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Theorem dgrsub 21738
Description: The degree of a difference of polynomials is at most the maximum of the degrees. (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypotheses
Ref Expression
dgrsub.1  |-  M  =  (deg `  F )
dgrsub.2  |-  N  =  (deg `  G )
Assertion
Ref Expression
dgrsub  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  ( F  oF  -  G
) )  <_  if ( M  <_  N ,  N ,  M )
)

Proof of Theorem dgrsub
StepHypRef Expression
1 plyssc 21667 . . . 4  |-  (Poly `  S )  C_  (Poly `  CC )
21sseli 3351 . . 3  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
3 ssid 3374 . . . . 5  |-  CC  C_  CC
4 neg1cn 10424 . . . . 5  |-  -u 1  e.  CC
5 plyconst 21673 . . . . 5  |-  ( ( CC  C_  CC  /\  -u 1  e.  CC )  ->  ( CC  X.  { -u 1 } )  e.  (Poly `  CC ) )
63, 4, 5mp2an 672 . . . 4  |-  ( CC 
X.  { -u 1 } )  e.  (Poly `  CC )
71sseli 3351 . . . 4  |-  ( G  e.  (Poly `  S
)  ->  G  e.  (Poly `  CC ) )
8 plymulcl 21688 . . . 4  |-  ( ( ( CC  X.  { -u 1 } )  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC ) )  -> 
( ( CC  X.  { -u 1 } )  oF  x.  G
)  e.  (Poly `  CC ) )
96, 7, 8sylancr 663 . . 3  |-  ( G  e.  (Poly `  S
)  ->  ( ( CC  X.  { -u 1 } )  oF  x.  G )  e.  (Poly `  CC )
)
10 dgrsub.1 . . . 4  |-  M  =  (deg `  F )
11 eqid 2442 . . . 4  |-  (deg `  ( ( CC  X.  { -u 1 } )  oF  x.  G
) )  =  (deg
`  ( ( CC 
X.  { -u 1 } )  oF  x.  G ) )
1210, 11dgradd 21733 . . 3  |-  ( ( F  e.  (Poly `  CC )  /\  (
( CC  X.  { -u 1 } )  oF  x.  G )  e.  (Poly `  CC ) )  ->  (deg `  ( F  oF  +  ( ( CC 
X.  { -u 1 } )  oF  x.  G ) ) )  <_  if ( M  <_  (deg `  (
( CC  X.  { -u 1 } )  oF  x.  G ) ) ,  (deg `  ( ( CC  X.  { -u 1 } )  oF  x.  G
) ) ,  M
) )
132, 9, 12syl2an 477 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  ( F  oF  +  ( ( CC  X.  { -u 1 } )  oF  x.  G ) ) )  <_  if ( M  <_  (deg `  ( ( CC  X.  { -u 1 } )  oF  x.  G
) ) ,  (deg
`  ( ( CC 
X.  { -u 1 } )  oF  x.  G ) ) ,  M ) )
14 plyf 21665 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
15 plyf 21665 . . . 4  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
16 cnex 9362 . . . . 5  |-  CC  e.  _V
17 ofnegsub 10319 . . . . 5  |-  ( ( CC  e.  _V  /\  F : CC --> CC  /\  G : CC --> CC )  ->  ( F  oF  +  ( ( CC  X.  { -u 1 } )  oF  x.  G ) )  =  ( F  oF  -  G )
)
1816, 17mp3an1 1301 . . . 4  |-  ( ( F : CC --> CC  /\  G : CC --> CC )  ->  ( F  oF  +  ( ( CC  X.  { -u 1 } )  oF  x.  G ) )  =  ( F  oF  -  G )
)
1914, 15, 18syl2an 477 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  oF  +  (
( CC  X.  { -u 1 } )  oF  x.  G ) )  =  ( F  oF  -  G
) )
2019fveq2d 5694 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  ( F  oF  +  ( ( CC  X.  { -u 1 } )  oF  x.  G ) ) )  =  (deg
`  ( F  oF  -  G )
) )
21 neg1ne0 10426 . . . . . . 7  |-  -u 1  =/=  0
22 dgrmulc 21737 . . . . . . 7  |-  ( (
-u 1  e.  CC  /\  -u 1  =/=  0  /\  G  e.  (Poly `  S ) )  -> 
(deg `  ( ( CC  X.  { -u 1 } )  oF  x.  G ) )  =  (deg `  G
) )
234, 21, 22mp3an12 1304 . . . . . 6  |-  ( G  e.  (Poly `  S
)  ->  (deg `  (
( CC  X.  { -u 1 } )  oF  x.  G ) )  =  (deg `  G ) )
24 dgrsub.2 . . . . . 6  |-  N  =  (deg `  G )
2523, 24syl6eqr 2492 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  (deg `  (
( CC  X.  { -u 1 } )  oF  x.  G ) )  =  N )
2625adantl 466 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  (
( CC  X.  { -u 1 } )  oF  x.  G ) )  =  N )
2726breq2d 4303 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( M  <_  (deg `  ( ( CC  X.  { -u 1 } )  oF  x.  G ) )  <-> 
M  <_  N )
)
2827, 26ifbieq1d 3811 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  if ( M  <_  (deg `  (
( CC  X.  { -u 1 } )  oF  x.  G ) ) ,  (deg `  ( ( CC  X.  { -u 1 } )  oF  x.  G
) ) ,  M
)  =  if ( M  <_  N ,  N ,  M )
)
2913, 20, 283brtr3d 4320 1  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  ( F  oF  -  G
) )  <_  if ( M  <_  N ,  N ,  M )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2605   _Vcvv 2971    C_ wss 3327   ifcif 3790   {csn 3876   class class class wbr 4291    X. cxp 4837   -->wf 5413   ` cfv 5417  (class class class)co 6090    oFcof 6317   CCcc 9279   0cc0 9281   1c1 9282    + caddc 9284    x. cmul 9286    <_ cle 9418    - cmin 9594   -ucneg 9595  Polycply 21651  degcdgr 21654
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 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359  ax-addf 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-map 7215  df-pm 7216  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-sup 7690  df-oi 7723  df-card 8108  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-n0 10579  df-z 10646  df-uz 10861  df-rp 10991  df-fz 11437  df-fzo 11548  df-fl 11641  df-seq 11806  df-exp 11865  df-hash 12103  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-clim 12965  df-rlim 12966  df-sum 13163  df-0p 21147  df-ply 21655  df-coe 21657  df-dgr 21658
This theorem is referenced by:  dgrcolem2  21740  plydivlem4  21761  plydiveu  21763  dgrsub2  29489
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