Theorem r1pval 24753

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

Hypotheses

Ref	Expression
r1pval.e	⊢ 𝐸 = (rem1p‘𝑅)
r1pval.p	⊢ 𝑃 = (Poly1‘𝑅)
r1pval.b	⊢ 𝐵 = (Base‘𝑃)
r1pval.q	⊢ 𝑄 = (quot1p‘𝑅)
r1pval.t	⊢ · = (.r‘𝑃)
r1pval.m	⊢ − = (-g‘𝑃)

Assertion

Ref	Expression
r1pval	⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹𝐸𝐺) = (𝐹 − ((𝐹𝑄𝐺) · 𝐺)))

Proof of Theorem r1pval

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

Step	Hyp	Ref	Expression
1		r1pval.p	. . . . 5 ⊢ 𝑃 = (Poly1‘𝑅)
2		r1pval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑃)
3	1, 2	elbasfv 16547	. . . 4 ⊢ (𝐹 ∈ 𝐵 → 𝑅 ∈ V)
4	3	adantr 483	. . 3 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝑅 ∈ V)
5		r1pval.e	. . . 4 ⊢ 𝐸 = (rem1p‘𝑅)
6		fveq2 6673	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
7	6, 1	syl6eqr 2877	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = 𝑃)
8	7	fveq2d 6677	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘(Poly1‘𝑟)) = (Base‘𝑃))
9	8, 2	syl6eqr 2877	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘(Poly1‘𝑟)) = 𝐵)
10	9	csbeq1d 3890	. . . . . 6 ⊢ (𝑟 = 𝑅 → ⦋(Base‘(Poly1‘𝑟)) / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓(-g‘(Poly1‘𝑟))((𝑓(quot1p‘𝑟)𝑔)(.r‘(Poly1‘𝑟))𝑔))) = ⦋𝐵 / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓(-g‘(Poly1‘𝑟))((𝑓(quot1p‘𝑟)𝑔)(.r‘(Poly1‘𝑟))𝑔))))
11	2	fvexi 6687	. . . . . . . 8 ⊢ 𝐵 ∈ V
12	11	a1i 11	. . . . . . 7 ⊢ (𝑟 = 𝑅 → 𝐵 ∈ V)
13		simpr 487	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
14	7	fveq2d 6677	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (-g‘(Poly1‘𝑟)) = (-g‘𝑃))
15		r1pval.m	. . . . . . . . . . 11 ⊢ − = (-g‘𝑃)
16	14, 15	syl6eqr 2877	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (-g‘(Poly1‘𝑟)) = − )
17		eqidd 2825	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → 𝑓 = 𝑓)
18	7	fveq2d 6677	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (.r‘(Poly1‘𝑟)) = (.r‘𝑃))
19		r1pval.t	. . . . . . . . . . . 12 ⊢ · = (.r‘𝑃)
20	18, 19	syl6eqr 2877	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (.r‘(Poly1‘𝑟)) = · )
21		fveq2 6673	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (quot1p‘𝑟) = (quot1p‘𝑅))
22		r1pval.q	. . . . . . . . . . . . 13 ⊢ 𝑄 = (quot1p‘𝑅)
23	21, 22	syl6eqr 2877	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (quot1p‘𝑟) = 𝑄)
24	23	oveqd 7176	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (𝑓(quot1p‘𝑟)𝑔) = (𝑓𝑄𝑔))
25		eqidd 2825	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → 𝑔 = 𝑔)
26	20, 24, 25	oveq123d 7180	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → ((𝑓(quot1p‘𝑟)𝑔)(.r‘(Poly1‘𝑟))𝑔) = ((𝑓𝑄𝑔) · 𝑔))
27	16, 17, 26	oveq123d 7180	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑓(-g‘(Poly1‘𝑟))((𝑓(quot1p‘𝑟)𝑔)(.r‘(Poly1‘𝑟))𝑔)) = (𝑓 − ((𝑓𝑄𝑔) · 𝑔)))
