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Theorem plyrem 21656
Description: The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 13709). 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  =  ( Xp  oF  -  ( CC 
X.  { A }
) )
plyrem.2  |-  R  =  ( F  oF  -  ( G  oF  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 21553 . . . . . . . 8  |-  (Poly `  S )  C_  (Poly `  CC )
2 simpl 454 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  F  e.  (Poly `  S )
)
31, 2sseldi 3342 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  F  e.  (Poly `  CC )
)
4 plyrem.1 . . . . . . . . . 10  |-  G  =  ( Xp  oF  -  ( CC 
X.  { A }
) )
54plyremlem 21655 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( G  e.  (Poly `  CC )  /\  (deg `  G
)  =  1  /\  ( `' G " { 0 } )  =  { A }
) )
65adantl 463 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( G  e.  (Poly `  CC )  /\  (deg `  G
)  =  1  /\  ( `' G " { 0 } )  =  { A }
) )
76simp1d 993 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  G  e.  (Poly `  CC )
)
86simp2d 994 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (deg `  G )  =  1 )
9 ax-1ne0 9339 . . . . . . . . . 10  |-  1  =/=  0
109a1i 11 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  1  =/=  0 )
118, 10eqnetrd 2616 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (deg `  G )  =/=  0
)
12 fveq2 5679 . . . . . . . . . 10  |-  ( G  =  0p  -> 
(deg `  G )  =  (deg `  0p
) )
13 dgr0 21614 . . . . . . . . . 10  |-  (deg ` 
0p )  =  0
1412, 13syl6eq 2481 . . . . . . . . 9  |-  ( G  =  0p  -> 
(deg `  G )  =  0 )
1514necon3i 2640 . . . . . . . 8  |-  ( (deg
`  G )  =/=  0  ->  G  =/=  0p )
1611, 15syl 16 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  G  =/=  0p )
17 plyrem.2 . . . . . . . 8  |-  R  =  ( F  oF  -  ( G  oF  x.  ( F quot  G ) ) )
1817quotdgr 21654 . . . . . . 7  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )  /\  G  =/=  0p )  ->  ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) )
193, 7, 16, 18syl3anc 1211 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) )
20 0lt1 9850 . . . . . . . 8  |-  0  <  1
2120, 8syl5breqr 4316 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  0  <  (deg `  G )
)
22 fveq2 5679 . . . . . . . . 9  |-  ( R  =  0p  -> 
(deg `  R )  =  (deg `  0p
) )
2322, 13syl6eq 2481 . . . . . . . 8  |-  ( R  =  0p  -> 
(deg `  R )  =  0 )
2423breq1d 4290 . . . . . . 7  |-  ( R  =  0p  -> 
( (deg `  R
)  <  (deg `  G
)  <->  0  <  (deg `  G ) ) )
2521, 24syl5ibrcom 222 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( R  =  0p 
->  (deg `  R )  <  (deg `  G )
) )
26 pm2.62 409 . . . . . 6  |-  ( ( R  =  0p  \/  (deg `  R
)  <  (deg `  G
) )  ->  (
( R  =  0p  ->  (deg `  R
)  <  (deg `  G
) )  ->  (deg `  R )  <  (deg `  G ) ) )
2719, 25, 26sylc 60 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (deg `  R )  <  (deg `  G ) )
2827, 8breqtrd 4304 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (deg `  R )  <  1
)
29 quotcl2 21653 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )  /\  G  =/=  0p )  ->  ( F quot  G )  e.  (Poly `  CC ) )
303, 7, 16, 29syl3anc 1211 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F quot  G )  e.  (Poly `  CC ) )
31 plymulcl 21574 . . . . . . . . 9  |-  ( ( G  e.  (Poly `  CC )  /\  ( F quot  G )  e.  (Poly `  CC ) )  -> 
( G  oF  x.  ( F quot  G
) )  e.  (Poly `  CC ) )
327, 30, 31syl2anc 654 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( G  oF  x.  ( F quot  G ) )  e.  (Poly `  CC )
)
