Theorem q1pval 25671

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 17150	. . . 4 ⊢ (𝐺 ∈ 𝐵 → 𝑅 ∈ V)
4		q1pval.q	. . . . 5 ⊢ 𝑄 = (quot1p‘𝑅)
5		fveq2 6892	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
6	5, 1	eqtr4di 2791	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = 𝑃)
7	6	csbeq1d 3898	. . . . . . 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 6906	. . . . . . . . 9 ⊢ 𝑃 ∈ V
9	8	a1i 11	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → 𝑃 ∈ V)
10		fveq2 6892	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑃 → (Base‘𝑝) = (Base‘𝑃))
11	10	adantl 483	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) → (Base‘𝑝) = (Base‘𝑃))
12	11, 2	eqtr4di 2791	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) → (Base‘𝑝) = 𝐵)
13	12	csbeq1d 3898	. . . . . . . . 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 6906	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
15	14	a1i 11	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) → 𝐵 ∈ V)
16		simpr 486	. . . . . . . . . . 11 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
17		fveq2 6892	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑅 → ( deg1 ‘𝑟) = ( deg1 ‘𝑅))
18	17	ad2antrr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → ( deg1 ‘𝑟) = ( deg1 ‘𝑅))
19		q1pval.d	. . . . . . . . . . . . . . 15 ⊢ 𝐷 = ( deg1 ‘𝑅)
20	18, 19	eqtr4di 2791	. . . . . . . . . . . . . 14 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → ( deg1 ‘𝑟) = 𝐷)
21		fveq2 6892	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = 𝑃 → (-g‘𝑝) = (-g‘𝑃))
22	21	ad2antlr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (-g‘𝑝) = (-g‘𝑃))
23		q1pval.m	. . . . . . . . . . . . . . . 16 ⊢ − = (-g‘𝑃)
24	22, 23	eqtr4di 2791	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (-g‘𝑝) = − )
25		eqidd 2734	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → 𝑓 = 𝑓)
26		fveq2 6892	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = 𝑃 → (.r‘𝑝) = (.r‘𝑃))
27	26	ad2antlr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (.r‘𝑝) = (.r‘𝑃))
28		q1pval.t	. . . . . . . . . . . . . . . . 17 ⊢ · = (.r‘𝑃)
29	27, 28	eqtr4di 2791	. . . . . . . . . . . . . . . 16 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (.r‘𝑝) = · )
30	29	oveqd 7426	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (𝑞(.r‘𝑝)𝑔) = (𝑞 · 𝑔))
31	24, 25, 30	oveq123d 7430	. . . . . . . . . . . . . 14 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (𝑓(-g‘𝑝)(𝑞(.r‘𝑝)𝑔)) = (𝑓 − (𝑞 · 𝑔)))
32	20, 31	fveq12d 6899	. . . . . . . . . . . . 13 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (( deg1 ‘𝑟)‘(𝑓(-g‘𝑝)(𝑞(.r‘𝑝)𝑔))) = (𝐷‘(𝑓 − (𝑞 · 𝑔))))
33	20	fveq1d 6894	. . . . . . . . . . . . 13 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (( deg1 ‘𝑟)‘𝑔) = (𝐷‘𝑔))
34	32, 33	breq12d 5162	. . . . . . . . . . . 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 7368	. . . . . . . . . . 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 7484	. . . . . . . . . 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 3932	. . . . . . . . 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 2773	. . . . . . . 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 3932	. . . . . . 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 2773	. . . . . 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 25650	. . . . . 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 8066	. . . . . 6 ⊢ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔))) ∈ V
43	40, 41, 42	fvmpt 6999	. . . . 5 ⊢ (𝑅 ∈ V → (quot1p‘𝑅) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔))))
44	4, 43	eqtrid 2785	. . . 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 483	. 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 7417	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑞 · 𝑔) = (𝑞 · 𝐺))
49	47, 48	oveqan12d 7428	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓 − (𝑞 · 𝑔)) = (𝐹 − (𝑞 · 𝐺)))
50	49	fveq2d 6896	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝐷‘(𝑓 − (𝑞 · 𝑔))) = (𝐷‘(𝐹 − (𝑞 · 𝐺))))
51		fveq2 6892	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝐷‘𝑔) = (𝐷‘𝐺))
52	51	adantl 483	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝐷‘𝑔) = (𝐷‘𝐺))
53	50, 52	breq12d 5162	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔) ↔ (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺)))
54	53	riotabidv 7367	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔)) = (℩𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺)))
55	54	adantl 483	. 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 484	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐵)
57		simpr 486	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵)
58		riotaex 7369	. . 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 7560	1 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹𝑄𝐺) = (℩𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺)))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475  ⦋csb 3894   class class class wbr 5149  ‘cfv 6544  ℩crio 7364  (class class class)co 7409   ∈ cmpo 7411   < clt 11248  Basecbs 17144  ._rcmulr 17198  -_gcsg 18821  Poly₁cpl1 21701   deg₁ cdg1 25569  quot_1pcq1p 25645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-1cn 11168  ax-addcl 11170

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-nn 12213  df-slot 17115  df-ndx 17127  df-base 17145  df-q1p 25650

This theorem is referenced by:  q1peqb  25672