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

Theorem plyrem 20179
Description: The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 13001). If a polynomial  F is divided by the linear factor  x  -  A, the remainder is equal to  F ( A ), the evaluation of the polynomial at  A (interpreted as a constant polynomial). (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypotheses
Ref Expression
plyrem.1  |-  G  =  ( X p  o F  -  ( CC  X.  { A } ) )
plyrem.2  |-  R  =  ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )
Assertion
Ref Expression
plyrem  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  R  =  ( CC  X.  { ( F `  A ) } ) )

Proof of Theorem plyrem
StepHypRef Expression
1 plyssc 20076 . . . . . . . 8  |-  (Poly `  S )  C_  (Poly `  CC )
2 simpl 444 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  F  e.  (Poly `  S )
)
31, 2sseldi 3310 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  F  e.  (Poly `  CC )
)
4 plyrem.1 . . . . . . . . . 10  |-  G  =  ( X p  o F  -  ( CC  X.  { A } ) )
54plyremlem 20178 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( G  e.  (Poly `  CC )  /\  (deg `  G
)  =  1  /\  ( `' G " { 0 } )  =  { A }
) )
65adantl 453 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( G  e.  (Poly `  CC )  /\  (deg `  G
)  =  1  /\  ( `' G " { 0 } )  =  { A }
) )
76simp1d 969 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  G  e.  (Poly `  CC )
)
86simp2d 970 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (deg `  G )  =  1 )
9 ax-1ne0 9019 . . . . . . . . . 10  |-  1  =/=  0
109a1i 11 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  1  =/=  0 )
118, 10eqnetrd 2589 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (deg `  G )  =/=  0
)
12 fveq2 5691 . . . . . . . . . 10  |-  ( G  =  0 p  -> 
(deg `  G )  =  (deg `  0 p
) )
13 dgr0 20137 . . . . . . . . . 10  |-  (deg ` 
0 p )  =  0
1412, 13syl6eq 2456 . . . . . . . . 9  |-  ( G  =  0 p  -> 
(deg `  G )  =  0 )
1514necon3i 2610 . . . . . . . 8  |-  ( (deg
`  G )  =/=  0  ->  G  =/=  0 p )
1611, 15syl 16 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  G  =/=  0 p )
17 plyrem.2 . . . . . . . 8  |-  R  =  ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )
1817quotdgr 20177 . . . . . . 7  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )  /\  G  =/=  0 p )  ->  ( R  =  0 p  \/  (deg `  R )  <  (deg `  G )
) )
193, 7, 16, 18syl3anc 1184 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( R  =  0 p  \/  (deg `  R )  <  (deg `  G )
) )
20 0lt1 9510 . . . . . . . 8  |-  0  <  1
2120, 8syl5breqr 4212 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  0  <  (deg `  G )
)
22 fveq2 5691 . . . . . . . . 9  |-  ( R  =  0 p  -> 
(deg `  R )  =  (deg `  0 p
) )
2322, 13syl6eq 2456 . . . . . . . 8  |-  ( R  =  0 p  -> 
(deg `  R )  =  0 )
2423breq1d 4186 . . . . . . 7  |-  ( R  =  0 p  -> 
( (deg `  R
)  <  (deg `  G
)  <->  0  <  (deg `  G ) ) )
2521, 24syl5ibrcom 214 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( R  =  0 p  ->  (deg `  R )  <  (deg `  G )
) )
26 pm2.62 399 . . . . . 6  |-  ( ( R  =  0 p  \/  (deg `  R
)  <  (deg `  G
) )  ->  (
( R  =  0 p  ->  (deg `  R
)  <  (deg `  G
) )  ->  (deg `  R )  <  (deg `  G ) ) )
2719, 25, 26sylc 58 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (deg `  R )  <  (deg `  G ) )
2827, 8breqtrd 4200 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (deg `  R )  <  1
)
29 quotcl2 20176 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )  /\  G  =/=  0 p )  ->  ( F quot  G )  e.  (Poly `  CC ) )
303, 7, 16, 29syl3anc 1184 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F quot  G )  e.  (Poly `  CC ) )
31 plymulcl 20097 . . . . . . . . 9  |-  ( ( G  e.  (Poly `  CC )  /\  ( F quot  G )  e.  (Poly `  CC ) )  -> 
( G  o F  x.  ( F quot  G
) )  e.  (Poly `  CC ) )
327, 30, 31syl2anc 643 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( G  o F  x.  ( F quot  G ) )  e.  (Poly `  CC )
)
33 plysubcl 20098 . . . . . . . 8  |-  ( ( F  e.  (Poly `  CC )  /\  ( G  o F  x.  ( F quot  G ) )  e.  (Poly `  CC )
)  ->  ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )  e.  (Poly `  CC ) )
343, 32, 33syl2anc 643 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )  e.  (Poly `  CC ) )
3517, 34syl5eqel 2492 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  R  e.  (Poly `  CC )
)
36 dgrcl 20109 . . . . . 6  |-  ( R  e.  (Poly `  CC )  ->  (deg `  R
)  e.  NN0 )
3735, 36syl 16 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (deg `  R )  e.  NN0 )
38 nn0lt10b 10296 . . . . 5  |-  ( (deg
`  R )  e. 
