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Theorem dgraaval 27020
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 5670 . . . . . . 7  |-  ( a  =  A  ->  (
p `  a )  =  ( p `  A ) )
21eqeq1d 2397 . . . . . 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 2672 . . . 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 2893 . . 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 7388 . 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 27018 . 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 9091 . . . 4  |-  <  Or  RR
9 cnvso 5353 . . . 4  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
108, 9mpbi 200 . . 3  |-  `'  <  Or  RR
1110supex 7403 . 2  |-  sup ( { d  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0 p }
) ( (deg `  p )  =  d  /\  ( p `  A )  =  0 ) } ,  RR ,  `'  <  )  e. 
_V
126, 7, 11fvmpt 5747 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 1649    e. wcel 1717   E.wrex 2652   {crab 2655    \ cdif 3262   {csn 3759    Or wor 4445   `'ccnv 4819   ` cfv 5396   supcsup 7382   RRcr 8924   0cc0 8925    < clt 9055   NNcn 9934   QQcq 10508   0 pc0p 19430  Polycply 19972  degcdgr 19975   AAcaa 20100  degAAcdgraa 27016
This theorem is referenced by:  dgraalem  27021  dgraaub  27024
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-resscn 8982  ax-pre-lttri 8999  ax-pre-lttrn 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-po 4446  df-so 4447  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-sup 7383  df-pnf 9057  df-mnf 9058  df-ltxr 9060  df-dgraa 27018
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