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Theorem itgocn 36753
 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 36748 . . . . 5 IntgOver = (𝑎 ∈ 𝒫 ℂ ↦ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ (Poly‘𝑎)((𝑐𝑏) = 0 ∧ ((coeff‘𝑐)‘(deg‘𝑐)) = 1)})
21dmmptss 5548 . . . 4 dom IntgOver ⊆ 𝒫 ℂ
32sseli 3564 . . 3 (𝑆 ∈ dom IntgOver → 𝑆 ∈ 𝒫 ℂ)
4 cnex 9896 . . . . 5 ℂ ∈ V
54elpw2 4755 . . . 4 (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
6 itgoval 36750 . . . . 5 (𝑆 ⊆ ℂ → (IntgOver‘𝑆) = {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)})
7 ssrab2 3650 . . . . 5 {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)} ⊆ ℂ
86, 7syl6eqss 3618 . . . 4 (𝑆 ⊆ ℂ → (IntgOver‘𝑆) ⊆ ℂ)
95, 8sylbi 206 . . 3 (𝑆 ∈ 𝒫 ℂ → (IntgOver‘𝑆) ⊆ ℂ)
103, 9syl 17 . 2 (𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) ⊆ ℂ)
11 ndmfv 6128 . . 3 𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) = ∅)
12 0ss 3924 . . 3 ∅ ⊆ ℂ
1311, 12syl6eqss 3618 . 2 𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) ⊆ ℂ)
1410, 13pm2.61i 175 1 (IntgOver‘𝑆) ⊆ ℂ
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {crab 2900   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  dom cdm 5038  ‘cfv 5804  ℂcc 9813  0cc0 9815  1c1 9816  Polycply 23744  coeffccoe 23746  degcdgr 23747  IntgOvercitgo 36746 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-cnex 9871 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-itgo 36748 This theorem is referenced by: (None)
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