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Theorem r1pval 26212
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 17251 . . . 4 (𝐹𝐵𝑅 ∈ V)
43adantr 480 . . 3 ((𝐹𝐵𝐺𝐵) → 𝑅 ∈ V)
5 r1pval.e . . . 4 𝐸 = (rem1p𝑅)
6 fveq2 6907 . . . . . . . . . 10 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
76, 1eqtr4di 2793 . . . . . . . . 9 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
87fveq2d 6911 . . . . . . . 8 (𝑟 = 𝑅 → (Base‘(Poly1𝑟)) = (Base‘𝑃))
98, 2eqtr4di 2793 . . . . . . 7 (𝑟 = 𝑅 → (Base‘(Poly1𝑟)) = 𝐵)
109csbeq1d 3912 . . . . . 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))))
112fvexi 6921 . . . . . . . 8 𝐵 ∈ V
1211a1i 11 . . . . . . 7 (𝑟 = 𝑅𝐵 ∈ V)
13 simpr 484 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = 𝐵) → 𝑏 = 𝐵)
147fveq2d 6911 . . . . . . . . . . 11 (𝑟 = 𝑅 → (-g‘(Poly1𝑟)) = (-g𝑃))
15 r1pval.m . . . . . . . . . . 11 = (-g𝑃)
1614, 15eqtr4di 2793 . . . . . . . . . 10 (𝑟 = 𝑅 → (-g‘(Poly1𝑟)) = )
17 eqidd 2736 . . . . . . . . . 10 (𝑟 = 𝑅𝑓 = 𝑓)
187fveq2d 6911 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (.r‘(Poly1𝑟)) = (.r𝑃))
19 r1pval.t . . . . . . . . . . . 12 · = (.r𝑃)
2018, 19eqtr4di 2793 . . . . . . . . . . 11 (𝑟 = 𝑅 → (.r‘(Poly1𝑟)) = · )
21 fveq2 6907 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (quot1p𝑟) = (quot1p𝑅))
22 r1pval.q . . . . . . . . . . . . 13 𝑄 = (quot1p𝑅)
2321, 22eqtr4di 2793 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (quot1p𝑟) = 𝑄)
2423oveqd 7448 . . . . . . . . . . 11 (𝑟 = 𝑅 → (𝑓(quot1p𝑟)𝑔) = (𝑓𝑄𝑔))
25 eqidd 2736 . . . . . . . . . . 11 (𝑟 = 𝑅𝑔 = 𝑔)
2620, 24, 25oveq123d 7452 . . . . . . . . . 10 (𝑟 = 𝑅 → ((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔) = ((𝑓𝑄𝑔) · 𝑔))
2716, 17, 26oveq123d 7452 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔)) = (𝑓 ((𝑓𝑄𝑔) · 𝑔)))
2827adantr 480 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = 𝐵) → (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔)) = (𝑓 ((𝑓𝑄𝑔) · 𝑔)))
2913, 13, 28mpoeq123dv 7508 . . . . . . 7 ((𝑟 = 𝑅𝑏 = 𝐵) → (𝑓𝑏, 𝑔𝑏 ↦ (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))))
3012, 29csbied 3946 . . . . . 6 (𝑟 = 𝑅𝐵 / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))))
3110, 30eqtrd 2775 . . . . 5 (𝑟 = 𝑅(Base‘(Poly1𝑟)) / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))))
32 df-r1p 26188 . . . . 5 rem1p = (𝑟 ∈ V ↦ (Base‘(Poly1𝑟)) / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔))))
3311, 11mpoex 8103 . . . . 5 (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))) ∈ V
3431, 32, 33fvmpt 7016 . . . 4 (𝑅 ∈ V → (rem1p𝑅) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))))
355, 34eqtrid 2787 . . 3 (𝑅 ∈ V → 𝐸 = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))))
364, 35syl 17 . 2 ((𝐹𝐵𝐺𝐵) → 𝐸 = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))))
37 simpl 482 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑓 = 𝐹)
38 oveq12 7440 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓𝑄𝑔) = (𝐹𝑄𝐺))
39 simpr 484 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑔 = 𝐺)
4038, 39oveq12d 7449 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑄𝑔) · 𝑔) = ((𝐹𝑄𝐺) · 𝐺))
4137, 40oveq12d 7449 . . 3 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓 ((𝑓𝑄𝑔) · 𝑔)) = (𝐹 ((𝐹𝑄𝐺) · 𝐺)))
4241adantl 481 . 2 (((𝐹𝐵𝐺𝐵) ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → (𝑓 ((𝑓𝑄𝑔) · 𝑔)) = (𝐹 ((𝐹𝑄𝐺) · 𝐺)))
43 simpl 482 . 2 ((𝐹𝐵𝐺𝐵) → 𝐹𝐵)
44 simpr 484 . 2 ((𝐹𝐵𝐺𝐵) → 𝐺𝐵)
45 ovexd 7466 . 2 ((𝐹𝐵𝐺𝐵) → (𝐹 ((𝐹𝑄𝐺) · 𝐺)) ∈ V)
4636, 42, 43, 44, 45ovmpod 7585 1 ((𝐹𝐵𝐺𝐵) → (𝐹𝐸𝐺) = (𝐹 ((𝐹𝑄𝐺) · 𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  csb 3908  cfv 6563  (class class class)co 7431  cmpo 7433  Basecbs 17245  .rcmulr 17299  -gcsg 18966  Poly1cpl1 22194  quot1pcq1p 26182  rem1pcr1p 26183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-1cn 11211  ax-addcl 11213
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-nn 12265  df-slot 17216  df-ndx 17228  df-base 17246  df-r1p 26188
This theorem is referenced by:  r1pcl  26213  r1pdeglt  26214  r1pid  26215  dvdsr1p  26218  ig1pdvds  26234  q1pdir  33603  q1pvsca  33604  r1pvsca  33605  r1pcyc  33607  r1padd1  33608  irredminply  33722
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