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Theorem plyrem 23270
Description: The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 14522). 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 23166 . . . . . . . 8  |-  (Poly `  S )  C_  (Poly `  CC )
2 simpl 459 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  F  e.  (Poly `  S )
)
31, 2sseldi 3432 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  F  e.  (Poly `  CC )
)
4 plyrem.1 . . . . . . . . . 10  |-  G  =  ( Xp  oF  -  ( CC 
X.  { A }
) )
54plyremlem 23269 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( G  e.  (Poly `  CC )  /\  (deg `  G
)  =  1  /\  ( `' G " { 0 } )  =  { A }
) )
65adantl 468 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( G  e.  (Poly `  CC )  /\  (deg `  G
)  =  1  /\  ( `' G " { 0 } )  =  { A }
) )
76simp1d 1021 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  G  e.  (Poly `  CC )
)
86simp2d 1022 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (deg `  G )  =  1 )
9 ax-1ne0 9613 . . . . . . . . . 10  |-  1  =/=  0
109a1i 11 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  1  =/=  0 )
118, 10eqnetrd 2693 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (deg `  G )  =/=  0
)
12 fveq2 5870 . . . . . . . . . 10  |-  ( G  =  0p  -> 
(deg `  G )  =  (deg `  0p
) )
13 dgr0 23228 . . . . . . . . . 10  |-  (deg ` 
0p )  =  0
1412, 13syl6eq 2503 . . . . . . . . 9  |-  ( G  =  0p  -> 
(deg `  G )  =  0 )
1514necon3i 2658 . . . . . . . 8  |-  ( (deg
`  G )  =/=  0  ->  G  =/=  0p )
1611, 15syl 17 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  G  =/=  0p )
17 plyrem.2 . . . . . . . 8  |-  R  =  ( F  oF  -  ( G  oF  x.  ( F quot  G ) ) )
1817quotdgr 23268 . . . . . . 7  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )  /\  G  =/=  0p )  ->  ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) )
193, 7, 16, 18syl3anc 1269 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) )
20 0lt1 10143 . . . . . . . 8  |-  0  <  1
2120, 8syl5breqr 4442 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  0  <  (deg `  G )
)
22 fveq2 5870 . . . . . . . . 9  |-  ( R  =  0p  -> 
(deg `  R )  =  (deg `  0p
) )
2322, 13syl6eq 2503 . . . . . . . 8  |-  ( R  =  0p  -> 
(deg `  R )  =  0 )
2423breq1d 4415 . . . . . . 7  |-  ( R  =  0p  -> 
( (deg `  R
)  <  (deg `  G
)  <->  0  <  (deg `  G ) ) )
2521, 24syl5ibrcom 226 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( R  =  0p 
->  (deg `  R )  <  (deg `  G )
) )
26 pm2.62 411 . . . . . 6  |-  ( ( R  =  0p  \/  (deg `  R
)  <  (deg `  G
) )  ->  (
( R  =  0p  ->  (deg `  R
)  <  (deg `  G
) )  ->  (deg `  R )  <  (deg `  G ) ) )
2719, 25, 26sylc 62 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (deg `  R )  <  (deg `  G ) )
2827, 8breqtrd 4430 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (deg `  R )  <  1
)
29 quotcl2 23267 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )  /\  G  =/=  0p )  ->  ( F quot  G )  e.  (Poly `  CC ) )
303, 7, 16, 29syl3anc 1269 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F quot  G )  e.  (Poly `  CC ) )
31 plymulcl 23187 . . . . . . . . 9  |-  ( ( G  e.  (Poly `  CC )  /\  ( F quot  G )  e.  (Poly `  CC ) )  -> 
( G  oF  x.  ( F quot  G
) )  e.  (Poly `  CC ) )
327, 30, 31syl2anc 667 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( G  oF  x.  ( F quot  G ) )  e.  (Poly `  CC )
)
33 plysubcl 23188 . . . . . . . 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 667 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F  oF  -  ( G  oF  x.  ( F quot  G ) ) )  e.  (Poly `  CC ) )
3517, 34syl5eqel 2535 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  R  e.  (Poly `  CC )
)
36 dgrcl 23199 . . . . . 6  |-  ( R  e.  (Poly `  CC )  ->  (deg `  R
)  e.  NN0 )
3735, 36syl 17 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (deg `  R )  e.  NN0 )
38 nn0lt10b 11005 . . . . 5  |-  ( (deg
`  R )  e. 
