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Theorem dgrval 21701
Description: Value of the degree function. (Contributed by Mario Carneiro, 22-Jul-2014.)
Hypothesis
Ref Expression
dgrval.1  |-  A  =  (coeff `  F )
Assertion
Ref Expression
dgrval  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )

Proof of Theorem dgrval
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 plyssc 21673 . . 3  |-  (Poly `  S )  C_  (Poly `  CC )
21sseli 3357 . 2  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
3 fveq2 5696 . . . . . . 7  |-  ( f  =  F  ->  (coeff `  f )  =  (coeff `  F ) )
4 dgrval.1 . . . . . . 7  |-  A  =  (coeff `  F )
53, 4syl6eqr 2493 . . . . . 6  |-  ( f  =  F  ->  (coeff `  f )  =  A )
65cnveqd 5020 . . . . 5  |-  ( f  =  F  ->  `' (coeff `  f )  =  `' A )
76imaeq1d 5173 . . . 4  |-  ( f  =  F  ->  ( `' (coeff `  f ) " ( CC  \  { 0 } ) )  =  ( `' A " ( CC 
\  { 0 } ) ) )
87supeq1d 7701 . . 3  |-  ( f  =  F  ->  sup ( ( `' (coeff `  f ) " ( CC  \  { 0 } ) ) ,  NN0 ,  <  )  =  sup ( ( `' A " ( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
9 df-dgr 21664 . . 3  |- deg  =  ( f  e.  (Poly `  CC )  |->  sup (
( `' (coeff `  f ) " ( CC  \  { 0 } ) ) ,  NN0 ,  <  ) )
10 nn0ssre 10588 . . . . 5  |-  NN0  C_  RR
11 ltso 9460 . . . . 5  |-  <  Or  RR
12 soss 4664 . . . . 5  |-  ( NN0  C_  RR  ->  (  <  Or  RR  ->  <  Or  NN0 ) )
1310, 11, 12mp2 9 . . . 4  |-  <  Or  NN0
1413supex 7718 . . 3  |-  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  )  e.  _V
158, 9, 14fvmpt 5779 . 2  |-  ( F  e.  (Poly `  CC )  ->  (deg `  F
)  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
162, 15syl 16 1  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756    \ cdif 3330    C_ wss 3333   {csn 3882    Or wor 4645   `'ccnv 4844   "cima 4848   ` cfv 5423   supcsup 7695   CCcc 9285   RRcr 9286   0cc0 9287    < clt 9423   NN0cn0 10584  Polycply 21657  coeffccoe 21659  degcdgr 21660
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 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-i2m1 9355  ax-1ne0 9356  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  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 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-sup 7696  df-pnf 9425  df-mnf 9426  df-ltxr 9428  df-nn 10328  df-n0 10585  df-ply 21661  df-dgr 21664
This theorem is referenced by:  dgrcl  21706  dgrub  21707  dgrlb  21709  coe11  21725
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