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Theorem dgraaval 29426
Description: Value of the degree function on an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
Assertion
Ref Expression
dgraaval  |-  ( A  e.  AA  ->  (degAA `  A )  =  sup ( { d  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0p }
) ( (deg `  p )  =  d  /\  ( p `  A )  =  0 ) } ,  RR ,  `'  <  ) )
Distinct variable group:    A, d, p

Proof of Theorem dgraaval
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 fveq2 5688 . . . . . . 7  |-  ( a  =  A  ->  (
p `  a )  =  ( p `  A ) )
21eqeq1d 2449 . . . . . 6  |-  ( a  =  A  ->  (
( p `  a
)  =  0  <->  (
p `  A )  =  0 ) )
32anbi2d 698 . . . . 5  |-  ( a  =  A  ->  (
( (deg `  p
)  =  d  /\  ( p `  a
)  =  0 )  <-> 
( (deg `  p
)  =  d  /\  ( p `  A
)  =  0 ) ) )
43rexbidv 2734 . . . 4  |-  ( a  =  A  ->  ( E. p  e.  (
(Poly `  QQ )  \  { 0p }
) ( (deg `  p )  =  d  /\  ( p `  a )  =  0 )  <->  E. p  e.  ( (Poly `  QQ )  \  { 0p }
) ( (deg `  p )  =  d  /\  ( p `  A )  =  0 ) ) )
54rabbidv 2962 . . 3  |-  ( a  =  A  ->  { d  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  {
0p } ) ( (deg `  p
)  =  d  /\  ( p `  a
)  =  0 ) }  =  { d  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  {
0p } ) ( (deg `  p
)  =  d  /\  ( p `  A
)  =  0 ) } )
65supeq1d 7692 . 2  |-  ( a  =  A  ->  sup ( { d  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0p }
) ( (deg `  p )  =  d  /\  ( p `  a )  =  0 ) } ,  RR ,  `'  <  )  =  sup ( { d  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  {
0p } ) ( (deg `  p
)  =  d  /\  ( p `  A
)  =  0 ) } ,  RR ,  `'  <  ) )
7 df-dgraa 29424 . 2  |- degAA  =  (
a  e.  AA  |->  sup ( { d  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0p } ) ( (deg
`  p )  =  d  /\  ( p `
 a )  =  0 ) } ,  RR ,  `'  <  ) )
8 ltso 9451 . . . 4  |-  <  Or  RR
9 cnvso 5373 . . . 4  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
108, 9mpbi 208 . . 3  |-  `'  <  Or  RR
1110supex 7709 . 2  |-  sup ( { d  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0p }
) ( (deg `  p )  =  d  /\  ( p `  A )  =  0 ) } ,  RR ,  `'  <  )  e. 
_V
126, 7, 11fvmpt 5771 1  |-  ( A  e.  AA  ->  (degAA `  A )  =  sup ( { d  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0p }
) ( (deg `  p )  =  d  /\  ( p `  A )  =  0 ) } ,  RR ,  `'  <  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   E.wrex 2714   {crab 2717    \ cdif 3322   {csn 3874    Or wor 4636   `'ccnv 4835   ` cfv 5415   supcsup 7686   RRcr 9277   0cc0 9278    < clt 9414   NNcn 10318   QQcq 10949   0pc0p 21106  Polycply 21611  degcdgr 21614   AAcaa 21739  degAAcdgraa 29422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-resscn 9335  ax-pre-lttri 9352  ax-pre-lttrn 9353
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-ltxr 9419  df-dgraa 29424
This theorem is referenced by:  dgraalem  29427  dgraaub  29430
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