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Theorem dgrsub 20182
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  o F  -  G
) )  <_  if ( M  <_  N ,  N ,  M )
)

Proof of Theorem dgrsub
StepHypRef Expression
1 plyssc 20111 . . . 4  |-  (Poly `  S )  C_  (Poly `  CC )
21sseli 3336 . . 3  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
3 ssid 3359 . . . . 5  |-  CC  C_  CC
4 neg1cn 10059 . . . . 5  |-  -u 1  e.  CC
5 plyconst 20117 . . . . 5  |-  ( ( CC  C_  CC  /\  -u 1  e.  CC )  ->  ( CC  X.  { -u 1 } )  e.  (Poly `  CC ) )
63, 4, 5mp2an 654 . . . 4  |-  ( CC 
X.  { -u 1 } )  e.  (Poly `  CC )
71sseli 3336 . . . 4  |-  ( G  e.  (Poly `  S
)  ->  G  e.  (Poly `  CC ) )
8 plymulcl 20132 . . . 4  |-  ( ( ( CC  X.  { -u 1 } )  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC ) )  -> 
( ( CC  X.  { -u 1 } )  o F  x.  G
)  e.  (Poly `  CC ) )
96, 7, 8sylancr 645 . . 3  |-  ( G  e.  (Poly `  S
)  ->  ( ( CC  X.  { -u 1 } )  o F  x.  G )  e.  (Poly `  CC )
)
10 dgrsub.1 . . . 4  |-  M  =  (deg `  F )
11 eqid 2435 . . . 4  |-  (deg `  ( ( CC  X.  { -u 1 } )  o F  x.  G
) )  =  (deg
`  ( ( CC 
X.  { -u 1 } )  o F  x.  G ) )
1210, 11dgradd 20177 . . 3  |-  ( ( F  e.  (Poly `  CC )  /\  (
( CC  X.  { -u 1 } )  o F  x.  G )  e.  (Poly `  CC ) )  ->  (deg `  ( F  o F  +  ( ( CC 
X.  { -u 1 } )  o F  x.  G ) ) )  <_  if ( M  <_  (deg `  (
( CC  X.  { -u 1 } )  o F  x.  G ) ) ,  (deg `  ( ( CC  X.  { -u 1 } )  o F  x.  G
) ) ,  M
) )
132, 9, 12syl2an 464 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  ( F  o F  +  ( ( CC  X.  { -u 1 } )  o F  x.  G ) ) )  <_  if ( M  <_  (deg `  ( ( CC  X.  { -u 1 } )  o F  x.  G
) ) ,  (deg
`  ( ( CC 
X.  { -u 1 } )  o F  x.  G ) ) ,  M ) )
14 plyf 20109 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
15 plyf 20109 . . . 4  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
16 cnex 9063 . . . . 5  |-  CC  e.  _V
17 ofnegsub 9990 . . . . 5  |-  ( ( CC  e.  _V  /\  F : CC --> CC  /\  G : CC --> CC )  ->  ( F  o F  +  ( ( CC  X.  { -u 1 } )  o F  x.  G ) )  =  ( F  o F  -  G )
)
1816, 17mp3an1 1266 . . . 4  |-  ( ( F : CC --> CC  /\  G : CC --> CC )  ->  ( F  o F  +  ( ( CC  X.  { -u 1 } )  o F  x.  G ) )  =  ( F  o F  -  G )
)
1914, 15, 18syl2an 464 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  o F  +  (
( CC  X.  { -u 1 } )  o F  x.  G ) )  =  ( F  o F  -  G
) )
2019fveq2d 5724 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  ( F  o F  +  ( ( CC  X.  { -u 1 } )  o F  x.  G ) ) )  =  (deg
`  ( F  o F  -  G )
) )
21 ax-1cn 9040 . . . . . . . 8  |-  1  e.  CC
22 ax-1ne0 9051 . . . . . . . 8  |-  1  =/=  0
2321, 22negne0i 9367 . . . . . . 7  |-  -u 1  =/=  0
24 dgrmulc 20181 . . . . . . 7  |-  ( (
-u 1  e.  CC  /\  -u 1  =/=  0  /\  G  e.  (Poly `  S ) )  -> 
(deg `  ( ( CC  X.  { -u 1 } )  o F  x.  G ) )  =  (deg `  G
) )
254, 23, 24mp3an12 1269 . . . . . 6  |-  ( G  e.  (Poly `  S
)  ->  (deg `  (
( CC  X.  { -u 1 } )  o F  x.  G ) )  =  (deg `  G ) )
26 dgrsub.2 . . . . . 6  |-  N  =  (deg `  G )
2725, 26syl6eqr 2485 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  (deg `  (
( CC  X.  { -u 1 } )  o F  x.  G ) )  =  N )
2827adantl 453 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  (
( CC  X.  { -u 1 } )  o F  x.  G ) )  =  N )
2928breq2d 4216 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( M  <_  (deg `  ( ( CC  X.  { -u 1 } )  o F  x.  G ) )  <-> 
M  <_  N )
)
30 eqidd 2436 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  =  M )
3129, 28, 30ifbieq12d 3753 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  if ( M  <_  (deg `  (
( CC  X.  { -u 1 } )  o F  x.  G ) ) ,  (deg `  ( ( CC  X.  { -u 1 } )  o F  x.  G
) ) ,  M
)  =  if ( M  <_  N ,  N ,  M )
)
3213, 20, 313brtr3d 4233 1  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  ( F  o F  -  G
) )  <_  if ( M  <_  N ,  N ,  M )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948    C_ wss 3312   ifcif 3731   {csn 3806   class class class wbr 4204    X. cxp 4868   -->wf 5442   ` cfv 5446  (class class class)co 6073    o Fcof 6295   CCcc 8980   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    <_ cle 9113    - cmin 9283   -ucneg 9284  Polycply 20095  degcdgr 20098
This theorem is referenced by:  dgrcolem2  20184  plydivlem4  20205  plydiveu  20207  dgrsub2  27307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-rlim 12275  df-sum 12472  df-0p 19554  df-ply 20099  df-coe 20101  df-dgr 20102
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