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Theorem itgoval 35438
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 9521 . . 3  |-  CC  e.  _V
21elpw2 4555 . 2  |-  ( S  e.  ~P CC  <->  S  C_  CC )
3 fveq2 5803 . . . . 5  |-  ( s  =  S  ->  (Poly `  s )  =  (Poly `  S ) )
43rexeqdv 3008 . . . 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 3048 . . 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 35436 . . 3  |- IntgOver  =  ( s  e.  ~P CC  |->  { x  e.  CC  |  E. p  e.  (Poly `  s ) ( ( p `  x )  =  0  /\  (
(coeff `  p ) `  (deg `  p )
)  =  1 ) } )
71rabex 4542 . . 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 367    = wceq 1403    e. wcel 1840   E.wrex 2752   {crab 2755    C_ wss 3411   ~Pcpw 3952   ` cfv 5523   CCcc 9438   0cc0 9440   1c1 9441  Polycply 22763  coeffccoe 22765  degcdgr 22766  IntgOvercitgo 35434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pr 4627  ax-cnex 9496
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-iota 5487  df-fun 5525  df-fv 5531  df-itgo 35436
This theorem is referenced by:  aaitgo  35439  itgoss  35440  itgocn  35441
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