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Theorem dgraaval 27317
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 5720 . . . . . . 7  |-  ( a  =  A  ->  (
p `  a )  =  ( p `  A ) )
21eqeq1d 2443 . . . . . 6  |-  ( a  =  A  ->  (
( p `  a
)  =  0  <->  (
p `  A )  =  0 ) )
32anbi2d 685 . . . . 5  |-  ( a  =  A  ->  (
( (deg `  p
)  =  d  /\  ( p `  a
)  =  0 )  <-> 
( (deg `  p
)  =  d  /\  ( p `  A
)  =  0 ) ) )
43rexbidv 2718 . . . 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 2940 . . 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 7443 . 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 27315 . 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 9148 . . . 4  |-  <  Or  RR
9 cnvso 5403 . . . 4  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
108, 9mpbi 200 . . 3  |-  `'  <  Or  RR
1110supex 7460 . 2  |-  sup ( { d  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0 p }
) ( (deg `  p )  =  d  /\  ( p `  A )  =  0 ) } ,  RR ,  `'  <  )  e. 
_V
126, 7, 11fvmpt 5798 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 359    = wceq 1652    e. wcel 1725   E.wrex 2698   {crab 2701    \ cdif 3309   {csn 3806    Or wor 4494   `'ccnv 4869   ` cfv 5446   supcsup 7437   RRcr 8981   0cc0 8982    < clt 9112   NNcn 9992   QQcq 10566   0 pc0p 19553  Polycply 20095  degcdgr 20098   AAcaa 20223  degAAcdgraa 27313
This theorem is referenced by:  dgraalem  27318  dgraaub  27321
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-pre-lttri 9056  ax-pre-lttrn 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-ltxr 9117  df-dgraa 27315
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