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Theorem dgrval 19708
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 19680 . . 3  |-  (Poly `  S )  C_  (Poly `  CC )
21sseli 3252 . 2  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
3 fveq2 5605 . . . . . . 7  |-  ( f  =  F  ->  (coeff `  f )  =  (coeff `  F ) )
4 dgrval.1 . . . . . . 7  |-  A  =  (coeff `  F )
53, 4syl6eqr 2408 . . . . . 6  |-  ( f  =  F  ->  (coeff `  f )  =  A )
65cnveqd 4936 . . . . 5  |-  ( f  =  F  ->  `' (coeff `  f )  =  `' A )
76imaeq1d 5090 . . . 4  |-  ( f  =  F  ->  ( `' (coeff `  f ) " ( CC  \  { 0 } ) )  =  ( `' A " ( CC 
\  { 0 } ) ) )
87supeq1d 7286 . . 3  |-  ( f  =  F  ->  sup ( ( `' (coeff `  f ) " ( CC  \  { 0 } ) ) ,  NN0 ,  <  )  =  sup ( ( `' A " ( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
9 df-dgr 19671 . . 3  |- deg  =  ( f  e.  (Poly `  CC )  |->  sup (
( `' (coeff `  f ) " ( CC  \  { 0 } ) ) ,  NN0 ,  <  ) )
10 nn0ssre 10058 . . . . 5  |-  NN0  C_  RR
11 ltso 8990 . . . . 5  |-  <  Or  RR
12 soss 4411 . . . . 5  |-  ( NN0  C_  RR  ->  (  <  Or  RR  ->  <  Or  NN0 ) )
1310, 11, 12mp2 17 . . . 4  |-  <  Or  NN0
1413supex 7301 . . 3  |-  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  )  e.  _V
158, 9, 14fvmpt 5682 . 2  |-  ( F  e.  (Poly `  CC )  ->  (deg `  F
)  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
162, 15syl 15 1  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642    e. wcel 1710    \ cdif 3225    C_ wss 3228   {csn 3716    Or wor 4392   `'ccnv 4767   "cima 4771   ` cfv 5334   supcsup 7280   CCcc 8822   RRcr 8823   0cc0 8824    < clt 8954   NN0cn0 10054  Polycply 19664  coeffccoe 19666  degcdgr 19667
This theorem is referenced by:  dgrcl  19713  dgrub  19714  dgrlb  19716  coe11  19732
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-i2m1 8892  ax-1ne0 8893  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-recs 6472  df-rdg 6507  df-er 6744  df-map 6859  df-en 6949  df-dom 6950  df-sdom 6951  df-sup 7281  df-pnf 8956  df-mnf 8957  df-ltxr 8959  df-nn 9834  df-n0 10055  df-ply 19668  df-dgr 19671
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