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Theorem r1pval 26086
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 17179 . . . 4 (𝐹𝐵𝑅 ∈ V)
43adantr 480 . . 3 ((𝐹𝐵𝐺𝐵) → 𝑅 ∈ V)
5 r1pval.e . . . 4 𝐸 = (rem1p𝑅)
6 fveq2 6891 . . . . . . . . . 10 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
76, 1eqtr4di 2785 . . . . . . . . 9 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
87fveq2d 6895 . . . . . . . 8 (𝑟 = 𝑅 → (Base‘(Poly1𝑟)) = (Base‘𝑃))
98, 2eqtr4di 2785 . . . . . . 7 (𝑟 = 𝑅 → (Base‘(Poly1𝑟)) = 𝐵)
109csbeq1d 3893 . . . . . 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 6905 . . . . . . . 8 𝐵 ∈ V
1211a1i 11 . . . . . . 7 (𝑟 = 𝑅𝐵 ∈ V)
13 simpr 484 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = 𝐵) → 𝑏 = 𝐵)
147fveq2d 6895 . . . . . . . . . . 11 (𝑟 = 𝑅 → (-g‘(Poly1𝑟)) = (-g𝑃))
15 r1pval.m . . . . . . . . . . 11 = (-g𝑃)
1614, 15eqtr4di 2785 . . . . . . . . . 10 (𝑟 = 𝑅 → (-g‘(Poly1𝑟)) = )
17 eqidd 2728 . . . . . . . . . 10 (𝑟 = 𝑅𝑓 = 𝑓)
187fveq2d 6895 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (.r‘(Poly1𝑟)) = (.r𝑃))
19 r1pval.t . . . . . . . . . . . 12 · = (.r𝑃)
2018, 19eqtr4di 2785 . . . . . . . . . . 11 (𝑟 = 𝑅 → (.r‘(Poly1𝑟)) = · )
21 fveq2 6891 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (quot1p𝑟) = (quot1p𝑅))
22 r1pval.q . . . . . . . . . . . . 13 𝑄 = (quot1p𝑅)
2321, 22eqtr4di 2785 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (quot1p𝑟) = 𝑄)
2423oveqd 7431 . . . . . . . . . . 11 (𝑟 = 𝑅 → (𝑓(quot1p𝑟)𝑔) = (𝑓𝑄𝑔))
25 eqidd 2728 . . . . . . . . . . 11 (𝑟 = 𝑅𝑔 = 𝑔)
2620, 24, 25oveq123d 7435 . . . . . . . . . 10 (𝑟 = 𝑅 → ((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔) = ((𝑓𝑄𝑔) · 𝑔))
2716, 17, 26oveq123d 7435 . . . . . . . . 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 7489 . . . . . . 7 ((𝑟 = 𝑅𝑏 = 𝐵) → (𝑓𝑏, 𝑔𝑏 ↦ (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))))
3012, 29csbied 3927 . . . . . 6 (𝑟 = 𝑅𝐵 / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))))
3110, 30eqtrd 2767 . . . . 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 26062 . . . . 5 rem1p = (𝑟 ∈ V ↦ (Base‘(Poly1𝑟)) / 𝑏(𝑓𝑏, 𝑔𝑏 ↦ (𝑓(-g‘(Poly1𝑟))((𝑓(quot1p𝑟)𝑔)(.r‘(Poly1𝑟))𝑔))))
3311, 11mpoex 8078 . . . . 5 (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))) ∈ V
3431, 32, 33fvmpt 6999 . . . 4 (𝑅 ∈ V → (rem1p𝑅) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓 ((𝑓𝑄𝑔) · 𝑔))))
355, 34eqtrid 2779 . . 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 7423 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓𝑄𝑔) = (𝐹𝑄𝐺))
39 simpr 484 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑔 = 𝐺)
4038, 39oveq12d 7432 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑄𝑔) · 𝑔) = ((𝐹𝑄𝐺) · 𝐺))
4137, 40oveq12d 7432 . . 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 7449 . 2 ((𝐹𝐵𝐺𝐵) → (𝐹 ((𝐹𝑄𝐺) · 𝐺)) ∈ V)
4636, 42, 43, 44, 45ovmpod 7567 1 ((𝐹𝐵𝐺𝐵) → (𝐹𝐸𝐺) = (𝐹 ((𝐹𝑄𝐺) · 𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  Vcvv 3469  csb 3889  cfv 6542  (class class class)co 7414  cmpo 7416  Basecbs 17173  .rcmulr 17227  -gcsg 18885  Poly1cpl1 22089  quot1pcq1p 26056  rem1pcr1p 26057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11188  ax-1cn 11190  ax-addcl 11192
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-nn 12237  df-slot 17144  df-ndx 17156  df-base 17174  df-r1p 26062
This theorem is referenced by:  r1pcl  26087  r1pdeglt  26088  r1pid  26089  dvdsr1p  26091  ig1pdvds  26107  q1pdir  33259  q1pvsca  33260  r1pvsca  33261  r1pcyc  33263  r1padd1  33264  irredminply  33374
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