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Theorem cytpval 41579
Description: Substitutions for the Nth cyclotomic polynomial. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypotheses
Ref Expression
cytpval.t 𝑇 = ((mulGrpβ€˜β„‚fld) β†Ύs (β„‚ βˆ– {0}))
cytpval.o 𝑂 = (odβ€˜π‘‡)
cytpval.p 𝑃 = (Poly1β€˜β„‚fld)
cytpval.x 𝑋 = (var1β€˜β„‚fld)
cytpval.q 𝑄 = (mulGrpβ€˜π‘ƒ)
cytpval.m βˆ’ = (-gβ€˜π‘ƒ)
cytpval.a 𝐴 = (algScβ€˜π‘ƒ)
Assertion
Ref Expression
cytpval (𝑁 ∈ β„• β†’ (CytPβ€˜π‘) = (𝑄 Ξ£g (π‘Ÿ ∈ (◑𝑂 β€œ {𝑁}) ↦ (𝑋 βˆ’ (π΄β€˜π‘Ÿ)))))
Distinct variable group:   𝑁,π‘Ÿ
Allowed substitution hints:   𝐴(π‘Ÿ)   𝑃(π‘Ÿ)   𝑄(π‘Ÿ)   𝑇(π‘Ÿ)   βˆ’ (π‘Ÿ)   𝑂(π‘Ÿ)   𝑋(π‘Ÿ)

Proof of Theorem cytpval
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 cytpval.p . . . . . . 7 𝑃 = (Poly1β€˜β„‚fld)
21eqcomi 2742 . . . . . 6 (Poly1β€˜β„‚fld) = 𝑃
32fveq2i 6846 . . . . 5 (mulGrpβ€˜(Poly1β€˜β„‚fld)) = (mulGrpβ€˜π‘ƒ)
4 cytpval.q . . . . 5 𝑄 = (mulGrpβ€˜π‘ƒ)
53, 4eqtr4i 2764 . . . 4 (mulGrpβ€˜(Poly1β€˜β„‚fld)) = 𝑄
65a1i 11 . . 3 (𝑛 = 𝑁 β†’ (mulGrpβ€˜(Poly1β€˜β„‚fld)) = 𝑄)
7 cytpval.o . . . . . . . 8 𝑂 = (odβ€˜π‘‡)
8 cytpval.t . . . . . . . . 9 𝑇 = ((mulGrpβ€˜β„‚fld) β†Ύs (β„‚ βˆ– {0}))
98fveq2i 6846 . . . . . . . 8 (odβ€˜π‘‡) = (odβ€˜((mulGrpβ€˜β„‚fld) β†Ύs (β„‚ βˆ– {0})))
107, 9eqtri 2761 . . . . . . 7 𝑂 = (odβ€˜((mulGrpβ€˜β„‚fld) β†Ύs (β„‚ βˆ– {0})))
1110cnveqi 5831 . . . . . 6 ◑𝑂 = β—‘(odβ€˜((mulGrpβ€˜β„‚fld) β†Ύs (β„‚ βˆ– {0})))
1211imaeq1i 6011 . . . . 5 (◑𝑂 β€œ {𝑛}) = (β—‘(odβ€˜((mulGrpβ€˜β„‚fld) β†Ύs (β„‚ βˆ– {0}))) β€œ {𝑛})
13 sneq 4597 . . . . . 6 (𝑛 = 𝑁 β†’ {𝑛} = {𝑁})
1413imaeq2d 6014 . . . . 5 (𝑛 = 𝑁 β†’ (◑𝑂 β€œ {𝑛}) = (◑𝑂 β€œ {𝑁}))
1512, 14eqtr3id 2787 . . . 4 (𝑛 = 𝑁 β†’ (β—‘(odβ€˜((mulGrpβ€˜β„‚fld) β†Ύs (β„‚ βˆ– {0}))) β€œ {𝑛}) = (◑𝑂 β€œ {𝑁}))
16 cytpval.x . . . . . . 7 𝑋 = (var1β€˜β„‚fld)
17 cytpval.a . . . . . . . . 9 𝐴 = (algScβ€˜π‘ƒ)
