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Theorem plydivalg 20084
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 20082 . 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 2388 . . . . 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 20083 . . . 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 2742 . 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 6029 . . . . . . 7  |-  ( q  =  p  ->  ( G  o F  x.  q
)  =  ( G  o F  x.  p
) )
2726oveq2d 6037 . . . . . 6  |-  ( q  =  p  ->  ( F  o F  -  ( G  o F  x.  q
) )  =  ( F  o F  -  ( G  o F  x.  p ) ) )
288, 27syl5eq 2432 . . . . 5  |-  ( q  =  p  ->  R  =  ( F  o F  -  ( G  o F  x.  p
) ) )
2928eqeq1d 2396 . . . 4  |-  ( q  =  p  ->  ( R  =  0 p  <->  ( F  o F  -  ( G  o F  x.  p ) )  =  0 p ) )
3028fveq2d 5673 . . . . 5  |-  ( q  =  p  ->  (deg `  R )  =  (deg
`  ( F  o F  -  ( G  o F  x.  p
) ) ) )
3130breq1d 4164 . . . 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 3072 . 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 1649    e. wcel 1717    =/= wne 2551   A.wral 2650   E.wrex 2651   E!wreu 2652   class class class wbr 4154   ` cfv 5395  (class class class)co 6021    o Fcof 6243   0cc0 8924   1c1 8925    + caddc 8927    x. cmul 8929    < clt 9054    - cmin 9224   -ucneg 9225    / cdiv 9610   0 pc0p 19429  Polycply 19971  degcdgr 19974
This theorem is referenced by:  quotlem  20085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002  ax-addf 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-map 6957  df-pm 6958  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-sup 7382  df-oi 7413  df-card 7760  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-rp 10546  df-fz 10977  df-fzo 11067  df-fl 11130  df-seq 11252  df-exp 11311  df-hash 11547  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-clim 12210  df-rlim 12211  df-sum 12408  df-0p 19430  df-ply 19975  df-coe 19977  df-dgr 19978
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