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Theorem plyrem 19685
Description: The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 12721). 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 19582 . . . . . . . 8  |-  (Poly `  S )  C_  (Poly `  CC )
2 simpl 443 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  F  e.  (Poly `  S )
)
31, 2sseldi 3178 . . . . . . 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 19684 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( G  e.  (Poly `  CC )  /\  (deg `  G
)  =  1  /\  ( `' G " { 0 } )  =  { A }
) )
65adantl 452 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( G  e.  (Poly `  CC )  /\  (deg `  G
)  =  1  /\  ( `' G " { 0 } )  =  { A }
) )
76simp1d 967 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  G  e.  (Poly `  CC )
)
86simp2d 968 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (deg `  G )  =  1 )
9 ax-1ne0 8806 . . . . . . . . . 10  |-  1  =/=  0
109a1i 10 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  1  =/=  0 )
118, 10eqnetrd 2464 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (deg `  G )  =/=  0
)
12 fveq2 5525 . . . . . . . . . 10  |-  ( G  =  0 p  -> 
(deg `  G )  =  (deg `  0 p
) )
13 dgr0 19643 . . . . . . . . . 10  |-  (deg ` 
0 p )  =  0
1412, 13syl6eq 2331 . . . . . . . . 9  |-  ( G  =  0 p  -> 
(deg `  G )  =  0 )
1514necon3i 2485 . . . . . . . 8  |-  ( (deg
`  G )  =/=  0  ->  G  =/=  0 p )
1611, 15syl 15 . . . . . . 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 19683 . . . . . . 7  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )  /\  G  =/=  0 p )  ->  ( R  =  0 p  \/  (deg `  R )  <  (deg `  G )
) )
193, 7, 16, 18syl3anc 1182 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( R  =  0 p  \/  (deg `  R )  <  (deg `  G )
) )
20 0lt1 9296 . . . . . . . 8  |-  0  <  1
2120, 8syl5breqr 4059 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  0  <  (deg `  G )
)
22 fveq2 5525 . . . . . . . . 9  |-  ( R  =  0 p  -> 
(deg `  R )  =  (deg `  0 p
) )
2322, 13syl6eq 2331 . . . . . . . 8  |-  ( R  =  0 p  -> 
(deg `  R )  =  0 )
2423breq1d 4033 . . . . . . 7  |-  ( R  =  0 p  -> 
( (deg `  R
)  <  (deg `  G
)  <->  0  <  (deg `  G ) ) )
2521, 24syl5ibrcom 213 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( R  =  0 p  ->  (deg `  R )  <  (deg `  G )
) )
26 pm2.62 398 . . . . . 6  |-  ( ( R  =  0 p  \/  (deg `  R
)  <  (deg `  G
) )  ->  (
( R  =  0 p  ->  (deg `  R
)  <  (deg `  G
) )  ->  (deg `  R )  <  (deg `  G ) ) )
2719, 25, 26sylc 56 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (deg `  R )  <  (deg `  G ) )
2827, 8breqtrd 4047 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (deg `  R )  <  1
)
29 quotcl2 19682 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )  /\  G  =/=  0 p )  ->  ( F quot  G )  e.  (Poly `  CC ) )
303, 7, 16, 29syl3anc 1182 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F quot  G )  e.  (Poly `  CC ) )
31 plymulcl 19603 . . . . . . . . 9  |-  ( ( G  e.  (Poly `  CC )  /\  ( F quot  G )  e.  (Poly `  CC ) )  -> 
( G  o F  x.  ( F quot  G
) )  e.  (Poly `  CC ) )
327, 30, 31syl2anc 642 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( G  o F  x.  ( F quot  G ) )  e.  (Poly `  CC )
)
33 plysubcl 19604 . . . . . . . 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 642 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )  e.  (Poly `  CC ) )
3517, 34syl5eqel 2367 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  R  e.  (Poly `  CC )
)
36 dgrcl 19615 . . . . . 6  |-  ( R  e.  (Poly `  CC )  ->  (deg `  R
)  e.  NN0 )
3735, 36syl 15 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (deg `  R )  e.  NN0 )
38 nn0lt10b 10078 . . . . 5  |-  ( (deg
`  R )  e. 
