Theorem q1pval 23913

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 15920	. . . 4 ⊢ (𝐺 ∈ 𝐵 → 𝑅 ∈ V)
4		q1pval.q	. . . . 5 ⊢ 𝑄 = (quot1p‘𝑅)
5		fveq2 6191	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
6	5, 1	syl6eqr 2674	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = 𝑃)
7	6	csbeq1d 3540	. . . . . . 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		fvex 6201	. . . . . . . . . 10 ⊢ (Poly1‘𝑅) ∈ V
9	1, 8	eqeltri 2697	. . . . . . . . 9 ⊢ 𝑃 ∈ V
10	9	a1i 11	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → 𝑃 ∈ V)
11		fveq2 6191	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑃 → (Base‘𝑝) = (Base‘𝑃))
12	11	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) → (Base‘𝑝) = (Base‘𝑃))
13	12, 2	syl6eqr 2674	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) → (Base‘𝑝) = 𝐵)
14	13	csbeq1d 3540	. . . . . . . . 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))))
15		fvex 6201	. . . . . . . . . . . 12 ⊢ (Base‘𝑃) ∈ V
16	2, 15	eqeltri 2697	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
17	16	a1i 11	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) → 𝐵 ∈ V)
18		simpr 477	. . . . . . . . . . 11 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
19		fveq2 6191	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑅 → ( deg1 ‘𝑟) = ( deg1 ‘𝑅))
20	19	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → ( deg1 ‘𝑟) = ( deg1 ‘𝑅))
21		q1pval.d	. . . . . . . . . . . . . . 15 ⊢ 𝐷 = ( deg1 ‘𝑅)
22	20, 21	syl6eqr 2674	. . . . . . . . . . . . . 14 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → ( deg1 ‘𝑟) = 𝐷)
23		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = 𝑃 → (-g‘𝑝) = (-g‘𝑃))
24	23	ad2antlr 763	. . . . . . . . . . . . . . . 16 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (-g‘𝑝) = (-g‘𝑃))
25		q1pval.m	. . . . . . . . . . . . . . . 16 ⊢ − = (-g‘𝑃)
26	24, 25	syl6eqr 2674	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (-g‘𝑝) = − )
27		eqidd 2623	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → 𝑓 = 𝑓)
28		fveq2 6191	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = 𝑃 → (.r‘𝑝) = (.r‘𝑃))
29	28	ad2antlr 763	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (.r‘𝑝) = (.r‘𝑃))
30		q1pval.t	. . . . . . . . . . . . . . . . 17 ⊢ · = (.r‘𝑃)
31	29, 30	syl6eqr 2674	. . . . . . . . . . . . . . . 16 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (.r‘𝑝) = · )
32	31	oveqd 6667	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (𝑞(.r‘𝑝)𝑔) = (𝑞 · 𝑔))
33	26, 27, 32	oveq123d 6671	. . . . . . . . . . . . . 14 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (𝑓(-g‘𝑝)(𝑞(.r‘𝑝)𝑔)) = (𝑓 − (𝑞 · 𝑔)))
34	22, 33	fveq12d 6197	. . . . . . . . . . . . 13 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (( deg1 ‘𝑟)‘(𝑓(-g‘𝑝)(𝑞(.r‘𝑝)𝑔))) = (𝐷‘(𝑓 − (𝑞 · 𝑔))))
35	22	fveq1d 6193	. . . . . . . . . . . . 13 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (( deg1 ‘𝑟)‘𝑔) = (𝐷‘𝑔))
36	34, 35	breq12d 4666	. . . . . . . . . . . 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)))
37	18, 36	riotaeqbidv 6614	. . . . . . . . . . 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)))
38	18, 18, 37	mpt2eq123dv 6717	. . . . . . . . . 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))))
39	17, 38	csbied 3560	. . . . . . . . 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))))
40	14, 39	eqtrd 2656	. . . . . . . 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))))
41	10, 40	csbied 3560	. . . . . . 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))))
42	7, 41	eqtrd 2656	. . . . . 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))))
43		df-q1p 23892	. . . . . 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))))
44	16, 16	mpt2ex 7247	. . . . . 6 ⊢ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔))) ∈ V
45	42, 43, 44	fvmpt 6282	. . . . 5 ⊢ (𝑅 ∈ V → (quot1p‘𝑅) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔))))
46	4, 45	syl5eq 2668	. . . 4 ⊢ (𝑅 ∈ V → 𝑄 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔))))
47	3, 46	syl 17	. . 3 ⊢ (𝐺 ∈ 𝐵 → 𝑄 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔))))
48	47	adantl 482	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝑄 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔))))
49		id 22	. . . . . . 7 ⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹)
50		oveq2 6658	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑞 · 𝑔) = (𝑞 · 𝐺))
51	49, 50	oveqan12d 6669	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓 − (𝑞 · 𝑔)) = (𝐹 − (𝑞 · 𝐺)))
52	51	fveq2d 6195	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝐷‘(𝑓 − (𝑞 · 𝑔))) = (𝐷‘(𝐹 − (𝑞 · 𝐺))))
53		fveq2 6191	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝐷‘𝑔) = (𝐷‘𝐺))
54	53	adantl 482	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝐷‘𝑔) = (𝐷‘𝐺))
55	52, 54	breq12d 4666	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔) ↔ (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺)))
56	55	riotabidv 6613	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔)) = (℩𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺)))
57	56	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)))
58		simpl 473	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐵)
59		simpr 477	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵)
60		riotaex 6615	. . 3 ⊢ (℩𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺)) ∈ V
61	60	a1i 11	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (℩𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺)) ∈ V)
62	48, 57, 58, 59, 61	ovmpt2d 6788	1 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹𝑄𝐺) = (℩𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ⦋csb 3533   class class class wbr 4653  ‘cfv 5888  ℩crio 6610  (class class class)co 6650   ↦ cmpt2 6652   < clt 10074  Basecbs 15857  ._rcmulr 15942  -_gcsg 17424  Poly₁cpl1 19547   deg₁ cdg1 23814  quot_1pcq1p 23887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-slot 15861  df-base 15863  df-q1p 23892

This theorem is referenced by:  q1peqb  23914