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Theorem quotval 21763
Description: Value of the quotient function. (Contributed by Mario Carneiro, 23-Jul-2014.)
Hypothesis
Ref Expression
quotval.1  |-  R  =  ( F  oF  -  ( G  oF  x.  q )
)
Assertion
Ref Expression
quotval  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( F quot  G )  =  (
iota_ q  e.  (Poly `  CC ) ( R  =  0p  \/  (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 21673 . . 3  |-  (Poly `  S )  C_  (Poly `  CC )
21sseli 3357 . 2  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
31sseli 3357 . . 3  |-  ( G  e.  (Poly `  S
)  ->  G  e.  (Poly `  CC ) )
4 eldifsn 4005 . . . . 5  |-  ( G  e.  ( (Poly `  CC )  \  { 0p } )  <->  ( G  e.  (Poly `  CC )  /\  G  =/=  0p ) )
5 oveq1 6103 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
g  oF  x.  q )  =  ( G  oF  x.  q ) )
6 oveq12 6105 . . . . . . . . . . 11  |-  ( ( f  =  F  /\  ( g  oF  x.  q )  =  ( G  oF  x.  q ) )  ->  ( f  oF  -  ( g  oF  x.  q
) )  =  ( F  oF  -  ( G  oF  x.  q ) ) )
75, 6sylan2 474 . . . . . . . . . 10  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f  oF  -  ( g  oF  x.  q ) )  =  ( F  oF  -  ( G  oF  x.  q
) ) )
8 quotval.1 . . . . . . . . . 10  |-  R  =  ( F  oF  -  ( G  oF  x.  q )
)
97, 8syl6eqr 2493 . . . . . . . . 9  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f  oF  -  ( g  oF  x.  q ) )  =  R )
10 dfsbcq 3193 . . . . . . . . 9  |-  ( ( f  oF  -  ( g  oF  x.  q ) )  =  R  ->  ( [. ( f  oF  -  ( g  oF  x.  q ) )  /  r ]. ( r  =  0p  \/  (deg `  r )  <  (deg `  g ) )  <->  [. R  / 
r ]. ( r  =  0p  \/  (deg `  r )  <  (deg `  g ) ) ) )
119, 10syl 16 . . . . . . . 8  |-  ( ( f  =  F  /\  g  =  G )  ->  ( [. ( f  oF  -  (
g  oF  x.  q ) )  / 
r ]. ( r  =  0p  \/  (deg `  r )  <  (deg `  g ) )  <->  [. R  / 
r ]. ( r  =  0p  \/  (deg `  r )  <  (deg `  g ) ) ) )
12 ovex 6121 . . . . . . . . . . 11  |-  ( F  oF  -  ( G  oF  x.  q
) )  e.  _V
138, 12eqeltri 2513 . . . . . . . . . 10  |-  R  e. 
