Theorem r1pval 23916

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 15920	. . . 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 6191	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
7	6, 1	syl6eqr 2674	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = 𝑃)
8	7	fveq2d 6195	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘(Poly1‘𝑟)) = (Base‘𝑃))
9	8, 2	syl6eqr 2674	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘(Poly1‘𝑟)) = 𝐵)
10	9	csbeq1d 3540	. . . . . 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		fvex 6201	. . . . . . . . 9 ⊢ (Base‘𝑃) ∈ V
12	2, 11	eqeltri 2697	. . . . . . . 8 ⊢ 𝐵 ∈ V
13	12	a1i 11	. . . . . . 7 ⊢ (𝑟 = 𝑅 → 𝐵 ∈ V)
14		simpr 477	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
15	7	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (-g‘(Poly1‘𝑟)) = (-g‘𝑃))
16		r1pval.m	. . . . . . . . . . 11 ⊢ − = (-g‘𝑃)
17	15, 16	syl6eqr 2674	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (-g‘(Poly1‘𝑟)) = − )
18		eqidd 2623	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → 𝑓 = 𝑓)
19	7	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (.r‘(Poly1‘𝑟)) = (.r‘𝑃))
20		r1pval.t	. . . . . . . . . . . 12 ⊢ · = (.r‘𝑃)
21	19, 20	syl6eqr 2674	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (.r‘(Poly1‘𝑟)) = · )
22		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → (quot1p‘𝑟) = (quot1p‘𝑅))
23		r1pval.q	. . . . . . . . . . . . 13 ⊢ 𝑄 = (quot1p‘𝑅)
24	22, 23	syl6eqr 2674	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (quot1p‘𝑟) = 𝑄)
25	24	oveqd 6667	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (𝑓(quot1p‘𝑟)𝑔) = (𝑓𝑄𝑔))
26		eqidd 2623	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → 𝑔 = 𝑔)
27	21, 25, 26	oveq123d 6671	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → ((𝑓(quot1p‘𝑟)𝑔)(.r‘(Poly1‘𝑟))𝑔) = ((𝑓𝑄𝑔) · 𝑔))
28	17, 18, 27	oveq123d 6671	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑓(-g‘(Poly1‘𝑟))((𝑓(quot1p‘𝑟)𝑔)(.r‘(Poly1‘𝑟))𝑔)) = (𝑓 − ((𝑓𝑄𝑔) · 𝑔)))
29	28	adantr 481	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = 𝐵) → (𝑓(-g‘(Poly1‘𝑟))((𝑓(quot1p‘𝑟)𝑔)(.r‘(Poly1‘𝑟))𝑔)) = (𝑓 − ((𝑓𝑄𝑔) · 𝑔)))
30	14, 14, 29	mpt2eq123dv 6717	. . . . . . 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))))
31	13, 30	csbied 3560	. . . . . 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))))
32	10, 31	eqtrd 2656	. . . . 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))))
33		df-r1p 23893	. . . . 5 ⊢ rem1p = (𝑟 ∈ V ↦ ⦋(Base‘(Poly1‘𝑟)) / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓(-g‘(Poly1‘𝑟))((𝑓(quot1p‘𝑟)𝑔)(.r‘(Poly1‘𝑟))𝑔))))
34	12, 12	mpt2ex 7247	. . . . 5 ⊢ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 − ((𝑓𝑄𝑔) · 𝑔))) ∈ V
35	32, 33, 34	fvmpt 6282	. . . 4 ⊢ (𝑅 ∈ V → (rem1p‘𝑅) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 − ((𝑓𝑄𝑔) · 𝑔))))
36	5, 35	syl5eq 2668	. . 3 ⊢ (𝑅 ∈ V → 𝐸 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 − ((𝑓𝑄𝑔) · 𝑔))))
37	4, 36	syl 17	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐸 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 − ((𝑓𝑄𝑔) · 𝑔))))
38		simpl 473	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
39		oveq12 6659	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓𝑄𝑔) = (𝐹𝑄𝐺))
40		simpr 477	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
41	39, 40	oveq12d 6668	. . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓𝑄𝑔) · 𝑔) = ((𝐹𝑄𝐺) · 𝐺))
42	38, 41	oveq12d 6668	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓 − ((𝑓𝑄𝑔) · 𝑔)) = (𝐹 − ((𝐹𝑄𝐺) · 𝐺)))
43	42	adantl 482	. 2 ⊢ (((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (𝑓 − ((𝑓𝑄𝑔) · 𝑔)) = (𝐹 − ((𝐹𝑄𝐺) · 𝐺)))
44		simpl 473	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐵)
45		simpr 477	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵)
46		ovexd 6680	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 − ((𝐹𝑄𝐺) · 𝐺)) ∈ V)
47	37, 43, 44, 45, 46	ovmpt2d 6788	1 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹𝐸𝐺) = (𝐹 − ((𝐹𝑄𝐺) · 𝐺)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ⦋csb 3533  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  Basecbs 15857  ._rcmulr 15942  -_gcsg 17424  Poly₁cpl1 19547  quot_1pcq1p 23887  rem_1pcr1p 23888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-slot 15861  df-base 15863  df-r1p 23893

This theorem is referenced by:  r1pcl  23917  r1pdeglt  23918  r1pid  23919  dvdsr1p  23921  ig1pdvds  23936