Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dgraalem Unicode version

Theorem dgraalem 27350
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 )  \  { 0 p } ) ( (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 27349 . . 3  |-  ( A  e.  AA  ->  (degAA `  A )  =  sup ( { a  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0 p }
) ( (deg `  p )  =  a  /\  ( p `  A )  =  0 ) } ,  RR ,  `'  <  ) )
2 ssrab2 3258 . . . . 5  |-  { a  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  {
0 p } ) ( (deg `  p
)  =  a  /\  ( p `  A
)  =  0 ) }  C_  NN
3 nnuz 10263 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
42, 3sseqtri 3210 . . . 4  |-  { a  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  {
0 p } ) ( (deg `  p
)  =  a  /\  ( p `  A
)  =  0 ) }  C_  ( ZZ>= ` 
1 )
5 eldifsn 3749 . . . . . . . . . . . 12  |-  ( b  e.  ( (Poly `  QQ )  \  { 0 p } )  <->  ( b  e.  (Poly `  QQ )  /\  b  =/=  0 p ) )
65biimpi 186 . . . . . . . . . . 11  |-  ( b  e.  ( (Poly `  QQ )  \  { 0 p } )  -> 
( b  e.  (Poly `  QQ )  /\  b  =/=  0 p ) )
76ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( b  e.  ( (Poly `  QQ )  \  { 0 p }
)  /\  ( b `  A )  =  0 )  /\  A  e.  CC )  ->  (
b  e.  (Poly `  QQ )  /\  b  =/=  0 p ) )
8 simpr 447 . . . . . . . . . 10  |-  ( ( ( b  e.  ( (Poly `  QQ )  \  { 0 p }
)  /\  ( b `  A )  =  0 )  /\  A  e.  CC )  ->  A  e.  CC )
9 simplr 731 . . . . . . . . . 10  |-  ( ( ( b  e.  ( (Poly `  QQ )  \  { 0 p }
)  /\  ( b `  A )  =  0 )  /\  A  e.  CC )  ->  (
b `  A )  =  0 )
10 dgrnznn 27340 . . . . . . . . . 10  |-  ( ( ( b  e.  (Poly `  QQ )  /\  b  =/=  0 p )  /\  ( A  e.  CC  /\  ( b `  A
)  =  0 ) )  ->  (deg `  b
)  e.  NN )
117, 8, 9, 10syl12anc 1180 . . . . . . . . 9  |-  ( ( ( b  e.  ( (Poly `  QQ )  \  { 0 p }
)  /\  ( b `  A )  =  0 )  /\  A  e.  CC )  ->  (deg `  b )  e.  NN )
12 simpll 730 . . . . . . . . 9  |-  ( ( ( b  e.  ( (Poly `  QQ )  \  { 0 p }
)  /\  ( b `  A )  =  0 )  /\  A  e.  CC )  ->  b  e.  ( (Poly `  QQ )  \  { 0 p } ) )
13 eqid 2283 . . . . . . . . . 10  |-  (deg `  b )  =  (deg
`  b )
149, 13jctil 523 . . . . . . . . 9  |-  ( ( ( b  e.  ( (Poly `  QQ )  \  { 0 p }
)  /\  ( b `  A )  =  0 )  /\  A  e.  CC )  ->  (
(deg `  b )  =  (deg `  b )  /\  ( b `  A
)  =  0 ) )
15 eqeq2 2292 . . . . . . . . . . 11  |-  ( a  =  (deg `  b
)  ->  ( (deg `  p )  =  a  <-> 
(deg `  p )  =  (deg `  b )
) )
1615anbi1d 685 . . . . . . . . . 10  |-  ( a  =  (deg `  b
)  ->  ( (
(deg `  p )  =  a  /\  (
p `  A )  =  0 )  <->  ( (deg `  p )  =  (deg
`  b )  /\  ( p `  A
)  =  0 ) ) )
17 fveq2 5525 . . . . . . . . . . . 12  |-  ( p  =  b  ->  (deg `  p )  =  (deg
`  b ) )
1817eqeq1d 2291 . . . . . . . . . . 11  |-  ( p  =  b  ->  (
(deg `  p )  =  (deg `  b )  <->  (deg
`  b )  =  (deg `  b )
) )
