Theorem cytpval 44396

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 2772	. . . . . 6 ⊢ (Poly1‘ℂfld) = 𝑃
3	2	fveq2i 6911	. . . . 5 ⊢ (mulGrp‘(Poly1‘ℂfld)) = (mulGrp‘𝑃)
4		cytpval.q	. . . . 5 ⊢ 𝑄 = (mulGrp‘𝑃)
5	3, 4	eqtr4i 2789	. . . 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 6911	. . . . . . . 8 ⊢ (od‘𝑇) = (od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})))
10	7, 9	eqtri 2786	. . . . . . 7 ⊢ 𝑂 = (od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})))
11	10	cnveqi 5859	. . . . . 6 ⊢ ◡𝑂 = ◡(od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})))
12	11	imaeq1i 6061	. . . . 5 ⊢ (◡𝑂 “ {𝑛}) = (◡(od‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) “ {𝑛})
13		sneq 4594	. . . . . 6 ⊢ (𝑛 = 𝑁 → {𝑛} = {𝑁})
14	13	imaeq2d 6064	. . . . 5 ⊢ (𝑛 = 𝑁 → (◡𝑂 “ {𝑛}) = (◡𝑂 “ {𝑁}))
15	12, 14	eqtr3id 2812	. . . 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 6911	. . . . . . . . 9 ⊢ (algSc‘𝑃) = (algSc‘(Poly1‘ℂfld))
19	17, 18	eqtri 2786	. . . . . . . 8 ⊢ 𝐴 = (algSc‘(Poly1‘ℂfld))
20	19	fveq1i 6909	. . . . . . 7 ⊢ (𝐴‘𝑟) = ((algSc‘(Poly1‘ℂfld))‘𝑟)
21		cytpval.m	. . . . . . . 8 ⊢ − = (-g‘𝑃)
22	1	fveq2i 6911	. . . . . . . 8 ⊢ (-g‘𝑃) = (-g‘(Poly1‘ℂfld))
23	21, 22	eqtri 2786	. . . . . . 7 ⊢ − = (-g‘(Poly1‘ℂfld))
24	16, 20, 23	oveq123i 7457	. . . . . 6 ⊢ (𝑋 − (𝐴‘𝑟)) = ((var1‘ℂfld)(-g‘(Poly1‘ℂfld))((algSc‘(Poly1‘ℂfld))‘𝑟))
25	24	eqcomi 2772	. . . . 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 5195	. . 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 7461	. 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 44392	. 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 7476	. 2 ⊢ (𝑄 Σg (𝑟 ∈ (◡𝑂 “ {𝑁}) ↦ (𝑋 − (𝐴‘𝑟)))) ∈ V
31	28, 29, 30	fvmpt 7016	1 ⊢ (𝑁 ∈ ℕ → (CytP‘𝑁) = (𝑄 Σg (𝑟 ∈ (◡𝑂 “ {𝑁}) ↦ (𝑋 − (𝐴‘𝑟)))))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147   ∖ cdif 4048  {csn 4588   ↦ cmpt 5189  ^◡ccnv 5654   “ cima 5658  ‘cfv 6558  (class class class)co 7443  ℂcc 11225  0cc0 11227  ℕcn 12362   ↾_s cress 17437   Σ_g cgsu 17640  -_gcsg 19175  odcod 19768  mulGrpcmgp 20390  ℂ_fldccnfld 21708  algSccascl 22190  var₁cv1 22524  Poly₁cpl1 22525  CytPccytp 44391

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fv 6566  df-ov 7446  df-cytp 44392

This theorem is used by: (None)