NN0  ->  ( (deg `  R )  <  1  <->  (deg
`  R )  =  0 ) )
3937, 38syl 16 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (
(deg `  R )  <  1  <->  (deg `  R )  =  0 ) )
4028, 39mpbid 202 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (deg `  R )  =  0 )
41 0dgrb 20122 . . . 4  |-  ( R  e.  (Poly `  CC )  ->  ( (deg `  R )  =  0  <-> 
R  =  ( CC 
X.  { ( R `
 0 ) } ) ) )
4235, 41syl 16 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (
(deg `  R )  =  0  <->  R  =  ( CC  X.  { ( R `  0 ) } ) ) )
4340, 42mpbid 202 . 2  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  R  =  ( CC  X.  { ( R ` 
0 ) } ) )
4443fveq1d 5693 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( R `  A )  =  ( ( CC 
X.  { ( R `
 0 ) } ) `  A ) )
4517fveq1i 5692 . . . . . . 7  |-  ( R `
 A )  =  ( ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) ) `
 A )
46 plyf 20074 . . . . . . . . . . 11  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
4746adantr 452 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  F : CC --> CC )
48 ffn 5554 . . . . . . . . . 10  |-  ( F : CC --> CC  ->  F  Fn  CC )
4947, 48syl 16 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  F  Fn  CC )
50 plyf 20074 . . . . . . . . . . . 12  |-  ( G  e.  (Poly `  CC )  ->  G : CC --> CC )
517, 50syl 16 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  G : CC --> CC )
52 ffn 5554 . . . . . . . . . . 11  |-  ( G : CC --> CC  ->  G  Fn  CC )
5351, 52syl 16 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  G  Fn  CC )
54 plyf 20074 . . . . . . . . . . . 12  |-  ( ( F quot  G )  e.  (Poly `  CC )  ->  ( F quot  G ) : CC --> CC )
5530, 54syl 16 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F quot  G ) : CC --> CC )
56 ffn 5554 . . . . . . . . . . 11  |-  ( ( F quot  G ) : CC --> CC  ->  ( F quot  G )  Fn  CC )
5755, 56syl 16 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F quot  G )  Fn  CC )
58 cnex 9031 . . . . . . . . . . 11  |-  CC  e.  _V
5958a1i 11 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  CC  e.  _V )
60 inidm 3514 . . . . . . . . . 10  |-  ( CC 
i^i  CC )  =  CC
6153, 57, 59, 59, 60offn 6279 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( G  o F  x.  ( F quot  G ) )  Fn  CC )
62 eqidd 2409 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  /\  A  e.  CC )  ->  ( F `  A )  =  ( F `  A ) )
636simp3d 971 . . . . . . . . . . . . . . 15  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( `' G " { 0 } )  =  { A } )
64 ssun1 3474 . . . . . . . . . . . . . . 15  |-  ( `' G " { 0 } )  C_  (
( `' G " { 0 } )  u.  ( `' ( F quot  G ) " { 0 } ) )
6563, 64syl6eqssr 3363 . . . . . . . . . . . . . 14  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  { A }  C_  ( ( `' G " { 0 } )  u.  ( `' ( F quot  G
) " { 0 } ) ) )
66 snssg 3896 . . . . . . . . . . . . . . 15  |-  ( A  e.  CC  ->  ( A  e.  ( ( `' G " { 0 } )  u.  ( `' ( F quot  G
) " { 0 } ) )  <->  { A }  C_  ( ( `' G " { 0 } )  u.  ( `' ( F quot  G
) " { 0 } ) ) ) )
6766adantl 453 . . . . . . . . . . . . . 14  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( A  e.  ( ( `' G " { 0 } )  u.  ( `' ( F quot  G
) " { 0 } ) )  <->  { A }  C_  ( ( `' G " { 0 } )  u.  ( `' ( F quot  G
) " { 0 } ) ) ) )
6865, 67mpbird 224 . . . . . . . . . . . . 13  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  A  e.  ( ( `' G " { 0 } )  u.  ( `' ( F quot  G ) " { 0 } ) ) )
69 ofmulrt 20156 . . . . . . . . . . . . . 14  |-  ( ( CC  e.  _V  /\  G : CC --> CC  /\  ( F quot  G ) : CC --> CC )  -> 
( `' ( G  o F  x.  ( F quot  G ) ) " { 0 } )  =  ( ( `' G " { 0 } )  u.  ( `' ( F quot  G
) " { 0 } ) ) )
7059, 51, 55, 69syl3anc 1184 . . . . . . . . . . . . 13  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( `' ( G  o F  x.  ( F quot  G ) ) " {
