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Theorem plydivalg 19695
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 19693 . 2  |-  ( ph  ->  E. q  e.  (Poly `  S ) ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
) )
10 simpll 730 . . . . . 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 457 . . . . 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 457 . . . . 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 457 . . . . 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 15 . . . . 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 15 . . . . 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 15 . . . . 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 15 . . . . 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 2296 . . . . 5  |-  ( F  o F  -  ( G  o F  x.  p
) )  =  ( F  o F  -  ( G  o F  x.  p ) )
19 simplrr 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 ) ) ) )  ->  p  e.  (Poly `  S ) )
20 simprr 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 ) ) ) )  ->  ( ( F  o F  -  ( G  o F  x.  p
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  p )
) )  <  (deg `  G ) ) )
21 simplrl 736 . . . . 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 732 . . . . 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 19694 . . . 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 423 . . 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 2648 . 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 5882 . . . . . . 7  |-  ( q  =  p  ->  ( G  o F  x.  q
)  =  ( G  o F  x.  p
) )
2726oveq2d 5890 . . . . . 6  |-  ( q  =  p  ->  ( F  o F  -  ( G  o F  x.  q
) )  =  ( F  o F  -  ( G  o F  x.  p ) ) )
288, 27syl5eq 2340 . . . . 5  |-  ( q  =  p  ->  R  =  ( F  o F  -  ( G  o F  x.  p
) ) )
2928eqeq1d 2304 . . . 4  |-  ( q  =  p  ->  ( R  =  0 p  <->  ( F  o F  -  ( G  o F  x.  p ) )  =  0 p ) )
3028fveq2d 5545 . . . . 5  |-  ( q  =  p  ->  (deg `  R )  =  (deg
`  ( F  o F  -  ( G  o F  x.  p
) ) ) )
3130breq1d 4049 . . . 4  |-  ( q  =  p  ->  (
(deg `  R )  <  (deg `  G )  <->  (deg
`  ( F  o F  -  ( G  o F  x.  p
) ) )  < 
(deg `  G )
) )
3229, 31orbi12d 690 . . 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 2972 . 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 645 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 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   E!wreu 2558   class class class wbr 4039   ` cfv 5271  (class class class)co 5874    o Fcof 6092   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    - cmin 9053   -ucneg 9054    / cdiv 9439   0 pc0p 19040  Polycply 19582  degcdgr 19585
This theorem is referenced by:  quotlem  19696
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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