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Theorem plydivalg 22879
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  =/=  0p )
plydiv.r  |-  R  =  ( F  oF  -  ( G  oF  x.  q )
)
Assertion
Ref Expression
plydivalg  |-  ( ph  ->  E! q  e.  (Poly `  S ) ( R  =  0p  \/  (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  =/=  0p )
8 plydiv.r . . 3  |-  R  =  ( F  oF  -  ( G  oF  x.  q )
)
91, 2, 3, 4, 5, 6, 7, 8plydivex 22877 . 2  |-  ( ph  ->  E. q  e.  (Poly `  S ) ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) )
10 simpll 752 . . . . . 6  |-  ( ( ( ph  /\  (
q  e.  (Poly `  S )  /\  p  e.  (Poly `  S )
) )  /\  (
( R  =  0p  \/  (deg `  R )  <  (deg `  G ) )  /\  ( ( F  oF  -  ( G  oF  x.  p
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  p )
) )  <  (deg `  G ) ) ) )  ->  ph )
1110, 1sylan 469 . . . . 5  |-  ( ( ( ( ph  /\  ( q  e.  (Poly `  S )  /\  p  e.  (Poly `  S )
) )  /\  (
( R  =  0p  \/  (deg `  R )  <  (deg `  G ) )  /\  ( ( F  oF  -  ( G  oF  x.  p
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  p )
) )  <  (deg `  G ) ) ) )  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  +  y )  e.  S )
1210, 2sylan 469 . . . . 5  |-  ( ( ( ( ph  /\  ( q  e.  (Poly `  S )  /\  p  e.  (Poly `  S )
) )  /\  (
( R  =  0p  \/  (deg `  R )  <  (deg `  G ) )  /\  ( ( F  oF  -  ( G  oF  x.  p
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  p )
) )  <  (deg `  G ) ) ) )  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  x.  y
)  e.  S )
1310, 3sylan 469 . . . . 5  |-  ( ( ( ( ph  /\  ( q  e.  (Poly `  S )  /\  p  e.  (Poly `  S )
) )  /\  (
( R  =  0p  \/  (deg `  R )  <  (deg `  G ) )  /\  ( ( F  oF  -  ( G  oF  x.  p
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  p )
) )  <  (deg `  G ) ) ) )  /\  ( x  e.  S  /\  x  =/=  0 ) )  -> 
( 1  /  x
)  e.  S )
1410, 4syl 17 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  (Poly `  S )  /\  p  e.  (Poly `  S )
) )  /\  (
( R  =  0p  \/  (deg `  R )  <  (deg `  G ) )  /\  ( ( F  oF  -  ( G  oF  x.  p
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  p )
) )  <  (deg `  G ) ) ) )  ->  -u 1  e.  S )
1510, 5syl 17 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  (Poly `  S )  /\  p  e.  (Poly `  S )
) )  /\  (
( R  =  0p  \/  (deg `  R )  <  (deg `  G ) )  /\  ( ( F  oF  -  ( G  oF  x.  p
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  p )
) )  <  (deg `  G ) ) ) )  ->  F  e.  (Poly `  S ) )
1610, 6syl 17 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  (Poly `  S )  /\  p  e.  (Poly `  S )
) )  /\  (
( R  =  0p  \/  (deg `  R )  <  (deg `  G ) )  /\  ( ( F  oF  -  ( G  oF  x.  p
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  p )
) )  <  (deg `  G ) ) ) )  ->  G  e.  (Poly `  S ) )
1710, 7syl 17 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  (Poly `  S )  /\  p  e.  (Poly `  S )
) )  /\  (
( R  =  0p  \/  (deg `  R )  <  (deg `  G ) )  /\  ( ( F  oF  -  ( G  oF  x.  p
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  p )
) )  <  (deg `  G ) ) ) )  ->  G  =/=  0p )
18 eqid 2402 . . . . 5  |-  ( F  oF  -  ( G  oF  x.  p
) )  =  ( F  oF  -  ( G  oF  x.  p ) )
19 simplrr 763 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  (Poly `  S )  /\  p  e.  (Poly `  S )
) )  /\  (
( R  =  0p  \/  (deg `  R )  <  (deg `  G ) )  /\  ( ( F  oF  -  ( G  oF  x.  p
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  p )
) )  <  (deg `  G ) ) ) )  ->  p  e.  (Poly `  S ) )
20 simprr 758 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  (Poly `  S )  /\  p  e.  (Poly `  S )
) )  /\  (
( R  =  0p  \/  (deg `  R )  <  (deg `  G ) )  /\  ( ( F  oF  -  ( G  oF  x.  p
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  p )
) )  <  (deg `  G ) ) ) )  ->  ( ( F  oF  -  ( G  oF  x.  p
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  p )
) )  <  (deg `  G ) ) )
21 simplrl 762 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  (Poly `  S )  /\  p  e.  (Poly `  S )
) )  /\  (
