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Theorem itgocn 27369
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 27364 . . . . 5  |- IntgOver  =  ( a  e.  ~P CC  |->  { b  e.  CC  |  E. c  e.  (Poly `  a ) ( ( c `  b )  =  0  /\  (
(coeff `  c ) `  (deg `  c )
)  =  1 ) } )
21dmmptss 5169 . . . 4  |-  dom IntgOver  C_  ~P CC
32sseli 3176 . . 3  |-  ( S  e.  dom IntgOver  ->  S  e. 
~P CC )
4 cnex 8818 . . . . 5  |-  CC  e.  _V
54elpw2 4175 . . . 4  |-  ( S  e.  ~P CC  <->  S  C_  CC )
6 itgoval 27366 . . . . 5  |-  ( S 
C_  CC  ->  (IntgOver `  S
)  =  { a  e.  CC  |  E. b  e.  (Poly `  S
) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) } )
7 ssrab2 3258 . . . . . 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 3212 . . . 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 5552 . . 3  |-  ( -.  S  e.  dom IntgOver  ->  (IntgOver `  S )  =  (/) )
13 0ss 3483 . . . 4  |-  (/)  C_  CC
1413a1i 10 . . 3  |-  ( -.  S  e.  dom IntgOver  ->  (/)  C_  CC )
1512, 14eqsstrd 3212 . 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 1623    e. wcel 1684   E.wrex 2544   {crab 2547    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   dom cdm 4689   ` cfv 5255   CCcc 8735   0cc0 8737   1c1 8738  Polycply 19566  coeffccoe 19568  degcdgr 19569  IntgOvercitgo 27362
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-cnex 8793
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-itgo 27364
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