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Theorem itgocn 41394
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 41389 . . . . 5 IntgOver = (𝑎 ∈ 𝒫 ℂ ↦ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ (Poly‘𝑎)((𝑐𝑏) = 0 ∧ ((coeff‘𝑐)‘(deg‘𝑐)) = 1)})
21dmmptss 6192 . . . 4 dom IntgOver ⊆ 𝒫 ℂ
32sseli 3939 . . 3 (𝑆 ∈ dom IntgOver → 𝑆 ∈ 𝒫 ℂ)
4 cnex 11091 . . . . 5 ℂ ∈ V
54elpw2 5301 . . . 4 (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
6 itgoval 41391 . . . . 5 (𝑆 ⊆ ℂ → (IntgOver‘𝑆) = {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)})
7 ssrab2 4036 . . . . 5 {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)} ⊆ ℂ
86, 7eqsstrdi 3997 . . . 4 (𝑆 ⊆ ℂ → (IntgOver‘𝑆) ⊆ ℂ)
95, 8sylbi 216 . . 3 (𝑆 ∈ 𝒫 ℂ → (IntgOver‘𝑆) ⊆ ℂ)
103, 9syl 17 . 2 (𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) ⊆ ℂ)
11 ndmfv 6875 . . 3 𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) = ∅)
12 0ss 4355 . . 3 ∅ ⊆ ℂ
1311, 12eqsstrdi 3997 . 2 𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) ⊆ ℂ)
1410, 13pm2.61i 182 1 (IntgOver‘𝑆) ⊆ ℂ
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 397   = wceq 1542  wcel 2107  wrex 3072  {crab 3406  wss 3909  c0 4281  𝒫 cpw 4559  dom cdm 5632  cfv 6494  cc 11008  0cc0 11010  1c1 11011  Polycply 25497  coeffccoe 25499  degcdgr 25500  IntgOvercitgo 41387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383  ax-cnex 11066
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fv 6502  df-itgo 41389
This theorem is referenced by: (None)
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