( R  =  0p  \/  (deg `  R )  <  (deg `  G ) )  /\  ( ( F  oF  -  ( G  oF  x.  p
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  p )
) )  <  (deg `  G ) ) ) )  ->  q  e.  (Poly `  S ) )
22 simprl 756 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  (Poly `  S )  /\  p  e.  (Poly `  S )
) )  /\  (
( R  =  0p  \/  (deg `  R )  <  (deg `  G ) )  /\  ( ( F  oF  -  ( G  oF  x.  p
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  p )
) )  <  (deg `  G ) ) ) )  ->  ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) )
2311, 12, 13, 14, 15, 16, 17, 18, 19, 20, 8, 21, 22plydiveu 22878 . . . 4  |-  ( ( ( ph  /\  (
q  e.  (Poly `  S )  /\  p  e.  (Poly `  S )
) )  /\  (
( R  =  0p  \/  (deg `  R )  <  (deg `  G ) )  /\  ( ( F  oF  -  ( G  oF  x.  p
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  p )
) )  <  (deg `  G ) ) ) )  ->  q  =  p )
2423ex 432 . . 3  |-  ( (
ph  /\  ( q  e.  (Poly `  S )  /\  p  e.  (Poly `  S ) ) )  ->  ( ( ( R  =  0p  \/  (deg `  R
)  <  (deg `  G
) )  /\  (
( F  oF  -  ( G  oF  x.  p )
)  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  p
) ) )  < 
(deg `  G )
) )  ->  q  =  p ) )
2524ralrimivva 2824 . 2  |-  ( ph  ->  A. q  e.  (Poly `  S ) A. p  e.  (Poly `  S )
( ( ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
)  /\  ( ( F  oF  -  ( G  oF  x.  p
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  p )
) )  <  (deg `  G ) ) )  ->  q  =  p ) )
26 oveq2 6242 . . . . . . 7  |-  ( q  =  p  ->  ( G  oF  x.  q
)  =  ( G  oF  x.  p
) )
2726oveq2d 6250 . . . . . 6  |-  ( q  =  p  ->  ( F  oF  -  ( G  oF  x.  q
) )  =  ( F  oF  -  ( G  oF  x.  p ) ) )
288, 27syl5eq 2455 . . . . 5  |-  ( q  =  p  ->  R  =  ( F  oF  -  ( G  oF  x.  p
) ) )
2928eqeq1d 2404 . . . 4  |-  ( q  =  p  ->  ( R  =  0p  <->  ( F  oF  -  ( G  oF  x.  p ) )  =  0p ) )
3028fveq2d 5809 . . . . 5  |-  ( q  =  p  ->  (deg `  R )  =  (deg
`  ( F  oF  -  ( G  oF  x.  p
) ) ) )
3130breq1d 4404 . . . 4  |-  ( q  =  p  ->  (
(deg `  R )  <  (deg `  G )  <->  (deg
`  ( F  oF  -  ( G  oF  x.  p
) ) )  < 
(deg `  G )
) )
3229, 31orbi12d 708 . . 3  |-  ( q  =  p  ->  (
( R  =  0p  \/  (deg `  R )  <  (deg `  G ) )  <->  ( ( F  oF  -  ( G  oF  x.  p
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  p )
) )  <  (deg `  G ) ) ) )
3332reu4 3242 . 2  |-  ( E! q  e.  (Poly `  S ) ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
)  <->  ( E. q  e.  (Poly `  S )
( R  =  0p  \/  (deg `  R )  <  (deg `  G ) )  /\  A. q  e.  (Poly `  S ) A. p  e.  (Poly `  S )
( ( ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
)  /\  ( ( F  oF  -  ( G  oF  x.  p
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  p )
) )  <  (deg `  G ) ) )  ->  q  =  p ) ) )
349, 25, 33sylanbrc 662 1  |-  ( ph  ->  E! q  e.  (Poly `  S ) ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2753   E.wrex 2754   E!wreu 2755   class class class wbr 4394   ` cfv 5525  (class class class)co 6234    oFcof 6475   0cc0 9442   1c1 9443    + caddc 9445    x. cmul 9447    < clt 9578    - cmin 9761   -ucneg 9762    / cdiv 10167   0pc0p 22260  Polycply 22765  degcdgr 22768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-inf2 8011  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519  ax-pre-sup 9520  ax-addf 9521
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-of 6477  df-om 6639  df-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-1o 7087  df-oadd 7091  df-er 7268  df-map 7379  df-pm 7380  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-sup 7855  df-oi 7889  df-card 8272  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-div 10168  df-nn 10497  df-2 10555  df-3 10556  df-n0 10757  df-z 10826  df-uz 11046  df-rp 11184  df-fz 11644  df-fzo 11768  df-fl 11879  df-seq 12062  df-exp 12121  df-hash 12360  df-cj 12988  df-re 12989  df-im 12990  df-sqrt 13124  df-abs 13125  df-clim 13367  df-rlim 13368  df-sum 13565  df-0p 22261  df-ply 22769  df-coe 22771  df-dgr 22772
This theorem is referenced by:  quotlem  22880
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