19 fveq1 5524 . . . . . . . . . . . 12  |-  ( p  =  b  ->  (
p `  A )  =  ( b `  A ) )
2019eqeq1d 2291 . . . . . . . . . . 11  |-  ( p  =  b  ->  (
( p `  A
)  =  0  <->  (
b `  A )  =  0 ) )
2118, 20anbi12d 691 . . . . . . . . . 10  |-  ( p  =  b  ->  (
( (deg `  p
)  =  (deg `  b )  /\  (
p `  A )  =  0 )  <->  ( (deg `  b )  =  (deg
`  b )  /\  ( b `  A
)  =  0 ) ) )
2216, 21rspc2ev 2892 . . . . . . . . 9  |-  ( ( (deg `  b )  e.  NN  /\  b  e.  ( (Poly `  QQ )  \  { 0 p } )  /\  (
(deg `  b )  =  (deg `  b )  /\  ( b `  A
)  =  0 ) )  ->  E. a  e.  NN  E. p  e.  ( (Poly `  QQ )  \  { 0 p } ) ( (deg
`  p )  =  a  /\  ( p `
 A )  =  0 ) )
2311, 12, 14, 22syl3anc 1182 . . . . . . . 8  |-  ( ( ( b  e.  ( (Poly `  QQ )  \  { 0 p }
)  /\  ( b `  A )  =  0 )  /\  A  e.  CC )  ->  E. a  e.  NN  E. p  e.  ( (Poly `  QQ )  \  { 0 p } ) ( (deg
`  p )  =  a  /\  ( p `
 A )  =  0 ) )
2423ex 423 . . . . . . 7  |-  ( ( b  e.  ( (Poly `  QQ )  \  {
0 p } )  /\  ( b `  A )  =  0 )  ->  ( A  e.  CC  ->  E. a  e.  NN  E. p  e.  ( (Poly `  QQ )  \  { 0 p } ) ( (deg
`  p )  =  a  /\  ( p `
 A )  =  0 ) ) )
2524rexlimiva 2662 . . . . . 6  |-  ( E. b  e.  ( (Poly `  QQ )  \  {
0 p } ) ( b `  A
)  =  0  -> 
( A  e.  CC  ->  E. a  e.  NN  E. p  e.  ( (Poly `  QQ )  \  {
0 p } ) ( (deg `  p
)  =  a  /\  ( p `  A
)  =  0 ) ) )
2625impcom 419 . . . . 5  |-  ( ( A  e.  CC  /\  E. b  e.  ( (Poly `  QQ )  \  {
0 p } ) ( b `  A
)  =  0 )  ->  E. a  e.  NN  E. p  e.  ( (Poly `  QQ )  \  {
0 p } ) ( (deg `  p
)  =  a  /\  ( p `  A
)  =  0 ) )
27 elqaa 19702 . . . . 5  |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. b  e.  ( (Poly `  QQ )  \  { 0 p } ) ( b `
 A )  =  0 ) )
28 rabn0 3474 . . . . 5  |-  ( { a  e.  NN  |  E. p  e.  (
(Poly `  QQ )  \  { 0 p }
) ( (deg `  p )  =  a  /\  ( p `  A )  =  0 ) }  =/=  (/)  <->  E. a  e.  NN  E. p  e.  ( (Poly `  QQ )  \  { 0 p } ) ( (deg
`  p )  =  a  /\  ( p `
 A )  =  0 ) )
2926, 27, 283imtr4i 257 . . . 4  |-  ( A  e.  AA  ->  { a  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  {
0 p } ) ( (deg `  p
)  =  a  /\  ( p `  A
)  =  0 ) }  =/=  (/) )
30 infmssuzcl 10301 . . . 4  |-  ( ( { a  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0 p }
) ( (deg `  p )  =  a  /\  ( p `  A )  =  0 ) }  C_  ( ZZ>=
`  1 )  /\  { a  e.  NN  |  E. p  e.  (
(Poly `  QQ )  \  { 0 p }
) ( (deg `  p )  =  a  /\  ( p `  A )  =  0 ) }  =/=  (/) )  ->  sup ( { a  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0 p } ) ( (deg
`  p )  =  a  /\  ( p `
 A )  =  0 ) } ,  RR ,  `'  <  )  e.  { a  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0 p } ) ( (deg
`  p )  =  a  /\  ( p `
 A )  =  0 ) } )
314, 29, 30sylancr 644 . . 3  |-  ( A  e.  AA  ->  sup ( { a  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0 p }
) ( (deg `  p )  =  a  /\  ( p `  A )  =  0 ) } ,  RR ,  `'  <  )  e. 
