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Theorem quotval 20170
Description: Value of the quotient function. (Contributed by Mario Carneiro, 23-Jul-2014.)
Hypothesis
Ref Expression
quotval.1  |-  R  =  ( F  o F  -  ( G  o F  x.  q )
)
Assertion
Ref Expression
quotval  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( F quot  G )  =  (
iota_ q  e.  (Poly `  CC ) ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
) ) )
Distinct variable groups:    F, q    G, q
Allowed substitution hints:    R( q)    S( q)

Proof of Theorem quotval
Dummy variables  f 
g  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plyssc 20080 . . 3  |-  (Poly `  S )  C_  (Poly `  CC )
21sseli 3312 . 2  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
31sseli 3312 . . 3  |-  ( G  e.  (Poly `  S
)  ->  G  e.  (Poly `  CC ) )
4 eldifsn 3895 . . . . 5  |-  ( G  e.  ( (Poly `  CC )  \  { 0 p } )  <->  ( G  e.  (Poly `  CC )  /\  G  =/=  0 p ) )
5 oveq1 6055 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
g  o F  x.  q )  =  ( G  o F  x.  q ) )
6 oveq12 6057 . . . . . . . . . . 11  |-  ( ( f  =  F  /\  ( g  o F  x.  q )  =  ( G  o F  x.  q ) )  ->  ( f  o F  -  ( g  o F  x.  q
) )  =  ( F  o F  -  ( G  o F  x.  q ) ) )
75, 6sylan2 461 . . . . . . . . . 10  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f  o F  -  ( g  o F  x.  q ) )  =  ( F  o F  -  ( G  o F  x.  q
) ) )
8 quotval.1 . . . . . . . . . 10  |-  R  =  ( F  o F  -  ( G  o F  x.  q )
)
97, 8syl6eqr 2462 . . . . . . . . 9  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f  o F  -  ( g  o F  x.  q ) )  =  R )
10 dfsbcq 3131 . . . . . . . . 9  |-  ( ( f  o F  -  ( g  o F  x.  q ) )  =  R  ->  ( [. ( f  o F  -  ( g  o F  x.  q ) )  /  r ]. ( r  =  0 p  \/  (deg `  r )  <  (deg `  g ) )  <->  [. R  / 
r ]. ( r  =  0 p  \/  (deg `  r )  <  (deg `  g ) ) ) )
119, 10syl 16 . . . . . . . 8  |-  ( ( f  =  F  /\  g  =  G )  ->  ( [. ( f  o F  -  (
g  o F  x.  q ) )  / 
r ]. ( r  =  0 p  \/  (deg `  r )  <  (deg `  g ) )  <->  [. R  / 
r ]. ( r  =  0 p  \/  (deg `  r )  <  (deg `  g ) ) ) )
12 ovex 6073 . . . . . . . . . . 11  |-  ( F  o F  -  ( G  o F  x.  q
) )  e.  _V
138, 12eqeltri 2482 . . . . . . . . . 10  |-  R  e. 
