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Theorem itgocn 42458
Description: All integral elements are complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
Assertion
Ref Expression
itgocn (IntgOverβ€˜π‘†) βŠ† β„‚

Proof of Theorem itgocn
Dummy variables π‘Ž 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-itgo 42453 . . . . 5 IntgOver = (π‘Ž ∈ 𝒫 β„‚ ↦ {𝑏 ∈ β„‚ ∣ βˆƒπ‘ ∈ (Polyβ€˜π‘Ž)((π‘β€˜π‘) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1)})
21dmmptss 6231 . . . 4 dom IntgOver βŠ† 𝒫 β„‚
32sseli 3971 . . 3 (𝑆 ∈ dom IntgOver β†’ 𝑆 ∈ 𝒫 β„‚)
4 cnex 11188 . . . . 5 β„‚ ∈ V
54elpw2 5336 . . . 4 (𝑆 ∈ 𝒫 β„‚ ↔ 𝑆 βŠ† β„‚)
6 itgoval 42455 . . . . 5 (𝑆 βŠ† β„‚ β†’ (IntgOverβ€˜π‘†) = {π‘Ž ∈ β„‚ ∣ βˆƒπ‘ ∈ (Polyβ€˜π‘†)((π‘β€˜π‘Ž) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1)})
7 ssrab2 4070 . . . . 5 {π‘Ž ∈ β„‚ ∣ βˆƒπ‘ ∈ (Polyβ€˜π‘†)((π‘β€˜π‘Ž) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1)} βŠ† β„‚
86, 7eqsstrdi 4029 . . . 4 (𝑆 βŠ† β„‚ β†’ (IntgOverβ€˜π‘†) βŠ† β„‚)
95, 8sylbi 216 . . 3 (𝑆 ∈ 𝒫 β„‚ β†’ (IntgOverβ€˜π‘†) βŠ† β„‚)
103, 9syl 17 . 2 (𝑆 ∈ dom IntgOver β†’ (IntgOverβ€˜π‘†) βŠ† β„‚)
11 ndmfv 6917 . . 3 (Β¬ 𝑆 ∈ dom IntgOver β†’ (IntgOverβ€˜π‘†) = βˆ…)
12 0ss 4389 . . 3 βˆ… βŠ† β„‚
1311, 12eqsstrdi 4029 . 2 (Β¬ 𝑆 ∈ dom IntgOver β†’ (IntgOverβ€˜π‘†) βŠ† β„‚)
1410, 13pm2.61i 182 1 (IntgOverβ€˜π‘†) βŠ† β„‚
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3062  {crab 3424   βŠ† wss 3941  βˆ…c0 4315  π’« cpw 4595  dom cdm 5667  β€˜cfv 6534  β„‚cc 11105  0cc0 11107  1c1 11108  Polycply 26062  coeffccoe 26064  degcdgr 26065  IntgOvercitgo 42451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-cnex 11163
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fv 6542  df-itgo 42453
This theorem is referenced by: (None)
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