Theorem cytpval 43191

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 2738	. . . . . 6 ⊢ (Poly1‘ℂfld) = 𝑃
3	2	fveq2i 6861	. . . . 5 ⊢ (mulGrp‘(Poly1‘ℂfld)) = (mulGrp‘𝑃)
4		cytpval.q	. . . . 5 ⊢ 𝑄 = (mulGrp‘𝑃)
5	3, 4	eqtr4i 2755	. . . 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 6861	. . . . . . . 8 ⊢ (od‘𝑇) = (od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})))
10	7, 9	eqtri 2752	. . . . . . 7 ⊢ 𝑂 = (od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})))
11	10	cnveqi 5838	. . . . . 6 ⊢ ◡𝑂 = ◡(od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})))
12	11	imaeq1i 6028	. . . . 5 ⊢ (◡𝑂 “ {𝑛}) = (◡(od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) “ {𝑛})
13		sneq 4599	. . . . . 6 ⊢ (𝑛 = 𝑁 → {𝑛} = {𝑁})
14	13	imaeq2d 6031	. . . . 5 ⊢ (𝑛 = 𝑁 → (◡𝑂 “ {𝑛}) = (◡𝑂 “ {𝑁}))
15	12, 14	eqtr3id 2778	. . . 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 6861	. . . . . . . . 9 ⊢ (algSc‘𝑃) = (algSc‘(Poly1‘ℂfld))
19	17, 18	eqtri 2752	. . . . . . . 8 ⊢ 𝐴 = (algSc‘(Poly1‘ℂfld))
20	19	fveq1i 6859	. . . . . . 7 ⊢ (𝐴‘𝑟) = ((algSc‘(Poly1‘ℂfld))‘𝑟)
21		cytpval.m	. . . . . . . 8 ⊢ − = (-g‘𝑃)
22	1	fveq2i 6861	. . . . . . . 8 ⊢ (-g‘𝑃) = (-g‘(Poly1‘ℂfld))
23	21, 22	eqtri 2752	. . . . . . 7 ⊢ − = (-g‘(Poly1‘ℂfld))
24	16, 20, 23	oveq123i 7401	. . . . . 6 ⊢ (𝑋 − (𝐴‘𝑟)) = ((var1‘ℂfld)(-g‘(Poly1‘ℂfld))((algSc‘(Poly1‘ℂfld))‘𝑟))
25	24	eqcomi 2738	. . . . 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 5194	. . 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 7405	. 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 43187	. 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 7420	. 2 ⊢ (𝑄 Σg (𝑟 ∈ (◡𝑂 “ {𝑁}) ↦ (𝑋 − (𝐴‘𝑟)))) ∈ V
31	28, 29, 30	fvmpt 6968	1 ⊢ (𝑁 ∈ ℕ → (CytP‘𝑁) = (𝑄 Σg (𝑟 ∈ (◡𝑂 “ {𝑁}) ↦ (𝑋 − (𝐴‘𝑟)))))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ∖ cdif 3911  {csn 4589   ↦ cmpt 5188  ^◡ccnv 5637   “ cima 5641  ‘cfv 6511  (class class class)co 7387  ℂcc 11066  0cc0 11068  ℕcn 12186   ↾_s cress 17200   Σ_g cgsu 17403  -_gcsg 18867  odcod 19454  mulGrpcmgp 20049  ℂ_fldccnfld 21264  algSccascl 21761  var₁cv1 22060  Poly₁cpl1 22061  CytPccytp 43186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-cytp 43187

This theorem is referenced by: (None)