Theorem cytpfn 43204

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 7471	. 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 43201	. 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 6719	1 ⊢ CytP Fn ℕ

Syntax hints:   ∖ cdif 3963  {csn 4634   ↦ cmpt 5234  ^◡ccnv 5692   “ cima 5696   Fn wfn 6564  ‘cfv 6569  (class class class)co 7438  ℂcc 11160  0cc0 11162  ℕcn 12273   ↾_s cress 17283   Σ_g cgsu 17496  -_gcsg 18975  odcod 19566  mulGrpcmgp 20161  ℂ_fldccnfld 21391  algSccascl 21899  var₁cv1 22202  Poly₁cpl1 22203  CytPccytp 43200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pr 5441

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-iota 6522  df-fun 6571  df-fn 6572  df-fv 6577  df-ov 7441  df-cytp 43201

This theorem is referenced by: (None)