_V
14 eqeq1 2418 . . . . . . . . . . 11  |-  ( r  =  R  ->  (
r  =  0 p  <-> 
R  =  0 p ) )
15 fveq2 5695 . . . . . . . . . . . 12  |-  ( r  =  R  ->  (deg `  r )  =  (deg
`  R ) )
1615breq1d 4190 . . . . . . . . . . 11  |-  ( r  =  R  ->  (
(deg `  r )  <  (deg `  g )  <->  (deg
`  R )  < 
(deg `  g )
) )
1714, 16orbi12d 691 . . . . . . . . . 10  |-  ( r  =  R  ->  (
( r  =  0 p  \/  (deg `  r )  <  (deg `  g ) )  <->  ( R  =  0 p  \/  (deg `  R )  < 
(deg `  g )
) ) )
1813, 17sbcie 3163 . . . . . . . . 9  |-  ( [. R  /  r ]. (
r  =  0 p  \/  (deg `  r
)  <  (deg `  g
) )  <->  ( R  =  0 p  \/  (deg `  R )  < 
(deg `  g )
) )
19 simpr 448 . . . . . . . . . . . 12  |-  ( ( f  =  F  /\  g  =  G )  ->  g  =  G )
2019fveq2d 5699 . . . . . . . . . . 11  |-  ( ( f  =  F  /\  g  =  G )  ->  (deg `  g )  =  (deg `  G )
)
2120breq2d 4192 . . . . . . . . . 10  |-  ( ( f  =  F  /\  g  =  G )  ->  ( (deg `  R
)  <  (deg `  g
)  <->  (deg `  R )  <  (deg `  G )
) )
2221orbi2d 683 . . . . . . . . 9  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( R  =  0 p  \/  (deg `  R )  <  (deg `  g ) )  <->  ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
) ) )
2318, 22syl5bb 249 . . . . . . . 8  |-  ( ( f  =  F  /\  g  =  G )  ->  ( [. R  / 
r ]. ( r  =  0 p  \/  (deg `  r )  <  (deg `  g ) )  <->  ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
) ) )
2411, 23bitrd 245 . . . . . . 7  |-  ( ( f  =  F  /\  g  =  G )  ->  ( [. ( f  o F  -  (
g  o F  x.  q ) )  / 
r ]. ( r  =  0 p  \/  (deg `  r )  <  (deg `  g ) )  <->  ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
) ) )
2524riotabidv 6518 . . . . . 6  |-  ( ( f  =  F  /\  g  =  G )  ->  ( iota_ q  e.  (Poly `  CC ) [. (
f  o F  -  ( g  o F  x.  q ) )  /  r ]. (
r  =  0 p  \/  (deg `  r
)  <  (deg `  g
) ) )  =  ( iota_ q  e.  (Poly `  CC ) ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
) ) )
26 df-quot 20169 . . . . . 6  |- quot  =  ( f  e.  (Poly `  CC ) ,  g  e.  ( (Poly `  CC )  \  { 0 p } )  |->  ( iota_ q  e.  (Poly `  CC ) [. ( f  o F  -  ( g  o F  x.  q
) )  /  r ]. ( r  =  0 p  \/  (deg `  r )  <  (deg `  g ) ) ) )
27 riotaex 6520 . . . . . 6  |-  ( iota_ q  e.  (Poly `  CC ) ( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) ) )  e.  _V
2825, 26, 27ovmpt2a 6171 . . . . 5  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  ( (Poly `  CC )  \  { 0 p } ) )  -> 
( F quot  G )  =  ( iota_ q  e.  (Poly `  CC )
( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) ) ) )
294, 28sylan2br 463 . . . 4  |-  ( ( F  e.  (Poly `  CC )  /\  ( G  e.  (Poly `  CC )  /\  G  =/=  0 p ) )  -> 
( F quot  G )  =  ( iota_ q  e.  (Poly `  CC )
( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) ) ) )
30293impb 1149 . . 3  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )  /\  G  =/=  0 p )  ->  ( F quot  G )  =  (
iota_ q  e.  (Poly `  CC ) ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
) ) )
313, 30syl3an2 1218 . 2  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( F quot  G )  =  (
iota_ q  e.  (Poly `  CC ) ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
) ) )
322, 31syl3an1 1217 1  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0 p )  ->  ( F quot  G )  =  (
iota_ q  e.  (Poly `  CC ) ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   _Vcvv 2924   [.wsbc 3129    \ cdif 3285   {csn 3782   class class class wbr 4180   ` cfv 5421  (class class class)co 6048    o Fcof 6270   iota_crio 6509   CCcc 8952    x. cmul 8959    < clt 9084    - cmin 9255   0 pc0p 19522  Polycply 20064  degcdgr 20067   quot cquot 20168
This theorem is referenced by:  quotlem  20178
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-i2m1 9022  ax-1ne0 9023  ax-rrecex 9026  ax-cnre 9027
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-map 6987  df-nn 9965  df-n0 10186  df-ply 20068  df-quot 20169
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