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Theorem dgraalem 29502
Description: Properties of the degree of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
Assertion
Ref Expression
dgraalem  |-  ( A  e.  AA  ->  (
(degAA `
 A )  e.  NN  /\  E. p  e.  ( (Poly `  QQ )  \  { 0p } ) ( (deg
`  p )  =  (degAA `  A )  /\  ( p `  A
)  =  0 ) ) )
Distinct variable group:    A, p

Proof of Theorem dgraalem
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dgraaval 29501 . . 3  |-  ( A  e.  AA  ->  (degAA `  A )  =  sup ( { a  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0p }
) ( (deg `  p )  =  a  /\  ( p `  A )  =  0 ) } ,  RR ,  `'  <  ) )
2 ssrab2 3437 . . . . 5  |-  { a  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  {
0p } ) ( (deg `  p
)  =  a  /\  ( p `  A
)  =  0 ) }  C_  NN
3 nnuz 10896 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
42, 3sseqtri 3388 . . . 4  |-  { a  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  {
0p } ) ( (deg `  p
)  =  a  /\  ( p `  A
)  =  0 ) }  C_  ( ZZ>= ` 
1 )
5 eldifsn 4000 . . . . . . . . . . . 12  |-  ( b  e.  ( (Poly `  QQ )  \  { 0p } )  <->  ( b  e.  (Poly `  QQ )  /\  b  =/=  0p ) )
65biimpi 194 . . . . . . . . . . 11  |-  ( b  e.  ( (Poly `  QQ )  \  { 0p } )  -> 
( b  e.  (Poly `  QQ )  /\  b  =/=  0p ) )
76ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( b  e.  ( (Poly `  QQ )  \  { 0p }
)  /\  ( b `  A )  =  0 )  /\  A  e.  CC )  ->  (
b  e.  (Poly `  QQ )  /\  b  =/=  0p ) )
8 simpr 461 . . . . . . . . . 10  |-  ( ( ( b  e.  ( (Poly `  QQ )  \  { 0p }
)  /\  ( b `  A )  =  0 )  /\  A  e.  CC )  ->  A  e.  CC )
9 simplr 754 . . . . . . . . . 10  |-  ( ( ( b  e.  ( (Poly `  QQ )  \  { 0p }
)  /\  ( b `  A )  =  0 )  /\  A  e.  CC )  ->  (
b `  A )  =  0 )
10 dgrnznn 29492 . . . . . . . . . 10  |-  ( ( ( b  e.  (Poly `  QQ )  /\  b  =/=  0p )  /\  ( A  e.  CC  /\  ( b `  A
)  =  0 ) )  ->  (deg `  b
)  e.  NN )
117, 8, 9, 10syl12anc 1216 . . . . . . . . 9  |-  ( ( ( b  e.  ( (Poly `  QQ )  \  { 0p }
)  /\  ( b `  A )  =  0 )  /\  A  e.  CC )  ->  (deg `  b )  e.  NN )
12 simpll 753 . . . . . . . . 9  |-  ( ( ( b  e.  ( (Poly `  QQ )  \  { 0p }
)  /\  ( b `  A )  =  0 )  /\  A  e.  CC )  ->  b  e.  ( (Poly `  QQ )  \  { 0p } ) )
13 eqid 2443 . . . . . . . . . 10  |-  (deg `  b )  =  (deg
`  b )
149, 13jctil 537 . . . . . . . . 9  |-  ( ( ( b  e.  ( (Poly `  QQ )  \  { 0p }
)  /\  ( b `  A )  =  0 )  /\  A  e.  CC )  ->  (
(deg `  b )  =  (deg `  b )  /\  ( b `  A
)  =  0 ) )
15 eqeq2 2452 . . . . . . . . . . 11  |-  ( a  =  (deg `  b
)  ->  ( (deg `  p )  =  a  <-> 
(deg `  p )  =  (deg `  b )
) )
1615anbi1d 704 . . . . . . . . . 10  |-  ( a  =  (deg `  b
)  ->  ( (
(deg `  p )  =  a  /\  (
p `  A )  =  0 )  <->  ( (deg `  p )  =  (deg
`  b )  /\  ( p `  A
)  =  0 ) ) )
17 fveq2 5691 . . . . . . . . . . . 12  |-  ( p  =  b  ->  (deg `  p )  =  (deg
`  b ) )