33 plysubcl 21575 . . . . . . . 8  |-  ( ( F  e.  (Poly `  CC )  /\  ( G  oF  x.  ( F quot  G ) )  e.  (Poly `  CC )
)  ->  ( F  oF  -  ( G  oF  x.  ( F quot  G ) ) )  e.  (Poly `  CC ) )
343, 32, 33syl2anc 654 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F  oF  -  ( G  oF  x.  ( F quot  G ) ) )  e.  (Poly `  CC ) )
3517, 34syl5eqel 2517 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  R  e.  (Poly `  CC )
)
36 dgrcl 21586 . . . . . 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 10694 . . . . 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 210 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (deg `  R )  =  0 )
41 0dgrb 21599 . . . 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 210 . 2  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  R  =  ( CC  X.  { ( R ` 
0 ) } ) )
4443fveq1d 5681 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( R `  A )  =  ( ( CC 
X.  { ( R `
 0 ) } ) `  A ) )
4517fveq1i 5680 . . . . . . 7  |-  ( R `
 A )  =  ( ( F  oF  -  ( G  oF  x.  ( F quot  G ) ) ) `
 A )
46 plyf 21551 . . . . . . . . . . 11  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
4746adantr 462 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  F : CC --> CC )
48 ffn 5547 . . . . . . . . . 10  |-  ( F : CC --> CC  ->  F  Fn  CC )
4947, 48syl 16 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  F  Fn  CC )
50 plyf 21551 . . . . . . . . . . . 12  |-  ( G  e.  (Poly `  CC )  ->  G : CC --> CC )
517, 50syl 16 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  G : CC --> CC )
52 ffn 5547 . . . . . . . . . . 11  |-  ( G : CC --> CC  ->  G  Fn  CC )
5351, 52syl 16 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  G  Fn  CC )
54 plyf 21551 . . . . . . . . . . . 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 5547 . . . . . . . . . . 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 9351 . . . . . . . . . . 11  |-  CC  e.  _V
5958a1i 11 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  CC  e.  _V )
60 inidm 3547 . . . . . . . . . 10  |-  ( CC 
i^i  CC )  =  CC
6153, 57, 59, 59, 60offn 6320 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( G  oF  x.  ( F quot  G ) )  Fn  CC )
62 eqidd 2434 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  /\  A  e.  CC )  ->  ( F `  A )  =  ( F `  A ) )
636simp3d 995 . . . . . . . . . . . . . . 15  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( `' G " { 0 } )  =  { A } )
64 ssun1 3507 . . . . . . . . . . . . . . 15  |-  ( `' G " { 0 } )  C_  (
( `' G " { 0 } )  u.  ( `' ( F quot  G ) " { 0 } ) )
6563, 64syl6eqssr 3395 . . . . . . . . . . . . . 14  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  { A }  C_  ( ( `' G " { 0 } )  u.  ( `' ( F quot  G
) " { 0 } ) ) )
66 snssg 3995 . . . . . . . . . . . . . . 15  |-  ( A  e.  CC  ->  ( A  e.  ( ( `' G " { 0 } )  u.  ( `' ( F quot  G
) " { 0 } ) )  <->  { A }  C_  ( ( `' G " { 0 } )  u.  ( `' ( F quot  G
) " { 0 } ) ) ) )
6766adantl 463 . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . 13  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  A  e.  ( ( `' G " { 0 } )  u.  ( `' ( F quot  G ) " { 0 } ) ) )
69 ofmulrt 21633 . . . . . . . . . . . . . 14  |-  ( ( CC  e.  _V  /\  G : CC --> CC  /\  ( F quot  G ) : CC --> CC )  -> 
( `' ( G  oF  x.  ( F quot  G ) ) " { 0 } )  =  ( ( `' G " { 0 } )  u.  ( `' ( F quot  G
) " { 0 } ) ) )
7059, 51, 55, 69syl3anc 1211 . . . . . . . . . . . . 13  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( `' ( G  oF  x.  ( F quot  G ) ) " {
