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 37268
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 2630 . . . . . 6 (Poly1‘ℂfld) = 𝑃
32fveq2i 6151 . . . . 5 (mulGrp‘(Poly1‘ℂfld)) = (mulGrp‘𝑃)
4 cytpval.q . . . . 5 𝑄 = (mulGrp‘𝑃)
53, 4eqtr4i 2646 . . . 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 6151 . . . . . . . 8 (od‘𝑇) = (od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})))
107, 9eqtri 2643 . . . . . . 7 𝑂 = (od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})))
1110cnveqi 5257 . . . . . 6 𝑂 = (od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})))
1211imaeq1i 5422 . . . . 5 (𝑂 “ {𝑛}) = ((od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) “ {𝑛})
13 sneq 4158 . . . . . 6 (𝑛 = 𝑁 → {𝑛} = {𝑁})
1413imaeq2d 5425 . . . . 5 (𝑛 = 𝑁 → (𝑂 “ {𝑛}) = (𝑂 “ {𝑁}))
1512, 14syl5eqr 2669 . . . 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 6151 . . . . . . . . 9 (algSc‘𝑃) = (algSc‘(Poly1‘ℂfld))
1917, 18eqtri 2643 . . . . . . . 8 𝐴 = (algSc‘(Poly1‘ℂfld))
2019fveq1i 6149 . . . . . . 7 (𝐴𝑟) = ((algSc‘(Poly1‘ℂfld))‘𝑟)
21 cytpval.m . . . . . . . 8 = (-g𝑃)
221fveq2i 6151 . . . . . . . 8 (-g𝑃) = (-g‘(Poly1‘ℂfld))
2321, 22eqtri 2643 . . . . . . 7 = (-g‘(Poly1‘ℂfld))
2416, 20, 23oveq123i 6618 . . . . . 6 (𝑋 (𝐴𝑟)) = ((var1‘ℂfld)(-g‘(Poly1‘ℂfld))((algSc‘(Poly1‘ℂfld))‘𝑟))
2524eqcomi 2630 . . . . 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 4693 . . 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 6622 . 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 37262 . 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 6632 . 2 (𝑄 Σg (𝑟 ∈ (𝑂 “ {𝑁}) ↦ (𝑋 (𝐴𝑟)))) ∈ V
3128, 29, 30fvmpt 6239 1 (𝑁 ∈ ℕ → (CytP‘𝑁) = (𝑄 Σg (𝑟 ∈ (𝑂 “ {𝑁}) ↦ (𝑋 (𝐴𝑟)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  cdif 3552  {csn 4148  cmpt 4673  ccnv 5073  cima 5077  cfv 5847  (class class class)co 6604  cc 9878  0cc0 9880  cn 10964  s cress 15782   Σg cgsu 16022  -gcsg 17345  odcod 17865  mulGrpcmgp 18410  algSccascl 19230  var1cv1 19465  Poly1cpl1 19466  fldccnfld 19665  CytPccytp 37261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-cytp 37262
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator