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Theorem quotlem 22423
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 2460 . . . . . 6  |-  ( F  oF  -  ( G  oF  x.  q
) )  =  ( F  oF  -  ( G  oF  x.  q ) )
54quotval 22415 . . . . 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 1223 . . . 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 22422 . . . . . 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 3071 . . . . . 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 9563 . . . . . . 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 9565 . . . . . . 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 10203 . . . . . . 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 10628 . . . . . . 7  |-  -u 1  e.  CC
2120a1i 11 . . . . . 6  |-  ( ph  -> 
-u 1  e.  CC )
22 plyssc 22325 . . . . . . 7  |-  (Poly `  S )  C_  (Poly `  CC )
2322, 1sseldi 3495 . . . . . 6  |-  ( ph  ->  F  e.  (Poly `  CC ) )
2422, 2sseldi 3495 . . . . . 6  |-  ( ph  ->  G  e.  (Poly `  CC ) )
2515, 17, 19, 21, 23, 24, 3, 4plydivalg 22422 . . . . 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 2818 . . . . . 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 6263 . . . . . 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 2504 . . 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 6250 . . . 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 2548 . 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 6283 . . . . . . 7  |-  ( q  =  ( F quot  G
)  ->  ( G  oF  x.  q
)  =  ( G  oF  x.  ( F quot  G ) ) )
3635oveq2d 6291 . . . . . 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 2519 . . . . 5  |-  ( q  =  ( F quot  G
)  ->  ( F  oF  -  ( G  oF  x.  q
) )  =  R )
3938eqeq1d 2462 . . . 4  |-  ( q  =  ( F quot  G
)  ->  ( ( F  oF  -  ( G  oF  x.  q
) )  =  0p  <->  R  =  0p ) )
4038fveq2d 5861 . . . . 5  |-  ( q  =  ( F quot  G
)  ->  (deg `  ( F  oF  -  ( G  oF  x.  q
) ) )  =  (deg `  R )
)
4140breq1d 4450 . . . 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 3254 . 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 1374    e. wcel 1762    =/= wne 2655   A.wral 2807   E.wrex 2808   E!wreu 2809   {crab 2811    C_ wss 3469   class class class wbr 4440   ` cfv 5579   iota_crio 6235  (class class class)co 6275    oFcof 6513   CCcc 9479   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    < clt 9617    - cmin 9794   -ucneg 9795    / cdiv 10195   0pc0p 21804  Polycply 22309  degcdgr 22312   quot cquot 22413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-fz 11662  df-fzo 11782  df-fl 11886  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-rlim 13261  df-sum 13458  df-0p 21805  df-ply 22313  df-coe 22315  df-dgr 22316  df-quot 22414
This theorem is referenced by:  quotcl  22424  quotdgr  22426
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