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Theorem quotlem 21766
Description: Lemma for properties of the polynomial quotient function. (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 )
quotlem.8  |-  R  =  ( F  oF  -  ( G  oF  x.  ( F quot  G ) ) )
Assertion
Ref Expression
quotlem  |-  ( ph  ->  ( ( F quot  G
)  e.  (Poly `  S )  /\  ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) ) )
Distinct variable groups:    x, y, F    ph, x, y    x, G, y    x, R, y   
x, S, y

Proof of Theorem quotlem
Dummy variable  q is distinct from all other variables.
StepHypRef Expression
1 plydiv.f . . . . 5  |-  ( ph  ->  F  e.  (Poly `  S ) )
2 plydiv.g . . . . 5  |-  ( ph  ->  G  e.  (Poly `  S ) )
3 plydiv.z . . . . 5  |-  ( ph  ->  G  =/=  0p )
4 eqid 2443 . . . . . 6  |-  ( F  oF  -  ( G  oF  x.  q
) )  =  ( F  oF  -  ( G  oF  x.  q ) )
54quotval 21758 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( F quot  G )  =  (
iota_ q  e.  (Poly `  CC ) ( ( F  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) )
61, 2, 3, 5syl3anc 1218 . . . 4  |-  ( ph  ->  ( F quot  G )  =  ( iota_ q  e.  (Poly `  CC )
( ( F  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) )
7 plydiv.pl . . . . . . 7  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  +  y )  e.  S )
8 plydiv.tm . . . . . . 7  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  x.  y
)  e.  S )
9 plydiv.rc . . . . . . 7  |-  ( (
ph  /\  ( x  e.  S  /\  x  =/=  0 ) )  -> 
( 1  /  x
)  e.  S )
10 plydiv.m1 . . . . . . 7  |-  ( ph  -> 
-u 1  e.  S
)
117, 8, 9, 10, 1, 2, 3, 4plydivalg 21765 . . . . . 6  |-  ( ph  ->  E! q  e.  (Poly `  S ) ( ( F  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )
12 reurex 2937 . . . . . 6  |-  ( E! q  e.  (Poly `  S ) ( ( F  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) )  ->  E. q  e.  (Poly `  S ) ( ( F  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )
1311, 12syl 16 . . . . 5  |-  ( ph  ->  E. q  e.  (Poly `  S ) ( ( F  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )
14 addcl 9364 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  e.  CC )
1514adantl 466 . . . . . 6  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  +  y )  e.  CC )
16 mulcl 9366 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
1716adantl 466 . . . . . 6  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
18 reccl 10001 . . . . . . 7  |-  ( ( x  e.  CC  /\  x  =/=  0 )  -> 
( 1  /  x
)  e.  CC )
1918adantl 466 . . . . . 6  |-  ( (
ph  /\  ( x  e.  CC  /\  x  =/=  0 ) )  -> 
( 1  /  x
)  e.  CC )
20 neg1cn 10425 . . . . . . 7  |-  -u 1  e.  CC
2120a1i 11 . . . . . 6  |-  ( ph  -> 
-u 1  e.  CC )
22 plyssc 21668 . . . . . . 7  |-  (Poly `  S )  C_  (Poly `  CC )
2322, 1sseldi 3354 . . . . . 6  |-  ( ph  ->  F  e.  (Poly `  CC ) )
2422, 2sseldi 3354 . . . . . 6  |-  ( ph  ->  G  e.  (Poly `  CC ) )
2515, 17, 19, 21, 23, 24, 3, 4plydivalg 21765 . . . . 5  |-  ( ph  ->  E! q  e.  (Poly `  CC ) ( ( F  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )
26 id 22 . . . . . . 7  |-  ( ( ( F  oF  -  ( G  oF  x.  q )
)  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q
) ) )  < 
(deg `  G )
)  ->  ( ( F  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )
2726rgenw 2783 . . . . . 6  |-  A. q  e.  (Poly `  S )
( ( ( F  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) )  -> 
( ( F  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )
28 riotass2 6079 . . . . . 6  |-  ( ( ( (Poly `  S
)  C_  (Poly `  CC )  /\  A. q  e.  (Poly `  S )
( ( ( F  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) )  -> 
( ( F  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) )  /\  ( E. q  e.  (Poly `  S ) ( ( F  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) )  /\  E! q  e.  (Poly `  CC ) ( ( F  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) )  ->  ( iota_ q  e.  (Poly `  S
