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Theorem dgrsub2 29491
Description: Subtracting two polynomials with the same degree and top coefficient gives a polynomial of strictly lower degree. (Contributed by Stefan O'Rear, 25-Nov-2014.)
Hypothesis
Ref Expression
dgrsub2.a  |-  N  =  (deg `  F )
Assertion
Ref Expression
dgrsub2  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
(deg `  ( F  oF  -  G
) )  <  N
)

Proof of Theorem dgrsub2
StepHypRef Expression
1 simpr2 995 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  ->  N  e.  NN )
2 dgr0 21729 . . . . 5  |-  (deg ` 
0p )  =  0
3 nngt0 10351 . . . . 5  |-  ( N  e.  NN  ->  0  <  N )
42, 3syl5eqbr 4325 . . . 4  |-  ( N  e.  NN  ->  (deg `  0p )  < 
N )
5 fveq2 5691 . . . . 5  |-  ( ( F  oF  -  G )  =  0p  ->  (deg `  ( F  oF  -  G
) )  =  (deg
`  0p ) )
65breq1d 4302 . . . 4  |-  ( ( F  oF  -  G )  =  0p  ->  ( (deg `  ( F  oF  -  G ) )  <  N  <->  (deg `  0p )  <  N
) )
74, 6syl5ibrcom 222 . . 3  |-  ( N  e.  NN  ->  (
( F  oF  -  G )  =  0p  ->  (deg `  ( F  oF  -  G ) )  <  N ) )
81, 7syl 16 . 2  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
( ( F  oF  -  G )  =  0p  -> 
(deg `  ( F  oF  -  G
) )  <  N
) )
9 plyssc 21668 . . . . . . . 8  |-  (Poly `  S )  C_  (Poly `  CC )
109sseli 3352 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
11 plyssc 21668 . . . . . . . 8  |-  (Poly `  T )  C_  (Poly `  CC )
1211sseli 3352 . . . . . . 7  |-  ( G  e.  (Poly `  T
)  ->  G  e.  (Poly `  CC ) )
13 eqid 2443 . . . . . . . 8  |-  (deg `  F )  =  (deg
`  F )
14 eqid 2443 . . . . . . . 8  |-  (deg `  G )  =  (deg
`  G )
1513, 14dgrsub 21739 . . . . . . 7  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )
)  ->  (deg `  ( F  oF  -  G
) )  <_  if ( (deg `  F )  <_  (deg `  G ) ,  (deg `  G ) ,  (deg `  F )
) )
1610, 12, 15syl2an 477 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  ->  (deg `  ( F  oF  -  G
) )  <_  if ( (deg `  F )  <_  (deg `  G ) ,  (deg `  G ) ,  (deg `  F )
) )
1716adantr 465 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
(deg `  ( F  oF  -  G
) )  <_  if ( (deg `  F )  <_  (deg `  G ) ,  (deg `  G ) ,  (deg `  F )
) )
18 simpr1 994 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
(deg `  G )  =  N )
19 dgrsub2.a . . . . . . . . 9  |-  N  =  (deg `  F )
2019eqcomi 2447 . . . . . . . 8  |-  (deg `  F )  =  N
2120a1i 11 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
(deg `  F )  =  N )
2218, 21ifeq12d 3809 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  ->  if ( (deg `  F
)  <_  (deg `  G
) ,  (deg `  G ) ,  (deg
`  F ) )  =  if ( (deg
`  F )  <_ 
(deg `  G ) ,  N ,  N ) )
23 ifid 3826 . . . . . 6  |-  if ( (deg `  F )  <_  (deg `  G ) ,  N ,  N )  =  N
2422, 23syl6eq 2491 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  ->  if ( (deg `  F
)  <_  (deg `  G
) ,  (deg `  G ) ,  (deg
`  F ) )  =  N )
2517, 24breqtrd 4316 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
(deg `  ( F  oF  -  G
) )  <_  N
)
26 eqid 2443 . . . . . . . . 9  |-  (coeff `  F )  =  (coeff `  F )
27 eqid 2443 . . . . . . . . 9  |-  (coeff `  G )  =  (coeff `  G )
2826, 27coesub 21724 . . . . . . . 8  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )
)  ->  (coeff `  ( F  oF  -  G
) )  =  ( (coeff `  F )  oF  -  (coeff `  G ) ) )
2910, 12, 28syl2an 477 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  ->  (coeff `  ( F  oF  -  G
) )  =  ( (coeff `  F )  oF  -  (coeff `  G ) ) )
3029adantr 465 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
(coeff `  ( F  oF  -  G
) )  =  ( (coeff `  F )  oF  -  (coeff `  G ) ) )
3130fveq1d 5693 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
( (coeff `  ( F  oF  -  G
) ) `  N
)  =  ( ( (coeff `  F )  oF  -  (coeff `  G ) ) `  N ) )
321nnnn0d 10636 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  ->  N  e.  NN0 )
3326coef3 21700 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
3433ad2antrr 725 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
(coeff `  F ) : NN0 --> CC )
35 ffn 5559 . . . . . . . 8  |-  ( (coeff `  F ) : NN0 --> CC 
->  (coeff `  F )  Fn  NN0 )
3634, 35syl 16 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
