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Theorem itgocn 43107
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 43102 . . . . 5 IntgOver = (𝑎 ∈ 𝒫 ℂ ↦ {𝑏 ∈ ℂ ∣ ∃𝑐 ∈ (Poly‘𝑎)((𝑐𝑏) = 0 ∧ ((coeff‘𝑐)‘(deg‘𝑐)) = 1)})
21dmmptss 6257 . . . 4 dom IntgOver ⊆ 𝒫 ℂ
32sseli 3991 . . 3 (𝑆 ∈ dom IntgOver → 𝑆 ∈ 𝒫 ℂ)
4 cnex 11227 . . . . 5 ℂ ∈ V
54elpw2 5335 . . . 4 (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
6 itgoval 43104 . . . . 5 (𝑆 ⊆ ℂ → (IntgOver‘𝑆) = {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)})
7 ssrab2 4090 . . . . 5 {𝑎 ∈ ℂ ∣ ∃𝑏 ∈ (Poly‘𝑆)((𝑏𝑎) = 0 ∧ ((coeff‘𝑏)‘(deg‘𝑏)) = 1)} ⊆ ℂ
86, 7eqsstrdi 4050 . . . 4 (𝑆 ⊆ ℂ → (IntgOver‘𝑆) ⊆ ℂ)
95, 8sylbi 217 . . 3 (𝑆 ∈ 𝒫 ℂ → (IntgOver‘𝑆) ⊆ ℂ)
103, 9syl 17 . 2 (𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) ⊆ ℂ)
11 ndmfv 6936 . . 3 𝑆 ∈ dom IntgOver → (IntgOver‘𝑆) = ∅)
12 0ss 4405 . . 3 ∅ ⊆ ℂ
1311, 12eqsstrdi 4050 . 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 395   = wceq 1535  wcel 2104  wrex 3066  {crab 3432  wss 3963  c0 4339  𝒫 cpw 4604  dom cdm 5683  cfv 6558  cc 11144  0cc0 11146  1c1 11147  Polycply 26219  coeffccoe 26221  degcdgr 26222  IntgOvercitgo 43100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5430  ax-cnex 11202
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6510  df-fun 6560  df-fv 6566  df-itgo 43102
This theorem is referenced by: (None)
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