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Theorem quotval 23139
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 23048 . . 3  |-  (Poly `  S )  C_  (Poly `  CC )
21sseli 3457 . 2  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
31sseli 3457 . . 3  |-  ( G  e.  (Poly `  S
)  ->  G  e.  (Poly `  CC ) )
4 eldifsn 4119 . . . . 5  |-  ( G  e.  ( (Poly `  CC )  \  { 0p } )  <->  ( G  e.  (Poly `  CC )  /\  G  =/=  0p ) )
5 oveq1 6303 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
g  oF  x.  q )  =  ( G  oF  x.  q ) )
6 oveq12 6305 . . . . . . . . . . 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 476 . . . . . . . . . 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 2479 . . . . . . . . 9  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f  oF  -  ( g  oF  x.  q ) )  =  R )
109sbceq1d 3301 . . . . . . . 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 ) ) ) )
11 ovex 6324 . . . . . . . . . . 11  |-  ( F  oF  -  ( G  oF  x.  q
) )  e.  _V
128, 11eqeltri 2504 . . . . . . . . . 10  |-  R  e. 
_V
13 eqeq1 2424 . . . . . . . . . . 11  |-  ( r  =  R  ->  (
r  =  0p  <-> 
R  =  0p ) )
14 fveq2 5872 . . . . . . . . . . . 12  |-  ( r  =  R  ->  (deg `  r )  =  (deg
`  R ) )
1514breq1d 4427 . . . . . . . . . . 11  |-  ( r  =  R  ->  (
(deg `  r )  <  (deg `  g )  <->  (deg
`  R )  < 
(deg `  g )
) )
1613, 15orbi12d 714 . . . . . . . . . 10  |-  ( r  =  R  ->  (
( r  =  0p  \/  (deg `  r )  <  (deg `  g ) )  <->  ( R  =  0p  \/  (deg `  R )  <  (deg `  g )
) ) )
1712, 16sbcie 3331 . . . . . . . . 9  |-  ( [. R  /  r ]. (
r  =  0p  \/  (deg `  r
)  <  (deg `  g
) )  <->  ( R  =  0p  \/  (deg `  R )  <  (deg `  g )
) )
18 simpr 462 . . . . . . . . . . . 12  |-  ( ( f  =  F  /\  g  =  G )  ->  g  =  G )
1918fveq2d 5876 . . . . . . . . . . 11  |-  ( ( f  =  F  /\  g  =  G )  ->  (deg `  g )  =  (deg `  G )
)
2019breq2d 4429 . . . . . . . . . 10  |-  ( ( f  =  F  /\  g  =  G )  ->  ( (deg `  R
)  <  (deg `  g
)  <->  (deg `  R )  <  (deg `  G )
) )
2120orbi2d 706 . . . . . . . . 9  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( R  =  0p  \/  (deg `  R )  <  (deg `  g ) )  <->  ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) ) )
2217, 21syl5bb 260 . . . . . . . 8  |-  ( ( f  =  F  /\  g  =  G )  ->  ( [. R  / 
r ]. ( r  =  0p  \/  (deg `  r )  <  (deg `  g ) )  <->  ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) ) )
2310, 22bitrd 256 . . . . . . 7  |-  ( ( f  =  F  /\  g  =  G )  ->  ( [. ( f  oF  -  (
g  oF  x.  q ) )  / 
r ]. ( r  =  0p  \/  (deg `  r )  <  (deg `  g ) )  <->  ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) ) )
2423riotabidv 6260 . . . . . 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 )
) ) )
25 df-quot 23138 . . . . . 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 ) ) ) )
26 riotaex 6262 . . . . . 6  |-  ( iota_ q  e.  (Poly `  CC ) ( R  =  0p  \/  (deg `  R )  <  (deg `  G ) ) )  e.  _V
2724, 25, 26ovmpt2a 6432 . . . . 5  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  ( (Poly `  CC )  \  { 0p } ) )  -> 
( F quot  G )  =  ( iota_ q  e.  (Poly `  CC )
( R  =  0p  \/  (deg `  R )  <  (deg `  G ) ) ) )
284, 27sylan2br 478 . . . 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 ) ) ) )
29283impb 1201 . . 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 )
) ) )
303, 29syl3an2 1298 . 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 )
) ) )
312, 30syl3an1 1297 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    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867    =/= wne 2616   _Vcvv 3078   [.wsbc 3296    \ cdif 3430   {csn 3993   class class class wbr 4417   ` cfv 5592   iota_crio 6257  (class class class)co 6296    oFcof 6534   CCcc 9526    x. cmul 9533    < clt 9664    - cmin 9849   0pc0p 22521  Polycply 23032  degcdgr 23035   quot cquot 23137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-i2m1 9596  ax-1ne0 9597  ax-rrecex 9600  ax-cnre 9601
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  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-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-map 7473  df-nn 10599  df-n0 10859  df-ply 23036  df-quot 23138
This theorem is referenced by:  quotlem  23147
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