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Theorem itgoval 29686
Description: Value of the integral-over function. (Contributed by Stefan O'Rear, 27-Nov-2014.)
Assertion
Ref Expression
itgoval  |-  ( S 
C_  CC  ->  (IntgOver `  S
)  =  { x  e.  CC  |  E. p  e.  (Poly `  S )
( ( p `  x )  =  0  /\  ( (coeff `  p ) `  (deg `  p ) )  =  1 ) } )
Distinct variable group:    x, p, S

Proof of Theorem itgoval
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 cnex 9477 . . 3  |-  CC  e.  _V
21elpw2 4567 . 2  |-  ( S  e.  ~P CC  <->  S  C_  CC )
3 fveq2 5802 . . . . 5  |-  ( s  =  S  ->  (Poly `  s )  =  (Poly `  S ) )
43rexeqdv 3030 . . . 4  |-  ( s  =  S  ->  ( E. p  e.  (Poly `  s ) ( ( p `  x )  =  0  /\  (
(coeff `  p ) `  (deg `  p )
)  =  1 )  <->  E. p  e.  (Poly `  S ) ( ( p `  x )  =  0  /\  (
(coeff `  p ) `  (deg `  p )
)  =  1 ) ) )
54rabbidv 3070 . . 3  |-  ( s  =  S  ->  { x  e.  CC  |  E. p  e.  (Poly `  s )
( ( p `  x )  =  0  /\  ( (coeff `  p ) `  (deg `  p ) )  =  1 ) }  =  { x  e.  CC  |  E. p  e.  (Poly `  S ) ( ( p `  x )  =  0  /\  (
(coeff `  p ) `  (deg `  p )
)  =  1 ) } )
6 df-itgo 29684 . . 3  |- IntgOver  =  ( s  e.  ~P CC  |->  { x  e.  CC  |  E. p  e.  (Poly `  s ) ( ( p `  x )  =  0  /\  (
(coeff `  p ) `  (deg `  p )
)  =  1 ) } )
71rabex 4554 . . 3  |-  { x  e.  CC  |  E. p  e.  (Poly `  S )
( ( p `  x )  =  0  /\  ( (coeff `  p ) `  (deg `  p ) )  =  1 ) }  e.  _V
85, 6, 7fvmpt 5886 . 2  |-  ( S  e.  ~P CC  ->  (IntgOver `  S )  =  {
x  e.  CC  |  E. p  e.  (Poly `  S ) ( ( p `  x )  =  0  /\  (
(coeff `  p ) `  (deg `  p )
)  =  1 ) } )
92, 8sylbir 213 1  |-  ( S 
C_  CC  ->  (IntgOver `  S
)  =  { x  e.  CC  |  E. p  e.  (Poly `  S )
( ( p `  x )  =  0  /\  ( (coeff `  p ) `  (deg `  p ) )  =  1 ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2800   {crab 2803    C_ wss 3439   ~Pcpw 3971   ` cfv 5529   CCcc 9394   0cc0 9396   1c1 9397  Polycply 21788  coeffccoe 21790  degcdgr 21791  IntgOvercitgo 29682
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-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-cnex 9452
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-iota 5492  df-fun 5531  df-fv 5537  df-itgo 29684
This theorem is referenced by:  aaitgo  29687  itgoss  29688  itgocn  29689
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