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Theorem dgrval 23050
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 23022 . . 3  |-  (Poly `  S )  C_  (Poly `  CC )
21sseli 3466 . 2  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
3 fveq2 5881 . . . . . . 7  |-  ( f  =  F  ->  (coeff `  f )  =  (coeff `  F ) )
4 dgrval.1 . . . . . . 7  |-  A  =  (coeff `  F )
53, 4syl6eqr 2488 . . . . . 6  |-  ( f  =  F  ->  (coeff `  f )  =  A )
65cnveqd 5030 . . . . 5  |-  ( f  =  F  ->  `' (coeff `  f )  =  `' A )
76imaeq1d 5187 . . . 4  |-  ( f  =  F  ->  ( `' (coeff `  f ) " ( CC  \  { 0 } ) )  =  ( `' A " ( CC 
\  { 0 } ) ) )
87supeq1d 7966 . . 3  |-  ( f  =  F  ->  sup ( ( `' (coeff `  f ) " ( CC  \  { 0 } ) ) ,  NN0 ,  <  )  =  sup ( ( `' A " ( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
9 df-dgr 23013 . . 3  |- deg  =  ( f  e.  (Poly `  CC )  |->  sup (
( `' (coeff `  f ) " ( CC  \  { 0 } ) ) ,  NN0 ,  <  ) )
10 nn0ssre 10873 . . . . 5  |-  NN0  C_  RR
11 ltso 9713 . . . . 5  |-  <  Or  RR
12 soss 4793 . . . . 5  |-  ( NN0  C_  RR  ->  (  <  Or  RR  ->  <  Or  NN0 ) )
1310, 11, 12mp2 9 . . . 4  |-  <  Or  NN0
1413supex 7983 . . 3  |-  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  )  e.  _V
158, 9, 14fvmpt 5964 . 2  |-  ( F  e.  (Poly `  CC )  ->  (deg `  F
)  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
162, 15syl 17 1  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  =  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1870    \ cdif 3439    C_ wss 3442   {csn 4002    Or wor 4774   `'ccnv 4853   "cima 4857   ` cfv 5601   supcsup 7960   CCcc 9536   RRcr 9537   0cc0 9538    < clt 9674   NN0cn0 10869  Polycply 23006  coeffccoe 23008  degcdgr 23009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-i2m1 9606  ax-1ne0 9607  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-sup 7962  df-pnf 9676  df-mnf 9677  df-ltxr 9679  df-nn 10610  df-n0 10870  df-ply 23010  df-dgr 23013
This theorem is referenced by:  dgrcl  23055  dgrub  23056  dgrlb  23058  coe11  23075
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