) ( ( F  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  =  ( iota_ q  e.  (Poly `  CC )
( ( F  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) )
2922, 27, 28mpanl12 682 . . . . 5  |-  ( ( E. q  e.  (Poly `  S ) ( ( F  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) )  /\  E! q  e.  (Poly `  CC ) ( ( F  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  ->  ( iota_ q  e.  (Poly `  S )
( ( F  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  =  ( iota_ q  e.  (Poly `  CC )
( ( F  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) )
3013, 25, 29syl2anc 661 . . . 4  |-  ( ph  ->  ( iota_ q  e.  (Poly `  S ) ( ( F  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  =  ( iota_ q  e.  (Poly `  CC )
( ( F  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) )
316, 30eqtr4d 2478 . . 3  |-  ( ph  ->  ( F quot  G )  =  ( iota_ q  e.  (Poly `  S )
( ( F  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) ) )
32 riotacl2 6066 . . . 4  |-  ( E! q  e.  (Poly `  S ) ( ( F  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) )  -> 
( iota_ q  e.  (Poly `  S ) ( ( F  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  e.  { q  e.  (Poly `  S )  |  ( ( F  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) } )
3311, 32syl 16 . . 3  |-  ( ph  ->  ( iota_ q  e.  (Poly `  S ) ( ( F  oF  -  ( G  oF  x.  q ) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) )  e.  { q  e.  (Poly `  S )  |  ( ( F  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) } )
3431, 33eqeltrd 2517 . 2  |-  ( ph  ->  ( F quot  G )  e.  { q  e.  (Poly `  S )  |  ( ( F  oF  -  ( G  oF  x.  q
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G ) ) } )
35 oveq2 6099 . . . . . . 7  |-  ( q  =  ( F quot  G
)  ->  ( G  oF  x.  q
)  =  ( G  oF  x.  ( F quot  G ) ) )
3635oveq2d 6107 . . . . . 6  |-  ( q  =  ( F quot  G
)  ->  ( F  oF  -  ( G  oF  x.  q
) )  =  ( F  oF  -  ( G  oF  x.  ( F quot  G ) ) ) )
37 quotlem.8 . . . . . 6  |-  R  =  ( F  oF  -  ( G  oF  x.  ( F quot  G ) ) )
3836, 37syl6eqr 2493 . . . . 5  |-  ( q  =  ( F quot  G
)  ->  ( F  oF  -  ( G  oF  x.  q
) )  =  R )
3938eqeq1d 2451 . . . 4  |-  ( q  =  ( F quot  G
)  ->  ( ( F  oF  -  ( G  oF  x.  q
) )  =  0p  <->  R  =  0p ) )
4038fveq2d 5695 . . . . 5  |-  ( q  =  ( F quot  G
)  ->  (deg `  ( F  oF  -  ( G  oF  x.  q
) ) )  =  (deg `  R )
)
4140breq1d 4302 . . . 4  |-  ( q  =  ( F quot  G
)  ->  ( (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G )  <->  (deg `  R
)  <  (deg `  G
) ) )
4239, 41orbi12d 709 . . 3  |-  ( q  =  ( F quot  G
)  ->  ( (
( F  oF  -  ( G  oF  x.  q )
)  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q
) ) )  < 
(deg `  G )
)  <->  ( R  =  0p  \/  (deg `  R )  <  (deg `  G ) ) ) )
4342elrab 3117 . 2  |-  ( ( F quot  G )  e. 
{ q  e.  (Poly `  S )  |  ( ( F  oF  -  ( G  oF  x.  q )
)  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  q
) ) )  < 
(deg `  G )
) }  <->  ( ( F quot  G )  e.  (Poly `  S )  /\  ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) ) )
4434, 43sylib 196 1  |-  ( ph  ->  ( ( F quot  G
)  e.  (Poly `  S )  /\  ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   E.wrex 2716   E!wreu 2717   {crab 2719    C_ wss 3328   class class class wbr 4292   ` cfv 5418   iota_crio 6051  (class class class)co 6091    oFcof 6318   CCcc 9280   0cc0 9282   1c1 9283    + caddc 9285    x. cmul 9287    < clt 9418    - cmin 9595   -ucneg 9596    / cdiv 9993   0pc0p 21147  Polycply 21652  degcdgr 21655   quot cquot 21756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fz 11438  df-fzo 11549  df-fl 11642  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-clim 12966  df-rlim 12967  df-sum 13164  df-0p 21148  df-ply 21656  df-coe 21658  df-dgr 21659  df-quot 21757
This theorem is referenced by:  quotcl  21767  quotdgr  21769
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