MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  quotval Structured version   Unicode version

Theorem quotval 22450
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 22360 . . 3  |-  (Poly `  S )  C_  (Poly `  CC )
21sseli 3500 . 2  |-  ( F  e.  (Poly `  S
)  ->  F  e.  (Poly `  CC ) )
31sseli 3500 . . 3  |-  ( G  e.  (Poly `  S
)  ->  G  e.  (Poly `  CC ) )
4 eldifsn 4152 . . . . 5  |-  ( G  e.  ( (Poly `  CC )  \  { 0p } )  <->  ( G  e.  (Poly `  CC )  /\  G  =/=  0p ) )
5 oveq1 6291 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
g  oF  x.  q )  =  ( G  oF  x.  q ) )
6 oveq12 6293 . . . . . . . . . . 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 2526 . . . . . . . . 9  |-  ( ( f  =  F  /\  g  =  G )  ->  ( f  oF  -  ( g  oF  x.  q ) )  =  R )
10 dfsbcq 3333 . . . . . . . . 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 6309 . . . . . . . . . . 11  |-  ( F  oF  -  ( G  oF  x.  q
) )  e.  _V
138, 12eqeltri 2551 . . . . . . . . . 10  |-  R  e. 
_V
14 eqeq1 2471 . . . . . . . . . . 11  |-  ( r  =  R  ->  (
r  =  0p  <-> 
R  =  0p ) )
15 fveq2 5866 . . . . . . . . . . . 12  |-  ( r  =  R  ->  (deg `  r )  =  (deg
`  R ) )
1615breq1d 4457 . . . . . . . . . . 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 3366 . . . . . . . . 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 5870 . . . . . . . . . . 11  |-  ( ( f  =  F  /\  g  =  G )  ->  (deg `  g )  =  (deg `  G )
)
2120breq2d 4459 . . . . . . . . . 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 6247 . . . . . 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 22449 . . . . . 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 6249 . . . . . 6  |-  ( iota_ q  e.  (Poly `  CC ) ( R  =  0p  \/  (deg `  R )  <  (deg `  G ) ) )  e.  _V
2825, 26, 27ovmpt2a 6417 . . . . 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 1192 . . 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 1262 . 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 1261 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113   [.wsbc 3331    \ cdif 3473   {csn 4027   class class class wbr 4447   ` cfv 5588   iota_crio 6244  (class class class)co 6284    oFcof 6522   CCcc 9490    x. cmul 9497    < clt 9628    - cmin 9805   0pc0p 21839  Polycply 22344  degcdgr 22347   quot cquot 22448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-i2m1 9560  ax-1ne0 9561  ax-rrecex 9564  ax-cnre 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-map 7422  df-nn 10537  df-n0 10796  df-ply 22348  df-quot 22449
This theorem is referenced by:  quotlem  22458
  Copyright terms: Public domain W3C validator