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Theorem plydivalg 20208
Description: The division algorithm on polynomials over a subfield  S of the complex numbers. If  F and  G  =/=  0 are polynomials over  S, then there is a unique quotient polynomial  q such that the remainder  F  -  G  x.  q is either zero or has degree less than  G. (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypotheses
Ref Expression
plydiv.pl  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  +  y )  e.  S )
plydiv.tm  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  x.  y
)  e.  S )
plydiv.rc  |-  ( (
ph  /\  ( x  e.  S  /\  x  =/=  0 ) )  -> 
( 1  /  x
)  e.  S )
plydiv.m1  |-  ( ph  -> 
-u 1  e.  S
)
plydiv.f  |-  ( ph  ->  F  e.  (Poly `  S ) )
plydiv.g  |-  ( ph  ->  G  e.  (Poly `  S ) )
plydiv.z  |-  ( ph  ->  G  =/=  0 p )
plydiv.r  |-  R  =  ( F  o F  -  ( G  o F  x.  q )
)
Assertion
Ref Expression
plydivalg  |-  ( ph  ->  E! q  e.  (Poly `  S ) ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
) )
Distinct variable groups:    x, y,
q, F    ph, x, y    G, q, x, y    x, R, y    S, q, x, y    ph, q
Allowed substitution hint:    R( q)

Proof of Theorem plydivalg
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 plydiv.pl . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  +  y )  e.  S )
2 plydiv.tm . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  x.  y
)  e.  S )
3 plydiv.rc . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  x  =/=  0 ) )  -> 
( 1  /  x
)  e.  S )
4 plydiv.m1 . . 3  |-  ( ph  -> 
-u 1  e.  S
)
5 plydiv.f . . 3  |-  ( ph  ->  F  e.  (Poly `  S ) )
6 plydiv.g . . 3  |-  ( ph  ->  G  e.  (Poly `  S ) )
7 plydiv.z . . 3  |-  ( ph  ->  G  =/=  0 p )
8 plydiv.r . . 3  |-  R  =  ( F  o F  -  ( G  o F  x.  q )
)
91, 2, 3, 4, 5, 6, 7, 8plydivex 20206 . 2  |-  ( ph  ->  E. q  e.  (Poly `  S ) ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
) )
10 simpll 731 . . . . . 6  |-  ( ( ( ph  /\  (
q  e.  (Poly `  S )  /\  p  e.  (Poly `  S )
) )  /\  (
( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) )  /\  ( ( F  o F  -  ( G  o F  x.  p
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  p )
) )  <  (deg `  G ) ) ) )  ->  ph )
1110, 1sylan 458 . . . . 5  |-  ( ( ( ( ph  /\  ( q  e.  (Poly `  S )  /\  p  e.  (Poly `  S )
) )  /\  (
( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) )  /\  ( ( F  o F  -  ( G  o F  x.  p
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  p )
) )  <  (deg `  G ) ) ) )  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  +  y )  e.  S )
1210, 2sylan 458 . . . . 5  |-  ( ( ( ( ph  /\  ( q  e.  (Poly `  S )  /\  p  e.  (Poly `  S )
) )  /\  (
( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) )  /\  ( ( F  o F  -  ( G  o F  x.  p
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  p )
) )  <  (deg `  G ) ) ) )  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  x.  y
)  e.  S )
1310, 3sylan 458 . . . . 5  |-  ( ( ( ( ph  /\  ( q  e.  (Poly `  S )  /\  p  e.  (Poly `  S )
) )  /\  (
( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) )  /\  ( ( F  o F  -  ( G  o F  x.  p
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  p )
) )  <  (deg `  G ) ) ) )  /\  ( x  e.  S  /\  x  =/=  0 ) )  -> 
( 1  /  x
)  e.  S )
1410, 4syl 16 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  (Poly `  S )  /\  p  e.  (Poly `  S )
) )  /\  (
( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) )  /\  ( ( F  o F  -  ( G  o F  x.  p
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  p )
) )  <  (deg `  G ) ) ) )  ->  -u 1  e.  S )
1510, 5syl 16 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  (Poly `  S )  /\  p  e.  (Poly `  S )
) )  /\  (
( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) )  /\  ( ( F  o F  -  ( G  o F  x.  p
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  p )
) )  <  (deg `  G ) ) ) )  ->  F  e.  (Poly `  S ) )
1610, 6syl 16 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  (Poly `  S )  /\  p  e.  (Poly `  S )
) )  /\  (
( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) )  /\  ( ( F  o F  -  ( G  o F  x.  p
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  p )
) )  <  (deg `  G ) ) ) )  ->  G  e.  (Poly `  S ) )
1710, 7syl 16 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  (Poly `  S )  /\  p  e.  (Poly `  S )
) )  /\  (
( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) )  /\  ( ( F  o F  -  ( G  o F  x.  p
