Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cytpval Structured version   Visualization version   GIF version

Theorem cytpval 38313
Description: Substitutions for the Nth cyclotomic polynomial. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypotheses
Ref Expression
cytpval.t 𝑇 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))
cytpval.o 𝑂 = (od‘𝑇)
cytpval.p 𝑃 = (Poly1‘ℂfld)
cytpval.x 𝑋 = (var1‘ℂfld)
cytpval.q 𝑄 = (mulGrp‘𝑃)
cytpval.m = (-g𝑃)
cytpval.a 𝐴 = (algSc‘𝑃)
Assertion
Ref Expression
cytpval (𝑁 ∈ ℕ → (CytP‘𝑁) = (𝑄 Σg (𝑟 ∈ (𝑂 “ {𝑁}) ↦ (𝑋 (𝐴𝑟)))))
Distinct variable group:   𝑁,𝑟
Allowed substitution hints:   𝐴(𝑟)   𝑃(𝑟)   𝑄(𝑟)   𝑇(𝑟)   (𝑟)   𝑂(𝑟)   𝑋(𝑟)

Proof of Theorem cytpval
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 cytpval.p . . . . . . 7 𝑃 = (Poly1‘ℂfld)
21eqcomi 2780 . . . . . 6 (Poly1‘ℂfld) = 𝑃
32fveq2i 6335 . . . . 5 (mulGrp‘(Poly1‘ℂfld)) = (mulGrp‘𝑃)
4 cytpval.q . . . . 5 𝑄 = (mulGrp‘𝑃)
53, 4eqtr4i 2796 . . . 4 (mulGrp‘(Poly1‘ℂfld)) = 𝑄
65a1i 11 . . 3 (𝑛 = 𝑁 → (mulGrp‘(Poly1‘ℂfld)) = 𝑄)
7 cytpval.o . . . . . . . 8 𝑂 = (od‘𝑇)
8 cytpval.t . . . . . . . . 9 𝑇 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))
98fveq2i 6335 . . . . . . . 8 (od‘𝑇) = (od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})))
107, 9eqtri 2793 . . . . . . 7 𝑂 = (od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})))
1110cnveqi 5435 . . . . . 6 𝑂 = (od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})))
1211imaeq1i 5604 . . . . 5 (𝑂 “ {𝑛}) = ((od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) “ {𝑛})
13 sneq 4326 . . . . . 6 (𝑛 = 𝑁 → {𝑛} = {𝑁})
1413imaeq2d 5607 . . . . 5 (𝑛 = 𝑁 → (𝑂 “ {𝑛}) = (𝑂 “ {𝑁}))
1512, 14syl5eqr 2819 . . . 4 (𝑛 = 𝑁 → ((od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) “ {𝑛}) = (𝑂 “ {𝑁}))
16 cytpval.x . . . . . . 7 𝑋 = (var1‘ℂfld)
17 cytpval.a . . . . . . . . 9 𝐴 = (algSc‘𝑃)
181fveq2i 6335 . . . . . . . . 9 (algSc‘𝑃) = (algSc‘(Poly1‘ℂfld))
1917, 18eqtri 2793 . . . . . . . 8 𝐴 = (algSc‘(Poly1‘ℂfld))
2019fveq1i 6333 . . . . . . 7 (𝐴𝑟) = ((algSc‘(Poly1‘ℂfld))‘𝑟)
21 cytpval.m . . . . . . . 8 = (-g𝑃)
221fveq2i 6335 . . . . . . . 8 (-g𝑃) = (-g‘(Poly1‘ℂfld))
2321, 22eqtri 2793 . . . . . . 7 = (-g‘(Poly1‘ℂfld))
2416, 20, 23oveq123i 6807 . . . . . 6 (𝑋 (𝐴𝑟)) = ((var1‘ℂfld)(-g‘(Poly1‘ℂfld))((algSc‘(Poly1‘ℂfld))‘𝑟))
2524eqcomi 2780 . . . . 5 ((var1‘ℂfld)(-g‘(Poly1‘ℂfld))((algSc‘(Poly1‘ℂfld))‘𝑟)) = (𝑋 (𝐴𝑟))
2625a1i 11 . . . 4 (𝑛 = 𝑁 → ((var1‘ℂfld)(-g‘(Poly1‘ℂfld))((algSc‘(Poly1‘ℂfld))‘𝑟)) = (𝑋 (𝐴𝑟)))
2715, 26mpteq12dv 4867 . . 3 (𝑛 = 𝑁 → (𝑟 ∈ ((od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) “ {𝑛}) ↦ ((var1‘ℂfld)(-g‘(Poly1‘ℂfld))((algSc‘(Poly1‘ℂfld))‘𝑟))) = (𝑟 ∈ (𝑂 “ {𝑁}) ↦ (𝑋 (𝐴𝑟))))
286, 27oveq12d 6811 . 2 (𝑛 = 𝑁 → ((mulGrp‘(Poly1‘ℂfld)) Σg (𝑟 ∈ ((od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) “ {𝑛}) ↦ ((var1‘ℂfld)(-g‘(Poly1‘ℂfld))((algSc‘(Poly1‘ℂfld))‘𝑟)))) = (𝑄 Σg (𝑟 ∈ (𝑂 “ {𝑁}) ↦ (𝑋 (𝐴𝑟)))))
29 df-cytp 38307 . 2 CytP = (𝑛 ∈ ℕ ↦ ((mulGrp‘(Poly1‘ℂfld)) Σg (𝑟 ∈ ((od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) “ {𝑛}) ↦ ((var1‘ℂfld)(-g‘(Poly1‘ℂfld))((algSc‘(Poly1‘ℂfld))‘𝑟)))))
30 ovex 6823 . 2 (𝑄 Σg (𝑟 ∈ (𝑂 “ {𝑁}) ↦ (𝑋 (𝐴𝑟)))) ∈ V
3128, 29, 30fvmpt 6424 1 (𝑁 ∈ ℕ → (CytP‘𝑁) = (𝑄 Σg (𝑟 ∈ (𝑂 “ {𝑁}) ↦ (𝑋 (𝐴𝑟)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  cdif 3720  {csn 4316  cmpt 4863  ccnv 5248  cima 5252  cfv 6031  (class class class)co 6793  cc 10136  0cc0 10138  cn 11222  s cress 16065   Σg cgsu 16309  -gcsg 17632  odcod 18151  mulGrpcmgp 18697  algSccascl 19526  var1cv1 19761  Poly1cpl1 19762  fldccnfld 19961  CytPccytp 38306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-cytp 38307
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator