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Theorem dgrsub2 27442
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 19659 . . . . 5  |-  (deg ` 
0 p )  =  0
3 nngt0 9791 . . . . 5  |-  ( N  e.  NN  ->  0  <  N )
42, 3syl5eqbr 4072 . . . 4  |-  ( N  e.  NN  ->  (deg `  0 p )  < 
N )
5 fveq2 5541 . . . . 5  |-  ( ( F  o F  -  G )  =  0 p  ->  (deg `  ( F  o F  -  G
) )  =  (deg
`  0 p ) )
65breq1d 4049 . . . 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 19598 . . . . . . . 8  |-  (Poly `  S )  C_  (Poly `  CC )
109sseli 3189 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
11 plyssc 19598 . . . . . . . 8  |-  (Poly `  T )  C_  (Poly `  CC )
1211sseli 3189 . . . . . . 7  |-  ( G  e.  (Poly `  T
)  ->  G  e.  (Poly `  CC ) )
13 eqid 2296 . . . . . . . 8  |-  (deg `  F )  =  (deg
`  F )
14 eqid 2296 . . . . . . . 8  |-  (deg `  G )  =  (deg
`  G )
1513, 14dgrsub 19669 . . . . . . 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 2300 . . . . . . . 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 3594 . . . . . 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 3610 . . . . . 6  |-  if ( (deg `  F )  <_  (deg `  G ) ,  N ,  N )  =  N
2422, 23syl6eq 2344 . . . . 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 4063 . . . 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 2296 . . . . . . . . 9  |-  (coeff `  F )  =  (coeff `  F )
27 eqid 2296 . . . . . . . . 9  |-  (coeff `  G )  =  (coeff `  G )
2826, 27coesub 19654 . . . . . . . 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 5543 . . . . 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 10034 . . . . . 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 19630 . . . . . . . . 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 5405 . . . . . . . 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 19630 . . . . . . . . 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 5405 . . . . . . . 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 9987 . . . . . . . 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 3391 . . . . . . 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 2297 . . . . . . 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 6103 . . . . . 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 5679 . . . . . . 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 9161 . . . . 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 2332 . . . 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 19620 . . . . . . 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 2296 . . . . . 6  |-  (deg `  ( F  o F  -  G ) )  =  (deg `  ( F  o F  -  G
) )
56 eqid 2296 . . . . . 6  |-  (coeff `  ( F  o F  -  G ) )  =  (coeff `  ( F  o F  -  G
) )
5755, 56dgrlt 19663 . . . . 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 1632    e. wcel 1696   _Vcvv 2801   ifcif 3578   class class class wbr 4039    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   CCcc 8751   0cc0 8753    < clt 8883    <_ cle 8884    - cmin 9053   NNcn 9762   NN0cn0 9981   0 pc0p 19040  Polycply 19582  coeffccoe 19584  degcdgr 19585
This theorem is referenced by:  mpaaeu  27458
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-0p 19041  df-ply 19586  df-coe 19588  df-dgr 19589
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