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Theorem itgocn 42588
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 42583 . . . . 5 IntgOver = (π‘Ž ∈ 𝒫 β„‚ ↦ {𝑏 ∈ β„‚ ∣ βˆƒπ‘ ∈ (Polyβ€˜π‘Ž)((π‘β€˜π‘) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1)})
21dmmptss 6245 . . . 4 dom IntgOver βŠ† 𝒫 β„‚
32sseli 3976 . . 3 (𝑆 ∈ dom IntgOver β†’ 𝑆 ∈ 𝒫 β„‚)
4 cnex 11220 . . . . 5 β„‚ ∈ V
54elpw2 5347 . . . 4 (𝑆 ∈ 𝒫 β„‚ ↔ 𝑆 βŠ† β„‚)
6 itgoval 42585 . . . . 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 4075 . . . . 5 {π‘Ž ∈ β„‚ ∣ βˆƒπ‘ ∈ (Polyβ€˜π‘†)((π‘β€˜π‘Ž) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1)} βŠ† β„‚
86, 7eqsstrdi 4034 . . . 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 6932 . . 3 (Β¬ 𝑆 ∈ dom IntgOver β†’ (IntgOverβ€˜π‘†) = βˆ…)
12 0ss 4397 . . 3 βˆ… βŠ† β„‚
1311, 12eqsstrdi 4034 . 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 1534   ∈ wcel 2099  βˆƒwrex 3067  {crab 3429   βŠ† wss 3947  βˆ…c0 4323  π’« cpw 4603  dom cdm 5678  β€˜cfv 6548  β„‚cc 11137  0cc0 11139  1c1 11140  Polycply 26131  coeffccoe 26133  degcdgr 26134  IntgOvercitgo 42581
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 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-cnex 11195
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fv 6556  df-itgo 42583
This theorem is referenced by: (None)
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