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Theorem dgrsub2 35448
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 1004 . . 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 22951 . . . . 5  |-  (deg ` 
0p )  =  0
3 nngt0 10605 . . . . 5  |-  ( N  e.  NN  ->  0  <  N )
42, 3syl5eqbr 4428 . . . 4  |-  ( N  e.  NN  ->  (deg `  0p )  < 
N )
5 fveq2 5849 . . . . 5  |-  ( ( F  oF  -  G )  =  0p  ->  (deg `  ( F  oF  -  G
) )  =  (deg
`  0p ) )
65breq1d 4405 . . . 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 17 . 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 22889 . . . . . . . 8  |-  (Poly `  S )  C_  (Poly `  CC )
109sseli 3438 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
11 plyssc 22889 . . . . . . . 8  |-  (Poly `  T )  C_  (Poly `  CC )
1211sseli 3438 . . . . . . 7  |-  ( G  e.  (Poly `  T
)  ->  G  e.  (Poly `  CC ) )
13 eqid 2402 . . . . . . . 8  |-  (deg `  F )  =  (deg
`  F )
14 eqid 2402 . . . . . . . 8  |-  (deg `  G )  =  (deg
`  G )
1513, 14dgrsub 22961 . . . . . . 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 475 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  ->  (deg `  ( F  oF  -  G
) )  <_  if ( (deg `  F )  <_  (deg `  G ) ,  (deg `  G ) ,  (deg `  F )
) )
1716adantr 463 . . . . 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 1003 . . . . . . 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 2415 . . . . . . . 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 3905 . . . . . 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 3922 . . . . . 6  |-  if ( (deg `  F )  <_  (deg `  G ) ,  N ,  N )  =  N
2422, 23syl6eq 2459 . . . . 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 4419 . . . 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 2402 . . . . . . . . 9  |-  (coeff `  F )  =  (coeff `  F )
27 eqid 2402 . . . . . . . . 9  |-  (coeff `  G )  =  (coeff `  G )
2826, 27coesub 22946 . . . . . . . 8  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )
)  ->  (coeff `  ( F  oF  -  G
) )  =  ( (coeff `  F )  oF  -  (coeff `  G ) ) )
2910, 12, 28syl2an 475 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  ->  (coeff `  ( F  oF  -  G
) )  =  ( (coeff `  F )  oF  -  (coeff `  G ) ) )
3029adantr 463 . . . . . 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 5851 . . . . 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 10893 . . . . . 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 22921 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
3433ad2antrr 724 . . . . . . . 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 5714 . . . . . . . 8  |-  ( (coeff `  F ) : NN0 --> CC 
->  (coeff `  F )  Fn  NN0 )
3634, 35syl 17 . . . . . . 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 22921 . . . . . . . . 9  |-  ( G  e.  (Poly `  T
)  ->  (coeff `  G
) : NN0 --> CC )
3837ad2antlr 725 . . . . . . . 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 5714 . . . . . . . 8  |-  ( (coeff `  G ) : NN0 --> CC 
->  (coeff `  G )  Fn  NN0 )
4038, 39syl 17 . . . . . . 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 10842 . . . . . . . 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 3648 . . . . . . 7  |-  ( NN0 
i^i  NN0 )  =  NN0
44 simplr3 1041 . . . . . . 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 2403 . . . . . . 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 6530 . . . . . 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 666 . . . . 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 6010 . . . . . 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 9955 . . . . 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 2447 . . . 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 22911 . . . . . . 7  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )
)  ->  ( F  oF  -  G
)  e.  (Poly `  CC ) )
5210, 12, 51syl2an 475 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  ->  ( F  oF  -  G
)  e.  (Poly `  CC ) )
5352adantr 463 . . . . 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 2402 . . . . . 6  |-  (deg `  ( F  oF  -  G ) )  =  (deg `  ( F  oF  -  G
) )
55 eqid 2402 . . . . . 6  |-  (coeff `  ( F  oF  -  G ) )  =  (coeff `  ( F  oF  -  G
) )
5654, 55dgrlt 22955 . . . . 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 659 . . . 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 923 . . 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 375 . 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 366    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   _Vcvv 3059   ifcif 3885   class class class wbr 4395    Fn wfn 5564   -->wf 5565   ` cfv 5569  (class class class)co 6278    oFcof 6519   CCcc 9520   0cc0 9522    < clt 9658    <_ cle 9659    - cmin 9841   NNcn 10576   NN0cn0 10836   0pc0p 22368  Polycply 22873  coeffccoe 22875  degcdgr 22876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-oi 7969  df-card 8352  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-fz 11727  df-fzo 11855  df-fl 11966  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-clim 13460  df-rlim 13461  df-sum 13658  df-0p 22369  df-ply 22877  df-coe 22879  df-dgr 22880
This theorem is referenced by:  mpaaeu  35463
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