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Theorem quotval 19672
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 19582 . . 3  |-  (Poly `  S )  C_  (Poly `  CC )
21sseli 3176 . 2  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
31sseli 3176 . . 3  |-  ( G  e.  (Poly `  S
)  ->  G  e.  (Poly `  CC ) )
4 eldifsn 3749 . . . . 5  |-  ( G  e.  ( (Poly `  CC )  \  { 0 p } )  <->  ( G  e.  (Poly `  CC )  /\  G  =/=  0 p ) )
5 oveq1 5865 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
g  o F  x.  q )  =  ( G  o F  x.  q ) )
6 oveq12 5867 . . . . . . . . . . 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 2333 . . . . . . . . 9  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f  o F  -  ( g  o F  x.  q ) )  =  R )
10 dfsbcq 2993 . . . . . . . . 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 5883 . . . . . . . . . . 11  |-  ( F  o F  -  ( G  o F  x.  q
) )  e.  _V
138, 12eqeltri 2353 . . . . . . . . . 10  |-  R  e. 
_V
14 eqeq1 2289 . . . . . . . . . . 11  |-  ( r  =  R  ->  (
r  =  0 p  <-> 
R  =  0 p ) )
15 fveq2 5525 . . . . . . . . . . . 12  |-  ( r  =  R  ->  (deg `  r )  =  (deg
`  R ) )
1615breq1d 4033 . . . . . . . . . . 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 3025 . . . . . . . . 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 5529 . . . . . . . . . . 11  |-  ( ( f  =  F  /\  g  =  G )  ->  (deg `  g )  =  (deg `  G )
)
2120breq2d 4035 . . . . . . . . . 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 6306 . . . . . 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 19671 . . . . . 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 6308 . . . . . 6  |-  ( iota_ q  e.  (Poly `  CC ) ( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) ) )  e.  _V
2825, 26, 27ovmpt2a 5978 . . . . 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 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   [.wsbc 2991    \ cdif 3149   {csn 3640   class class class wbr 4023   ` cfv 5255  (class class class)co 5858    o Fcof 6076   iota_crio 6297   CCcc 8735    x. cmul 8742    < clt 8867    - cmin 9037   0 pc0p 19024  Polycply 19566  degcdgr 19569   quot cquot 19670
This theorem is referenced by:  quotlem  19680
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-map 6774  df-nn 9747  df-n0 9966  df-ply 19570  df-quot 19671
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