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Theorem dgraaval 30689
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 5859 . . . . . . 7  |-  ( a  =  A  ->  (
p `  a )  =  ( p `  A ) )
21eqeq1d 2464 . . . . . 6  |-  ( a  =  A  ->  (
( p `  a
)  =  0  <->  (
p `  A )  =  0 ) )
32anbi2d 703 . . . . 5  |-  ( a  =  A  ->  (
( (deg `  p
)  =  d  /\  ( p `  a
)  =  0 )  <-> 
( (deg `  p
)  =  d  /\  ( p `  A
)  =  0 ) ) )
43rexbidv 2968 . . . 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 3100 . . 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 7897 . 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 30687 . 2  |- degAA  =  (
a  e.  AA  |->  sup ( { d  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0p } ) ( (deg
`  p )  =  d  /\  ( p `
 a )  =  0 ) } ,  RR ,  `'  <  ) )
8 ltso 9656 . . . 4  |-  <  Or  RR
9 cnvso 5539 . . . 4  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
108, 9mpbi 208 . . 3  |-  `'  <  Or  RR
1110supex 7914 . 2  |-  sup ( { d  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0p }
) ( (deg `  p )  =  d  /\  ( p `  A )  =  0 ) } ,  RR ,  `'  <  )  e. 
_V
126, 7, 11fvmpt 5943 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 1374    e. wcel 1762   E.wrex 2810   {crab 2813    \ cdif 3468   {csn 4022    Or wor 4794   `'ccnv 4993   ` cfv 5581   supcsup 7891   RRcr 9482   0cc0 9483    < clt 9619   NNcn 10527   QQcq 11173   0pc0p 21806  Polycply 22311  degcdgr 22314   AAcaa 22439  degAAcdgraa 30685
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-resscn 9540  ax-pre-lttri 9557  ax-pre-lttrn 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-po 4795  df-so 4796  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-sup 7892  df-pnf 9621  df-mnf 9622  df-ltxr 9624  df-dgraa 30687
This theorem is referenced by:  dgraalem  30690  dgraaub  30693
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