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Theorem cytpfn 42629
Description: Functionality of the cyclotomic polynomial sequence. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Assertion
Ref Expression
cytpfn CytP Fn β„•

Proof of Theorem cytpfn
Dummy variables 𝑛 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 7453 . 2 ((mulGrpβ€˜(Poly1β€˜β„‚fld)) Ξ£g (π‘Ÿ ∈ (β—‘(odβ€˜((mulGrpβ€˜β„‚fld) β†Ύs (β„‚ βˆ– {0}))) β€œ {𝑛}) ↦ ((var1β€˜β„‚fld)(-gβ€˜(Poly1β€˜β„‚fld))((algScβ€˜(Poly1β€˜β„‚fld))β€˜π‘Ÿ)))) ∈ V
2 df-cytp 42626 . 2 CytP = (𝑛 ∈ β„• ↦ ((mulGrpβ€˜(Poly1β€˜β„‚fld)) Ξ£g (π‘Ÿ ∈ (β—‘(odβ€˜((mulGrpβ€˜β„‚fld) β†Ύs (β„‚ βˆ– {0}))) β€œ {𝑛}) ↦ ((var1β€˜β„‚fld)(-gβ€˜(Poly1β€˜β„‚fld))((algScβ€˜(Poly1β€˜β„‚fld))β€˜π‘Ÿ)))))
31, 2fnmpti 6698 1 CytP Fn β„•
Colors of variables: wff setvar class
Syntax hints:   βˆ– cdif 3944  {csn 4629   ↦ cmpt 5231  β—‘ccnv 5677   β€œ cima 5681   Fn wfn 6543  β€˜cfv 6548  (class class class)co 7420  β„‚cc 11137  0cc0 11139  β„•cn 12243   β†Ύs cress 17209   Ξ£g cgsu 17422  -gcsg 18892  odcod 19479  mulGrpcmgp 20074  β„‚fldccnfld 21279  algSccascl 21786  var1cv1 22095  Poly1cpl1 22096  CytPccytp 42625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fn 6551  df-fv 6556  df-ov 7423  df-cytp 42626
This theorem is referenced by: (None)
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