28	27	adantr 483	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → (𝑓(-g‘(Poly1‘𝑟))((𝑓(quot1p‘𝑟)𝑔)(.r‘(Poly1‘𝑟))𝑔)) = (𝑓 − ((𝑓𝑄𝑔) · 𝑔)))
29	13, 13, 28	mpoeq123dv 7232	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓(-g‘(Poly1‘𝑟))((𝑓(quot1p‘𝑟)𝑔)(.r‘(Poly1‘𝑟))𝑔))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 − ((𝑓𝑄𝑔) · 𝑔))))
30	12, 29	csbied 3922	. . . . . 6 ⊢ (𝑟 = 𝑅 → ⦋𝐵 / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓(-g‘(Poly1‘𝑟))((𝑓(quot1p‘𝑟)𝑔)(.r‘(Poly1‘𝑟))𝑔))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 − ((𝑓𝑄𝑔) · 𝑔))))
31	10, 30	eqtrd 2859	. . . . 5 ⊢ (𝑟 = 𝑅 → ⦋(Base‘(Poly1‘𝑟)) / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓(-g‘(Poly1‘𝑟))((𝑓(quot1p‘𝑟)𝑔)(.r‘(Poly1‘𝑟))𝑔))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 − ((𝑓𝑄𝑔) · 𝑔))))
32		df-r1p 24730	. . . . 5 ⊢ rem1p = (𝑟 ∈ V ↦ ⦋(Base‘(Poly1‘𝑟)) / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓(-g‘(Poly1‘𝑟))((𝑓(quot1p‘𝑟)𝑔)(.r‘(Poly1‘𝑟))𝑔))))
33	11, 11	mpoex 7780	. . . . 5 ⊢ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 − ((𝑓𝑄𝑔) · 𝑔))) ∈ V
34	31, 32, 33	fvmpt 6771	. . . 4 ⊢ (𝑅 ∈ V → (rem1p‘𝑅) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 − ((𝑓𝑄𝑔) · 𝑔))))
35	5, 34	syl5eq 2871	. . 3 ⊢ (𝑅 ∈ V → 𝐸 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 − ((𝑓𝑄𝑔) · 𝑔))))
36	4, 35	syl 17	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐸 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 − ((𝑓𝑄𝑔) · 𝑔))))
37		simpl 485	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
38		oveq12 7168	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓𝑄𝑔) = (𝐹𝑄𝐺))
39		simpr 487	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
40	38, 39	oveq12d 7177	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓𝑄𝑔) · 𝑔) = ((𝐹𝑄𝐺) · 𝐺))
41	37, 40	oveq12d 7177	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓 − ((𝑓𝑄𝑔) · 𝑔)) = (𝐹 − ((𝐹𝑄𝐺) · 𝐺)))
42	41	adantl 484	. 2 ⊢ (((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (𝑓 − ((𝑓𝑄𝑔) · 𝑔)) = (𝐹 − ((𝐹𝑄𝐺) · 𝐺)))
43		simpl 485	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐵)
44		simpr 487	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵)
45		ovexd 7194	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 − ((𝐹𝑄𝐺) · 𝐺)) ∈ V)
46	36, 42, 43, 44, 45	ovmpod 7305	1 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹𝐸𝐺) = (𝐹 − ((𝐹𝑄𝐺) · 𝐺)))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113  Vcvv 3497  ⦋csb 3886  ‘cfv 6358  (class class class)co 7159   ∈ cmpo 7161  Basecbs 16486  ._rcmulr 16569  -_gcsg 18108  Poly₁cpl1 20348  quot_1pcq1p 24724  rem_1pcr1p 24725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-1st 7692  df-2nd 7693  df-slot 16490  df-base 16492  df-r1p 24730

This theorem is referenced by:  r1pcl  24754  r1pdeglt  24755  r1pid  24756  dvdsr1p  24758  ig1pdvds  24773