = (deg‘𝑅))
14	13	breq1d 5095	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → ((deg‘𝑟) < (deg‘𝑔) ↔ (deg‘𝑅) < (deg‘𝑔)))
15	12, 14	orbi12d 919	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → ((𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔)) ↔ (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝑔))))
16	11, 15	sbcie 3770	. . . . . . . . 9 ⊢ ([𝑅 / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔)) ↔ (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝑔)))
17		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
18	17	fveq2d 6844	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (deg‘𝑔) = (deg‘𝐺))
19	18	breq2d 5097	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((deg‘𝑅) < (deg‘𝑔) ↔ (deg‘𝑅) < (deg‘𝐺)))
20	19	orbi2d 916	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝑔)) ↔ (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
21	16, 20	bitrid 283	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ([𝑅 / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔)) ↔ (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
22	10, 21	bitrd 279	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ([(𝑓 ∘f − (𝑔 ∘f · 𝑞)) / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔)) ↔ (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
23	22	riotabidv 7326	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (℩𝑞 ∈ (Poly‘ℂ)[(𝑓 ∘f − (𝑔 ∘f · 𝑞)) / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔))) = (℩𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
24		df-quot 26257	. . . . . 6 ⊢ quot = (𝑓 ∈ (Poly‘ℂ), 𝑔 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↦ (℩𝑞 ∈ (Poly‘ℂ)[(𝑓 ∘f − (𝑔 ∘f · 𝑞)) / 𝑟](𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔))))
25		riotaex 7328	. . . . . 6 ⊢ (℩𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))) ∈ V
26	23, 24, 25	ovmpoa 7522	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ ((Poly‘ℂ) ∖ {0𝑝})) → (𝐹 quot 𝐺) = (℩𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
27	4, 26	sylan2br 596	. . . 4 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ (𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝)) → (𝐹 quot 𝐺) = (℩𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
28	27	3impb 1115	. . 3 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) = (℩𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
29	3, 28	syl3an2 1165	. 2 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) = (℩𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
30	2, 29	syl3an1 1164	1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐺 ∈ (Poly‘𝑆) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) = (℩𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  [wsbc 3728   ∖ cdif 3886  {csn 4567   class class class wbr 5085  ‘cfv 6498  ℩crio 7323  (class class class)co 7367   ∘_f cof 7629  ℂcc 11036   · cmul 11043   < clt 11179   − cmin 11377  0_𝑝c0p 25636  Polycply 26149  degcdgr 26152   quot cquot 26256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-1cn 11096  ax-addcl 11098

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-map 8775  df-nn 12175  df-n0 12438  df-ply 26153  df-quot 26257

This theorem is referenced by:  quotlem  26266