(coeff `  F )  Fn  NN0 )
3727coef3 21700 . . . . . . . . 9  |-  ( G  e.  (Poly `  T
)  ->  (coeff `  G
) : NN0 --> CC )
3837ad2antlr 726 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
(coeff `  G ) : NN0 --> CC )
39 ffn 5559 . . . . . . . 8  |-  ( (coeff `  G ) : NN0 --> CC 
->  (coeff `  G )  Fn  NN0 )
4038, 39syl 16 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
(coeff `  G )  Fn  NN0 )
41 nn0ex 10585 . . . . . . . 8  |-  NN0  e.  _V
4241a1i 11 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  ->  NN0  e.  _V )
43 inidm 3559 . . . . . . 7  |-  ( NN0 
i^i  NN0 )  =  NN0
44 simplr3 1032 . . . . . . 7  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T ) )  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F ) `  N
)  =  ( (coeff `  G ) `  N
) ) )  /\  N  e.  NN0 )  -> 
( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) )
45 eqidd 2444 . . . . . . 7  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T ) )  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F ) `  N
)  =  ( (coeff `  G ) `  N
) ) )  /\  N  e.  NN0 )  -> 
( (coeff `  G
) `  N )  =  ( (coeff `  G ) `  N
) )
4636, 40, 42, 42, 43, 44, 45ofval 6329 . . . . . 6  |-  ( ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T ) )  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F ) `  N
)  =  ( (coeff `  G ) `  N
) ) )  /\  N  e.  NN0 )  -> 
( ( (coeff `  F )  oF  -  (coeff `  G
) ) `  N
)  =  ( ( (coeff `  G ) `  N )  -  (
(coeff `  G ) `  N ) ) )
4732, 46mpdan 668 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
( ( (coeff `  F )  oF  -  (coeff `  G
) ) `  N
)  =  ( ( (coeff `  G ) `  N )  -  (
(coeff `  G ) `  N ) ) )
4838, 32ffvelrnd 5844 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
( (coeff `  G
) `  N )  e.  CC )
4948subidd 9707 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
( ( (coeff `  G ) `  N
)  -  ( (coeff `  G ) `  N
) )  =  0 )
5031, 47, 493eqtrd 2479 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
( (coeff `  ( F  oF  -  G
) ) `  N
)  =  0 )
51 plysubcl 21690 . . . . . . 7  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )
)  ->  ( F  oF  -  G
)  e.  (Poly `  CC ) )
5210, 12, 51syl2an 477 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  ->  ( F  oF  -  G
)  e.  (Poly `  CC ) )
5352adantr 465 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
( F  oF  -  G )  e.  (Poly `  CC )
)
54 eqid 2443 . . . . . 6  |-  (deg `  ( F  oF  -  G ) )  =  (deg `  ( F  oF  -  G
) )
55 eqid 2443 . . . . . 6  |-  (coeff `  ( F  oF  -  G ) )  =  (coeff `  ( F  oF  -  G
) )
5654, 55dgrlt 21733 . . . . 5  |-  ( ( ( F  oF  -  G )  e.  (Poly `  CC )  /\  N  e.  NN0 )  ->  ( ( ( F  oF  -  G )  =  0p  \/  (deg `  ( F  oF  -  G ) )  < 
N )  <->  ( (deg `  ( F  oF  -  G ) )  <_  N  /\  (
(coeff `  ( F  oF  -  G
) ) `  N
)  =  0 ) ) )
5753, 32, 56syl2anc 661 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
( ( ( F  oF  -  G
)  =  0p  \/  (deg `  ( F  oF  -  G
) )  <  N
)  <->  ( (deg `  ( F  oF  -  G ) )  <_  N  /\  ( (coeff `  ( F  oF  -  G ) ) `  N )  =  0 ) ) )
5825, 50, 57mpbir2and 913 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
( ( F  oF  -  G )  =  0p  \/  (deg `  ( F  oF  -  G
) )  <  N
) )
5958ord 377 . 2  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
( -.  ( F  oF  -  G
)  =  0p  ->  (deg `  ( F  oF  -  G
) )  <  N
) )
608, 59pm2.61d 158 1  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
(deg `  ( F  oF  -  G
) )  <  N
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2972   ifcif 3791   class class class wbr 4292    Fn wfn 5413   -->wf 5414   ` cfv 5418  (class class class)co 6091    oFcof 6318   CCcc 9280   0cc0 9282    < clt 9418    <_ cle 9419    - cmin 9595   NNcn 10322   NN0cn0 10579   0pc0p 21147  Polycply 21652  coeffccoe 21654  degcdgr 21655
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fz 11438  df-fzo 11549  df-fl 11642  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-clim 12966  df-rlim 12967  df-sum 13164  df-0p 21148  df-ply 21656  df-coe 21658  df-dgr 21659
This theorem is referenced by:  mpaaeu  29507
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