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  p )
) )  <  (deg `  G ) ) ) )  ->  G  =/=  0 p )
18 eqid 2435 . . . . 5  |-  ( F  o F  -  ( G  o F  x.  p
) )  =  ( F  o F  -  ( G  o F  x.  p ) )
19 simplrr 738 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  (Poly `  S )  /\  p  e.  (Poly `  S )
) )  /\  (
( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) )  /\  ( ( F  o F  -  ( G  o F  x.  p
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  p )
) )  <  (deg `  G ) ) ) )  ->  p  e.  (Poly `  S ) )
20 simprr 734 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  (Poly `  S )  /\  p  e.  (Poly `  S )
) )  /\  (
( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) )  /\  ( ( F  o F  -  ( G  o F  x.  p
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  p )
) )  <  (deg `  G ) ) ) )  ->  ( ( F  o F  -  ( G  o F  x.  p
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  p )
) )  <  (deg `  G ) ) )
21 simplrl 737 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  (Poly `  S )  /\  p  e.  (Poly `  S )
) )  /\  (
( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) )  /\  ( ( F  o F  -  ( G  o F  x.  p
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  p )
) )  <  (deg `  G ) ) ) )  ->  q  e.  (Poly `  S ) )
22 simprl 733 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  (Poly `  S )  /\  p  e.  (Poly `  S )
) )  /\  (
( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) )  /\  ( ( F  o F  -  ( G  o F  x.  p
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  p )
) )  <  (deg `  G ) ) ) )  ->  ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
) )
2311, 12, 13, 14, 15, 16, 17, 18, 19, 20, 8, 21, 22plydiveu 20207 . . . 4  |-  ( ( ( ph  /\  (
q  e.  (Poly `  S )  /\  p  e.  (Poly `  S )
) )  /\  (
( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) )  /\  ( ( F  o F  -  ( G  o F  x.  p
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  p )
) )  <  (deg `  G ) ) ) )  ->  q  =  p )
2423ex 424 . . 3  |-  ( (
ph  /\  ( q  e.  (Poly `  S )  /\  p  e.  (Poly `  S ) ) )  ->  ( ( ( R  =  0 p  \/  (deg `  R
)  <  (deg `  G
) )  /\  (
( F  o F  -  ( G  o F  x.  p )
)  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  p
) ) )  < 
(deg `  G )
) )  ->  q  =  p ) )
2524ralrimivva 2790 . 2  |-  ( ph  ->  A. q  e.  (Poly `  S ) A. p  e.  (Poly `  S )
( ( ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
)  /\  ( ( F  o F  -  ( G  o F  x.  p
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  p )
) )  <  (deg `  G ) ) )  ->  q  =  p ) )
26 oveq2 6081 . . . . . . 7  |-  ( q  =  p  ->  ( G  o F  x.  q
)  =  ( G  o F  x.  p
) )
2726oveq2d 6089 . . . . . 6  |-  ( q  =  p  ->  ( F  o F  -  ( G  o F  x.  q
) )  =  ( F  o F  -  ( G  o F  x.  p ) ) )
288, 27syl5eq 2479 . . . . 5  |-  ( q  =  p  ->  R  =  ( F  o F  -  ( G  o F  x.  p
) ) )
2928eqeq1d 2443 . . . 4  |-  ( q  =  p  ->  ( R  =  0 p  <->  ( F  o F  -  ( G  o F  x.  p ) )  =  0 p ) )
3028fveq2d 5724 . . . . 5  |-  ( q  =  p  ->  (deg `  R )  =  (deg
`  ( F  o F  -  ( G  o F  x.  p
) ) ) )
3130breq1d 4214 . . . 4  |-  ( q  =  p  ->  (
(deg `  R )  <  (deg `  G )  <->  (deg
`  ( F  o F  -  ( G  o F  x.  p
) ) )  < 
(deg `  G )
) )
3229, 31orbi12d 691 . . 3  |-  ( q  =  p  ->  (
( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) )  <->  ( ( F  o F  -  ( G  o F  x.  p
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  p )
) )  <  (deg `  G ) ) ) )
3332reu4 3120 . 2  |-  ( E! q  e.  (Poly `  S ) ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
)  <->  ( E. q  e.  (Poly `  S )
( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) )  /\  A. q  e.  (Poly `  S ) A. p  e.  (Poly `  S )
( ( ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
)  /\  ( ( F  o F  -  ( G  o F  x.  p
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  p )
) )  <  (deg `  G ) ) )  ->  q  =  p ) ) )
349, 25, 33sylanbrc 646 1  |-  ( ph  ->  E! q  e.  (Poly `  S ) ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   E!wreu 2699   class class class wbr 4204   ` cfv 5446  (class class class)co 6073    o Fcof 6295   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    < clt 9112    - cmin 9283   -ucneg 9284    / cdiv 9669   0 pc0p 19553  Polycply 20095  degcdgr 20098
This theorem is referenced by:  quotlem  20209
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-rlim 12275  df-sum 12472  df-0p 19554  df-ply 20099  df-coe 20101  df-dgr 20102
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