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Theorem r1pval 25521
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 17089 . . . 4 (𝐹𝐵𝑅 ∈ V)
43adantr 481 . . 3 ((𝐹𝐵𝐺𝐵) → 𝑅 ∈ V)
5 r1pval.e . . . 4 𝐸 = (rem1p𝑅)
6 fveq2 6842 . . . . . . . . . 10 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
76, 1eqtr4di 2794 . . . . . . . . 9 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
87fveq2d 6846 . . . . . . . 8 (𝑟 = 𝑅 → (Base‘(Poly1𝑟)) = (Base‘𝑃))
98, 2eqtr4di 2794 . . . . . . 7 (𝑟 = 𝑅 → (Base‘(Poly1𝑟)) = 𝐵)
109csbeq1d 3859 . . . . . 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 6856 . . . . . . . 8 𝐵 ∈ V
1211a1i 11 . . . . . . 7 (𝑟 = 𝑅𝐵 ∈ V)
13 simpr 485 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = 𝐵) → 𝑏 = 𝐵)
147fveq2d 6846 . . . . . . . . . . 11 (𝑟 = 𝑅 → (-g‘(Poly1𝑟)) = (-g𝑃))
15 r1pval.m . . . . . . . . . . 11 = (-g𝑃)
1614, 15eqtr4di 2794 . . . . . . . . . 10 (𝑟 = 𝑅 → (-g‘(Poly1𝑟)) = )
17 eqidd 2737 . . . . . . . . . 10 (𝑟 = 𝑅𝑓 = 𝑓)
187fveq2d 6846 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (.r‘(Poly1𝑟)) = (.r𝑃))
19 r1pval.t . . . . . . . . . . . 12 · = (.r𝑃)
2018, 19eqtr4di 2794 . . . . . . . . . . 11 (𝑟 = 𝑅 → (.r‘(Poly1𝑟)) = · )
21 fveq2 6842 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (quot1p𝑟) = (quot1p𝑅))
22 r1pval.q . . . . . . . . . . . . 13 𝑄 = (quot1p𝑅)
2321, 22eqtr4di 2794 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (quot1p𝑟) = 𝑄)
2423oveqd 7374 . . . . . . . . . . 11 (𝑟 = 𝑅 → (𝑓(quot1p𝑟)𝑔) = (𝑓𝑄𝑔))
25 eqidd 2737 . . . . . . . . . . 11 (𝑟 = 𝑅𝑔 = 𝑔)
2620, 24, 25oveq123d 7378 . . . . . . . . . 10 (𝑟 = 𝑅 → ((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔) = ((𝑓𝑄𝑔) · 𝑔))
2716, 17, 26oveq123d 7378 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔)) = (𝑓 ((𝑓𝑄𝑔) · 𝑔)))
2827adantr 481 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = 𝐵) → (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔)) = (𝑓 ((𝑓𝑄𝑔) · 𝑔)))
2913, 13, 28mpoeq123dv 7432 . . . . . . 7 ((𝑟 = 𝑅𝑏 = 𝐵) → (𝑓𝑏, 𝑔𝑏 ↦ (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))))
3012, 29csbied 3893 . . . . . 6 (𝑟 = 𝑅𝐵 / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))))
3110, 30eqtrd 2776 . . . . 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 25498 . . . . 5 rem1p = (𝑟 ∈ V ↦ (Base‘(Poly1𝑟)) / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔))))
3311, 11mpoex 8012 . . . . 5 (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))) ∈ V
3431, 32, 33fvmpt 6948 . . . 4 (𝑅 ∈ V → (rem1p𝑅) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))))
355, 34eqtrid 2788 . . 3 (𝑅 ∈ V → 𝐸 = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))))
364, 35syl 17 . 2 ((𝐹𝐵𝐺𝐵) → 𝐸 = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))))
37 simpl 483 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑓 = 𝐹)
38 oveq12 7366 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓𝑄𝑔) = (𝐹𝑄𝐺))
39 simpr 485 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑔 = 𝐺)
4038, 39oveq12d 7375 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑄𝑔) · 𝑔) = ((𝐹𝑄𝐺) · 𝐺))
4137, 40oveq12d 7375 . . 3 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓 ((𝑓𝑄𝑔) · 𝑔)) = (𝐹 ((𝐹𝑄𝐺) · 𝐺)))
4241adantl 482 . 2 (((𝐹𝐵𝐺𝐵) ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → (𝑓 ((𝑓𝑄𝑔) · 𝑔)) = (𝐹 ((𝐹𝑄𝐺) · 𝐺)))
43 simpl 483 . 2 ((𝐹𝐵𝐺𝐵) → 𝐹𝐵)
44 simpr 485 . 2 ((𝐹𝐵𝐺𝐵) → 𝐺𝐵)
45 ovexd 7392 . 2 ((𝐹𝐵𝐺𝐵) → (𝐹 ((𝐹𝑄𝐺) · 𝐺)) ∈ V)
4636, 42, 43, 44, 45ovmpod 7507 1 ((𝐹𝐵𝐺𝐵) → (𝐹𝐸𝐺) = (𝐹 ((𝐹𝑄𝐺) · 𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3445  csb 3855  cfv 6496  (class class class)co 7357  cmpo 7359  Basecbs 17083  .rcmulr 17134  -gcsg 18750  Poly1cpl1 21548  quot1pcq1p 25492  rem1pcr1p 25493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-1cn 11109  ax-addcl 11111
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-nn 12154  df-slot 17054  df-ndx 17066  df-base 17084  df-r1p 25498
This theorem is referenced by:  r1pcl  25522  r1pdeglt  25523  r1pid  25524  dvdsr1p  25526  ig1pdvds  25541
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