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Theorem itgocn 27472
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 27467 . . . . 5  |- IntgOver  =  ( a  e.  ~P CC  |->  { b  e.  CC  |  E. c  e.  (Poly `  a ) ( ( c `  b )  =  0  /\  (
(coeff `  c ) `  (deg `  c )
)  =  1 ) } )
21dmmptss 5185 . . . 4  |-  dom IntgOver  C_  ~P CC
32sseli 3189 . . 3  |-  ( S  e.  dom IntgOver  ->  S  e. 
~P CC )
4 cnex 8834 . . . . 5  |-  CC  e.  _V
54elpw2 4191 . . . 4  |-  ( S  e.  ~P CC  <->  S  C_  CC )
6 itgoval 27469 . . . . 5  |-  ( S 
C_  CC  ->  (IntgOver `  S
)  =  { a  e.  CC  |  E. b  e.  (Poly `  S
) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) } )
7 ssrab2 3271 . . . . . 6  |-  { a  e.  CC  |  E. b  e.  (Poly `  S
) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) }  C_  CC
87a1i 10 . . . . 5  |-  ( S 
C_  CC  ->  { a  e.  CC  |  E. b  e.  (Poly `  S
) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) }  C_  CC )
96, 8eqsstrd 3225 . . . 4  |-  ( S 
C_  CC  ->  (IntgOver `  S
)  C_  CC )
105, 9sylbi 187 . . 3  |-  ( S  e.  ~P CC  ->  (IntgOver `  S )  C_  CC )
113, 10syl 15 . 2  |-  ( S  e.  dom IntgOver  ->  (IntgOver `  S
)  C_  CC )
12 ndmfv 5568 . . 3  |-  ( -.  S  e.  dom IntgOver  ->  (IntgOver `  S )  =  (/) )
13 0ss 3496 . . . 4  |-  (/)  C_  CC
1413a1i 10 . . 3  |-  ( -.  S  e.  dom IntgOver  ->  (/)  C_  CC )
1512, 14eqsstrd 3225 . 2  |-  ( -.  S  e.  dom IntgOver  ->  (IntgOver `  S )  C_  CC )
1611, 15pm2.61i 156 1  |-  (IntgOver `  S
)  C_  CC
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   {crab 2560    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   dom cdm 4705   ` cfv 5271   CCcc 8751   0cc0 8753   1c1 8754  Polycply 19582  coeffccoe 19584  degcdgr 19585  IntgOvercitgo 27465
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-cnex 8809
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-itgo 27467
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