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Theorem itgocn 29692
Description: All integral elements are complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
Assertion
Ref Expression
itgocn  |-  (IntgOver `  S
)  C_  CC

Proof of Theorem itgocn
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-itgo 29687 . . . . 5  |- IntgOver  =  ( a  e.  ~P CC  |->  { b  e.  CC  |  E. c  e.  (Poly `  a ) ( ( c `  b )  =  0  /\  (
(coeff `  c ) `  (deg `  c )
)  =  1 ) } )
21dmmptss 5445 . . . 4  |-  dom IntgOver  C_  ~P CC
32sseli 3463 . . 3  |-  ( S  e.  dom IntgOver  ->  S  e. 
~P CC )
4 cnex 9478 . . . . 5  |-  CC  e.  _V
54elpw2 4567 . . . 4  |-  ( S  e.  ~P CC  <->  S  C_  CC )
6 itgoval 29689 . . . . 5  |-  ( S 
C_  CC  ->  (IntgOver `  S
)  =  { a  e.  CC  |  E. b  e.  (Poly `  S
) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) } )
7 ssrab2 3548 . . . . 5  |-  { a  e.  CC  |  E. b  e.  (Poly `  S
) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) }  C_  CC
86, 7syl6eqss 3517 . . . 4  |-  ( S 
C_  CC  ->  (IntgOver `  S
)  C_  CC )
95, 8sylbi 195 . . 3  |-  ( S  e.  ~P CC  ->  (IntgOver `  S )  C_  CC )
103, 9syl 16 . 2  |-  ( S  e.  dom IntgOver  ->  (IntgOver `  S
)  C_  CC )
11 ndmfv 5826 . . 3  |-  ( -.  S  e.  dom IntgOver  ->  (IntgOver `  S )  =  (/) )
12 0ss 3777 . . 3  |-  (/)  C_  CC
1311, 12syl6eqss 3517 . 2  |-  ( -.  S  e.  dom IntgOver  ->  (IntgOver `  S )  C_  CC )
1410, 13pm2.61i 164 1  |-  (IntgOver `  S
)  C_  CC
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2800   {crab 2803    C_ wss 3439   (/)c0 3748   ~Pcpw 3971   dom cdm 4951   ` cfv 5529   CCcc 9395   0cc0 9397   1c1 9398  Polycply 21795  coeffccoe 21797  degcdgr 21798  IntgOvercitgo 29685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-cnex 9453
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fv 5537  df-itgo 29687
This theorem is referenced by: (None)
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