1817eqeq1d 2451 . . . . . . . . . . 11  |-  ( p  =  b  ->  (
(deg `  p )  =  (deg `  b )  <->  (deg
`  b )  =  (deg `  b )
) )
19 fveq1 5690 . . . . . . . . . . . 12  |-  ( p  =  b  ->  (
p `  A )  =  ( b `  A ) )
2019eqeq1d 2451 . . . . . . . . . . 11  |-  ( p  =  b  ->  (
( p `  A
)  =  0  <->  (
b `  A )  =  0 ) )
2118, 20anbi12d 710 . . . . . . . . . 10  |-  ( p  =  b  ->  (
( (deg `  p
)  =  (deg `  b )  /\  (
p `  A )  =  0 )  <->  ( (deg `  b )  =  (deg
`  b )  /\  ( b `  A
)  =  0 ) ) )
2216, 21rspc2ev 3081 . . . . . . . . 9  |-  ( ( (deg `  b )  e.  NN  /\  b  e.  ( (Poly `  QQ )  \  { 0p } )  /\  (
(deg `  b )  =  (deg `  b )  /\  ( b `  A
)  =  0 ) )  ->  E. a  e.  NN  E. p  e.  ( (Poly `  QQ )  \  { 0p } ) ( (deg
`  p )  =  a  /\  ( p `
 A )  =  0 ) )
2311, 12, 14, 22syl3anc 1218 . . . . . . . 8  |-  ( ( ( b  e.  ( (Poly `  QQ )  \  { 0p }
)  /\  ( b `  A )  =  0 )  /\  A  e.  CC )  ->  E. a  e.  NN  E. p  e.  ( (Poly `  QQ )  \  { 0p } ) ( (deg
`  p )  =  a  /\  ( p `
 A )  =  0 ) )
2423ex 434 . . . . . . 7  |-  ( ( b  e.  ( (Poly `  QQ )  \  {
0p } )  /\  ( b `  A )  =  0 )  ->  ( A  e.  CC  ->  E. a  e.  NN  E. p  e.  ( (Poly `  QQ )  \  { 0p } ) ( (deg
`  p )  =  a  /\  ( p `
 A )  =  0 ) ) )
2524rexlimiva 2836 . . . . . 6  |-  ( E. b  e.  ( (Poly `  QQ )  \  {
0p } ) ( b `  A
)  =  0  -> 
( A  e.  CC  ->  E. a  e.  NN  E. p  e.  ( (Poly `  QQ )  \  {
0p } ) ( (deg `  p
)  =  a  /\  ( p `  A
)  =  0 ) ) )
2625impcom 430 . . . . 5  |-  ( ( A  e.  CC  /\  E. b  e.  ( (Poly `  QQ )  \  {
0p } ) ( b `  A
)  =  0 )  ->  E. a  e.  NN  E. p  e.  ( (Poly `  QQ )  \  {
0p } ) ( (deg `  p
)  =  a  /\  ( p `  A
)  =  0 ) )
27 elqaa 21788 . . . . 5  |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. b  e.  ( (Poly `  QQ )  \  { 0p } ) ( b `
 A )  =  0 ) )
28 rabn0 3657 . . . . 5  |-  ( { a  e.  NN  |  E. p  e.  (
(Poly `  QQ )  \  { 0p }
) ( (deg `  p )  =  a  /\  ( p `  A )  =  0 ) }  =/=  (/)  <->  E. a  e.  NN  E. p  e.  ( (Poly `  QQ )  \  { 0p } ) ( (deg
`  p )  =  a  /\  ( p `
 A )  =  0 ) )
2926, 27, 283imtr4i 266 . . . 4  |-  ( A  e.  AA  ->  { a  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  {
0p } ) ( (deg `  p
)  =  a  /\  ( p `  A
)  =  0 ) }  =/=  (/) )
30 infmssuzcl 10938 . . . 4  |-  ( ( { a  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0p }
) ( (deg `  p )  =  a  /\  ( p `  A )  =  0 ) }  C_  ( ZZ>=
`  1 )  /\  { a  e.  NN  |  E. p  e.  (
(Poly `  QQ )  \  { 0p }
) ( (deg `  p )  =  a  /\  ( p `  A )  =  0 ) }  =/=  (/) )  ->  sup ( { a  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0p } ) ( (deg
`  p )  =  a  /\  ( p `
 A )  =  0 ) } ,  RR ,  `'  <  )  e.  { a  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0p } ) ( (deg
`  p )  =  a  /\  ( p `
 A )  =  0 ) } )
314, 29, 30sylancr 663 . . 3  |-  ( A  e.  AA  ->  sup ( { a  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0p }
) ( (deg `  p )  =  a  /\  ( p `  A )  =  0 ) } ,  RR ,  `'  <  )  e. 
