Theorem r1pval 26531

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 17437	. . . 4 ⊢ (𝐹 ∈ 𝐵 → 𝑅 ∈ V)
4	3	adantr 486	. . 3 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝑅 ∈ V)
5		r1pval.e	. . . 4 ⊢ 𝐸 = (rem1p‘𝑅)
6		fveq2 6916	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
7	6, 1	eqtr4di 2816	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = 𝑃)
8	7	fveq2d 6920	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘(Poly1‘𝑟)) = (Base‘𝑃))
9	8, 2	eqtr4di 2816	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘(Poly1‘𝑟)) = 𝐵)
10	9	csbeq1d 3855	. . . . . 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 6930	. . . . . . . 8 ⊢ 𝐵 ∈ V
12	11	a1i 11	. . . . . . 7 ⊢ (𝑟 = 𝑅 → 𝐵 ∈ V)
13		simpr 490	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
14	7	fveq2d 6920	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (-g‘(Poly1‘𝑟)) = (-g‘𝑃))
15		r1pval.m	. . . . . . . . . . 11 ⊢ − = (-g‘𝑃)
16	14, 15	eqtr4di 2816	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (-g‘(Poly1‘𝑟)) = − )
17		eqidd 2764	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → 𝑓 = 𝑓)
18	7	fveq2d 6920	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (.r‘(Poly1‘𝑟)) = (.r‘𝑃))
19		r1pval.t	. . . . . . . . . . . 12 ⊢ · = (.r‘𝑃)
20	18, 19	eqtr4di 2816	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (.r‘(Poly1‘𝑟)) = · )
21		fveq2 6916	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (quot1p‘𝑟) = (quot1p‘𝑅))
22		r1pval.q	. . . . . . . . . . . . 13 ⊢ 𝑄 = (quot1p‘𝑅)
23	21, 22	eqtr4di 2816	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (quot1p‘𝑟) = 𝑄)
24	23	oveqd 7468	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (𝑓(quot1p‘𝑟)𝑔) = (𝑓𝑄𝑔))
25		eqidd 2764	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → 𝑔 = 𝑔)
26	20, 24, 25	oveq123d 7472	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → ((𝑓(quot1p‘𝑟)𝑔)(.r‘(Poly1‘𝑟))𝑔) = ((𝑓𝑄𝑔) · 𝑔))
27	16, 17, 26	oveq123d 7472	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑓(-g‘(Poly1‘𝑟))((𝑓(quot1p‘𝑟)𝑔)(.r‘(Poly1‘𝑟))𝑔)) = (𝑓 − ((𝑓𝑄𝑔) · 𝑔)))
28	27	adantr 486	. . . . . . . 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 7526	. . . . . . 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 3887	. . . . . 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 2798	. . . . 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 26507	. . . . 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 8127	. . . . 5 ⊢ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 − ((𝑓𝑄𝑔) · 𝑔))) ∈ V
34	31, 32, 33	fvmpt 7024	. . . 4 ⊢ (𝑅 ∈ V → (rem1p‘𝑅) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 − ((𝑓𝑄𝑔) · 𝑔))))
35	5, 34	eqtrid 2810	. . 3 ⊢ (𝑅 ∈ V → 𝐸 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 − ((𝑓𝑄𝑔) · 𝑔))))
36	4, 35	syl 18	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐸 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 − ((𝑓𝑄𝑔) · 𝑔))))
37		simpl 488	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
38		oveq12 7460	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓𝑄𝑔) = (𝐹𝑄𝐺))
39		simpr 490	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
40	38, 39	oveq12d 7469	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓𝑄𝑔) · 𝑔) = ((𝐹𝑄𝐺) · 𝐺))
41	37, 40	oveq12d 7469	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓 − ((𝑓𝑄𝑔) · 𝑔)) = (𝐹 − ((𝐹𝑄𝐺) · 𝐺)))
42	41	adantl 487	. 2 ⊢ (((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (𝑓 − ((𝑓𝑄𝑔) · 𝑔)) = (𝐹 − ((𝐹𝑄𝐺) · 𝐺)))
43		simpl 488	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐵)
44		simpr 490	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵)
45		ovexd 7486	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 − ((𝐹𝑄𝐺) · 𝐺)) ∈ V)
46	36, 42, 43, 44, 45	ovmpod 7604	1 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹𝐸𝐺) = (𝐹 − ((𝐹𝑄𝐺) · 𝐺)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ⦋csb 3851  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  Basecbs 17431  ._rcmulr 17473  -_gcsg 19196  Poly₁cpl1 22550  quot_1pcq1p 26501  rem_1pcr1p 26502

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-1cn 11297  ax-addcl 11299

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-nn 12376  df-slot 17404  df-ndx 17416  df-base 17432  df-r1p 26507

This theorem is used by:  r1pcl  26532  r1pdeglt  26533  r1pid  26534  dvdsr1p  26537  ig1pdvds  26553  q1pdir  34240  q1pvsca  34241  r1pvsca  34242  r1pcyc  34244  r1padd1  34245  irredminply  34459