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Theorem r1pval 23961
Description: Value of the polynomial remainder function. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Hypotheses
Ref Expression
r1pval.e 𝐸 = (rem1p𝑅)
r1pval.p 𝑃 = (Poly1𝑅)
r1pval.b 𝐵 = (Base‘𝑃)
r1pval.q 𝑄 = (quot1p𝑅)
r1pval.t · = (.r𝑃)
r1pval.m = (-g𝑃)
Assertion
Ref Expression
r1pval ((𝐹𝐵𝐺𝐵) → (𝐹𝐸𝐺) = (𝐹 ((𝐹𝑄𝐺) · 𝐺)))

Proof of Theorem r1pval
Dummy variables 𝑏 𝑓 𝑔 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 r1pval.p . . . . 5 𝑃 = (Poly1𝑅)
2 r1pval.b . . . . 5 𝐵 = (Base‘𝑃)
31, 2elbasfv 15967 . . . 4 (𝐹𝐵𝑅 ∈ V)
43adantr 480 . . 3 ((𝐹𝐵𝐺𝐵) → 𝑅 ∈ V)
5 r1pval.e . . . 4 𝐸 = (rem1p𝑅)
6 fveq2 6229 . . . . . . . . . 10 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
76, 1syl6eqr 2703 . . . . . . . . 9 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
87fveq2d 6233 . . . . . . . 8 (𝑟 = 𝑅 → (Base‘(Poly1𝑟)) = (Base‘𝑃))
98, 2syl6eqr 2703 . . . . . . 7 (𝑟 = 𝑅 → (Base‘(Poly1𝑟)) = 𝐵)
109csbeq1d 3573 . . . . . 6 (𝑟 = 𝑅(Base‘(Poly1𝑟)) / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔))) = 𝐵 / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔))))
11 fvex 6239 . . . . . . . . 9 (Base‘𝑃) ∈ V
122, 11eqeltri 2726 . . . . . . . 8 𝐵 ∈ V
1312a1i 11 . . . . . . 7 (𝑟 = 𝑅𝐵 ∈ V)
14 simpr 476 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = 𝐵) → 𝑏 = 𝐵)
157fveq2d 6233 . . . . . . . . . . 11 (𝑟 = 𝑅 → (-g‘(Poly1𝑟)) = (-g𝑃))
16 r1pval.m . . . . . . . . . . 11 = (-g𝑃)
1715, 16syl6eqr 2703 . . . . . . . . . 10 (𝑟 = 𝑅 → (-g‘(Poly1𝑟)) = )
18 eqidd 2652 . . . . . . . . . 10 (𝑟 = 𝑅𝑓 = 𝑓)
197fveq2d 6233 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (.r‘(Poly1𝑟)) = (.r𝑃))
20 r1pval.t . . . . . . . . . . . 12 · = (.r𝑃)
2119, 20syl6eqr 2703 . . . . . . . . . . 11 (𝑟 = 𝑅 → (.r‘(Poly1𝑟)) = · )
22 fveq2 6229 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (quot1p𝑟) = (quot1p𝑅))
23 r1pval.q . . . . . . . . . . . . 13 𝑄 = (quot1p𝑅)
2422, 23syl6eqr 2703 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (quot1p𝑟) = 𝑄)
2524oveqd 6707 . . . . . . . . . . 11 (𝑟 = 𝑅 → (𝑓(quot1p𝑟)𝑔) = (𝑓𝑄𝑔))
26 eqidd 2652 . . . . . . . . . . 11 (𝑟 = 𝑅𝑔 = 𝑔)
2721, 25, 26oveq123d 6711 . . . . . . . . . 10 (𝑟 = 𝑅 → ((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔) = ((𝑓𝑄𝑔) · 𝑔))
2817, 18, 27oveq123d 6711 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔)) = (𝑓 ((𝑓𝑄𝑔) · 𝑔)))
2928adantr 480 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = 𝐵) → (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔)) = (𝑓 ((𝑓𝑄𝑔) · 𝑔)))
3014, 14, 29mpt2eq123dv 6759 . . . . . . 7 ((𝑟 = 𝑅𝑏 = 𝐵) → (𝑓𝑏, 𝑔𝑏 ↦ (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))))
3113, 30csbied 3593 . . . . . 6 (𝑟 = 𝑅𝐵 / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))))
3210, 31eqtrd 2685 . . . . 5 (𝑟 = 𝑅(Base‘(Poly1𝑟)) / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))))
33 df-r1p 23938 . . . . 5 rem1p = (𝑟 ∈ V ↦ (Base‘(Poly1𝑟)) / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔))))
3412, 12mpt2ex 7292 . . . . 5 (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))) ∈ V
3532, 33, 34fvmpt 6321 . . . 4 (𝑅 ∈ V → (rem1p𝑅) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))))
365, 35syl5eq 2697 . . 3 (𝑅 ∈ V → 𝐸 = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))))
374, 36syl 17 . 2 ((𝐹𝐵𝐺𝐵) → 𝐸 = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))))
38 simpl 472 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑓 = 𝐹)
39 oveq12 6699 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓𝑄𝑔) = (𝐹𝑄𝐺))
40 simpr 476 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑔 = 𝐺)
4139, 40oveq12d 6708 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑄𝑔) · 𝑔) = ((𝐹𝑄𝐺) · 𝐺))
4238, 41oveq12d 6708 . . 3 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓 ((𝑓𝑄𝑔) · 𝑔)) = (𝐹 ((𝐹𝑄𝐺) · 𝐺)))
4342adantl 481 . 2 (((𝐹𝐵𝐺𝐵) ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → (𝑓 ((𝑓𝑄𝑔) · 𝑔)) = (𝐹 ((𝐹𝑄𝐺) · 𝐺)))
44 simpl 472 . 2 ((𝐹𝐵𝐺𝐵) → 𝐹𝐵)
45 simpr 476 . 2 ((𝐹𝐵𝐺𝐵) → 𝐺𝐵)
46 ovexd 6720 . 2 ((𝐹𝐵𝐺𝐵) → (𝐹 ((𝐹𝑄𝐺) · 𝐺)) ∈ V)
4737, 43, 44, 45, 46ovmpt2d 6830 1 ((𝐹𝐵𝐺𝐵) → (𝐹𝐸𝐺) = (𝐹 ((𝐹𝑄𝐺) · 𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  csb 3566  cfv 5926  (class class class)co 6690  cmpt2 6692  Basecbs 15904  .rcmulr 15989  -gcsg 17471  Poly1cpl1 19595  quot1pcq1p 23932  rem1pcr1p 23933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-slot 15908  df-base 15910  df-r1p 23938
This theorem is referenced by:  r1pcl  23962  r1pdeglt  23963  r1pid  23964  dvdsr1p  23966  ig1pdvds  23981
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