Theorem quotval 26200

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 26105	. . 3 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
2	1	sseli 3942	. 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
3	1	sseli 3942	. . 3 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺 ∈ (Poly‘ℂ))
4		eldifsn 4750	. . . . 5 ⊢ (𝐺 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝))
5		oveq1 7394	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (𝑔 ∘f · 𝑞) = (𝐺 ∘f · 𝑞))
6		oveq12 7396	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ (𝑔 ∘f · 𝑞) = (𝐺 ∘f · 𝑞)) → (𝑓 ∘f − (𝑔 ∘f · 𝑞)) = (𝐹 ∘f − (𝐺 ∘f · 𝑞)))
7	5, 6	sylan2 593	. . . . . . . . . 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 2782	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓 ∘f − (𝑔 ∘f · 𝑞)) = 𝑅)
10	9	sbceq1d 3758	. . . . . . . 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 7421	. . . . . . . . . 10 ⊢ 𝑅 ∈ V
12		eqeq1 2733	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (𝑟 = 0𝑝 ↔ 𝑅 = 0𝑝))
13		fveq2 6858	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (deg‘𝑟) = (deg‘𝑅))
14	13	breq1d 5117	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → ((deg‘𝑟) < (deg‘𝑔) ↔ (deg‘𝑅) < (deg‘𝑔)))
15	12, 14	orbi12d 918	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → ((𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔)) ↔ (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝑔))))
16	11, 15	sbcie 3795	. . . . . . . . 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 6862	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (deg‘𝑔) = (deg‘𝐺))
19	18	breq2d 5119	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((deg‘𝑅) < (deg‘𝑔) ↔ (deg‘𝑅) < (deg‘𝐺)))
20	19	orbi2d 915	. . . . . . . . 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 7346	. . . . . 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 26199	. . . . . 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 7348	. . . . . 6 ⊢ (℩𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))) ∈ V
26	23, 24, 25	ovmpoa 7544	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ ((Poly‘ℂ) ∖ {0𝑝})) → (𝐹 quot 𝐺) = (℩𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
27	4, 26	sylan2br 595	. . . 4 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ (𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝)) → (𝐹 quot 𝐺) = (℩𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
28	27	3impb 1114	. . 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 1164	. 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 1163	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 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  [wsbc 3753   ∖ cdif 3911  {csn 4589   class class class wbr 5107  ‘cfv 6511  ℩crio 7343  (class class class)co 7387   ∘_f cof 7651  ℂcc 11066   · cmul 11073   < clt 11208   − cmin 11405  0_𝑝c0p 25570  Polycply 26089  degcdgr 26092   quot cquot 26198

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-1cn 11126  ax-addcl 11128

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-map 8801  df-nn 12187  df-n0 12443  df-ply 26093  df-quot 26199

This theorem is referenced by:  quotlem  26208