0 } )  =  ( ( `' G " { 0 } )  u.  ( `' ( F quot  G ) " { 0 } ) ) )
7168, 70eleqtrrd 2485 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  A  e.  ( `' ( G  o F  x.  ( F quot  G ) ) " { 0 } ) )
72 fniniseg 5814 . . . . . . . . . . . . 13  |-  ( ( G  o F  x.  ( F quot  G )
)  Fn  CC  ->  ( A  e.  ( `' ( G  o F  x.  ( F quot  G
) ) " {
0 } )  <->  ( A  e.  CC  /\  ( ( G  o F  x.  ( F quot  G )
) `  A )  =  0 ) ) )
7361, 72syl 16 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( A  e.  ( `' ( G  o F  x.  ( F quot  G ) ) " { 0 } )  <->  ( A  e.  CC  /\  ( ( G  o F  x.  ( F quot  G )
) `  A )  =  0 ) ) )
7471, 73mpbid 202 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( A  e.  CC  /\  (
( G  o F  x.  ( F quot  G
) ) `  A
)  =  0 ) )
7574simprd 450 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (
( G  o F  x.  ( F quot  G
) ) `  A
)  =  0 )
7675adantr 452 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  /\  A  e.  CC )  ->  (
( G  o F  x.  ( F quot  G
) ) `  A
)  =  0 )
7749, 61, 59, 59, 60, 62, 76ofval 6277 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  /\  A  e.  CC )  ->  (
( F  o F  -  ( G  o F  x.  ( F quot  G ) ) ) `  A )  =  ( ( F `  A
)  -  0 ) )
7877anabss3 797 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (
( F  o F  -  ( G  o F  x.  ( F quot  G ) ) ) `  A )  =  ( ( F `  A
)  -  0 ) )
7945, 78syl5eq 2452 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( R `  A )  =  ( ( F `
 A )  - 
0 ) )
8046ffvelrnda 5833 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F `  A )  e.  CC )
8180subid1d 9360 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (
( F `  A
)  -  0 )  =  ( F `  A ) )
8279, 81eqtrd 2440 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( R `  A )  =  ( F `  A ) )
83 fvex 5705 . . . . . . 7  |-  ( R `
 0 )  e. 
_V
8483fvconst2 5910 . . . . . 6  |-  ( A  e.  CC  ->  (
( CC  X.  {
( R `  0
) } ) `  A )  =  ( R `  0 ) )
8584adantl 453 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (
( CC  X.  {
( R `  0
) } ) `  A )  =  ( R `  0 ) )
8644, 82, 853eqtr3d 2448 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F `  A )  =  ( R ` 
0 ) )
8786sneqd 3791 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  { ( F `  A ) }  =  { ( R `  0 ) } )
8887xpeq2d 4865 . 2  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( CC  X.  { ( F `
 A ) } )  =  ( CC 
X.  { ( R `
 0 ) } ) )
8943, 88eqtr4d 2443 1  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  R  =  ( CC  X.  { ( F `  A ) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2571   _Vcvv 2920    u. cun 3282    C_ wss 3284   {csn 3778   class class class wbr 4176    X. cxp 4839   `'ccnv 4840   "cima 4844    Fn wfn 5412   -->wf 5413   ` cfv 5417  (class class class)co 6044    o Fcof 6266   CCcc 8948   0cc0 8950   1c1 8951    x. cmul 8955    < clt 9080    - cmin 9251   NN0cn0 10181   0 pc0p 19518  Polycply 20060   X pcidp 20061  degcdgr 20063   quot cquot 20164
This theorem is referenced by:  facth  20180
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028  ax-addf 9029
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-of 6268  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-map 6983  df-pm 6984  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-sup 7408  df-oi 7439  df-card 7786  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-n0 10182  df-z 10243  df-uz 10449  df-rp 10573  df-fz 11004  df-fzo 11095  df-fl 11161  df-seq 11283  df-exp 11342  df-hash 11578  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-clim 12241  df-rlim 12242  df-sum 12439  df-0p 19519  df-ply 20064  df-idp 20065  df-coe 20066  df-dgr 20067  df-quot 20165
  Copyright terms: Public domain W3C validator