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Theorem dgraalemOLD 36079
Description: Properties of the degree of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) Obsolete version of dgraalem 36078 as of 29-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
dgraalemOLD  |-  ( 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 dgraalemOLD
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dgraavalOLD 36077 . . 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 3500 . . . . 5  |-  { a  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  {
0p } ) ( (deg `  p
)  =  a  /\  ( p `  A
)  =  0 ) }  C_  NN
3 nnuz 11218 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
42, 3sseqtri 3450 . . . 4  |-  { a  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  {
0p } ) ( (deg `  p
)  =  a  /\  ( p `  A
)  =  0 ) }  C_  ( ZZ>= ` 
1 )
5 eldifsn 4088 . . . . . . . . . . . 12  |-  ( b  e.  ( (Poly `  QQ )  \  { 0p } )  <->  ( b  e.  (Poly `  QQ )  /\  b  =/=  0p ) )
65biimpi 199 . . . . . . . . . . 11  |-  ( b  e.  ( (Poly `  QQ )  \  { 0p } )  -> 
( b  e.  (Poly `  QQ )  /\  b  =/=  0p ) )
76ad2antrr 740 . . . . . . . . . 10  |-  ( ( ( b  e.  ( (Poly `  QQ )  \  { 0p }
)  /\  ( b `  A )  =  0 )  /\  A  e.  CC )  ->  (
b  e.  (Poly `  QQ )  /\  b  =/=  0p ) )
8 simpr 468 . . . . . . . . . 10  |-  ( ( ( b  e.  ( (Poly `  QQ )  \  { 0p }
)  /\  ( b `  A )  =  0 )  /\  A  e.  CC )  ->  A  e.  CC )
9 simplr 770 . . . . . . . . . 10  |-  ( ( ( b  e.  ( (Poly `  QQ )  \  { 0p }
)  /\  ( b `  A )  =  0 )  /\  A  e.  CC )  ->  (
b `  A )  =  0 )
10 dgrnznn 23280 . . . . . . . . . 10  |-  ( ( ( b  e.  (Poly `  QQ )  /\  b  =/=  0p )  /\  ( A  e.  CC  /\  ( b `  A
)  =  0 ) )  ->  (deg `  b
)  e.  NN )
117, 8, 9, 10syl12anc 1290 . . . . . . . . 9  |-  ( ( ( b  e.  ( (Poly `  QQ )  \  { 0p }
)  /\  ( b `  A )  =  0 )  /\  A  e.  CC )  ->  (deg `  b )  e.  NN )
12 simpll 768 . . . . . . . . 9  |-  ( ( ( b  e.  ( (Poly `  QQ )  \  { 0p }
)  /\  ( b `  A )  =  0 )  /\  A  e.  CC )  ->  b  e.  ( (Poly `  QQ )  \  { 0p } ) )
13 eqid 2471 . . . . . . . . . 10  |-  (deg `  b )  =  (deg
`  b )
149, 13jctil 546 . . . . . . . . 9  |-  ( ( ( b  e.  ( (Poly `  QQ )  \  { 0p }
)  /\  ( b `  A )  =  0 )  /\  A  e.  CC )  ->  (
(deg `  b )  =  (deg `  b )  /\  ( b `  A
)  =  0 ) )
15 eqeq2 2482 . . . . . . . . . . 11  |-  ( a  =  (deg `  b
)  ->  ( (deg `  p )  =  a  <-> 
(deg `  p )  =  (deg `  b )
) )
1615anbi1d 719 . . . . . . . . . 10  |-  ( a  =  (deg `  b
)  ->  ( (
(deg `  p )  =  a  /\  (
p `  A )  =  0 )  <->  ( (deg `  p )  =  (deg
`  b )  /\  ( p `  A
)  =  0 ) ) )
17 fveq2 5879 . . . . . . . . . . . 12  |-  ( p  =  b  ->  (deg `  p )  =  (deg
`  b ) )
1817eqeq1d 2473 . . . . . . . . . . 11  |-  ( p  =  b  ->  (
(deg `  p )  =  (deg `  b )  <->  (deg
`  b )  =  (deg `  b )
) )
19 fveq1 5878 . . . . . . . . . . . 12  |-  ( p  =  b  ->  (
p `  A )  =  ( b `  A ) )
2019eqeq1d 2473 . . . . . . . . . . 11  |-  ( p  =  b  ->  (
( p `  A
)  =  0  <->  (
b `  A )  =  0 ) )
2118, 20anbi12d 725 . . . . . . . . . 10  |-  ( p  =  b  ->  (
( (deg `  p
)  =  (deg `  b )  /\  (
p `  A )  =  0 )  <->  ( (deg `  b )  =  (deg
`  b )  /\  ( b `  A
)  =  0 ) ) )
2216, 21rspc2ev 3149 . . . . . . . . 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 1292 . . . . . . . 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 441 . . . . . . 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 2868 . . . . . 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 437 . . . . 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 23357 . . . . 5  |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. b  e.  ( (Poly `  QQ )  \  { 0p } ) ( b `
 A )  =  0 ) )
28 rabn0 3755 . . . . 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 274 . . . 4  |-  ( A  e.  AA  ->  { a  e.  NN  |  E. p  e.  ( (Poly `  QQ )  \  {
0p } ) ( (deg `  p
)  =  a  /\  ( p `  A
)  =  0 ) }  =/=  (/) )
30 infmssuzclOLD 11270 . . . 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 676 . . 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 2549 . 2  |-  ( A  e.  AA  ->  (degAA `  A )  e.  {
a  e.  NN  |  E. p  e.  (
(Poly `  QQ )  \  { 0p }
) ( (deg `  p )  =  a  /\  ( p `  A )  =  0 ) } )
33 eqeq2 2482 . . . . 5  |-  ( a  =  (degAA `  A )  -> 
( (deg `  p
)  =  a  <->  (deg `  p
)  =  (degAA `  A
) ) )
3433anbi1d 719 . . . 4  |-  ( a  =  (degAA `  A )  -> 
( ( (deg `  p )  =  a  /\  ( p `  A )  =  0 )  <->  ( (deg `  p )  =  (degAA `  A )  /\  (
p `  A )  =  0 ) ) )
3534rexbidv 2892 . . 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 3184 . 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 201 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 376    = wceq 1452    e. wcel 1904    =/= wne 2641   E.wrex 2757   {crab 2760    \ cdif 3387    C_ wss 3390   (/)c0 3722   {csn 3959   `'ccnv 4838   ` cfv 5589   supcsup 7972   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    < clt 9693   NNcn 10631   ZZ>=cuz 11182   QQcq 11287   0pc0p 22706  Polycply 23217  degcdgr 23220   AAcaa 23346  degAAcdgraaold 36071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-rlim 13630  df-sum 13830  df-0p 22707  df-ply 23221  df-coe 23223  df-dgr 23224  df-aa 23347  df-dgraaOLD 36074
This theorem is referenced by:  dgraaclOLD  36081
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