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Theorem cytpval 43163
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 2749 . . . . . 6 (Poly1‘ℂfld) = 𝑃
32fveq2i 6923 . . . . 5 (mulGrp‘(Poly1‘ℂfld)) = (mulGrp‘𝑃)
4 cytpval.q . . . . 5 𝑄 = (mulGrp‘𝑃)
53, 4eqtr4i 2771 . . . 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 6923 . . . . . . . 8 (od‘𝑇) = (od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})))
107, 9eqtri 2768 . . . . . . 7 𝑂 = (od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})))
1110cnveqi 5899 . . . . . 6 𝑂 = (od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})))
1211imaeq1i 6086 . . . . 5 (𝑂 “ {𝑛}) = ((od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) “ {𝑛})
13 sneq 4658 . . . . . 6 (𝑛 = 𝑁 → {𝑛} = {𝑁})
1413imaeq2d 6089 . . . . 5 (𝑛 = 𝑁 → (𝑂 “ {𝑛}) = (𝑂 “ {𝑁}))
1512, 14eqtr3id 2794 . . . 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 6923 . . . . . . . . 9 (algSc‘𝑃) = (algSc‘(Poly1‘ℂfld))
1917, 18eqtri 2768 . . . . . . . 8 𝐴 = (algSc‘(Poly1‘ℂfld))
2019fveq1i 6921 . . . . . . 7 (𝐴𝑟) = ((algSc‘(Poly1‘ℂfld))‘𝑟)
21 cytpval.m . . . . . . . 8 = (-g𝑃)
221fveq2i 6923 . . . . . . . 8 (-g𝑃) = (-g‘(Poly1‘ℂfld))
2321, 22eqtri 2768 . . . . . . 7 = (-g‘(Poly1‘ℂfld))
2416, 20, 23oveq123i 7462 . . . . . 6 (𝑋 (𝐴𝑟)) = ((var1‘ℂfld)(-g‘(Poly1‘ℂfld))((algSc‘(Poly1‘ℂfld))‘𝑟))
2524eqcomi 2749 . . . . 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 5257 . . 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 7466 . 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 43159 . 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 7481 . 2 (𝑄 Σg (𝑟 ∈ (𝑂 “ {𝑁}) ↦ (𝑋 (𝐴𝑟)))) ∈ V
3128, 29, 30fvmpt 7029 1 (𝑁 ∈ ℕ → (CytP‘𝑁) = (𝑄 Σg (𝑟 ∈ (𝑂 “ {𝑁}) ↦ (𝑋 (𝐴𝑟)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2108  cdif 3973  {csn 4648  cmpt 5249  ccnv 5699  cima 5703  cfv 6573  (class class class)co 7448  cc 11182  0cc0 11184  cn 12293  s cress 17287   Σg cgsu 17500  -gcsg 18975  odcod 19566  mulGrpcmgp 20161  fldccnfld 21387  algSccascl 21895  var1cv1 22198  Poly1cpl1 22199  CytPccytp 43158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-cytp 43159
This theorem is referenced by: (None)
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