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Theorem cytpfn 42507
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 7437 . 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 42503 . 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 6686 1 CytP Fn β„•
Colors of variables: wff setvar class
Syntax hints:   βˆ– cdif 3940  {csn 4623   ↦ cmpt 5224  β—‘ccnv 5668   β€œ cima 5672   Fn wfn 6531  β€˜cfv 6536  (class class class)co 7404  β„‚cc 11107  0cc0 11109  β„•cn 12213   β†Ύs cress 17180   Ξ£g cgsu 17393  -gcsg 18863  odcod 19442  mulGrpcmgp 20037  β„‚fldccnfld 21236  algSccascl 21743  var1cv1 22046  Poly1cpl1 22047  CytPccytp 42502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fn 6539  df-fv 6544  df-ov 7407  df-cytp 42503
This theorem is referenced by: (None)
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