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Theorem dgrsub 20058
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 19987 . . . 4  |-  (Poly `  S )  C_  (Poly `  CC )
21sseli 3288 . . 3  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
3 ssid 3311 . . . . 5  |-  CC  C_  CC
4 neg1cn 10000 . . . . 5  |-  -u 1  e.  CC
5 plyconst 19993 . . . . 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 3288 . . . 4  |-  ( G  e.  (Poly `  S
)  ->  G  e.  (Poly `  CC ) )
8 plymulcl 20008 . . . 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 2388 . . . 4  |-  (deg `  ( ( CC  X.  { -u 1 } )  o F  x.  G
) )  =  (deg
`  ( ( CC 
X.  { -u 1 } )  o F  x.  G ) )
1210, 11dgradd 20053 . . 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 19985 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
15 plyf 19985 . . . 4  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
16 cnex 9005 . . . . 5  |-  CC  e.  _V
17 ofnegsub 9931 . . . . 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 5673 . 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 8982 . . . . . . . 8  |-  1  e.  CC
22 ax-1ne0 8993 . . . . . . . 8  |-  1  =/=  0
2321, 22negne0i 9308 . . . . . . 7  |-  -u 1  =/=  0
24 dgrmulc 20057 . . . . . . 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 2438 . . . . 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 4166 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( M  <_  (deg `  ( ( CC  X.  { -u 1 } )  o F  x.  G ) )  <-> 
M  <_  N )
)
30 eqidd 2389 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  M  =  M )
3129, 28, 30ifbieq12d 3705 . 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 4183 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 1649    e. wcel 1717    =/= wne 2551   _Vcvv 2900    C_ wss 3264   ifcif 3683   {csn 3758   class class class wbr 4154    X. cxp 4817   -->wf 5391   ` cfv 5395  (class class class)co 6021    o Fcof 6243   CCcc 8922   0cc0 8924   1c1 8925    + caddc 8927    x. cmul 8929    <_ cle 9055    - cmin 9224   -ucneg 9225  Polycply 19971  degcdgr 19974
This theorem is referenced by:  dgrcolem2  20060  plydivlem4  20081  plydiveu  20083  dgrsub2  27009
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002  ax-addf 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-map 6957  df-pm 6958  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-sup 7382  df-oi 7413  df-card 7760  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-rp 10546  df-fz 10977  df-fzo 11067  df-fl 11130  df-seq 11252  df-exp 11311  df-hash 11547  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-clim 12210  df-rlim 12211  df-sum 12408  df-0p 19430  df-ply 19975  df-coe 19977  df-dgr 19978
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