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Theorem dgrsub2 30716
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 1003 . . 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 22421 . . . . 5  |-  (deg ` 
0p )  =  0
3 nngt0 10565 . . . . 5  |-  ( N  e.  NN  ->  0  <  N )
42, 3syl5eqbr 4480 . . . 4  |-  ( N  e.  NN  ->  (deg `  0p )  < 
N )
5 fveq2 5866 . . . . 5  |-  ( ( F  oF  -  G )  =  0p  ->  (deg `  ( F  oF  -  G
) )  =  (deg
`  0p ) )
65breq1d 4457 . . . 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 22360 . . . . . . . 8  |-  (Poly `  S )  C_  (Poly `  CC )
109sseli 3500 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
11 plyssc 22360 . . . . . . . 8  |-  (Poly `  T )  C_  (Poly `  CC )
1211sseli 3500 . . . . . . 7  |-  ( G  e.  (Poly `  T
)  ->  G  e.  (Poly `  CC ) )
13 eqid 2467 . . . . . . . 8  |-  (deg `  F )  =  (deg
`  F )
14 eqid 2467 . . . . . . . 8  |-  (deg `  G )  =  (deg
`  G )
1513, 14dgrsub 22431 . . . . . . 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 1002 . . . . . . 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 2480 . . . . . . . 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 3959 . . . . . 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 3976 . . . . . 6  |-  if ( (deg `  F )  <_  (deg `  G ) ,  N ,  N )  =  N
2422, 23syl6eq 2524 . . . . 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 4471 . . . 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 2467 . . . . . . . . 9  |-  (coeff `  F )  =  (coeff `  F )
27 eqid 2467 . . . . . . . . 9  |-  (coeff `  G )  =  (coeff `  G )
2826, 27coesub 22416 . . . . . . . 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 5868 . . . . 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 10852 . . . . . 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 22392 . . . . . . . . 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 5731 . . . . . . . 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 22392 . . . . . . . . 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 5731 . . . . . . . 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 10801 . . . . . . . 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 3707 . . . . . . 7  |-  ( NN0 
i^i  NN0 )  =  NN0
44 simplr3 1040 . . . . . . 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 2468 . . . . . . 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 6533 . . . . . 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 6022 . . . . . 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 9918 . . . . 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 2512 . . . 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 22382 . . . . . . 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 2467 . . . . . 6  |-  (deg `  ( F  oF  -  G ) )  =  (deg `  ( F  oF  -  G
) )
55 eqid 2467 . . . . . 6  |-  (coeff `  ( F  oF  -  G ) )  =  (coeff `  ( F  oF  -  G
) )
5654, 55dgrlt 22425 . . . . 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 920 . . 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 973    = wceq 1379    e. wcel 1767   _Vcvv 3113   ifcif 3939   class class class wbr 4447    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284    oFcof 6522   CCcc 9490   0cc0 9492    < clt 9628    <_ cle 9629    - cmin 9805   NNcn 10536   NN0cn0 10795   0pc0p 21839  Polycply 22344  coeffccoe 22346  degcdgr 22347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-oi 7935  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-rp 11221  df-fz 11673  df-fzo 11793  df-fl 11897  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-rlim 13275  df-sum 13472  df-0p 21840  df-ply 22348  df-coe 22350  df-dgr 22351
This theorem is referenced by:  mpaaeu  30732
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