181fveq2i 6846 . . . . . . . . 9 (algScβ€˜π‘ƒ) = (algScβ€˜(Poly1β€˜β„‚fld))
1917, 18eqtri 2761 . . . . . . . 8 𝐴 = (algScβ€˜(Poly1β€˜β„‚fld))
2019fveq1i 6844 . . . . . . 7 (π΄β€˜π‘Ÿ) = ((algScβ€˜(Poly1β€˜β„‚fld))β€˜π‘Ÿ)
21 cytpval.m . . . . . . . 8 βˆ’ = (-gβ€˜π‘ƒ)
221fveq2i 6846 . . . . . . . 8 (-gβ€˜π‘ƒ) = (-gβ€˜(Poly1β€˜β„‚fld))
2321, 22eqtri 2761 . . . . . . 7 βˆ’ = (-gβ€˜(Poly1β€˜β„‚fld))
2416, 20, 23oveq123i 7372 . . . . . 6 (𝑋 βˆ’ (π΄β€˜π‘Ÿ)) = ((var1β€˜β„‚fld)(-gβ€˜(Poly1β€˜β„‚fld))((algScβ€˜(Poly1β€˜β„‚fld))β€˜π‘Ÿ))
2524eqcomi 2742 . . . . 5 ((var1β€˜β„‚fld)(-gβ€˜(Poly1β€˜β„‚fld))((algScβ€˜(Poly1β€˜β„‚fld))β€˜π‘Ÿ)) = (𝑋 βˆ’ (π΄β€˜π‘Ÿ))
2625a1i 11 . . . 4 (𝑛 = 𝑁 β†’ ((var1β€˜β„‚fld)(-gβ€˜(Poly1β€˜β„‚fld))((algScβ€˜(Poly1β€˜β„‚fld))β€˜π‘Ÿ)) = (𝑋 βˆ’ (π΄β€˜π‘Ÿ)))
2715, 26mpteq12dv 5197 . . 3 (𝑛 = 𝑁 β†’ (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))) = (π‘Ÿ ∈ (◑𝑂 β€œ {𝑁}) ↦ (𝑋 βˆ’ (π΄β€˜π‘Ÿ))))
286, 27oveq12d 7376 . 2 (𝑛 = 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)))) = (𝑄 Ξ£g (π‘Ÿ ∈ (◑𝑂 β€œ {𝑁}) ↦ (𝑋 βˆ’ (π΄β€˜π‘Ÿ)))))
29 df-cytp 41573 . 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)))))
30 ovex 7391 . 2 (𝑄 Ξ£g (π‘Ÿ ∈ (◑𝑂 β€œ {𝑁}) ↦ (𝑋 βˆ’ (π΄β€˜π‘Ÿ)))) ∈ V
3128, 29, 30fvmpt 6949 1 (𝑁 ∈ β„• β†’ (CytPβ€˜π‘) = (𝑄 Ξ£g (π‘Ÿ ∈ (◑𝑂 β€œ {𝑁}) ↦ (𝑋 βˆ’ (π΄β€˜π‘Ÿ)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107   βˆ– cdif 3908  {csn 4587   ↦ cmpt 5189  β—‘ccnv 5633   β€œ cima 5637  β€˜cfv 6497  (class class class)co 7358  β„‚cc 11054  0cc0 11056  β„•cn 12158   β†Ύs cress 17117   Ξ£g cgsu 17327  -gcsg 18755  odcod 19311  mulGrpcmgp 19901  β„‚fldccnfld 20812  algSccascl 21274  var1cv1 21563  Poly1cpl1 21564  CytPccytp 41572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-cytp 41573
This theorem is referenced by: (None)
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