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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cytpfn | Structured version Visualization version GIF version |
Description: Functionality of the cyclotomic polynomial sequence. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Ref | Expression |
---|---|
cytpfn | ⊢ CytP Fn ℕ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7459 | . 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 42881 | . 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 6706 | 1 ⊢ CytP Fn ℕ |
Colors of variables: wff setvar class |
Syntax hints: ∖ cdif 3944 {csn 4633 ↦ cmpt 5238 ◡ccnv 5683 “ cima 5687 Fn wfn 6551 ‘cfv 6556 (class class class)co 7426 ℂcc 11158 0cc0 11160 ℕcn 12266 ↾s cress 17244 Σg cgsu 17457 -gcsg 18932 odcod 19524 mulGrpcmgp 20119 ℂfldccnfld 21345 algSccascl 21852 var1cv1 22167 Poly1cpl1 22168 CytPccytp 42880 |
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 2697 ax-sep 5306 ax-nul 5313 ax-pr 5435 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-br 5156 df-opab 5218 df-mpt 5239 df-id 5582 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6508 df-fun 6558 df-fn 6559 df-fv 6564 df-ov 7429 df-cytp 42881 |
This theorem is referenced by: (None) |
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