_V
14 eqeq1 2449 . . . . . . . . . . 11  |-  ( r  =  R  ->  (
r  =  0p  <-> 
R  =  0p ) )
15 fveq2 5696 . . . . . . . . . . . 12  |-  ( r  =  R  ->  (deg `  r )  =  (deg
`  R ) )
1615breq1d 4307 . . . . . . . . . . 11  |-  ( r  =  R  ->  (
(deg `  r )  <  (deg `  g )  <->  (deg
`  R )  < 
(deg `  g )
) )
1714, 16orbi12d 709 . . . . . . . . . 10  |-  ( r  =  R  ->  (
( r  =  0p  \/  (deg `  r )  <  (deg `  g ) )  <->  ( R  =  0p  \/  (deg `  R )  <  (deg `  g )
) ) )
1813, 17sbcie 3226 . . . . . . . . 9  |-  ( [. R  /  r ]. (
r  =  0p  \/  (deg `  r
)  <  (deg `  g
) )  <->  ( R  =  0p  \/  (deg `  R )  <  (deg `  g )
) )
19 simpr 461 . . . . . . . . . . . 12  |-  ( ( f  =  F  /\  g  =  G )  ->  g  =  G )
2019fveq2d 5700 . . . . . . . . . . 11  |-  ( ( f  =  F  /\  g  =  G )  ->  (deg `  g )  =  (deg `  G )
)
2120breq2d 4309 . . . . . . . . . 10  |-  ( ( f  =  F  /\  g  =  G )  ->  ( (deg `  R
)  <  (deg `  g
)  <->  (deg `  R )  <  (deg `  G )
) )
2221orbi2d 701 . . . . . . . . 9  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( R  =  0p  \/  (deg `  R )  <  (deg `  g ) )  <->  ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) ) )
2318, 22syl5bb 257 . . . . . . . 8  |-  ( ( f  =  F  /\  g  =  G )  ->  ( [. R  / 
r ]. ( r  =  0p  \/  (deg `  r )  <  (deg `  g ) )  <->  ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) ) )
2411, 23bitrd 253 . . . . . . 7  |-  ( ( f  =  F  /\  g  =  G )  ->  ( [. ( f  oF  -  (
g  oF  x.  q ) )  / 
r ]. ( r  =  0p  \/  (deg `  r )  <  (deg `  g ) )  <->  ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) ) )
2524riotabidv 6059 . . . . . 6  |-  ( ( f  =  F  /\  g  =  G )  ->  ( iota_ q  e.  (Poly `  CC ) [. (
f  oF  -  ( g  oF  x.  q ) )  /  r ]. (
r  =  0p  \/  (deg `  r
)  <  (deg `  g
) ) )  =  ( iota_ q  e.  (Poly `  CC ) ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) ) )
26 df-quot 21762 . . . . . 6  |- quot  =  ( f  e.  (Poly `  CC ) ,  g  e.  ( (Poly `  CC )  \  { 0p } )  |->  ( iota_ q  e.  (Poly `  CC ) [. ( f  oF  -  ( g  oF  x.  q
) )  /  r ]. ( r  =  0p  \/  (deg `  r )  <  (deg `  g ) ) ) )
27 riotaex 6061 . . . . . 6  |-  ( iota_ q  e.  (Poly `  CC ) ( R  =  0p  \/  (deg `  R )  <  (deg `  G ) ) )  e.  _V
2825, 26, 27ovmpt2a 6226 . . . . 5  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  ( (Poly `  CC )  \  { 0p } ) )  -> 
( F quot  G )  =  ( iota_ q  e.  (Poly `  CC )
( R  =  0p  \/  (deg `  R )  <  (deg `  G ) ) ) )
294, 28sylan2br 476 . . . 4  |-  ( ( F  e.  (Poly `  CC )  /\  ( G  e.  (Poly `  CC )  /\  G  =/=  0p ) )  -> 
( F quot  G )  =  ( iota_ q  e.  (Poly `  CC )
( R  =  0p  \/  (deg `  R )  <  (deg `  G ) ) ) )
30293impb 1183 . . 3  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )  /\  G  =/=  0p )  ->  ( F quot  G )  =  (
iota_ q  e.  (Poly `  CC ) ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) ) )
313, 30syl3an2 1252 . 2  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( F quot  G )  =  (
iota_ q  e.  (Poly `  CC ) ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) ) )
322, 31syl3an1 1251 1  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( F quot  G )  =  (
iota_ q  e.  (Poly `  CC ) ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   _Vcvv 2977   [.wsbc 3191    \ cdif 3330   {csn 3882   class class class wbr 4297   ` cfv 5423   iota_crio 6056  (class class class)co 6096    oFcof 6323   CCcc 9285    x. cmul 9292    < clt 9423    - cmin 9600   0pc0p 21152  Polycply 21657  degcdgr 21660   quot cquot 21761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-i2m1 9355  ax-1ne0 9356  ax-rrecex 9359  ax-cnre 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  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 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-map 7221  df-nn 10328  df-n0 10585  df-ply 21661  df-quot 21762
This theorem is referenced by:  quotlem  21771
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