Theorem q1pval 25518

Description: Value of the univariate polynomial quotient function. (Contributed by Stefan O'Rear, 28-Mar-2015.)

Hypotheses

Ref	Expression
q1pval.q	⊢ 𝑄 = (quot1p‘𝑅)
q1pval.p	⊢ 𝑃 = (Poly1‘𝑅)
q1pval.b	⊢ 𝐵 = (Base‘𝑃)
q1pval.d	⊢ 𝐷 = ( deg1 ‘𝑅)
q1pval.m	⊢ − = (-g‘𝑃)
q1pval.t	⊢ · = (.r‘𝑃)

Assertion

Ref	Expression
q1pval	⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹𝑄𝐺) = (℩𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺)))

Distinct variable groups:   𝐵,𝑞   𝐹,𝑞   𝐺,𝑞   𝑃,𝑞   𝑅,𝑞

Allowed substitution hints:   𝐷(𝑞)   𝑄(𝑞)   · (𝑞)   − (𝑞)

Proof of Theorem q1pval

Dummy variables 𝑏 𝑓 𝑔 𝑝 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		q1pval.p	. . . . 5 ⊢ 𝑃 = (Poly1‘𝑅)
2		q1pval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑃)
3	1, 2	elbasfv 17089	. . . 4 ⊢ (𝐺 ∈ 𝐵 → 𝑅 ∈ V)
4		q1pval.q	. . . . 5 ⊢ 𝑄 = (quot1p‘𝑅)
5		fveq2 6842	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
6	5, 1	eqtr4di 2794	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = 𝑃)
7	6	csbeq1d 3859	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ⦋(Poly1‘𝑟) / 𝑝⦌⦋(Base‘𝑝) / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (℩𝑞 ∈ 𝑏 (( deg1 ‘𝑟)‘(𝑓(-g‘𝑝)(𝑞(.r‘𝑝)𝑔))) < (( deg1 ‘𝑟)‘𝑔))) = ⦋𝑃 / 𝑝⦌⦋(Base‘𝑝) / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (℩𝑞 ∈ 𝑏 (( deg1 ‘𝑟)‘(𝑓(-g‘𝑝)(𝑞(.r‘𝑝)𝑔))) < (( deg1 ‘𝑟)‘𝑔))))
8	1	fvexi 6856	. . . . . . . . 9 ⊢ 𝑃 ∈ V
9	8	a1i 11	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → 𝑃 ∈ V)
10		fveq2 6842	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑃 → (Base‘𝑝) = (Base‘𝑃))
11	10	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) → (Base‘𝑝) = (Base‘𝑃))
12	11, 2	eqtr4di 2794	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) → (Base‘𝑝) = 𝐵)
13	12	csbeq1d 3859	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) → ⦋(Base‘𝑝) / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (℩𝑞 ∈ 𝑏 (( deg1 ‘𝑟)‘(𝑓(-g‘𝑝)(𝑞(.r‘𝑝)𝑔))) < (( deg1 ‘𝑟)‘𝑔))) = ⦋𝐵 / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (℩𝑞 ∈ 𝑏 (( deg1 ‘𝑟)‘(𝑓(-g‘𝑝)(𝑞(.r‘𝑝)𝑔))) < (( deg1 ‘𝑟)‘𝑔))))
14	2	fvexi 6856	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
15	14	a1i 11	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) → 𝐵 ∈ V)
16		simpr 485	. . . . . . . . . . 11 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
17		fveq2 6842	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑅 → ( deg1 ‘𝑟) = ( deg1 ‘𝑅))
18	17	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → ( deg1 ‘𝑟) = ( deg1 ‘𝑅))
19		q1pval.d	. . . . . . . . . . . . . . 15 ⊢ 𝐷 = ( deg1 ‘𝑅)
20	18, 19	eqtr4di 2794	. . . . . . . . . . . . . 14 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → ( deg1 ‘𝑟) = 𝐷)
21		fveq2 6842	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = 𝑃 → (-g‘𝑝) = (-g‘𝑃))
22	21	ad2antlr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (-g‘𝑝) = (-g‘𝑃))
23		q1pval.m	. . . . . . . . . . . . . . . 16 ⊢ − = (-g‘𝑃)
24	22, 23	eqtr4di 2794	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (-g‘𝑝) = − )
25		eqidd 2737	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → 𝑓 = 𝑓)
26		fveq2 6842	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = 𝑃 → (.r‘𝑝) = (.r‘𝑃))
27	26	ad2antlr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (.r‘𝑝) = (.r‘𝑃))
28		q1pval.t	. . . . . . . . . . . . . . . . 17 ⊢ · = (.r‘𝑃)
29	27, 28	eqtr4di 2794	. . . . . . . . . . . . . . . 16 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (.r‘𝑝) = · )
30	29	oveqd 7374	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (𝑞(.r‘𝑝)𝑔) = (𝑞 · 𝑔))
31	24, 25, 30	oveq123d 7378	. . . . . . . . . . . . . 14 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (𝑓(-g‘𝑝)(𝑞(.r‘𝑝)𝑔)) = (𝑓 − (𝑞 · 𝑔)))
32	20, 31	fveq12d 6849	. . . . . . . . . . . . 13 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (( deg1 ‘𝑟)‘(𝑓(-g‘𝑝)(𝑞(.r‘𝑝)𝑔))) = (𝐷‘(𝑓 − (𝑞 · 𝑔))))
33	20	fveq1d 6844	. . . . . . . . . . . . 13 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (( deg1 ‘𝑟)‘𝑔) = (𝐷‘𝑔))
34	32, 33	breq12d 5118	. . . . . . . . . . . 12 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → ((( deg1 ‘𝑟)‘(𝑓(-g‘𝑝)(𝑞(.r‘𝑝)𝑔))) < (( deg1 ‘𝑟)‘𝑔) ↔ (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔)))
