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Theorem cytpfn 26675
Description: Functionality of the cyclotomic polynomial sequence. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Assertion
Ref Expression
cytpfn  |- CytP  Fn  NN

Proof of Theorem cytpfn
Dummy variables  n  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 5925 . 2  |-  ( (mulGrp `  (Poly1 ` fld ) )  gsumg  ( r  e.  ( `' ( od `  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) ) ) " {
n } )  |->  ( (var1 ` fld ) ( -g `  (Poly1 ` fld )
) ( (algSc `  (Poly1 ` fld ) ) `  r
) ) ) )  e.  _V
2 df-cytp 26670 . 2  |- CytP  =  ( n  e.  NN  |->  ( (mulGrp `  (Poly1 ` fld ) )  gsumg  ( r  e.  ( `' ( od `  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) ) ) " {
n } )  |->  ( (var1 ` fld ) ( -g `  (Poly1 ` fld )
) ( (algSc `  (Poly1 ` fld ) ) `  r
) ) ) ) )
31, 2fnmpti 5409 1  |- CytP  Fn  NN
Colors of variables: wff set class
Syntax hints:    \ cdif 3183   {csn 3674    e. cmpt 4114   `'ccnv 4725   "cima 4729    Fn wfn 5287   ` cfv 5292  (class class class)co 5900   CCcc 8780   0cc0 8782   NNcn 9791   ↾s cress 13196    gsumg cgsu 13450   -gcsg 14414   odcod 14889  mulGrpcmgp 15374  algSccascl 16101  var1cv1 16300  Poly1cpl1 16301  ℂfldccnfld 16432  CytPccytp 26669
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-iota 5256  df-fun 5294  df-fn 5295  df-fv 5300  df-ov 5903  df-cytp 26670
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