Theorem q1pval 26453

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 17373	. . . 4 ⊢ (𝐺 ∈ 𝐵 → 𝑅 ∈ V)
4		q1pval.q	. . . . 5 ⊢ 𝑄 = (quot1p‘𝑅)
5		fveq2 6877	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
6	5, 1	eqtr4di 2814	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = 𝑃)
7	6	csbeq1d 3851	. . . . . . 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 6891	. . . . . . . . 9 ⊢ 𝑃 ∈ V
9	8	a1i 11	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → 𝑃 ∈ V)
10		fveq2 6877	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑃 → (Base‘𝑝) = (Base‘𝑃))
11	10	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) → (Base‘𝑝) = (Base‘𝑃))
12	11, 2	eqtr4di 2814	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) → (Base‘𝑝) = 𝐵)
13	12	csbeq1d 3851	. . . . . . . . 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 6891	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
15	14	a1i 11	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) → 𝐵 ∈ V)
16		simpr 490	. . . . . . . . . . 11 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
17		fveq2 6877	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑅 → (deg1‘𝑟) = (deg1‘𝑅))
18	17	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (deg1‘𝑟) = (deg1‘𝑅))
19		q1pval.d	. . . . . . . . . . . . . . 15 ⊢ 𝐷 = (deg1‘𝑅)
20	18, 19	eqtr4di 2814	. . . . . . . . . . . . . 14 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (deg1‘𝑟) = 𝐷)
21		fveq2 6877	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = 𝑃 → (-g‘𝑝) = (-g‘𝑃))
22	21	ad2antlr 740	. . . . . . . . . . . . . . . 16 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (-g‘𝑝) = (-g‘𝑃))
23		q1pval.m	. . . . . . . . . . . . . . . 16 ⊢ − = (-g‘𝑃)
24	22, 23	eqtr4di 2814	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (-g‘𝑝) = − )
25		eqidd 2762	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → 𝑓 = 𝑓)
26		fveq2 6877	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = 𝑃 → (.r‘𝑝) = (.r‘𝑃))
27	26	ad2antlr 740	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (.r‘𝑝) = (.r‘𝑃))
28		q1pval.t	. . . . . . . . . . . . . . . . 17 ⊢ · = (.r‘𝑃)
29	27, 28	eqtr4di 2814	. . . . . . . . . . . . . . . 16 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (.r‘𝑝) = · )
30	29	oveqd 7429	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (𝑞(.r‘𝑝)𝑔) = (𝑞 · 𝑔))
31	24, 25, 30	oveq123d 7433	. . . . . . . . . . . . . 14 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (𝑓(-g‘𝑝)(𝑞(.r‘𝑝)𝑔)) = (𝑓 − (𝑞 · 𝑔)))
32	20, 31	fveq12d 6884	. . . . . . . . . . . . 13 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → ((deg1‘𝑟)‘(𝑓(-g‘𝑝)(𝑞(.r‘𝑝)𝑔))) = (𝐷‘(𝑓 − (𝑞 · 𝑔))))
33	20	fveq1d 6879	. . . . . . . . . . . . 13 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → ((deg1‘𝑟)‘𝑔) = (𝐷‘𝑔))
34	32, 33	breq12d 5116	. . . . . . . . . . . 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 7372	. . . . . . . . . . 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 7487	. . . . . . . . . 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 3883	. . . . . . . . 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 2796	. . . . . . . 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 3883	. . . . . . 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 2796	. . . . . 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 26431	. . . . . 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 8081	. . . . . 6 ⊢ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔))) ∈ V
43	40, 41, 42	fvmpt 6985	. . . . 5 ⊢ (𝑅 ∈ V → (quot1p‘𝑅) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔))))
44	4, 43	eqtrid 2808	. . . 4 ⊢ (𝑅 ∈ V → 𝑄 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔))))
45	3, 44	syl 18	. . 3 ⊢ (𝐺 ∈ 𝐵 → 𝑄 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔))))
46	45	adantl 487	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝑄 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔))))
47		id 23	. . . . . . 7 ⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹)
48		oveq2 7420	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑞 · 𝑔) = (𝑞 · 𝐺))
49	47, 48	oveqan12d 7431	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓 − (𝑞 · 𝑔)) = (𝐹 − (𝑞 · 𝐺)))
50	49	fveq2d 6881	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝐷‘(𝑓 − (𝑞 · 𝑔))) = (𝐷‘(𝐹 − (𝑞 · 𝐺))))
51		fveq2 6877	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝐷‘𝑔) = (𝐷‘𝐺))
52	51	adantl 487	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝐷‘𝑔) = (𝐷‘𝐺))
53	50, 52	breq12d 5116	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔) ↔ (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺)))
54	53	riotabidv 7371	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔)) = (℩𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺)))
55	54	adantl 487	. 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 488	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐵)
57		simpr 490	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵)
58		riotaex 7373	. . 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 7564	1 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹𝑄𝐺) = (℩𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ⦋csb 3847   class class class wbr 5103  ‘cfv 6531  ℩crio 7368  (class class class)co 7412   ∈ cmpo 7414   < clt 11324  Basecbs 17367  ._rcmulr 17409  -_gcsg 19126  Poly₁cpl1 22475  deg₁cdg1 26352  quot_1pcq1p 26426

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-1cn 11239  ax-addcl 11241

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-nn 12317  df-slot 17340  df-ndx 17352  df-base 17368  df-q1p 26431

This theorem is used by:  q1peqb  26454