Theorem r1pval 26141

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 17176	. . . 4 ⊢ (𝐹 ∈ 𝐵 → 𝑅 ∈ V)
4	3	adantr 481	. . 3 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝑅 ∈ V)
5		r1pval.e	. . . 4 ⊢ 𝐸 = (rem1p‘𝑅)
6		fveq2 6827	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
7	6, 1	eqtr4di 2792	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = 𝑃)
8	7	fveq2d 6831	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘(Poly1‘𝑟)) = (Base‘𝑃))
9	8, 2	eqtr4di 2792	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘(Poly1‘𝑟)) = 𝐵)
10	9	csbeq1d 3835	. . . . . 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 6841	. . . . . . . 8 ⊢ 𝐵 ∈ V
12	11	a1i 11	. . . . . . 7 ⊢ (𝑟 = 𝑅 → 𝐵 ∈ V)
13		simpr 485	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
14	7	fveq2d 6831	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (-g‘(Poly1‘𝑟)) = (-g‘𝑃))
15		r1pval.m	. . . . . . . . . . 11 ⊢ − = (-g‘𝑃)
16	14, 15	eqtr4di 2792	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (-g‘(Poly1‘𝑟)) = − )
17		eqidd 2740	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → 𝑓 = 𝑓)
18	7	fveq2d 6831	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (.r‘(Poly1‘𝑟)) = (.r‘𝑃))
19		r1pval.t	. . . . . . . . . . . 12 ⊢ · = (.r‘𝑃)
20	18, 19	eqtr4di 2792	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (.r‘(Poly1‘𝑟)) = · )
21		fveq2 6827	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (quot1p‘𝑟) = (quot1p‘𝑅))
22		r1pval.q	. . . . . . . . . . . . 13 ⊢ 𝑄 = (quot1p‘𝑅)
23	21, 22	eqtr4di 2792	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (quot1p‘𝑟) = 𝑄)
24	23	oveqd 7373	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (𝑓(quot1p‘𝑟)𝑔) = (𝑓𝑄𝑔))
25		eqidd 2740	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → 𝑔 = 𝑔)
26	20, 24, 25	oveq123d 7377	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → ((𝑓(quot1p‘𝑟)𝑔)(.r‘(Poly1‘𝑟))𝑔) = ((𝑓𝑄𝑔) · 𝑔))
27	16, 17, 26	oveq123d 7377	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑓(-g‘(Poly1‘𝑟))((𝑓(quot1p‘𝑟)𝑔)(.r‘(Poly1‘𝑟))𝑔)) = (𝑓 − ((𝑓𝑄𝑔) · 𝑔)))
28	27	adantr 481	. . . . . . . 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 7431	. . . . . . 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 3867	. . . . . 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 2774	. . . . 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 26117	. . . . 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 8021	. . . . 5 ⊢ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 − ((𝑓𝑄𝑔) · 𝑔))) ∈ V
34	31, 32, 33	fvmpt 6935	. . . 4 ⊢ (𝑅 ∈ V → (rem1p‘𝑅) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 − ((𝑓𝑄𝑔) · 𝑔))))
35	5, 34	eqtrid 2786	. . 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 483	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
38		oveq12 7365	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓𝑄𝑔) = (𝐹𝑄𝐺))
39		simpr 485	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
40	38, 39	oveq12d 7374	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓𝑄𝑔) · 𝑔) = ((𝐹𝑄𝐺) · 𝐺))
41	37, 40	oveq12d 7374	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓 − ((𝑓𝑄𝑔) · 𝑔)) = (𝐹 − ((𝐹𝑄𝐺) · 𝐺)))
42	41	adantl 482	. 2 ⊢ (((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (𝑓 − ((𝑓𝑄𝑔) · 𝑔)) = (𝐹 − ((𝐹𝑄𝐺) · 𝐺)))
43		simpl 483	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐵)
44		simpr 485	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵)
45		ovexd 7391	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 − ((𝐹𝑄𝐺) · 𝐺)) ∈ V)
46	36, 42, 43, 44, 45	ovmpod 7508	1 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹𝐸𝐺) = (𝐹 − ((𝐹𝑄𝐺) · 𝐺)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3431  ⦋csb 3831  ‘cfv 6485  (class class class)co 7356   ∈ cmpo 7358  Basecbs 17170  ._rcmulr 17212  -_gcsg 18902  Poly₁cpl1 22162  quot_1pcq1p 26111  rem_1pcr1p 26112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-1cn 11087  ax-addcl 11089

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-nn 12166  df-slot 17143  df-ndx 17155  df-base 17171  df-r1p 26117

This theorem is referenced by:  r1pcl  26142  r1pdeglt  26143  r1pid  26144  dvdsr1p  26147  ig1pdvds  26163  q1pdir  33686  q1pvsca  33687  r1pvsca  33688  r1pcyc  33690  r1padd1  33691  irredminply  33900