{ a  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  { 0 p }
) ( (deg `  p )  =  a  /\  ( p `  A )  =  0 ) } )
321, 31eqeltrd 2357 . 2  |-  ( A  e.  AA  ->  (degAA `  A )  e.  {
a  e.  NN  |  E. p  e.  (
(Poly `  QQ )  \  { 0 p }
) ( (deg `  p )  =  a  /\  ( p `  A )  =  0 ) } )
33 eqeq2 2292 . . . . 5  |-  ( a  =  (degAA `  A )  -> 
( (deg `  p
)  =  a  <->  (deg `  p
)  =  (degAA `  A
) ) )
3433anbi1d 685 . . . 4  |-  ( a  =  (degAA `  A )  -> 
( ( (deg `  p )  =  a  /\  ( p `  A )  =  0 )  <->  ( (deg `  p )  =  (degAA `  A )  /\  (
p `  A )  =  0 ) ) )
3534rexbidv 2564 . . 3  |-  ( a  =  (degAA `  A )  -> 
( E. p  e.  ( (Poly `  QQ )  \  { 0 p } ) ( (deg
`  p )  =  a  /\  ( p `
 A )  =  0 )  <->  E. p  e.  ( (Poly `  QQ )  \  { 0 p } ) ( (deg
`  p )  =  (degAA `  A )  /\  ( p `  A
)  =  0 ) ) )
3635elrab 2923 . 2  |-  ( (degAA `  A )  e.  {
a  e.  NN  |  E. p  e.  (
(Poly `  QQ )  \  { 0 p }
) ( (deg `  p )  =  a  /\  ( p `  A )  =  0 ) }  <->  ( (degAA `  A )  e.  NN  /\ 
E. p  e.  ( (Poly `  QQ )  \  { 0 p }
) ( (deg `  p )  =  (degAA `  A )  /\  (
p `  A )  =  0 ) ) )
3732, 36sylib 188 1  |-  ( A  e.  AA  ->  (
(degAA `
 A )  e.  NN  /\  E. p  e.  ( (Poly `  QQ )  \  { 0 p } ) ( (deg
`  p )  =  (degAA `  A )  /\  ( p `  A
)  =  0 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   {crab 2547    \ cdif 3149    C_ wss 3152   (/)c0 3455   {csn 3640   `'ccnv 4688   ` cfv 5255   supcsup 7193   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    < clt 8867   NNcn 9746   ZZ>=cuz 10230   QQcq 10316   0 pc0p 19024  Polycply 19566  degcdgr 19569   AAcaa 19694  degAAcdgraa 27345
This theorem is referenced by:  dgraacl  27351  mpaaeu  27355
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816
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-reu 2550  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-0p 19025  df-ply 19570  df-coe 19572  df-dgr 19573  df-aa 19695  df-dgraa 27347
  Copyright terms: Public domain W3C validator