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Theorem itgocn 40029
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 40024 . . . . 5 IntgOver = (𝑎 ∈ 𝒫 ℂ ↦ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ (Poly‘𝑎)((𝑐𝑏) = 0 ∧ ((coeff‘𝑐)‘(deg‘𝑐)) = 1)})
21dmmptss 6082 . . . 4 dom IntgOver ⊆ 𝒫 ℂ
32sseli 3949 . . 3 (𝑆 ∈ dom IntgOver → 𝑆 ∈ 𝒫 ℂ)
4 cnex 10616 . . . . 5 ℂ ∈ V
54elpw2 5234 . . . 4 (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
6 itgoval 40026 . . . . 5 (𝑆 ⊆ ℂ → (IntgOver‘𝑆) = {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)})
7 ssrab2 4042 . . . . 5 {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)} ⊆ ℂ
86, 7eqsstrdi 4007 . . . 4 (𝑆 ⊆ ℂ → (IntgOver‘𝑆) ⊆ ℂ)
95, 8sylbi 220 . . 3 (𝑆 ∈ 𝒫 ℂ → (IntgOver‘𝑆) ⊆ ℂ)
103, 9syl 17 . 2 (𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) ⊆ ℂ)
11 ndmfv 6691 . . 3 𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) = ∅)
12 0ss 4333 . . 3 ∅ ⊆ ℂ
1311, 12eqsstrdi 4007 . 2 𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) ⊆ ℂ)
1410, 13pm2.61i 185 1 (IntgOver‘𝑆) ⊆ ℂ
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 399   = wceq 1538  wcel 2115  wrex 3134  {crab 3137  wss 3919  c0 4276  𝒫 cpw 4522  dom cdm 5542  cfv 6343  cc 10533  0cc0 10535  1c1 10536  Polycply 24788  coeffccoe 24790  degcdgr 24791  IntgOvercitgo 40022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-cnex 10591
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fv 6351  df-itgo 40024
This theorem is referenced by: (None)
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