Theorem cytpfn 43443

Description: Functionality of the cyclotomic polynomial sequence. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Assertion

Ref	Expression
cytpfn	⊢ CytP Fn ℕ

Proof of Theorem cytpfn

Dummy variables 𝑛 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 7391	. 2 ⊢ ((mulGrp‘(Poly1‘ℂfld)) Σg (𝑟 ∈ (◡(od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) “ {𝑛}) ↦ ((var1‘ℂfld)(-g‘(Poly1‘ℂfld))((algSc‘(Poly1‘ℂfld))‘𝑟)))) ∈ V
2		df-cytp 43440	. 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)))))
3	1, 2	fnmpti 6635	1 ⊢ CytP Fn ℕ

Syntax hints:   ∖ cdif 3898  {csn 4580   ↦ cmpt 5179  ^◡ccnv 5623   “ cima 5627   Fn wfn 6487  ‘cfv 6492  (class class class)co 7358  ℂcc 11024  0cc0 11026  ℕcn 12145   ↾_s cress 17157   Σ_g cgsu 17360  -_gcsg 18865  odcod 19453  mulGrpcmgp 20075  ℂ_fldccnfld 21309  algSccascl 21807  var₁cv1 22116  Poly₁cpl1 22117  CytPccytp 43439

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500  df-ov 7361  df-cytp 43440

This theorem is referenced by: (None)