35	16, 34	riotaeqbidv 7316	. . . . . . . . . . 11 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (℩𝑞 ∈ 𝑏 (( deg1 ‘𝑟)‘(𝑓(-g‘𝑝)(𝑞(.r‘𝑝)𝑔))) < (( deg1 ‘𝑟)‘𝑔)) = (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔)))
36	16, 16, 35	mpoeq123dv 7432	. . . . . . . . . 10 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (℩𝑞 ∈ 𝑏 (( deg1 ‘𝑟)‘(𝑓(-g‘𝑝)(𝑞(.r‘𝑝)𝑔))) < (( deg1 ‘𝑟)‘𝑔))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔))))
37	15, 36	csbied 3893	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) → ⦋𝐵 / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (℩𝑞 ∈ 𝑏 (( deg1 ‘𝑟)‘(𝑓(-g‘𝑝)(𝑞(.r‘𝑝)𝑔))) < (( deg1 ‘𝑟)‘𝑔))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔))))
38	13, 37	eqtrd 2776	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) → ⦋(Base‘𝑝) / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (℩𝑞 ∈ 𝑏 (( deg1 ‘𝑟)‘(𝑓(-g‘𝑝)(𝑞(.r‘𝑝)𝑔))) < (( deg1 ‘𝑟)‘𝑔))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔))))
39	9, 38	csbied 3893	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ⦋𝑃 / 𝑝⦌⦋(Base‘𝑝) / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (℩𝑞 ∈ 𝑏 (( deg1 ‘𝑟)‘(𝑓(-g‘𝑝)(𝑞(.r‘𝑝)𝑔))) < (( deg1 ‘𝑟)‘𝑔))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔))))
40	7, 39	eqtrd 2776	. . . . . 6 ⊢ (𝑟 = 𝑅 → ⦋(Poly1‘𝑟) / 𝑝⦌⦋(Base‘𝑝) / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (℩𝑞 ∈ 𝑏 (( deg1 ‘𝑟)‘(𝑓(-g‘𝑝)(𝑞(.r‘𝑝)𝑔))) < (( deg1 ‘𝑟)‘𝑔))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔))))
41		df-q1p 25497	. . . . . 6 ⊢ quot1p = (𝑟 ∈ V ↦ ⦋(Poly1‘𝑟) / 𝑝⦌⦋(Base‘𝑝) / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (℩𝑞 ∈ 𝑏 (( deg1 ‘𝑟)‘(𝑓(-g‘𝑝)(𝑞(.r‘𝑝)𝑔))) < (( deg1 ‘𝑟)‘𝑔))))
42	14, 14	mpoex 8012	. . . . . 6 ⊢ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔))) ∈ V
43	40, 41, 42	fvmpt 6948	. . . . 5 ⊢ (𝑅 ∈ V → (quot1p‘𝑅) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔))))
44	4, 43	eqtrid 2788	. . . 4 ⊢ (𝑅 ∈ V → 𝑄 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔))))
45	3, 44	syl 17	. . 3 ⊢ (𝐺 ∈ 𝐵 → 𝑄 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔))))
46	45	adantl 482	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝑄 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔))))
47		id 22	. . . . . . 7 ⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹)
48		oveq2 7365	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑞 · 𝑔) = (𝑞 · 𝐺))
49	47, 48	oveqan12d 7376	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓 − (𝑞 · 𝑔)) = (𝐹 − (𝑞 · 𝐺)))
50	49	fveq2d 6846	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝐷‘(𝑓 − (𝑞 · 𝑔))) = (𝐷‘(𝐹 − (𝑞 · 𝐺))))
51		fveq2 6842	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝐷‘𝑔) = (𝐷‘𝐺))
52	51	adantl 482	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝐷‘𝑔) = (𝐷‘𝐺))
53	50, 52	breq12d 5118	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔) ↔ (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺)))
54	53	riotabidv 7315	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔)) = (℩𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺)))
55	54	adantl 482	. 2 ⊢ (((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔)) = (℩𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺)))
56		simpl 483	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐵)
57		simpr 485	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵)
58		riotaex 7317	. . 3 ⊢ (℩𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺)) ∈ V
59	58	a1i 11	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (℩𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺)) ∈ V)
60	46, 55, 56, 57, 59	ovmpod 7507	1 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹𝑄𝐺) = (℩𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3445  ⦋csb 3855   class class class wbr 5105  ‘cfv 6496  ℩crio 7312  (class class class)co 7357   ∈ cmpo 7359   < clt 11189  Basecbs 17083  ._rcmulr 17134  -_gcsg 18750  Poly₁cpl1 21548   deg₁ cdg1 25416  quot_1pcq1p 25492

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-1cn 11109  ax-addcl 11111

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-nn 12154  df-slot 17054  df-ndx 17066  df-base 17084  df-q1p 25497

This theorem is referenced by:  q1peqb  25519