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Theorem dgrval 19610
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 19582 . . 3  |-  (Poly `  S )  C_  (Poly `  CC )
21sseli 3176 . 2  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
3 fveq2 5525 . . . . . . 7  |-  ( f  =  F  ->  (coeff `  f )  =  (coeff `  F ) )
4 dgrval.1 . . . . . . 7  |-  A  =  (coeff `  F )
53, 4syl6eqr 2333 . . . . . 6  |-  ( f  =  F  ->  (coeff `  f )  =  A )
65cnveqd 4857 . . . . 5  |-  ( f  =  F  ->  `' (coeff `  f )  =  `' A )
76imaeq1d 5011 . . . 4  |-  ( f  =  F  ->  ( `' (coeff `  f ) " ( CC  \  { 0 } ) )  =  ( `' A " ( CC 
\  { 0 } ) ) )
87supeq1d 7199 . . 3  |-  ( f  =  F  ->  sup ( ( `' (coeff `  f ) " ( CC  \  { 0 } ) ) ,  NN0 ,  <  )  =  sup ( ( `' A " ( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
9 df-dgr 19573 . . 3  |- deg  =  ( f  e.  (Poly `  CC )  |->  sup (
( `' (coeff `  f ) " ( CC  \  { 0 } ) ) ,  NN0 ,  <  ) )
10 nn0ssre 9969 . . . . 5  |-  NN0  C_  RR
11 ltso 8903 . . . . 5  |-  <  Or  RR
12 soss 4332 . . . . 5  |-  ( NN0  C_  RR  ->  (  <  Or  RR  ->  <  Or  NN0 ) )
1310, 11, 12mp2 17 . . . 4  |-  <  Or  NN0
1413supex 7214 . . 3  |-  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  )  e.  _V
158, 9, 14fvmpt 5602 . 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 1623    e. wcel 1684    \ cdif 3149    C_ wss 3152   {csn 3640    Or wor 4313   `'ccnv 4688   "cima 4692   ` cfv 5255   supcsup 7193   CCcc 8735   RRcr 8736   0cc0 8737    < clt 8867   NN0cn0 9965  Polycply 19566  coeffccoe 19568  degcdgr 19569
This theorem is referenced by:  dgrcl  19615  dgrub  19616  dgrlb  19618  coe11  19634
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-i2m1 8805  ax-1ne0 8806  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-nn 9747  df-n0 9966  df-ply 19570  df-dgr 19573
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