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Theorem quotlem 20217
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  =/=  0 p )
quotlem.8  |-  R  =  ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )
Assertion
Ref Expression
quotlem  |-  ( ph  ->  ( ( F quot  G
)  e.  (Poly `  S )  /\  ( R  =  0 p  \/  (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  =/=  0 p )
4 eqid 2436 . . . . . 6  |-  ( F  o F  -  ( G  o F  x.  q
) )  =  ( F  o F  -  ( G  o F  x.  q ) )
54quotval 20209 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( F quot  G )  =  (
iota_ q  e.  (Poly `  CC ) ( ( F  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) )
61, 2, 3, 5syl3anc 1184 . . . 4  |-  ( ph  ->  ( F quot  G )  =  ( iota_ q  e.  (Poly `  CC )
( ( F  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  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 20216 . . . . . 6  |-  ( ph  ->  E! q  e.  (Poly `  S ) ( ( F  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )
12 reurex 2922 . . . . . 6  |-  ( E! q  e.  (Poly `  S ) ( ( F  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) )  ->  E. q  e.  (Poly `  S ) ( ( F  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )
1311, 12syl 16 . . . . 5  |-  ( ph  ->  E. q  e.  (Poly `  S ) ( ( F  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )
14 addcl 9072 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  e.  CC )
1514adantl 453 . . . . . 6  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  +  y )  e.  CC )
16 mulcl 9074 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
1716adantl 453 . . . . . 6  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
18 reccl 9685 . . . . . . 7  |-  ( ( x  e.  CC  /\  x  =/=  0 )  -> 
( 1  /  x
)  e.  CC )
1918adantl 453 . . . . . 6  |-  ( (
ph  /\  ( x  e.  CC  /\  x  =/=  0 ) )  -> 
( 1  /  x
)  e.  CC )
20 neg1cn 10067 . . . . . . 7  |-  -u 1  e.  CC
2120a1i 11 . . . . . 6  |-  ( ph  -> 
-u 1  e.  CC )
22 plyssc 20119 . . . . . . 7  |-  (Poly `  S )  C_  (Poly `  CC )
2322, 1sseldi 3346 . . . . . 6  |-  ( ph  ->  F  e.  (Poly `  CC ) )
2422, 2sseldi 3346 . . . . . 6  |-  ( ph  ->  G  e.  (Poly `  CC ) )
2515, 17, 19, 21, 23, 24, 3, 4plydivalg 20216 . . . . 5  |-  ( ph  ->  E! q  e.  (Poly `  CC ) ( ( F  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )
26 id 20 . . . . . . 7  |-  ( ( ( F  o F  -  ( G  o F  x.  q )
)  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q
) ) )  < 
(deg `  G )
)  ->  ( ( F  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )
2726rgenw 2773 . . . . . 6  |-  A. q  e.  (Poly `  S )
( ( ( F  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) )  -> 
( ( F  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )
28 riotass2 6577 . . . . . 6  |-  ( ( ( (Poly `  S
)  C_  (Poly `  CC )  /\  A. q  e.  (Poly `  S )
( ( ( F  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) )  -> 
( ( F  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) )  /\  ( E. q  e.  (Poly `  S ) ( ( F  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) )  /\  E! q  e.  (Poly `  CC ) ( ( F  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) )  ->  ( iota_ q  e.  (Poly `  S
) ( ( F  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  =  ( iota_ q  e.  (Poly `  CC )
( ( F  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) )
2922, 27, 28mpanl12 664 . . . . 5  |-  ( ( E. q  e.  (Poly `  S ) ( ( F  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) )  /\  E! q  e.  (Poly `  CC ) ( ( F  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  ->  ( iota_ q  e.  (Poly `  S )
( ( F  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  =  ( iota_ q  e.  (Poly `  CC )
( ( F  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) )
3013, 25, 29syl2anc 643 . . . 4  |-  ( ph  ->  ( iota_ q  e.  (Poly `  S ) ( ( F  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  =  ( iota_ q  e.  (Poly `  CC )
( ( F  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) )
316, 30eqtr4d 2471 . . 3  |-  ( ph  ->  ( F quot  G )  =  ( iota_ q  e.  (Poly `  S )
( ( F  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) ) )
32 riotacl2 6563 . . . 4  |-  ( E! q  e.  (Poly `  S ) ( ( F  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) )  -> 
( iota_ q  e.  (Poly `  S ) ( ( F  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  e.  { q  e.  (Poly `  S )  |  ( ( F  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) } )
3311, 32syl 16 . . 3  |-  ( ph  ->  ( iota_ q  e.  (Poly `  S ) ( ( F  o F  -  ( G  o F  x.  q ) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) )  e.  { q  e.  (Poly `  S )  |  ( ( F  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) } )
3431, 33eqeltrd 2510 . 2  |-  ( ph  ->  ( F quot  G )  e.  { q  e.  (Poly `  S )  |  ( ( F  o F  -  ( G  o F  x.  q
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G ) ) } )
35 oveq2 6089 . . . . . . 7  |-  ( q  =  ( F quot  G
)  ->  ( G  o F  x.  q
)  =  ( G  o F  x.  ( F quot  G ) ) )
3635oveq2d 6097 . . . . . 6  |-  ( q  =  ( F quot  G
)  ->  ( F  o F  -  ( G  o F  x.  q
) )  =  ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) ) )
37 quotlem.8 . . . . . 6  |-  R  =  ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )
3836, 37syl6eqr 2486 . . . . 5  |-  ( q  =  ( F quot  G
)  ->  ( F  o F  -  ( G  o F  x.  q
) )  =  R )
3938eqeq1d 2444 . . . 4  |-  ( q  =  ( F quot  G
)  ->  ( ( F  o F  -  ( G  o F  x.  q
) )  =  0 p  <->  R  =  0 p ) )
4038fveq2d 5732 . . . . 5  |-  ( q  =  ( F quot  G
)  ->  (deg `  ( F  o F  -  ( G  o F  x.  q
) ) )  =  (deg `  R )
)
4140breq1d 4222 . . . 4  |-  ( q  =  ( F quot  G
)  ->  ( (deg `  ( F  o F  -  ( G  o F  x.  q )
) )  <  (deg `  G )  <->  (deg `  R
)  <  (deg `  G
) ) )
4239, 41orbi12d 691 . . 3  |-  ( q  =  ( F quot  G
)  ->  ( (
( F  o F  -  ( G  o F  x.  q )
)  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q
) ) )  < 
(deg `  G )
)  <->  ( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) ) ) )
4342elrab 3092 . 2  |-  ( ( F quot  G )  e. 
{ q  e.  (Poly `  S )  |  ( ( F  o F  -  ( G  o F  x.  q )
)  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  q
) ) )  < 
(deg `  G )
) }  <->  ( ( F quot  G )  e.  (Poly `  S )  /\  ( R  =  0 p  \/  (deg `  R )  <  (deg `  G )
) ) )
4434, 43sylib 189 1  |-  ( ph  ->  ( ( F quot  G
)  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 2599   A.wral 2705   E.wrex 2706   E!wreu 2707   {crab 2709    C_ wss 3320   class class class wbr 4212   ` cfv 5454  (class class class)co 6081    o Fcof 6303   iota_crio 6542   CCcc 8988   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    < clt 9120    - cmin 9291   -ucneg 9292    / cdiv 9677   0 pc0p 19561  Polycply 20103  degcdgr 20106   quot cquot 20207
This theorem is referenced by:  quotcl  20218  quotdgr  20220
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-rlim 12283  df-sum 12480  df-0p 19562  df-ply 20107  df-coe 20109  df-dgr 20110  df-quot 20208
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