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Theorem itgocn 41534
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 41529 . . . . 5 IntgOver = (π‘Ž ∈ 𝒫 β„‚ ↦ {𝑏 ∈ β„‚ ∣ βˆƒπ‘ ∈ (Polyβ€˜π‘Ž)((π‘β€˜π‘) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1)})
21dmmptss 6194 . . . 4 dom IntgOver βŠ† 𝒫 β„‚
32sseli 3941 . . 3 (𝑆 ∈ dom IntgOver β†’ 𝑆 ∈ 𝒫 β„‚)
4 cnex 11137 . . . . 5 β„‚ ∈ V
54elpw2 5303 . . . 4 (𝑆 ∈ 𝒫 β„‚ ↔ 𝑆 βŠ† β„‚)
6 itgoval 41531 . . . . 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 4038 . . . . 5 {π‘Ž ∈ β„‚ ∣ βˆƒπ‘ ∈ (Polyβ€˜π‘†)((π‘β€˜π‘Ž) = 0 ∧ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1)} βŠ† β„‚
86, 7eqsstrdi 3999 . . . 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 6878 . . 3 (Β¬ 𝑆 ∈ dom IntgOver β†’ (IntgOverβ€˜π‘†) = βˆ…)
12 0ss 4357 . . 3 βˆ… βŠ† β„‚
1311, 12eqsstrdi 3999 . 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 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3070  {crab 3406   βŠ† wss 3911  βˆ…c0 4283  π’« cpw 4561  dom cdm 5634  β€˜cfv 6497  β„‚cc 11054  0cc0 11056  1c1 11057  Polycply 25561  coeffccoe 25563  degcdgr 25564  IntgOvercitgo 41527
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 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-cnex 11112
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fv 6505  df-itgo 41529
This theorem is referenced by: (None)
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