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Theorem dgrsub 22398
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 22327 . . . 4  |-  (Poly `  S )  C_  (Poly `  CC )
21sseli 3495 . . 3  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
3 ssid 3518 . . . . 5  |-  CC  C_  CC
4 neg1cn 10630 . . . . 5  |-  -u 1  e.  CC
5 plyconst 22333 . . . . 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 3495 . . . 4  |-  ( G  e.  (Poly `  S
)  ->  G  e.  (Poly `  CC ) )
8 plymulcl 22348 . . . 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 2462 . . . 4  |-  (deg `  ( ( CC  X.  { -u 1 } )  oF  x.  G
) )  =  (deg
`  ( ( CC 
X.  { -u 1 } )  oF  x.  G ) )
1210, 11dgradd 22393 . . 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 22325 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
15 plyf 22325 . . . 4  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
16 cnex 9564 . . . . 5  |-  CC  e.  _V
17 ofnegsub 10525 . . . . 5  |-  ( ( CC  e.  _V  /\  F : CC --> CC  /\  G : CC --> CC )  ->  ( F  oF  +  ( ( CC  X.  { -u 1 } )  oF  x.  G ) )  =  ( F  oF  -  G )
)
1816, 17mp3an1 1306 . . . 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 5863 . 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 10632 . . . . . . 7  |-  -u 1  =/=  0
22 dgrmulc 22397 . . . . . . 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 1309 . . . . . 6  |-  ( G  e.  (Poly `  S
)  ->  (deg `  (
( CC  X.  { -u 1 } )  oF  x.  G ) )  =  (deg `  G ) )
24 dgrsub.2 . . . . . 6  |-  N  =  (deg `  G )
2523, 24syl6eqr 2521 . . . . 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 4454 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( M  <_  (deg `  ( ( CC  X.  { -u 1 } )  oF  x.  G ) )  <-> 
M  <_  N )
)
2827, 26ifbieq1d 3957 . 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 4471 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 1374    e. wcel 1762    =/= wne 2657   _Vcvv 3108    C_ wss 3471   ifcif 3934   {csn 4022   class class class wbr 4442    X. cxp 4992   -->wf 5577   ` cfv 5581  (class class class)co 6277    oFcof 6515   CCcc 9481   0cc0 9483   1c1 9484    + caddc 9486    x. cmul 9488    <_ cle 9620    - cmin 9796   -ucneg 9797  Polycply 22311  degcdgr 22314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561  ax-addf 9562
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-sup 7892  df-oi 7926  df-card 8311  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-n0 10787  df-z 10856  df-uz 11074  df-rp 11212  df-fz 11664  df-fzo 11784  df-fl 11888  df-seq 12066  df-exp 12125  df-hash 12363  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-clim 13262  df-rlim 13263  df-sum 13460  df-0p 21807  df-ply 22315  df-coe 22317  df-dgr 22318
This theorem is referenced by:  dgrcolem2  22400  plydivlem4  22421  plydiveu  22423  dgrsub2  30679
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