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Theorem itgocn 27348
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 27343 . . . . 5  |- IntgOver  =  ( a  e.  ~P CC  |->  { b  e.  CC  |  E. c  e.  (Poly `  a ) ( ( c `  b )  =  0  /\  (
(coeff `  c ) `  (deg `  c )
)  =  1 ) } )
21dmmptss 5368 . . . 4  |-  dom IntgOver  C_  ~P CC
32sseli 3346 . . 3  |-  ( S  e.  dom IntgOver  ->  S  e. 
~P CC )
4 cnex 9073 . . . . 5  |-  CC  e.  _V
54elpw2 4366 . . . 4  |-  ( S  e.  ~P CC  <->  S  C_  CC )
6 itgoval 27345 . . . . 5  |-  ( S 
C_  CC  ->  (IntgOver `  S
)  =  { a  e.  CC  |  E. b  e.  (Poly `  S
) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) } )
7 ssrab2 3430 . . . . 5  |-  { a  e.  CC  |  E. b  e.  (Poly `  S
) ( ( b `
 a )  =  0  /\  ( (coeff `  b ) `  (deg `  b ) )  =  1 ) }  C_  CC
86, 7syl6eqss 3400 . . . 4  |-  ( S 
C_  CC  ->  (IntgOver `  S
)  C_  CC )
95, 8sylbi 189 . . 3  |-  ( S  e.  ~P CC  ->  (IntgOver `  S )  C_  CC )
103, 9syl 16 . 2  |-  ( S  e.  dom IntgOver  ->  (IntgOver `  S
)  C_  CC )
11 ndmfv 5757 . . 3  |-  ( -.  S  e.  dom IntgOver  ->  (IntgOver `  S )  =  (/) )
12 0ss 3658 . . 3  |-  (/)  C_  CC
1311, 12syl6eqss 3400 . 2  |-  ( -.  S  e.  dom IntgOver  ->  (IntgOver `  S )  C_  CC )
1410, 13pm2.61i 159 1  |-  (IntgOver `  S
)  C_  CC
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708   {crab 2711    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   dom cdm 4880   ` cfv 5456   CCcc 8990   0cc0 8992   1c1 8993  Polycply 20105  coeffccoe 20107  degcdgr 20108  IntgOvercitgo 27341
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-cnex 9048
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fv 5464  df-itgo 27343
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