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Theorem r1pval 24322
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 16290 . . . 4 (𝐹𝐵𝑅 ∈ V)
43adantr 474 . . 3 ((𝐹𝐵𝐺𝐵) → 𝑅 ∈ V)
5 r1pval.e . . . 4 𝐸 = (rem1p𝑅)
6 fveq2 6437 . . . . . . . . . 10 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
76, 1syl6eqr 2879 . . . . . . . . 9 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
87fveq2d 6441 . . . . . . . 8 (𝑟 = 𝑅 → (Base‘(Poly1𝑟)) = (Base‘𝑃))
98, 2syl6eqr 2879 . . . . . . 7 (𝑟 = 𝑅 → (Base‘(Poly1𝑟)) = 𝐵)
109csbeq1d 3764 . . . . . 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 6451 . . . . . . . 8 𝐵 ∈ V
1211a1i 11 . . . . . . 7 (𝑟 = 𝑅𝐵 ∈ V)
13 simpr 479 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = 𝐵) → 𝑏 = 𝐵)
147fveq2d 6441 . . . . . . . . . . 11 (𝑟 = 𝑅 → (-g‘(Poly1𝑟)) = (-g𝑃))
15 r1pval.m . . . . . . . . . . 11 = (-g𝑃)
1614, 15syl6eqr 2879 . . . . . . . . . 10 (𝑟 = 𝑅 → (-g‘(Poly1𝑟)) = )
17 eqidd 2826 . . . . . . . . . 10 (𝑟 = 𝑅𝑓 = 𝑓)
187fveq2d 6441 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (.r‘(Poly1𝑟)) = (.r𝑃))
19 r1pval.t . . . . . . . . . . . 12 · = (.r𝑃)
2018, 19syl6eqr 2879 . . . . . . . . . . 11 (𝑟 = 𝑅 → (.r‘(Poly1𝑟)) = · )
21 fveq2 6437 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (quot1p𝑟) = (quot1p𝑅))
22 r1pval.q . . . . . . . . . . . . 13 𝑄 = (quot1p𝑅)
2321, 22syl6eqr 2879 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (quot1p𝑟) = 𝑄)
2423oveqd 6927 . . . . . . . . . . 11 (𝑟 = 𝑅 → (𝑓(quot1p𝑟)𝑔) = (𝑓𝑄𝑔))
25 eqidd 2826 . . . . . . . . . . 11 (𝑟 = 𝑅𝑔 = 𝑔)
2620, 24, 25oveq123d 6931 . . . . . . . . . 10 (𝑟 = 𝑅 → ((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔) = ((𝑓𝑄𝑔) · 𝑔))
2716, 17, 26oveq123d 6931 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔)) = (𝑓 ((𝑓𝑄𝑔) · 𝑔)))
2827adantr 474 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = 𝐵) → (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔)) = (𝑓 ((𝑓𝑄𝑔) · 𝑔)))
2913, 13, 28mpt2eq123dv 6982 . . . . . . 7 ((𝑟 = 𝑅𝑏 = 𝐵) → (𝑓𝑏, 𝑔𝑏 ↦ (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))))
3012, 29csbied 3784 . . . . . 6 (𝑟 = 𝑅𝐵 / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))))
3110, 30eqtrd 2861 . . . . 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 24299 . . . . 5 rem1p = (𝑟 ∈ V ↦ (Base‘(Poly1𝑟)) / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔))))
3311, 11mpt2ex 7515 . . . . 5 (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))) ∈ V
3431, 32, 33fvmpt 6533 . . . 4 (𝑅 ∈ V → (rem1p𝑅) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))))
355, 34syl5eq 2873 . . 3 (𝑅 ∈ V → 𝐸 = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))))
364, 35syl 17 . 2 ((𝐹𝐵𝐺𝐵) → 𝐸 = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))))
37 simpl 476 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑓 = 𝐹)
38 oveq12 6919 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓𝑄𝑔) = (𝐹𝑄𝐺))
39 simpr 479 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑔 = 𝐺)
4038, 39oveq12d 6928 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑄𝑔) · 𝑔) = ((𝐹𝑄𝐺) · 𝐺))
4137, 40oveq12d 6928 . . 3 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓 ((𝑓𝑄𝑔) · 𝑔)) = (𝐹 ((𝐹𝑄𝐺) · 𝐺)))
4241adantl 475 . 2 (((𝐹𝐵𝐺𝐵) ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → (𝑓 ((𝑓𝑄𝑔) · 𝑔)) = (𝐹 ((𝐹𝑄𝐺) · 𝐺)))
43 simpl 476 . 2 ((𝐹𝐵𝐺𝐵) → 𝐹𝐵)
44 simpr 479 . 2 ((𝐹𝐵𝐺𝐵) → 𝐺𝐵)
45 ovexd 6944 . 2 ((𝐹𝐵𝐺𝐵) → (𝐹 ((𝐹𝑄𝐺) · 𝐺)) ∈ V)
4636, 42, 43, 44, 45ovmpt2d 7053 1 ((𝐹𝐵𝐺𝐵) → (𝐹𝐸𝐺) = (𝐹 ((𝐹𝑄𝐺) · 𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164  Vcvv 3414  csb 3757  cfv 6127  (class class class)co 6910  cmpt2 6912  Basecbs 16229  .rcmulr 16313  -gcsg 17785  Poly1cpl1 19914  quot1pcq1p 24293  rem1pcr1p 24294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-slot 16233  df-base 16235  df-r1p 24299
This theorem is referenced by:  r1pcl  24323  r1pdeglt  24324  r1pid  24325  dvdsr1p  24327  ig1pdvds  24342
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