Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dgrsub2 Unicode version

Theorem dgrsub2 27339
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  o F  -  G
) )  <  N
)

Proof of Theorem dgrsub2
StepHypRef Expression
1 simpr2 962 . . 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 19643 . . . . 5  |-  (deg ` 
0 p )  =  0
3 nngt0 9775 . . . . 5  |-  ( N  e.  NN  ->  0  <  N )
42, 3syl5eqbr 4056 . . . 4  |-  ( N  e.  NN  ->  (deg `  0 p )  < 
N )
5 fveq2 5525 . . . . 5  |-  ( ( F  o F  -  G )  =  0 p  ->  (deg `  ( F  o F  -  G
) )  =  (deg
`  0 p ) )
65breq1d 4033 . . . 4  |-  ( ( F  o F  -  G )  =  0 p  ->  ( (deg `  ( F  o F  -  G ) )  <  N  <->  (deg `  0 p )  <  N
) )
74, 6syl5ibrcom 213 . . 3  |-  ( N  e.  NN  ->  (
( F  o F  -  G )  =  0 p  ->  (deg `  ( F  o F  -  G ) )  <  N ) )
81, 7syl 15 . 2  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
( ( F  o F  -  G )  =  0 p  -> 
(deg `  ( F  o F  -  G
) )  <  N
) )
9 plyssc 19582 . . . . . . . 8  |-  (Poly `  S )  C_  (Poly `  CC )
109sseli 3176 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
11 plyssc 19582 . . . . . . . 8  |-  (Poly `  T )  C_  (Poly `  CC )
1211sseli 3176 . . . . . . 7  |-  ( G  e.  (Poly `  T
)  ->  G  e.  (Poly `  CC ) )
13 eqid 2283 . . . . . . . 8  |-  (deg `  F )  =  (deg
`  F )
14 eqid 2283 . . . . . . . 8  |-  (deg `  G )  =  (deg
`  G )
1513, 14dgrsub 19653 . . . . . . 7  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )
)  ->  (deg `  ( F  o F  -  G
) )  <_  if ( (deg `  F )  <_  (deg `  G ) ,  (deg `  G ) ,  (deg `  F )
) )
1610, 12, 15syl2an 463 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  ->  (deg `  ( F  o F  -  G
) )  <_  if ( (deg `  F )  <_  (deg `  G ) ,  (deg `  G ) ,  (deg `  F )
) )
1716adantr 451 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
(deg `  ( F  o F  -  G
) )  <_  if ( (deg `  F )  <_  (deg `  G ) ,  (deg `  G ) ,  (deg `  F )
) )
18 simpr1 961 . . . . . . 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 2287 . . . . . . . 8  |-  (deg `  F )  =  N
2120a1i 10 . . . . . . 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 3581 . . . . . 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 3597 . . . . . 6  |-  if ( (deg `  F )  <_  (deg `  G ) ,  N ,  N )  =  N
2422, 23syl6eq 2331 . . . . 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 4047 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
(deg `  ( F  o F  -  G
) )  <_  N
)
26 eqid 2283 . . . . . . . . 9  |-  (coeff `  F )  =  (coeff `  F )
27 eqid 2283 . . . . . . . . 9  |-  (coeff `  G )  =  (coeff `  G )
2826, 27coesub 19638 . . . . . . . 8  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )
)  ->  (coeff `  ( F  o F  -  G
) )  =  ( (coeff `  F )  o F  -  (coeff `  G ) ) )
2910, 12, 28syl2an 463 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  ->  (coeff `  ( F  o F  -  G
) )  =  ( (coeff `  F )  o F  -  (coeff `  G ) ) )
3029adantr 451 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
(coeff `  ( F  o F  -  G
) )  =  ( (coeff `  F )  o F  -  (coeff `  G ) ) )
3130fveq1d 5527 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
( (coeff `  ( F  o F  -  G
) ) `  N
)  =  ( ( (coeff `  F )  o F  -  (coeff `  G ) ) `  N ) )
321nnnn0d 10018 . . . . . 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 19614 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
3433ad2antrr 706 . . . . . . . 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 5389 . . . . . . . 8  |-  ( (coeff `  F ) : NN0 --> CC 
->  (coeff `  F )  Fn  NN0 )
3634, 35syl 15 . . . . . . 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 19614 . . . . . . . . 9  |-  ( G  e.  (Poly `  T
)  ->  (coeff `  G
) : NN0 --> CC )
3837ad2antlr 707 . . . . . . . 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 5389 . . . . . . . 8  |-  ( (coeff `  G ) : NN0 --> CC 
->  (coeff `  G )  Fn  NN0 )
4038, 39syl 15 . . . . . . 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 9971 . . . . . . . 8  |-  NN0  e.  _V
4241a1i 10 . . . . . . 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 3378 . . . . . . 7  |-  ( NN0 
i^i  NN0 )  =  NN0
44 simplr3 999 . . . . . . 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 2284 . . . . . . 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 6087 . . . . . 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 )  o F  -  (coeff `  G
) ) `  N
)  =  ( ( (coeff `  G ) `  N )  -  (
(coeff `  G ) `  N ) ) )
4732, 46mpdan 649 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
( ( (coeff `  F )  o F  -  (coeff `  G
) ) `  N
)  =  ( ( (coeff `  G ) `  N )  -  (
(coeff `  G ) `  N ) ) )
48 ffvelrn 5663 . . . . . . 7  |-  ( ( (coeff `  G ) : NN0 --> CC  /\  N  e.  NN0 )  ->  (
(coeff `  G ) `  N )  e.  CC )
4938, 32, 48syl2anc 642 . . . . . 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 )
5049subidd 9145 . . . . 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 )
5131, 47, 503eqtrd 2319 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
( (coeff `  ( F  o F  -  G
) ) `  N
)  =  0 )
52 plysubcl 19604 . . . . . . 7  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )
)  ->  ( F  o F  -  G
)  e.  (Poly `  CC ) )
5310, 12, 52syl2an 463 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  ->  ( F  o F  -  G
)  e.  (Poly `  CC ) )
5453adantr 451 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
( F  o F  -  G )  e.  (Poly `  CC )
)
55 eqid 2283 . . . . . 6  |-  (deg `  ( F  o F  -  G ) )  =  (deg `  ( F  o F  -  G
) )
56 eqid 2283 . . . . . 6  |-  (coeff `  ( F  o F  -  G ) )  =  (coeff `  ( F  o F  -  G
) )
5755, 56dgrlt 19647 . . . . 5  |-  ( ( ( F  o F  -  G )  e.  (Poly `  CC )  /\  N  e.  NN0 )  ->  ( ( ( F  o F  -  G )  =  0 p  \/  (deg `  ( F  o F  -  G ) )  < 
N )  <->  ( (deg `  ( F  o F  -  G ) )  <_  N  /\  (
(coeff `  ( F  o F  -  G
) ) `  N
)  =  0 ) ) )
5854, 32, 57syl2anc 642 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
( ( ( F  o F  -  G
)  =  0 p  \/  (deg `  ( F  o F  -  G
) )  <  N
)  <->  ( (deg `  ( F  o F  -  G ) )  <_  N  /\  ( (coeff `  ( F  o F  -  G ) ) `  N )  =  0 ) ) )
5925, 51, 58mpbir2and 888 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
( ( F  o F  -  G )  =  0 p  \/  (deg `  ( F  o F  -  G )
)  <  N )
)
6059ord 366 . 2  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
( -.  ( F  o F  -  G
)  =  0 p  ->  (deg `  ( F  o F  -  G
) )  <  N
) )
618, 60pm2.61d 150 1  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  T )
)  /\  ( (deg `  G )  =  N  /\  N  e.  NN  /\  ( (coeff `  F
) `  N )  =  ( (coeff `  G ) `  N
) ) )  -> 
(deg `  ( F  o F  -  G
) )  <  N
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   ifcif 3565   class class class wbr 4023    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   CCcc 8735   0cc0 8737    < clt 8867    <_ cle 8868    - cmin 9037   NNcn 9746   NN0cn0 9965   0 pc0p 19024  Polycply 19566  coeffccoe 19568  degcdgr 19569
This theorem is referenced by:  mpaaeu  27355
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-0p 19025  df-ply 19570  df-coe 19572  df-dgr 19573
  Copyright terms: Public domain W3C validator