Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  itgocn Structured version   Unicode version

Theorem itgocn 31042
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 31037 . . . . 5  |- IntgOver  =  ( a  e.  ~P CC  |->  { b  e.  CC  |  E. c  e.  (Poly `  a ) ( ( c `  b )  =  0  /\  (
(coeff `  c ) `  (deg `  c )
)  =  1 ) } )
21dmmptss 5509 . . . 4  |-  dom IntgOver  C_  ~P CC
32sseli 3505 . . 3  |-  ( S  e.  dom IntgOver  ->  S  e. 
~P CC )
4 cnex 9585 . . . . 5  |-  CC  e.  _V
54elpw2 4617 . . . 4  |-  ( S  e.  ~P CC  <->  S  C_  CC )
6 itgoval 31039 . . . . 5  |-  ( S 
C_  CC  ->  (IntgOver `  S
)  =  { a  e.  CC  |  E. b  e.  (Poly `  S
) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) } )
7 ssrab2 3590 . . . . 5  |-  { a  e.  CC  |  E. b  e.  (Poly `  S
) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) }  C_  CC
86, 7syl6eqss 3559 . . . 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 5896 . . 3  |-  ( -.  S  e.  dom IntgOver  ->  (IntgOver `  S )  =  (/) )
12 0ss 3819 . . 3  |-  (/)  C_  CC
1311, 12syl6eqss 3559 . 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 1379    e. wcel 1767   E.wrex 2818   {crab 2821    C_ wss 3481   (/)c0 3790   ~Pcpw 4016   dom cdm 5005   ` cfv 5594   CCcc 9502   0cc0 9504   1c1 9505  Polycply 22449  coeffccoe 22451  degcdgr 22452  IntgOvercitgo 31035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-cnex 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-itgo 31037
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator