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Theorem dgraaub 27023
Description: Upper bound on degree of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
Assertion
Ref Expression
dgraaub  |-  ( ( ( P  e.  (Poly `  QQ )  /\  P  =/=  0 p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  (degAA `  A
)  <_  (deg `  P
) )

Proof of Theorem dgraaub
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprl 733 . . . 4  |-  ( ( ( P  e.  (Poly `  QQ )  /\  P  =/=  0 p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  A  e.  CC )
2 eldifsn 3871 . . . . . . 7  |-  ( P  e.  ( (Poly `  QQ )  \  { 0 p } )  <->  ( P  e.  (Poly `  QQ )  /\  P  =/=  0 p ) )
32biimpri 198 . . . . . 6  |-  ( ( P  e.  (Poly `  QQ )  /\  P  =/=  0 p )  ->  P  e.  ( (Poly `  QQ )  \  {
0 p } ) )
43adantr 452 . . . . 5  |-  ( ( ( P  e.  (Poly `  QQ )  /\  P  =/=  0 p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  P  e.  ( (Poly `  QQ )  \  { 0 p }
) )
5 simprr 734 . . . . 5  |-  ( ( ( P  e.  (Poly `  QQ )  /\  P  =/=  0 p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  ( P `  A )  =  0 )
6 fveq1 5668 . . . . . . 7  |-  ( a  =  P  ->  (
a `  A )  =  ( P `  A ) )
76eqeq1d 2396 . . . . . 6  |-  ( a  =  P  ->  (
( a `  A
)  =  0  <->  ( P `  A )  =  0 ) )
87rspcev 2996 . . . . 5  |-  ( ( P  e.  ( (Poly `  QQ )  \  {
0 p } )  /\  ( P `  A )  =  0 )  ->  E. a  e.  ( (Poly `  QQ )  \  { 0 p } ) ( a `
 A )  =  0 )
94, 5, 8syl2anc 643 . . . 4  |-  ( ( ( P  e.  (Poly `  QQ )  /\  P  =/=  0 p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  E. a  e.  ( (Poly `  QQ )  \  { 0 p } ) ( a `
 A )  =  0 )
10 elqaa 20107 . . . 4  |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. a  e.  ( (Poly `  QQ )  \  { 0 p } ) ( a `
 A )  =  0 ) )
111, 9, 10sylanbrc 646 . . 3  |-  ( ( ( P  e.  (Poly `  QQ )  /\  P  =/=  0 p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  A  e.  AA )
12 dgraaval 27019 . . 3  |-  ( A  e.  AA  ->  (degAA `  A )  =  sup ( { a  e.  NN  |  E. b  e.  ( (Poly `  QQ )  \  { 0 p }
) ( (deg `  b )  =  a  /\  ( b `  A )  =  0 ) } ,  RR ,  `'  <  ) )
1311, 12syl 16 . 2  |-  ( ( ( P  e.  (Poly `  QQ )  /\  P  =/=  0 p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  (degAA `  A
)  =  sup ( { a  e.  NN  |  E. b  e.  ( (Poly `  QQ )  \  { 0 p }
) ( (deg `  b )  =  a  /\  ( b `  A )  =  0 ) } ,  RR ,  `'  <  ) )
14 ssrab2 3372 . . . 4  |-  { a  e.  NN  |  E. b  e.  ( (Poly `  QQ )  \  {
0 p } ) ( (deg `  b
)  =  a  /\  ( b `  A
)  =  0 ) }  C_  NN
15 nnuz 10454 . . . 4  |-  NN  =  ( ZZ>= `  1 )
1614, 15sseqtri 3324 . . 3  |-  { a  e.  NN  |  E. b  e.  ( (Poly `  QQ )  \  {
0 p } ) ( (deg `  b
)  =  a  /\  ( b `  A
)  =  0 ) }  C_  ( ZZ>= ` 
1 )
17 dgrnznn 27010 . . . 4  |-  ( ( ( P  e.  (Poly `  QQ )  /\  P  =/=  0 p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  (deg `  P
)  e.  NN )
18 eqid 2388 . . . . . 6  |-  (deg `  P )  =  (deg
`  P )
195, 18jctil 524 . . . . 5  |-  ( ( ( P  e.  (Poly `  QQ )  /\  P  =/=  0 p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  ( (deg `  P )  =  (deg
`  P )  /\  ( P `  A )  =  0 ) )
20 fveq2 5669 . . . . . . . 8  |-  ( b  =  P  ->  (deg `  b )  =  (deg
`  P ) )
2120eqeq1d 2396 . . . . . . 7  |-  ( b  =  P  ->  (
(deg `  b )  =  (deg `  P )  <->  (deg
`  P )  =  (deg `  P )
) )
22 fveq1 5668 . . . . . . . 8  |-  ( b  =  P  ->  (
b `  A )  =  ( P `  A ) )
2322eqeq1d 2396 . . . . . . 7  |-  ( b  =  P  ->  (
( b `  A
)  =  0  <->  ( P `  A )  =  0 ) )
2421, 23anbi12d 692 . . . . . 6  |-  ( b  =  P  ->  (
( (deg `  b
)  =  (deg `  P )  /\  (
b `  A )  =  0 )  <->  ( (deg `  P )  =  (deg
`  P )  /\  ( P `  A )  =  0 ) ) )
2524rspcev 2996 . . . . 5  |-  ( ( P  e.  ( (Poly `  QQ )  \  {
0 p } )  /\  ( (deg `  P )  =  (deg
`  P )  /\  ( P `  A )  =  0 ) )  ->  E. b  e.  ( (Poly `  QQ )  \  { 0 p }
) ( (deg `  b )  =  (deg
`  P )  /\  ( b `  A
)  =  0 ) )
264, 19, 25syl2anc 643 . . . 4  |-  ( ( ( P  e.  (Poly `  QQ )  /\  P  =/=  0 p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  E. b  e.  ( (Poly `  QQ )  \  { 0 p } ) ( (deg
`  b )  =  (deg `  P )  /\  ( b `  A
)  =  0 ) )
27 eqeq2 2397 . . . . . . 7  |-  ( a  =  (deg `  P
)  ->  ( (deg `  b )  =  a  <-> 
(deg `  b )  =  (deg `  P )
) )
2827anbi1d 686 . . . . . 6  |-  ( a  =  (deg `  P
)  ->  ( (
(deg `  b )  =  a  /\  (
b `  A )  =  0 )  <->  ( (deg `  b )  =  (deg
`  P )  /\  ( b `  A
)  =  0 ) ) )
2928rexbidv 2671 . . . . 5  |-  ( a  =  (deg `  P
)  ->  ( E. b  e.  ( (Poly `  QQ )  \  {
0 p } ) ( (deg `  b
)  =  a  /\  ( b `  A
)  =  0 )  <->  E. b  e.  (
(Poly `  QQ )  \  { 0 p }
) ( (deg `  b )  =  (deg
`  P )  /\  ( b `  A
)  =  0 ) ) )
3029elrab 3036 . . . 4  |-  ( (deg
`  P )  e. 
{ a  e.  NN  |  E. b  e.  ( (Poly `  QQ )  \  { 0 p }
) ( (deg `  b )  =  a  /\  ( b `  A )  =  0 ) }  <->  ( (deg `  P )  e.  NN  /\ 
E. b  e.  ( (Poly `  QQ )  \  { 0 p }
) ( (deg `  b )  =  (deg
`  P )  /\  ( b `  A
)  =  0 ) ) )
3117, 26, 30sylanbrc 646 . . 3  |-  ( ( ( P  e.  (Poly `  QQ )  /\  P  =/=  0 p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  (deg `  P
)  e.  { a  e.  NN  |  E. b  e.  ( (Poly `  QQ )  \  {
0 p } ) ( (deg `  b
)  =  a  /\  ( b `  A
)  =  0 ) } )
32 infmssuzle 10491 . . 3  |-  ( ( { a  e.  NN  |  E. b  e.  ( (Poly `  QQ )  \  { 0 p }
) ( (deg `  b )  =  a  /\  ( b `  A )  =  0 ) }  C_  ( ZZ>=
`  1 )  /\  (deg `  P )  e. 
{ a  e.  NN  |  E. b  e.  ( (Poly `  QQ )  \  { 0 p }
) ( (deg `  b )  =  a  /\  ( b `  A )  =  0 ) } )  ->  sup ( { a  e.  NN  |  E. b  e.  ( (Poly `  QQ )  \  { 0 p } ) ( (deg
`  b )  =  a  /\  ( b `
 A )  =  0 ) } ,  RR ,  `'  <  )  <_  (deg `  P
) )
3316, 31, 32sylancr 645 . 2  |-  ( ( ( P  e.  (Poly `  QQ )  /\  P  =/=  0 p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  sup ( { a  e.  NN  |  E. b  e.  ( (Poly `  QQ )  \  { 0 p }
) ( (deg `  b )  =  a  /\  ( b `  A )  =  0 ) } ,  RR ,  `'  <  )  <_ 
(deg `  P )
)
3413, 33eqbrtrd 4174 1  |-  ( ( ( P  e.  (Poly `  QQ )  /\  P  =/=  0 p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  (degAA `  A
)  <_  (deg `  P
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2551   E.wrex 2651   {crab 2654    \ cdif 3261    C_ wss 3264   {csn 3758   class class class wbr 4154   `'ccnv 4818   ` cfv 5395   supcsup 7381   CCcc 8922   RRcr 8923   0cc0 8924   1c1 8925    < clt 9054    <_ cle 9055   NNcn 9933   ZZ>=cuz 10421   QQcq 10507   0 pc0p 19429  Polycply 19971  degcdgr 19974   AAcaa 20099  degAAcdgraa 27015
This theorem is referenced by:  dgraa0p  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 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002  ax-addf 9003
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-map 6957  df-pm 6958  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-sup 7382  df-oi 7413  df-card 7760  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-q 10508  df-rp 10546  df-fz 10977  df-fzo 11067  df-fl 11130  df-mod 11179  df-seq 11252  df-exp 11311  df-hash 11547  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-clim 12210  df-rlim 12211  df-sum 12408  df-0p 19430  df-ply 19975  df-coe 19977  df-dgr 19978  df-aa 20100  df-dgraa 27017
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