0 } )  =  ( ( `' G " { 0 } )  u.  ( `' ( F quot  G ) " { 0 } ) ) )
7168, 70eleqtrrd 2510 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  A  e.  ( `' ( G  oF  x.  ( F quot  G ) ) " { 0 } ) )
72 fniniseg 5812 . . . . . . . . . . . . 13  |-  ( ( G  oF  x.  ( F quot  G ) )  Fn  CC  ->  ( A  e.  ( `' ( G  oF  x.  ( F quot  G
) ) " {
0 } )  <->  ( A  e.  CC  /\  ( ( G  oF  x.  ( F quot  G ) ) `  A )  =  0 ) ) )
7361, 72syl 16 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( A  e.  ( `' ( G  oF  x.  ( F quot  G ) ) " { 0 } )  <->  ( A  e.  CC  /\  ( ( G  oF  x.  ( F quot  G ) ) `  A )  =  0 ) ) )
7471, 73mpbid 210 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( A  e.  CC  /\  (
( G  oF  x.  ( F quot  G
) ) `  A
)  =  0 ) )
7574simprd 460 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (
( G  oF  x.  ( F quot  G
) ) `  A
)  =  0 )
7675adantr 462 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  /\  A  e.  CC )  ->  (
( G  oF  x.  ( F quot  G
) ) `  A
)  =  0 )
7749, 61, 59, 59, 60, 62, 76ofval 6318 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  /\  A  e.  CC )  ->  (
( F  oF  -  ( G  oF  x.  ( F quot  G ) ) ) `  A )  =  ( ( F `  A
)  -  0 ) )
7877anabss3 812 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (
( F  oF  -  ( G  oF  x.  ( F quot  G ) ) ) `  A )  =  ( ( F `  A
)  -  0 ) )
7945, 78syl5eq 2477 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( R `  A )  =  ( ( F `
 A )  - 
0 ) )
8046ffvelrnda 5831 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F `  A )  e.  CC )
8180subid1d 9696 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (
( F `  A
)  -  0 )  =  ( F `  A ) )
8279, 81eqtrd 2465 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( R `  A )  =  ( F `  A ) )
83 fvex 5689 . . . . . . 7  |-  ( R `
 0 )  e. 
_V
8483fvconst2 5920 . . . . . 6  |-  ( A  e.  CC  ->  (
( CC  X.  {
( R `  0
) } ) `  A )  =  ( R `  0 ) )
8584adantl 463 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (
( CC  X.  {
( R `  0
) } ) `  A )  =  ( R `  0 ) )
8644, 82, 853eqtr3d 2473 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F `  A )  =  ( R ` 
0 ) )
8786sneqd 3877 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  { ( F `  A ) }  =  { ( R `  0 ) } )
8887xpeq2d 4851 . 2  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( CC  X.  { ( F `
 A ) } )  =  ( CC 
X.  { ( R `
 0 ) } ) )
8943, 88eqtr4d 2468 1  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  R  =  ( CC  X.  { ( F `  A ) } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 958    = wceq 1362    e. wcel 1755    =/= wne 2596   _Vcvv 2962    u. cun 3314    C_ wss 3316   {csn 3865   class class class wbr 4280    X. cxp 4825   `'ccnv 4826   "cima 4830    Fn wfn 5401   -->wf 5402   ` cfv 5406  (class class class)co 6080    oFcof 6307   CCcc 9268   0cc0 9270   1c1 9271    x. cmul 9275    < clt 9406    - cmin 9583   NN0cn0 10567   0pc0p 20989  Polycply 21537   Xpcidp 21538  degcdgr 21540   quot cquot 21641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-sup 7679  df-oi 7712  df-card 8097  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-n0 10568  df-z 10635  df-uz 10850  df-rp 10980  df-fz 11425  df-fzo 11533  df-fl 11626  df-seq 11791  df-exp 11850  df-hash 12088  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-clim 12950  df-rlim 12951  df-sum 13148  df-0p 20990  df-ply 21541  df-idp 21542  df-coe 21543  df-dgr 21544  df-quot 21642
This theorem is referenced by:  facth  21657
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