Theorem quotval 26352

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 26259	. . 3 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ)
2	1	sseli 4004	. 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ))
3	1	sseli 4004	. . 3 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺 ∈ (Poly‘ℂ))
4		eldifsn 4811	. . . . 5 ⊢ (𝐺 ∈ ((Poly‘ℂ) ∖ {0𝑝}) ↔ (𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝))
5		oveq1 7455	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (𝑔 ∘f · 𝑞) = (𝐺 ∘f · 𝑞))
6		oveq12 7457	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ (𝑔 ∘f · 𝑞) = (𝐺 ∘f · 𝑞)) → (𝑓 ∘f − (𝑔 ∘f · 𝑞)) = (𝐹 ∘f − (𝐺 ∘f · 𝑞)))
7	5, 6	sylan2 592	. . . . . . . . . 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 2798	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓 ∘f − (𝑔 ∘f · 𝑞)) = 𝑅)
10	9	sbceq1d 3809	. . . . . . . 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 7482	. . . . . . . . . 10 ⊢ 𝑅 ∈ V
12		eqeq1 2744	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (𝑟 = 0𝑝 ↔ 𝑅 = 0𝑝))
13		fveq2 6920	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (deg‘𝑟) = (deg‘𝑅))
14	13	breq1d 5176	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → ((deg‘𝑟) < (deg‘𝑔) ↔ (deg‘𝑅) < (deg‘𝑔)))
15	12, 14	orbi12d 917	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → ((𝑟 = 0𝑝 ∨ (deg‘𝑟) < (deg‘𝑔)) ↔ (𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝑔))))
16	11, 15	sbcie 3848	. . . . . . . . 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 6924	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (deg‘𝑔) = (deg‘𝐺))
19	18	breq2d 5178	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((deg‘𝑅) < (deg‘𝑔) ↔ (deg‘𝑅) < (deg‘𝐺)))
20	19	orbi2d 914	. . . . . . . . 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 7406	. . . . . 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 26351	. . . . . 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 7408	. . . . . 6 ⊢ (℩𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))) ∈ V
26	23, 24, 25	ovmpoa 7605	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ ((Poly‘ℂ) ∖ {0𝑝})) → (𝐹 quot 𝐺) = (℩𝑞 ∈ (Poly‘ℂ)(𝑅 = 0𝑝 ∨ (deg‘𝑅) < (deg‘𝐺))))
27	4, 26	sylan2br 594	. . . 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 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 846   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  [wsbc 3804   ∖ cdif 3973  {csn 4648   class class class wbr 5166  ‘cfv 6573  ℩crio 7403  (class class class)co 7448   ∘_f cof 7712  ℂcc 11182   · cmul 11189   < clt 11324   − cmin 11520  0_𝑝c0p 25723  Polycply 26243  degcdgr 26246   quot cquot 26350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-1cn 11242  ax-addcl 11244

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-map 8886  df-nn 12294  df-n0 12554  df-ply 26247  df-quot 26351

This theorem is referenced by:  quotlem  26360