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Theorem itgocn 37553
Description: All integral elements are complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
Assertion
Ref Expression
itgocn (IntgOver‘𝑆) ⊆ ℂ

Proof of Theorem itgocn
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-itgo 37548 . . . . 5 IntgOver = (𝑎 ∈ 𝒫 ℂ ↦ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ (Poly‘𝑎)((𝑐𝑏) = 0 ∧ ((coeff‘𝑐)‘(deg‘𝑐)) = 1)})
21dmmptss 5619 . . . 4 dom IntgOver ⊆ 𝒫 ℂ
32sseli 3591 . . 3 (𝑆 ∈ dom IntgOver → 𝑆 ∈ 𝒫 ℂ)
4 cnex 10002 . . . . 5 ℂ ∈ V
54elpw2 4819 . . . 4 (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
6 itgoval 37550 . . . . 5 (𝑆 ⊆ ℂ → (IntgOver‘𝑆) = {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)})
7 ssrab2 3679 . . . . 5 {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)} ⊆ ℂ
86, 7syl6eqss 3647 . . . 4 (𝑆 ⊆ ℂ → (IntgOver‘𝑆) ⊆ ℂ)
95, 8sylbi 207 . . 3 (𝑆 ∈ 𝒫 ℂ → (IntgOver‘𝑆) ⊆ ℂ)
103, 9syl 17 . 2 (𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) ⊆ ℂ)
11 ndmfv 6205 . . 3 𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) = ∅)
12 0ss 3963 . . 3 ∅ ⊆ ℂ
1311, 12syl6eqss 3647 . 2 𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) ⊆ ℂ)
1410, 13pm2.61i 176 1 (IntgOver‘𝑆) ⊆ ℂ
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1481  wcel 1988  wrex 2910  {crab 2913  wss 3567  c0 3907  𝒫 cpw 4149  dom cdm 5104  cfv 5876  cc 9919  0cc0 9921  1c1 9922  Polycply 23921  coeffccoe 23923  degcdgr 23924  IntgOvercitgo 37546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-cnex 9977
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fv 5884  df-itgo 37548
This theorem is referenced by: (None)
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