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Theorem dgraaval 27349
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 )  \  { 0 p }
) ( (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 5525 . . . . . . 7  |-  ( a  =  A  ->  (
p `  a )  =  ( p `  A ) )
21eqeq1d 2291 . . . . . 6  |-  ( a  =  A  ->  (
( p `  a
)  =  0  <->  (
p `  A )  =  0 ) )
32anbi2d 684 . . . . 5  |-  ( a  =  A  ->  (
( (deg `  p
)  =  d  /\  ( p `  a
)  =  0 )  <-> 
( (deg `  p
)  =  d  /\  ( p `  A
)  =  0 ) ) )
43rexbidv 2564 . . . 4  |-  ( a  =  A  ->  ( E. p  e.  (
(Poly `  QQ )  \  { 0 p }
) ( (deg `  p )  =  d  /\  ( p `  a )  =  0 )  <->  E. p  e.  ( (Poly `  QQ )  \  { 0 p }
) ( (deg `  p )  =  d  /\  ( p `  A )  =  0 ) ) )
54rabbidv 2780 . . 3  |-  ( a  =  A  ->  { d  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  {
0 p } ) ( (deg `  p
)  =  d  /\  ( p `  a
)  =  0 ) }  =  { d  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  {
0 p } ) ( (deg `  p
)  =  d  /\  ( p `  A
)  =  0 ) } )
65supeq1d 7199 . 2  |-  ( a  =  A  ->  sup ( { d  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0 p }
) ( (deg `  p )  =  d  /\  ( p `  a )  =  0 ) } ,  RR ,  `'  <  )  =  sup ( { d  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  {
0 p } ) ( (deg `  p
)  =  d  /\  ( p `  A
)  =  0 ) } ,  RR ,  `'  <  ) )
7 df-dgraa 27347 . 2  |- degAA  =  (
a  e.  AA  |->  sup ( { d  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0 p } ) ( (deg
`  p )  =  d  /\  ( p `
 a )  =  0 ) } ,  RR ,  `'  <  ) )
8 ltso 8903 . . . 4  |-  <  Or  RR
9 cnvso 5214 . . . 4  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
108, 9mpbi 199 . . 3  |-  `'  <  Or  RR
1110supex 7214 . 2  |-  sup ( { d  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0 p }
) ( (deg `  p )  =  d  /\  ( p `  A )  =  0 ) } ,  RR ,  `'  <  )  e. 
_V
126, 7, 11fvmpt 5602 1  |-  ( A  e.  AA  ->  (degAA `  A )  =  sup ( { d  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0 p }
) ( (deg `  p )  =  d  /\  ( p `  A )  =  0 ) } ,  RR ,  `'  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   {crab 2547    \ cdif 3149   {csn 3640    Or wor 4313   `'ccnv 4688   ` cfv 5255   supcsup 7193   RRcr 8736   0cc0 8737    < clt 8867   NNcn 9746   QQcq 10316   0 pc0p 19024  Polycply 19566  degcdgr 19569   AAcaa 19694  degAAcdgraa 27345
This theorem is referenced by:  dgraalem  27350  dgraaub  27353
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-pre-lttri 8811  ax-pre-lttrn 8812
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-dgraa 27347
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