{ a  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0p }
) ( (deg `  p )  =  a  /\  ( p `  A )  =  0 ) } )
321, 31eqeltrd 2517 . 2  |-  ( A  e.  AA  ->  (degAA `  A )  e.  {
a  e.  NN  |  E. p  e.  (
(Poly `  QQ )  \  { 0p }
) ( (deg `  p )  =  a  /\  ( p `  A )  =  0 ) } )
33 eqeq2 2452 . . . . 5  |-  ( a  =  (degAA `  A )  -> 
( (deg `  p
)  =  a  <->  (deg `  p
)  =  (degAA `  A
) ) )
3433anbi1d 704 . . . 4  |-  ( a  =  (degAA `  A )  -> 
( ( (deg `  p )  =  a  /\  ( p `  A )  =  0 )  <->  ( (deg `  p )  =  (degAA `  A )  /\  (
p `  A )  =  0 ) ) )
3534rexbidv 2736 . . 3  |-  ( a  =  (degAA `  A )  -> 
( E. p  e.  ( (Poly `  QQ )  \  { 0p } ) ( (deg
`  p )  =  a  /\  ( p `
 A )  =  0 )  <->  E. p  e.  ( (Poly `  QQ )  \  { 0p } ) ( (deg
`  p )  =  (degAA `  A )  /\  ( p `  A
)  =  0 ) ) )
3635elrab 3117 . 2  |-  ( (degAA `  A )  e.  {
a  e.  NN  |  E. p  e.  (
(Poly `  QQ )  \  { 0p }
) ( (deg `  p )  =  a  /\  ( p `  A )  =  0 ) }  <->  ( (degAA `  A )  e.  NN  /\ 
E. p  e.  ( (Poly `  QQ )  \  { 0p }
) ( (deg `  p )  =  (degAA `  A )  /\  (
p `  A )  =  0 ) ) )
3732, 36sylib 196 1  |-  ( A  e.  AA  ->  (
(degAA `
 A )  e.  NN  /\  E. p  e.  ( (Poly `  QQ )  \  { 0p } ) ( (deg
`  p )  =  (degAA `  A )  /\  ( p `  A
)  =  0 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606   E.wrex 2716   {crab 2719    \ cdif 3325    C_ wss 3328   (/)c0 3637   {csn 3877   `'ccnv 4839   ` cfv 5418   supcsup 7690   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283    < clt 9418   NNcn 10322   ZZ>=cuz 10861   QQcq 10953   0pc0p 21147  Polycply 21652  degcdgr 21655   AAcaa 21780  degAAcdgraa 29497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-q 10954  df-rp 10992  df-fz 11438  df-fzo 11549  df-fl 11642  df-mod 11709  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-clim 12966  df-rlim 12967  df-sum 13164  df-0p 21148  df-ply 21656  df-coe 21658  df-dgr 21659  df-aa 21781  df-dgraa 29499
This theorem is referenced by:  dgraacl  29503  mpaaeu  29507
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