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Theorem cytpval 40146
Description: Substitutions for the Nth cyclotomic polynomial. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypotheses
Ref Expression
cytpval.t 𝑇 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))
cytpval.o 𝑂 = (od‘𝑇)
cytpval.p 𝑃 = (Poly1‘ℂfld)
cytpval.x 𝑋 = (var1‘ℂfld)
cytpval.q 𝑄 = (mulGrp‘𝑃)
cytpval.m = (-g𝑃)
cytpval.a 𝐴 = (algSc‘𝑃)
Assertion
Ref Expression
cytpval (𝑁 ∈ ℕ → (CytP‘𝑁) = (𝑄 Σg (𝑟 ∈ (𝑂 “ {𝑁}) ↦ (𝑋 (𝐴𝑟)))))
Distinct variable group:   𝑁,𝑟
Allowed substitution hints:   𝐴(𝑟)   𝑃(𝑟)   𝑄(𝑟)   𝑇(𝑟)   (𝑟)   𝑂(𝑟)   𝑋(𝑟)

Proof of Theorem cytpval
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 cytpval.p . . . . . . 7 𝑃 = (Poly1‘ℂfld)
21eqcomi 2810 . . . . . 6 (Poly1‘ℂfld) = 𝑃
32fveq2i 6652 . . . . 5 (mulGrp‘(Poly1‘ℂfld)) = (mulGrp‘𝑃)
4 cytpval.q . . . . 5 𝑄 = (mulGrp‘𝑃)
53, 4eqtr4i 2827 . . . 4 (mulGrp‘(Poly1‘ℂfld)) = 𝑄
65a1i 11 . . 3 (𝑛 = 𝑁 → (mulGrp‘(Poly1‘ℂfld)) = 𝑄)
7 cytpval.o . . . . . . . 8 𝑂 = (od‘𝑇)
8 cytpval.t . . . . . . . . 9 𝑇 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))
98fveq2i 6652 . . . . . . . 8 (od‘𝑇) = (od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})))
107, 9eqtri 2824 . . . . . . 7 𝑂 = (od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})))
1110cnveqi 5713 . . . . . 6 𝑂 = (od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})))
1211imaeq1i 5897 . . . . 5 (𝑂 “ {𝑛}) = ((od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) “ {𝑛})
13 sneq 4538 . . . . . 6 (𝑛 = 𝑁 → {𝑛} = {𝑁})
1413imaeq2d 5900 . . . . 5 (𝑛 = 𝑁 → (𝑂 “ {𝑛}) = (𝑂 “ {𝑁}))
1512, 14syl5eqr 2850 . . . 4 (𝑛 = 𝑁 → ((od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) “ {𝑛}) = (𝑂 “ {𝑁}))
16 cytpval.x . . . . . . 7 𝑋 = (var1‘ℂfld)
17 cytpval.a . . . . . . . . 9 𝐴 = (algSc‘𝑃)
181fveq2i 6652 . . . . . . . . 9 (algSc‘𝑃) = (algSc‘(Poly1‘ℂfld))
1917, 18eqtri 2824 . . . . . . . 8 𝐴 = (algSc‘(Poly1‘ℂfld))
2019fveq1i 6650 . . . . . . 7 (𝐴𝑟) = ((algSc‘(Poly1‘ℂfld))‘𝑟)
21 cytpval.m . . . . . . . 8 = (-g𝑃)
221fveq2i 6652 . . . . . . . 8 (-g𝑃) = (-g‘(Poly1‘ℂfld))
2321, 22eqtri 2824 . . . . . . 7 = (-g‘(Poly1‘ℂfld))
2416, 20, 23oveq123i 7153 . . . . . 6 (𝑋 (𝐴𝑟)) = ((var1‘ℂfld)(-g‘(Poly1‘ℂfld))((algSc‘(Poly1‘ℂfld))‘𝑟))
2524eqcomi 2810 . . . . 5 ((var1‘ℂfld)(-g‘(Poly1‘ℂfld))((algSc‘(Poly1‘ℂfld))‘𝑟)) = (𝑋 (𝐴𝑟))
2625a1i 11 . . . 4 (𝑛 = 𝑁 → ((var1‘ℂfld)(-g‘(Poly1‘ℂfld))((algSc‘(Poly1‘ℂfld))‘𝑟)) = (𝑋 (𝐴𝑟)))
2715, 26mpteq12dv 5118 . . 3 (𝑛 = 𝑁 → (𝑟 ∈ ((od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) “ {𝑛}) ↦ ((var1‘ℂfld)(-g‘(Poly1‘ℂfld))((algSc‘(Poly1‘ℂfld))‘𝑟))) = (𝑟 ∈ (𝑂 “ {𝑁}) ↦ (𝑋 (𝐴𝑟))))
286, 27oveq12d 7157 . 2 (𝑛 = 𝑁 → ((mulGrp‘(Poly1‘ℂfld)) Σg (𝑟 ∈ ((od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) “ {𝑛}) ↦ ((var1‘ℂfld)(-g‘(Poly1‘ℂfld))((algSc‘(Poly1‘ℂfld))‘𝑟)))) = (𝑄 Σg (𝑟 ∈ (𝑂 “ {𝑁}) ↦ (𝑋 (𝐴𝑟)))))
29 df-cytp 40140 . 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)))))
30 ovex 7172 . 2 (𝑄 Σg (𝑟 ∈ (𝑂 “ {𝑁}) ↦ (𝑋 (𝐴𝑟)))) ∈ V
3128, 29, 30fvmpt 6749 1 (𝑁 ∈ ℕ → (CytP‘𝑁) = (𝑄 Σg (𝑟 ∈ (𝑂 “ {𝑁}) ↦ (𝑋 (𝐴𝑟)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  cdif 3881  {csn 4528  cmpt 5113  ccnv 5522  cima 5526  cfv 6328  (class class class)co 7139  cc 10528  0cc0 10530  cn 11629  s cress 16480   Σg cgsu 16710  -gcsg 18101  odcod 18648  mulGrpcmgp 19236  fldccnfld 20095  algSccascl 20545  var1cv1 20809  Poly1cpl1 20810  CytPccytp 40139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-cytp 40140
This theorem is referenced by: (None)
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