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Theorem dgrval 20152
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 20124 . . 3  |-  (Poly `  S )  C_  (Poly `  CC )
21sseli 3346 . 2  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
3 fveq2 5731 . . . . . . 7  |-  ( f  =  F  ->  (coeff `  f )  =  (coeff `  F ) )
4 dgrval.1 . . . . . . 7  |-  A  =  (coeff `  F )
53, 4syl6eqr 2488 . . . . . 6  |-  ( f  =  F  ->  (coeff `  f )  =  A )
65cnveqd 5051 . . . . 5  |-  ( f  =  F  ->  `' (coeff `  f )  =  `' A )
76imaeq1d 5205 . . . 4  |-  ( f  =  F  ->  ( `' (coeff `  f ) " ( CC  \  { 0 } ) )  =  ( `' A " ( CC 
\  { 0 } ) ) )
87supeq1d 7454 . . 3  |-  ( f  =  F  ->  sup ( ( `' (coeff `  f ) " ( CC  \  { 0 } ) ) ,  NN0 ,  <  )  =  sup ( ( `' A " ( CC  \  {
0 } ) ) ,  NN0 ,  <  ) )
9 df-dgr 20115 . . 3  |- deg  =  ( f  e.  (Poly `  CC )  |->  sup (
( `' (coeff `  f ) " ( CC  \  { 0 } ) ) ,  NN0 ,  <  ) )
10 nn0ssre 10230 . . . . 5  |-  NN0  C_  RR
11 ltso 9161 . . . . 5  |-  <  Or  RR
12 soss 4524 . . . . 5  |-  ( NN0  C_  RR  ->  (  <  Or  RR  ->  <  Or  NN0 ) )
1310, 11, 12mp2 9 . . . 4  |-  <  Or  NN0
1413supex 7471 . . 3  |-  sup (
( `' A "
( CC  \  {
0 } ) ) ,  NN0 ,  <  )  e.  _V
158, 9, 14fvmpt 5809 . 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 set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    \ cdif 3319    C_ wss 3322   {csn 3816    Or wor 4505   `'ccnv 4880   "cima 4884   ` cfv 5457   supcsup 7448   CCcc 8993   RRcr 8994   0cc0 8995    < clt 9125   NN0cn0 10226  Polycply 20108  coeffccoe 20110  degcdgr 20111
This theorem is referenced by:  dgrcl  20157  dgrub  20158  dgrlb  20160  coe11  20176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-i2m1 9063  ax-1ne0 9064  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-recs 6636  df-rdg 6671  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-sup 7449  df-pnf 9127  df-mnf 9128  df-ltxr 9130  df-nn 10006  df-n0 10227  df-ply 20112  df-dgr 20115
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