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Theorem plyrem 22435
Description: The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 14035). 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). This is part of Metamath 100 proof #89. (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 22332 . . . . . . . 8  |-  (Poly `  S )  C_  (Poly `  CC )
2 simpl 457 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  F  e.  (Poly `  S )
)
31, 2sseldi 3502 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  F  e.  (Poly `  CC )
)
4 plyrem.1 . . . . . . . . . 10  |-  G  =  ( Xp  oF  -  ( CC 
X.  { A }
) )
54plyremlem 22434 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( G  e.  (Poly `  CC )  /\  (deg `  G
)  =  1  /\  ( `' G " { 0 } )  =  { A }
) )
65adantl 466 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( G  e.  (Poly `  CC )  /\  (deg `  G
)  =  1  /\  ( `' G " { 0 } )  =  { A }
) )
76simp1d 1008 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  G  e.  (Poly `  CC )
)
86simp2d 1009 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (deg `  G )  =  1 )
9 ax-1ne0 9557 . . . . . . . . . 10  |-  1  =/=  0
109a1i 11 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  1  =/=  0 )
118, 10eqnetrd 2760 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (deg `  G )  =/=  0
)
12 fveq2 5864 . . . . . . . . . 10  |-  ( G  =  0p  -> 
(deg `  G )  =  (deg `  0p
) )
13 dgr0 22393 . . . . . . . . . 10  |-  (deg ` 
0p )  =  0
1412, 13syl6eq 2524 . . . . . . . . 9  |-  ( G  =  0p  -> 
(deg `  G )  =  0 )
1514necon3i 2707 . . . . . . . 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 22433 . . . . . . 7  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )  /\  G  =/=  0p )  ->  ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) )
193, 7, 16, 18syl3anc 1228 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) )
20 0lt1 10071 . . . . . . . 8  |-  0  <  1
2120, 8syl5breqr 4483 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  0  <  (deg `  G )
)
22 fveq2 5864 . . . . . . . . 9  |-  ( R  =  0p  -> 
(deg `  R )  =  (deg `  0p
) )
2322, 13syl6eq 2524 . . . . . . . 8  |-  ( R  =  0p  -> 
(deg `  R )  =  0 )
2423breq1d 4457 . . . . . . 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 4471 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (deg `  R )  <  1
)
29 quotcl2 22432 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )  /\  G  =/=  0p )  ->  ( F quot  G )  e.  (Poly `  CC ) )
303, 7, 16, 29syl3anc 1228 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F quot  G )  e.  (Poly `  CC ) )
31 plymulcl 22353 . . . . . . . . 9  |-  ( ( G  e.  (Poly `  CC )  /\  ( F quot  G )  e.  (Poly `  CC ) )  -> 
( G  oF  x.  ( F quot  G
) )  e.  (Poly `  CC ) )
327, 30, 31syl2anc 661 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( G  oF  x.  ( F quot  G ) )  e.  (Poly `  CC )
)
33 plysubcl 22354 . . . . . . . 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 661 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F  oF  -  ( G  oF  x.  ( F quot  G ) ) )  e.  (Poly `  CC ) )
3517, 34syl5eqel 2559 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  R  e.  (Poly `  CC )
)
36 dgrcl 22365 . . . . . 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 10920 . . . . 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 22378 . . . 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 5866 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( R `  A )  =  ( ( CC 
X.  { ( R `
 0 ) } ) `  A ) )
4517fveq1i 5865 . . . . . . 7  |-  ( R `
 A )  =  ( ( F  oF  -  ( G  oF  x.  ( F quot  G ) ) ) `
 A )
46 plyf 22330 . . . . . . . . . . 11  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
4746adantr 465 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  F : CC --> CC )
48 ffn 5729 . . . . . . . . . 10  |-  ( F : CC --> CC  ->  F  Fn  CC )
4947, 48syl 16 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  F  Fn  CC )
50 plyf 22330 . . . . . . . . . . . 12  |-  ( G  e.  (Poly `  CC )  ->  G : CC --> CC )
517, 50syl 16 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  G : CC --> CC )
52 ffn 5729 . . . . . . . . . . 11  |-  ( G : CC --> CC  ->  G  Fn  CC )
5351, 52syl 16 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  G  Fn  CC )
54 plyf 22330 . . . . . . . . . . . 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 5729 . . . . . . . . . . 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 9569 . . . . . . . . . . 11  |-  CC  e.  _V
