Theorem cytpval 43234

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	⊢ − = (-g‘𝑃)
cytpval.a	⊢ 𝐴 = (algSc‘𝑃)

Assertion

Ref	Expression
cytpval	⊢ (𝑁 ∈ ℕ → (CytP‘𝑁) = (𝑄 Σg (𝑟 ∈ (◡𝑂 “ {𝑁}) ↦ (𝑋 − (𝐴‘𝑟)))))

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.

Step	Hyp	Ref	Expression
1		cytpval.p	. . . . . . 7 ⊢ 𝑃 = (Poly1‘ℂfld)
2	1	eqcomi 2740	. . . . . 6 ⊢ (Poly1‘ℂfld) = 𝑃
3	2	fveq2i 6825	. . . . 5 ⊢ (mulGrp‘(Poly1‘ℂfld)) = (mulGrp‘𝑃)
4		cytpval.q	. . . . 5 ⊢ 𝑄 = (mulGrp‘𝑃)
5	3, 4	eqtr4i 2757	. . . 4 ⊢ (mulGrp‘(Poly1‘ℂfld)) = 𝑄
6	5	a1i 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}))
9	8	fveq2i 6825	. . . . . . . 8 ⊢ (od‘𝑇) = (od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})))
10	7, 9	eqtri 2754	. . . . . . 7 ⊢ 𝑂 = (od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})))
11	10	cnveqi 5814	. . . . . 6 ⊢ ◡𝑂 = ◡(od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})))
12	11	imaeq1i 6006	. . . . 5 ⊢ (◡𝑂 “ {𝑛}) = (◡(od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) “ {𝑛})
13		sneq 4586	. . . . . 6 ⊢ (𝑛 = 𝑁 → {𝑛} = {𝑁})
14	13	imaeq2d 6009	. . . . 5 ⊢ (𝑛 = 𝑁 → (◡𝑂 “ {𝑛}) = (◡𝑂 “ {𝑁}))
15	12, 14	eqtr3id 2780	. . . 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)
18	1	fveq2i 6825	. . . . . . . . 9 ⊢ (algSc‘𝑃) = (algSc‘(Poly1‘ℂfld))
19	17, 18	eqtri 2754	. . . . . . . 8 ⊢ 𝐴 = (algSc‘(Poly1‘ℂfld))
20	19	fveq1i 6823	. . . . . . 7 ⊢ (𝐴‘𝑟) = ((algSc‘(Poly1‘ℂfld))‘𝑟)
21		cytpval.m	. . . . . . . 8 ⊢ − = (-g‘𝑃)
22	1	fveq2i 6825	. . . . . . . 8 ⊢ (-g‘𝑃) = (-g‘(Poly1‘ℂfld))
23	21, 22	eqtri 2754	. . . . . . 7 ⊢ − = (-g‘(Poly1‘ℂfld))
24	16, 20, 23	oveq123i 7360	. . . . . 6 ⊢ (𝑋 − (𝐴‘𝑟)) = ((var1‘ℂfld)(-g‘(Poly1‘ℂfld))((algSc‘(Poly1‘ℂfld))‘𝑟))
25	24	eqcomi 2740	. . . . 5 ⊢ ((var1‘ℂfld)(-g‘(Poly1‘ℂfld))((algSc‘(Poly1‘ℂfld))‘𝑟)) = (𝑋 − (𝐴‘𝑟))
26	25	a1i 11	. . . 4 ⊢ (𝑛 = 𝑁 → ((var1‘ℂfld)(-g‘(Poly1‘ℂfld))((algSc‘(Poly1‘ℂfld))‘𝑟)) = (𝑋 − (𝐴‘𝑟)))
27	15, 26	mpteq12dv 5178	. . 3 ⊢ (𝑛 = 𝑁 → (𝑟 ∈ (◡(od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) “ {𝑛}) ↦ ((var1‘ℂfld)(-g‘(Poly1‘ℂfld))((algSc‘(Poly1‘ℂfld))‘𝑟))) = (𝑟 ∈ (◡𝑂 “ {𝑁}) ↦ (𝑋 − (𝐴‘𝑟))))
28	6, 27	oveq12d 7364	. 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 − (𝐴‘𝑟)))))
29		df-cytp 43230	. 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 7379	. 2 ⊢ (𝑄 Σg (𝑟 ∈ (◡𝑂 “ {𝑁}) ↦ (𝑋 − (𝐴‘𝑟)))) ∈ V
31	28, 29, 30	fvmpt 6929	1 ⊢ (𝑁 ∈ ℕ → (CytP‘𝑁) = (𝑄 Σg (𝑟 ∈ (◡𝑂 “ {𝑁}) ↦ (𝑋 − (𝐴‘𝑟)))))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111   ∖ cdif 3899  {csn 4576   ↦ cmpt 5172  ^◡ccnv 5615   “ cima 5619  ‘cfv 6481  (class class class)co 7346  ℂcc 11001  0cc0 11003  ℕcn 12122   ↾_s cress 17138   Σ_g cgsu 17341  -_gcsg 18845  odcod 19434  mulGrpcmgp 20056  ℂ_fldccnfld 21289  algSccascl 21787  var₁cv1 22086  Poly₁cpl1 22087  CytPccytp 43229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-cytp 43230

This theorem is referenced by: (None)