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Theorem quotval 20214
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 20124 . . 3  |-  (Poly `  S )  C_  (Poly `  CC )
21sseli 3346 . 2  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
31sseli 3346 . . 3  |-  ( G  e.  (Poly `  S
)  ->  G  e.  (Poly `  CC ) )
4 eldifsn 3929 . . . . 5  |-  ( G  e.  ( (Poly `  CC )  \  { 0 p } )  <->  ( G  e.  (Poly `  CC )  /\  G  =/=  0 p ) )
5 oveq1 6091 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
g  o F  x.  q )  =  ( G  o F  x.  q ) )
6 oveq12 6093 . . . . . . . . . . 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 462 . . . . . . . . . 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 2488 . . . . . . . . 9  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f  o F  -  ( g  o F  x.  q ) )  =  R )
10 dfsbcq 3165 . . . . . . . . 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 6109 . . . . . . . . . . 11  |-  ( F  o F  -  ( G  o F  x.  q
) )  e.  _V
138, 12eqeltri 2508 . . . . . . . . . 10  |-  R  e. 
_V
14 eqeq1 2444 . . . . . . . . . . 11  |-  ( r  =  R  ->  (
r  =  0 p  <-> 
R  =  0 p ) )
15 fveq2 5731 . . . . . . . . . . . 12  |-  ( r  =  R  ->  (deg `  r )  =  (deg
`  R ) )
1615breq1d 4225 . . . . . . . . . . 11  |-  ( r  =  R  ->  (
(deg `  r )  <  (deg `  g )  <->  (deg
`  R )  < 
(deg `  g )
) )
1714, 16orbi12d 692 . . . . . . . . . 10  |-  ( r  =  R  ->  (
( r  =  0 p  \/  (deg `  r )  <  (deg `  g ) )  <->  ( R  =  0 p  \/  (deg `  R )  < 
(deg `  g )
) ) )
1813, 17sbcie 3197 . . . . . . . . 9  |-  ( [. R  /  r ]. (
r  =  0 p  \/  (deg `  r
)  <  (deg `  g
) )  <->  ( R  =  0 p  \/  (deg `  R )  < 
(deg `  g )
) )
19 simpr 449 . . . . . . . . . . . 12  |-  ( ( f  =  F  /\  g  =  G )  ->  g  =  G )
2019fveq2d 5735 . . . . . . . . . . 11  |-  ( ( f  =  F  /\  g  =  G )  ->  (deg `  g )  =  (deg `  G )
)
2120breq2d 4227 . . . . . . . . . 10  |-  ( ( f  =  F  /\  g  =  G )  ->  ( (deg `  R
)  <  (deg `  g
)  <->  (deg `  R )  <  (deg `  G )
) )
2221orbi2d 684 . . . . . . . . 9  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( R  =  0 p  \/  (deg `  R )  <  (deg `  g ) )  <->  ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
) ) )
2318, 22syl5bb 250 . . . . . . . 8  |-  ( ( f  =  F  /\  g  =  G )  ->  ( [. R  / 
r ]. ( r  =  0 p  \/  (deg `  r )  <  (deg `  g ) )  <->  ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
) ) )
2411, 23bitrd 246 . . . . . . 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 6554 . . . . . 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 20213 . . . . . 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 6556 . . . . . 6  |-  ( iota_ q  e.  (Poly `  CC ) ( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) ) )  e.  _V
2825, 26, 27ovmpt2a 6207 . . . . 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 464 . . . 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 1150 . . 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 1219 . 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 1218 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 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   _Vcvv 2958   [.wsbc 3163    \ cdif 3319   {csn 3816   class class class wbr 4215   ` cfv 5457  (class class class)co 6084    o Fcof 6306   iota_crio 6545   CCcc 8993    x. cmul 9000    < clt 9125    - cmin 9296   0 pc0p 19564  Polycply 20108  degcdgr 20111   quot cquot 20212
This theorem is referenced by:  quotlem  20222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-i2m1 9063  ax-1ne0 9064  ax-rrecex 9067  ax-cnre 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-map 7023  df-nn 10006  df-n0 10227  df-ply 20112  df-quot 20213
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