Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  df-dgraa Structured version   Unicode version

Definition df-dgraa 31259
Description: Define the degree of an algebraic number as the smallest degree of any nonzero polynomial which has said number as a root. (Contributed by Stefan O'Rear, 25-Nov-2014.)
Assertion
Ref Expression
df-dgraa  |- degAA  =  (
x  e.  AA  |->  sup ( { d  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0p } ) ( (deg
`  p )  =  d  /\  ( p `
 x )  =  0 ) } ,  RR ,  `'  <  ) )
Distinct variable group:    x, d, p

Detailed syntax breakdown of Definition df-dgraa
StepHypRef Expression
1 cdgraa 31257 . 2  class degAA
2 vx . . 3  setvar  x
3 caa 22795 . . 3  class  AA
4 vp . . . . . . . . . 10  setvar  p
54cv 1398 . . . . . . . . 9  class  p
6 cdgr 22669 . . . . . . . . 9  class deg
75, 6cfv 5496 . . . . . . . 8  class  (deg `  p )
8 vd . . . . . . . . 9  setvar  d
98cv 1398 . . . . . . . 8  class  d
107, 9wceq 1399 . . . . . . 7  wff  (deg `  p )  =  d
112cv 1398 . . . . . . . . 9  class  x
1211, 5cfv 5496 . . . . . . . 8  class  ( p `
 x )
13 cc0 9403 . . . . . . . 8  class  0
1412, 13wceq 1399 . . . . . . 7  wff  ( p `
 x )  =  0
1510, 14wa 367 . . . . . 6  wff  ( (deg
`  p )  =  d  /\  ( p `
 x )  =  0 )
16 cq 11101 . . . . . . . 8  class  QQ
17 cply 22666 . . . . . . . 8  class Poly
1816, 17cfv 5496 . . . . . . 7  class  (Poly `  QQ )
19 c0p 22161 . . . . . . . 8  class  0p
2019csn 3944 . . . . . . 7  class  { 0p }
2118, 20cdif 3386 . . . . . 6  class  ( (Poly `  QQ )  \  {
0p } )
2215, 4, 21wrex 2733 . . . . 5  wff  E. p  e.  ( (Poly `  QQ )  \  { 0p } ) ( (deg
`  p )  =  d  /\  ( p `
 x )  =  0 )
23 cn 10452 . . . . 5  class  NN
2422, 8, 23crab 2736 . . . 4  class  { d  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  {
0p } ) ( (deg `  p
)  =  d  /\  ( p `  x
)  =  0 ) }
25 cr 9402 . . . 4  class  RR
26 clt 9539 . . . . 5  class  <
2726ccnv 4912 . . . 4  class  `'  <
2824, 25, 27csup 7815 . . 3  class  sup ( { d  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0p }
) ( (deg `  p )  =  d  /\  ( p `  x )  =  0 ) } ,  RR ,  `'  <  )
292, 3, 28cmpt 4425 . 2  class  ( x  e.  AA  |->  sup ( { d  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0p }
) ( (deg `  p )  =  d  /\  ( p `  x )  =  0 ) } ,  RR ,  `'  <  ) )
301, 29wceq 1399 1  wff degAA  =  (
x  e.  AA  |->  sup ( { d  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0p } ) ( (deg
`  p )  =  d  /\  ( p `
 x )  =  0 ) } ,  RR ,  `'  <  ) )
Colors of variables: wff setvar class
This definition is referenced by:  dgraaval  31261  dgraaf  31264
  Copyright terms: Public domain W3C validator