5958a1i 11 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  CC  e.  _V )
60 inidm 3707 . . . . . . . . . 10  |-  ( CC 
i^i  CC )  =  CC
6153, 57, 59, 59, 60offn 6533 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( G  oF  x.  ( F quot  G ) )  Fn  CC )
62 eqidd 2468 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  /\  A  e.  CC )  ->  ( F `  A )  =  ( F `  A ) )
636simp3d 1010 . . . . . . . . . . . . . . 15  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( `' G " { 0 } )  =  { A } )
64 ssun1 3667 . . . . . . . . . . . . . . 15  |-  ( `' G " { 0 } )  C_  (
( `' G " { 0 } )  u.  ( `' ( F quot  G ) " { 0 } ) )
6563, 64syl6eqssr 3555 . . . . . . . . . . . . . 14  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  { A }  C_  ( ( `' G " { 0 } )  u.  ( `' ( F quot  G
) " { 0 } ) ) )
66 snssg 4160 . . . . . . . . . . . . . . 15  |-  ( A  e.  CC  ->  ( A  e.  ( ( `' G " { 0 } )  u.  ( `' ( F quot  G
) " { 0 } ) )  <->  { A }  C_  ( ( `' G " { 0 } )  u.  ( `' ( F quot  G
) " { 0 } ) ) ) )
6766adantl 466 . . . . . . . . . . . . . 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 22412 . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . 13  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( `' ( G  oF  x.  ( F quot  G ) ) " {
0 } )  =  ( ( `' G " { 0 } )  u.  ( `' ( F quot  G ) " { 0 } ) ) )
7168, 70eleqtrrd 2558 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  A  e.  ( `' ( G  oF  x.  ( F quot  G ) ) " { 0 } ) )
72 fniniseg 6000 . . . . . . . . . . . . 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 463 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (
( G  oF  x.  ( F quot  G
) ) `  A
)  =  0 )
7675adantr 465 . . . . . . . . 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 6531 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  /\  A  e.  CC )  ->  (
( F  oF  -  ( G  oF  x.  ( F quot  G ) ) ) `  A )  =  ( ( F `  A
)  -  0 ) )
7877anabss3 821 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (
( F  oF  -  ( G  oF  x.  ( F quot  G ) ) ) `  A )  =  ( ( F `  A
)  -  0 ) )
7945, 78syl5eq 2520 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( R `  A )  =  ( ( F `
 A )  - 
0 ) )
8046ffvelrnda 6019 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F `  A )  e.  CC )
8180subid1d 9915 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (
( F `  A
)  -  0 )  =  ( F `  A ) )
8279, 81eqtrd 2508 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( R `  A )  =  ( F `  A ) )
83 fvex 5874 . . . . . . 7  |-  ( R `
 0 )  e. 
_V
8483fvconst2 6114 . . . . . 6  |-  ( A  e.  CC  ->  (
( CC  X.  {
( R `  0
) } ) `  A )  =  ( R `  0 ) )
8584adantl 466 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (
( CC  X.  {
( R `  0
) } ) `  A )  =  ( R `  0 ) )
8644, 82, 853eqtr3d 2516 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F `  A )  =  ( R ` 
0 ) )
8786sneqd 4039 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  { ( F `  A ) }  =  { ( R `  0 ) } )
8887xpeq2d 5023 . 2  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( CC  X.  { ( F `
 A ) } )  =  ( CC 
X.  { ( R `
 0 ) } ) )
8943, 88eqtr4d 2511 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113    u. cun 3474    C_ wss 3476   {csn 4027   class class class wbr 4447    X. cxp 4997   `'ccnv 4998   "cima 5002    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282    oFcof 6520   CCcc 9486   0cc0 9488   1c1 9489    x. cmul 9493    < clt 9624    - cmin 9801   NN0cn0 10791   0pc0p 21811  Polycply 22316   Xpcidp 22317  degcdgr 22319   quot cquot 22420
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 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  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-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  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-int 4283  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-se 4839  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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fz 11669  df-fzo 11789  df-fl 11893  df-seq 12072  df-exp 12131  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-rlim 13271  df-sum 13468  df-0p 21812  df-ply 22320  df-idp 22321  df-coe 22322  df-dgr 22323  df-quot 22421
This theorem is referenced by:  facth  22436
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