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Theorem quotval 19776
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 19686 . . 3  |-  (Poly `  S )  C_  (Poly `  CC )
21sseli 3252 . 2  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
31sseli 3252 . . 3  |-  ( G  e.  (Poly `  S
)  ->  G  e.  (Poly `  CC ) )
4 eldifsn 3825 . . . . 5  |-  ( G  e.  ( (Poly `  CC )  \  { 0 p } )  <->  ( G  e.  (Poly `  CC )  /\  G  =/=  0 p ) )
5 oveq1 5952 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
g  o F  x.  q )  =  ( G  o F  x.  q ) )
6 oveq12 5954 . . . . . . . . . . 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 460 . . . . . . . . . 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 2408 . . . . . . . . 9  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f  o F  -  ( g  o F  x.  q ) )  =  R )
10 dfsbcq 3069 . . . . . . . . 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 15 . . . . . . . 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 5970 . . . . . . . . . . 11  |-  ( F  o F  -  ( G  o F  x.  q
) )  e.  _V
138, 12eqeltri 2428 . . . . . . . . . 10  |-  R  e. 
_V
14 eqeq1 2364 . . . . . . . . . . 11  |-  ( r  =  R  ->  (
r  =  0 p  <-> 
R  =  0 p ) )
15 fveq2 5608 . . . . . . . . . . . 12  |-  ( r  =  R  ->  (deg `  r )  =  (deg
`  R ) )
1615breq1d 4114 . . . . . . . . . . 11  |-  ( r  =  R  ->  (
(deg `  r )  <  (deg `  g )  <->  (deg
`  R )  < 
(deg `  g )
) )
1714, 16orbi12d 690 . . . . . . . . . 10  |-  ( r  =  R  ->  (
( r  =  0 p  \/  (deg `  r )  <  (deg `  g ) )  <->  ( R  =  0 p  \/  (deg `  R )  < 
(deg `  g )
) ) )
1813, 17sbcie 3101 . . . . . . . . 9  |-  ( [. R  /  r ]. (
r  =  0 p  \/  (deg `  r
)  <  (deg `  g
) )  <->  ( R  =  0 p  \/  (deg `  R )  < 
(deg `  g )
) )
19 simpr 447 . . . . . . . . . . . 12  |-  ( ( f  =  F  /\  g  =  G )  ->  g  =  G )
2019fveq2d 5612 . . . . . . . . . . 11  |-  ( ( f  =  F  /\  g  =  G )  ->  (deg `  g )  =  (deg `  G )
)
2120breq2d 4116 . . . . . . . . . 10  |-  ( ( f  =  F  /\  g  =  G )  ->  ( (deg `  R
)  <  (deg `  g
)  <->  (deg `  R )  <  (deg `  G )
) )
2221orbi2d 682 . . . . . . . . 9  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( R  =  0 p  \/  (deg `  R )  <  (deg `  g ) )  <->  ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
) ) )
2318, 22syl5bb 248 . . . . . . . 8  |-  ( ( f  =  F  /\  g  =  G )  ->  ( [. R  / 
r ]. ( r  =  0 p  \/  (deg `  r )  <  (deg `  g ) )  <->  ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
) ) )
2411, 23bitrd 244 . . . . . . 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 6393 . . . . . 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 19775 . . . . . 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 6395 . . . . . 6  |-  ( iota_ q  e.  (Poly `  CC ) ( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) ) )  e.  _V
2825, 26, 27ovmpt2a 6065 . . . . 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 462 . . . 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 1147 . . 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 1216 . 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 1215 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521   _Vcvv 2864   [.wsbc 3067    \ cdif 3225   {csn 3716   class class class wbr 4104   ` cfv 5337  (class class class)co 5945    o Fcof 6163   iota_crio 6384   CCcc 8825    x. cmul 8832    < clt 8957    - cmin 9127   0 pc0p 19128  Polycply 19670  degcdgr 19673   quot cquot 19774
This theorem is referenced by:  quotlem  19784
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-i2m1 8895  ax-1ne0 8896  ax-rrecex 8899  ax-cnre 8900
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-map 6862  df-nn 9837  df-n0 10058  df-ply 19674  df-quot 19775
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