Theorem q1pval 26478

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 17398	. . . 4 ⊢ (𝐺 ∈ 𝐵 → 𝑅 ∈ V)
4		q1pval.q	. . . . 5 ⊢ 𝑄 = (quot1p‘𝑅)
5		fveq2 6885	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
6	5, 1	eqtr4di 2813	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = 𝑃)
7	6	csbeq1d 3850	. . . . . . 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 6899	. . . . . . . . 9 ⊢ 𝑃 ∈ V
9	8	a1i 11	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → 𝑃 ∈ V)
10		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑃 → (Base‘𝑝) = (Base‘𝑃))
11	10	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) → (Base‘𝑝) = (Base‘𝑃))
12	11, 2	eqtr4di 2813	. . . . . . . . . 10 ⊢ ((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) → (Base‘𝑝) = 𝐵)
13	12	csbeq1d 3850	. . . . . . . . 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 6899	. . . . . . . . . . 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 6885	. . . . . . . . . . . . . . . 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 2813	. . . . . . . . . . . . . 14 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (deg1‘𝑟) = 𝐷)
21		fveq2 6885	. . . . . . . . . . . . . . . . 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 2813	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (-g‘𝑝) = − )
25		eqidd 2761	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → 𝑓 = 𝑓)
26		fveq2 6885	. . . . . . . . . . . . . . . . . 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 2813	. . . . . . . . . . . . . . . 16 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (.r‘𝑝) = · )
30	29	oveqd 7437	. . . . . . . . . . . . . . 15 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (𝑞(.r‘𝑝)𝑔) = (𝑞 · 𝑔))
31	24, 25, 30	oveq123d 7441	. . . . . . . . . . . . . 14 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → (𝑓(-g‘𝑝)(𝑞(.r‘𝑝)𝑔)) = (𝑓 − (𝑞 · 𝑔)))
32	20, 31	fveq12d 6892	. . . . . . . . . . . . 13 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → ((deg1‘𝑟)‘(𝑓(-g‘𝑝)(𝑞(.r‘𝑝)𝑔))) = (𝐷‘(𝑓 − (𝑞 · 𝑔))))
33	20	fveq1d 6887	. . . . . . . . . . . . 13 ⊢ (((𝑟 = 𝑅 ∧ 𝑝 = 𝑃) ∧ 𝑏 = 𝐵) → ((deg1‘𝑟)‘𝑔) = (𝐷‘𝑔))
34	32, 33	breq12d 5115	. . . . . . . . . . . 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 7380	. . . . . . . . . . 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 7495	. . . . . . . . . 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 3882	. . . . . . . . 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 2795	. . . . . . . 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 3882	. . . . . . 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 2795	. . . . . 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 26456	. . . . . 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 8092	. . . . . 6 ⊢ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔))) ∈ V
43	40, 41, 42	fvmpt 6993	. . . . 5 ⊢ (𝑅 ∈ V → (quot1p‘𝑅) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (℩𝑞 ∈ 𝐵 (𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔))))
44	4, 43	eqtrid 2807	. . . 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 7428	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑞 · 𝑔) = (𝑞 · 𝐺))
49	47, 48	oveqan12d 7439	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓 − (𝑞 · 𝑔)) = (𝐹 − (𝑞 · 𝐺)))
50	49	fveq2d 6889	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝐷‘(𝑓 − (𝑞 · 𝑔))) = (𝐷‘(𝐹 − (𝑞 · 𝐺))))
51		fveq2 6885	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝐷‘𝑔) = (𝐷‘𝐺))
52	51	adantl 487	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝐷‘𝑔) = (𝐷‘𝐺))
53	50, 52	breq12d 5115	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝐷‘(𝑓 − (𝑞 · 𝑔))) < (𝐷‘𝑔) ↔ (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺)))
54	53	riotabidv 7379	. . 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 7381	. . 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 7572	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 1569   ∈ wcel 2144  Vcvv 3450  ⦋csb 3846   class class class wbr 5102  ‘cfv 6538  ℩crio 7376  (class class class)co 7420   ∈ cmpo 7422   < clt 11346  Basecbs 17392  ._rcmulr 17434  -_gcsg 19151  Poly₁cpl1 22500  deg₁cdg1 26377  quot_1pcq1p 26451

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11259  ax-1cn 11261  ax-addcl 11263

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-nn 12339  df-slot 17365  df-ndx 17377  df-base 17393  df-q1p 26456

This theorem is used by:  q1peqb  26479