NN0  ->  ( (deg `  R )  <  1  <->  (deg
`  R )  =  0 ) )
3937, 38syl 15 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (
(deg `  R )  <  1  <->  (deg `  R )  =  0 ) )
4028, 39mpbid 201 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (deg `  R )  =  0 )
41 0dgrb 19628 . . . 4  |-  ( R  e.  (Poly `  CC )  ->  ( (deg `  R )  =  0  <-> 
R  =  ( CC 
X.  { ( R `
 0 ) } ) ) )
4235, 41syl 15 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (
(deg `  R )  =  0  <->  R  =  ( CC  X.  { ( R `  0 ) } ) ) )
4340, 42mpbid 201 . 2  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  R  =  ( CC  X.  { ( R ` 
0 ) } ) )
4443fveq1d 5527 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( R `  A )  =  ( ( CC 
X.  { ( R `
 0 ) } ) `  A ) )
4517fveq1i 5526 . . . . . . 7  |-  ( R `
 A )  =  ( ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) ) `
 A )
46 plyf 19580 . . . . . . . . . . 11  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
4746adantr 451 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  F : CC --> CC )
48 ffn 5389 . . . . . . . . . 10  |-  ( F : CC --> CC  ->  F  Fn  CC )
4947, 48syl 15 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  F  Fn  CC )
50 plyf 19580 . . . . . . . . . . . 12  |-  ( G  e.  (Poly `  CC )  ->  G : CC --> CC )
517, 50syl 15 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  G : CC --> CC )
52 ffn 5389 . . . . . . . . . . 11  |-  ( G : CC --> CC  ->  G  Fn  CC )
5351, 52syl 15 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  G  Fn  CC )
54 plyf 19580 . . . . . . . . . . . 12  |-  ( ( F quot  G )  e.  (Poly `  CC )  ->  ( F quot  G ) : CC --> CC )
5530, 54syl 15 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F quot  G ) : CC --> CC )
56 ffn 5389 . . . . . . . . . . 11  |-  ( ( F quot  G ) : CC --> CC  ->  ( F quot  G )  Fn  CC )
5755, 56syl 15 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F quot  G )  Fn  CC )
58 cnex 8818 . . . . . . . . . . 11  |-  CC  e.  _V
5958a1i 10 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  CC  e.  _V )
60 inidm 3378 . . . . . . . . . 10  |-  ( CC 
i^i  CC )  =  CC
6153, 57, 59, 59, 60offn 6089 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( G  o F  x.  ( F quot  G ) )  Fn  CC )
62 eqidd 2284 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  /\  A  e.  CC )  ->  ( F `  A )  =  ( F `  A ) )
636simp3d 969 . . . . . . . . . . . . . . 15  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( `' G " { 0 } )  =  { A } )
64 ssun1 3338 . . . . . . . . . . . . . . . 16  |-  ( `' G " { 0 } )  C_  (
( `' G " { 0 } )  u.  ( `' ( F quot  G ) " { 0 } ) )
6564a1i 10 . . . . . . . . . . . . . . 15  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( `' G " { 0 } )  C_  (
( `' G " { 0 } )  u.  ( `' ( F quot  G ) " { 0 } ) ) )
6663, 65eqsstr3d 3213 . . . . . . . . . . . . . 14  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  { A }  C_  ( ( `' G " { 0 } )  u.  ( `' ( F quot  G
) " { 0 } ) ) )
67 snssg 3754 . . . . . . . . . . . . . . 15  |-  ( A  e.  CC  ->  ( A  e.  ( ( `' G " { 0 } )  u.  ( `' ( F quot  G
) " { 0 } ) )  <->  { A }  C_  ( ( `' G " { 0 } )  u.  ( `' ( F quot  G
) " { 0 } ) ) ) )
6867adantl 452 . . . . . . . . . . . . . 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 } ) ) ) )
6966, 68mpbird 223 . . . . . . . . . . . . 13  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  A  e.  ( ( `' G " { 0 } )  u.  ( `' ( F quot  G ) " { 0 } ) ) )
70 ofmulrt 19662 . . . . . . . . . . . . . 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 } ) ) )
7159, 51, 55, 70syl3anc 1182 . . . . . . . . . . . . 13  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( `' ( G  o F  x.  ( F quot  G ) ) " {
0 } )  =  ( ( `' G " { 0 } )  u.  ( `' ( F quot  G ) " { 0 } ) ) )
7269, 71eleqtrrd 2360 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  A  e.  ( `' ( G  o F  x.  ( F quot  G ) ) " { 0 } ) )
73 fniniseg 5646 . . . . . . . . . . . . 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 ) ) )
7461, 73syl 15 . . . . . . . . . . . 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 ) ) )
7572, 74mpbid 201 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( A  e.  CC  /\  (
( G  o F  x.  ( F quot  G
) ) `  A
)  =  0 ) )
7675simprd 449 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (
( G  o F  x.  ( F quot  G
) ) `  A
)  =  0 )
7776adantr 451 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  /\  A  e.  CC )  ->  (
( G  o F  x.  ( F quot  G
) ) `  A
)  =  0 )
7849, 61, 59, 59, 60, 62, 77ofval 6087 . . . . . . . 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 ) )
7978anabss3 796 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (
( F  o F  -  ( G  o F  x.  ( F quot  G ) ) ) `  A )  =  ( ( F `  A
)  -  0 ) )
8045, 79syl5eq 2327 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( R `  A )  =  ( ( F `
 A )  - 
0 ) )
81 ffvelrn 5663 . . . . . . . 8  |-  ( ( F : CC --> CC  /\  A  e.  CC )  ->  ( F `  A
)  e.  CC )
8246, 81sylan 457 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F `  A )  e.  CC )
8382subid1d 9146 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (
( F `  A
)  -  0 )  =  ( F `  A ) )
8480, 83eqtrd 2315 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( R `  A )  =  ( F `  A ) )
85 fvex 5539 . . . . . . 7  |-  ( R `
 0 )  e. 
_V
8685fvconst2 5729 . . . . . 6  |-  ( A  e.  CC  ->  (
( CC  X.  {
( R `  0
) } ) `  A )  =  ( R `  0 ) )
8786adantl 452 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (
( CC  X.  {
( R `  0
) } ) `  A )  =  ( R `  0 ) )
8844, 84, 873eqtr3d 2323 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F `  A )  =  ( R ` 
0 ) )
8988sneqd 3653 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  { ( F `  A ) }  =  { ( R `  0 ) } )
9089xpeq2d 4713 . 2  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( CC  X.  { ( F `
 A ) } )  =  ( CC 
X.  { ( R `
 0 ) } ) )
9143, 90eqtr4d 2318 1  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  R  =  ( CC  X.  { ( F `  A ) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    u. cun 3150    C_ wss 3152   {csn 3640   class class class wbr 4023    X. cxp 4687   `'ccnv 4688   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   CCcc 8735   0cc0 8737   1c1 8738    x. cmul 8742    < clt 8867    - cmin 9037   NN0cn0 9965   0 pc0p 19024  Polycply 19566   X pcidp 19567  degcdgr 19569   quot cquot 19670
This theorem is referenced by:  facth  19686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-0p 19025  df-ply 19570  df-idp 19571  df-coe 19572  df-dgr 19573  df-quot 19671
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