Theorem r1pval 26315

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 17270	. . . 4 ⊢ (𝐹 ∈ 𝐵 → 𝑅 ∈ V)
4	3	adantr 485	. . 3 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝑅 ∈ V)
5		r1pval.e	. . . 4 ⊢ 𝐸 = (rem1p‘𝑅)
6		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
7	6, 1	eqtr4di 2816	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = 𝑃)
8	7	fveq2d 6885	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘(Poly1‘𝑟)) = (Base‘𝑃))
9	8, 2	eqtr4di 2816	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘(Poly1‘𝑟)) = 𝐵)
10	9	csbeq1d 3857	. . . . . 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 6895	. . . . . . . 8 ⊢ 𝐵 ∈ V
12	11	a1i 11	. . . . . . 7 ⊢ (𝑟 = 𝑅 → 𝐵 ∈ V)
13		simpr 489	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
14	7	fveq2d 6885	. . . . . . . . . . 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 6885	. . . . . . . . . . . 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 6881	. . . . . . . . . . . . 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 7427	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (𝑓(quot1p‘𝑟)𝑔) = (𝑓𝑄𝑔))
25		eqidd 2764	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → 𝑔 = 𝑔)
26	20, 24, 25	oveq123d 7431	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → ((𝑓(quot1p‘𝑟)𝑔)(.r‘(Poly1‘𝑟))𝑔) = ((𝑓𝑄𝑔) · 𝑔))
27	16, 17, 26	oveq123d 7431	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑓(-g‘(Poly1‘𝑟))((𝑓(quot1p‘𝑟)𝑔)(.r‘(Poly1‘𝑟))𝑔)) = (𝑓 − ((𝑓𝑄𝑔) · 𝑔)))
28	27	adantr 485	. . . . . . . 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 7485	. . . . . . 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 3889	. . . . . 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 26291	. . . . 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 8072	. . . . 5 ⊢ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 − ((𝑓𝑄𝑔) · 𝑔))) ∈ V
34	31, 32, 33	fvmpt 6989	. . . 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 487	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
38		oveq12 7419	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓𝑄𝑔) = (𝐹𝑄𝐺))
39		simpr 489	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
40	38, 39	oveq12d 7428	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓𝑄𝑔) · 𝑔) = ((𝐹𝑄𝐺) · 𝐺))
41	37, 40	oveq12d 7428	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓 − ((𝑓𝑄𝑔) · 𝑔)) = (𝐹 − ((𝐹𝑄𝐺) · 𝐺)))
42	41	adantl 486	. 2 ⊢ (((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (𝑓 − ((𝑓𝑄𝑔) · 𝑔)) = (𝐹 − ((𝐹𝑄𝐺) · 𝐺)))
43		simpl 487	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐵)
44		simpr 489	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵)
45		ovexd 7445	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 − ((𝐹𝑄𝐺) · 𝐺)) ∈ V)
46	36, 42, 43, 44, 45	ovmpod 7562	1 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹𝐸𝐺) = (𝐹 − ((𝐹𝑄𝐺) · 𝐺)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ⦋csb 3853  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  Basecbs 17264  ._rcmulr 17306  -_gcsg 18997  Poly₁cpl1 22337  quot_1pcq1p 26285  rem_1pcr1p 26286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-1cn 11153  ax-addcl 11155

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-nn 12229  df-slot 17237  df-ndx 17249  df-base 17265  df-r1p 26291

This theorem is referenced by:  r1pcl  26316  r1pdeglt  26317  r1pid  26318  dvdsr1p  26321  ig1pdvds  26337  q1pdir  33893  q1pvsca  33894  r1pvsca  33895  r1pcyc  33897  r1padd1  33898  irredminply  34106