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Theorem quotlem 22986
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 2402 . . . . . 6  |-  ( F  oF  -  ( G  oF  x.  q
) )  =  ( F  oF  -  ( G  oF  x.  q ) )
54quotval 22978 . . . . 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 1230 . . . 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 22985 . . . . . 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 3023 . . . . . 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 17 . . . . 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 9603 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  e.  CC )
1514adantl 464 . . . . . 6  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  +  y )  e.  CC )
16 mulcl 9605 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
1716adantl 464 . . . . . 6  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
18 reccl 10254 . . . . . . 7  |-  ( ( x  e.  CC  /\  x  =/=  0 )  -> 
( 1  /  x
)  e.  CC )
1918adantl 464 . . . . . 6  |-  ( (
ph  /\  ( x  e.  CC  /\  x  =/=  0 ) )  -> 
( 1  /  x
)  e.  CC )
20 neg1cn 10679 . . . . . . 7  |-  -u 1  e.  CC
2120a1i 11 . . . . . 6  |-  ( ph  -> 
-u 1  e.  CC )
22 plyssc 22887 . . . . . . 7  |-  (Poly `  S )  C_  (Poly `  CC )
2322, 1sseldi 3439 . . . . . 6  |-  ( ph  ->  F  e.  (Poly `  CC ) )
2422, 2sseldi 3439 . . . . . 6  |-  ( ph  ->  G  e.  (Poly `  CC ) )
2515, 17, 19, 21, 23, 24, 3, 4plydivalg 22985 . . . . 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 2764 . . . . . 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 6265 . . . . . 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 680 . . . . 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 659 . . . 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 2446 . . 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 6252 . . . 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 17 . . 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 2490 . 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 6285 . . . . . . 7  |-  ( q  =  ( F quot  G
)  ->  ( G  oF  x.  q
)  =  ( G  oF  x.  ( F quot  G ) ) )
3635oveq2d 6293 . . . . . 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 2461 . . . . 5  |-  ( q  =  ( F quot  G
)  ->  ( F  oF  -  ( G  oF  x.  q
) )  =  R )
3938eqeq1d 2404 . . . 4  |-  ( q  =  ( F quot  G
)  ->  ( ( F  oF  -  ( G  oF  x.  q
) )  =  0p  <->  R  =  0p ) )
4038fveq2d 5852 . . . . 5  |-  ( q  =  ( F quot  G
)  ->  (deg `  ( F  oF  -  ( G  oF  x.  q
) ) )  =  (deg `  R )
)
4140breq1d 4404 . . . 4  |-  ( q  =  ( F quot  G
)  ->  ( (deg `  ( F  oF  -  ( G  oF  x.  q )
) )  <  (deg `  G )  <->  (deg `  R
)  <  (deg `  G
) ) )
4239, 41orbi12d 708 . . 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 3206 . 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 366    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2753   E.wrex 2754   E!wreu 2755   {crab 2757    C_ wss 3413   class class class wbr 4394   ` cfv 5568   iota_crio 6238  (class class class)co 6277    oFcof 6518   CCcc 9519   0cc0 9521   1c1 9522    + caddc 9524    x. cmul 9526    < clt 9657    - cmin 9840   -ucneg 9841    / cdiv 10246   0pc0p 22366  Polycply 22871  degcdgr 22874   quot cquot 22976
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 6573  ax-inf2 8090  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599  ax-addf 9600
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-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6520  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-pm 7459  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-sup 7934  df-oi 7968  df-card 8351  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-n0 10836  df-z 10905  df-uz 11127  df-rp 11265  df-fz 11725  df-fzo 11853  df-fl 11964  df-seq 12150  df-exp 12209  df-hash 12451  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-clim 13458  df-rlim 13459  df-sum 13656  df-0p 22367  df-ply 22875  df-coe 22877  df-dgr 22878  df-quot 22977
This theorem is referenced by:  quotcl  22987  quotdgr  22989
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