NN0  ->  ( (deg `  R )  <  1  <->  (deg
`  R )  =  0 ) )
3937, 38syl 17 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (
(deg `  R )  <  1  <->  (deg `  R )  =  0 ) )
4028, 39mpbid 214 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (deg `  R )  =  0 )
41 0dgrb 23212 . . . 4  |-  ( R  e.  (Poly `  CC )  ->  ( (deg `  R )  =  0  <-> 
R  =  ( CC 
X.  { ( R `
 0 ) } ) ) )
4235, 41syl 17 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (
(deg `  R )  =  0  <->  R  =  ( CC  X.  { ( R `  0 ) } ) ) )
4340, 42mpbid 214 . 2  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  R  =  ( CC  X.  { ( R ` 
0 ) } ) )
4443fveq1d 5872 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( R `  A )  =  ( ( CC 
X.  { ( R `
 0 ) } ) `  A ) )
4517fveq1i 5871 . . . . . . 7  |-  ( R `
 A )  =  ( ( F  oF  -  ( G  oF  x.  ( F quot  G ) ) ) `
 A )
46 plyf 23164 . . . . . . . . . . 11  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
4746adantr 467 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  F : CC --> CC )
48 ffn 5733 . . . . . . . . . 10  |-  ( F : CC --> CC  ->  F  Fn  CC )
4947, 48syl 17 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  F  Fn  CC )
50 plyf 23164 . . . . . . . . . . . 12  |-  ( G  e.  (Poly `  CC )  ->  G : CC --> CC )
517, 50syl 17 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  G : CC --> CC )
52 ffn 5733 . . . . . . . . . . 11  |-  ( G : CC --> CC  ->  G  Fn  CC )
5351, 52syl 17 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  G  Fn  CC )
54 plyf 23164 . . . . . . . . . . . 12  |-  ( ( F quot  G )  e.  (Poly `  CC )  ->  ( F quot  G ) : CC --> CC )
5530, 54syl 17 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F quot  G ) : CC --> CC )
56 ffn 5733 . . . . . . . . . . 11  |-  ( ( F quot  G ) : CC --> CC  ->  ( F quot  G )  Fn  CC )
5755, 56syl 17 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F quot  G )  Fn  CC )
58 cnex 9625 . . . . . . . . . . 11  |-  CC  e.  _V
5958a1i 11 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  CC  e.  _V )
60 inidm 3643 . . . . . . . . . 10  |-  ( CC 
i^i  CC )  =  CC
6153, 57, 59, 59, 60offn 6547 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( G  oF  x.  ( F quot  G ) )  Fn  CC )
62 eqidd 2454 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  /\  A  e.  CC )  ->  ( F `  A )  =  ( F `  A ) )
636simp3d 1023 . . . . . . . . . . . . . . 15  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( `' G " { 0 } )  =  { A } )
64 ssun1 3599 . . . . . . . . . . . . . . 15  |-  ( `' G " { 0 } )  C_  (
( `' G " { 0 } )  u.  ( `' ( F quot  G ) " { 0 } ) )
6563, 64syl6eqssr 3485 . . . . . . . . . . . . . 14  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  { A }  C_  ( ( `' G " { 0 } )  u.  ( `' ( F quot  G
) " { 0 } ) ) )
66 snssg 4108 . . . . . . . . . . . . . . 15  |-  ( A  e.  CC  ->  ( A  e.  ( ( `' G " { 0 } )  u.  ( `' ( F quot  G
) " { 0 } ) )  <->  { A }  C_  ( ( `' G " { 0 } )  u.  ( `' ( F quot  G
) " { 0 } ) ) ) )
6766adantl 468 . . . . . . . . . . . . . 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 236 . . . . . . . . . . . . 13  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  A  e.  ( ( `' G " { 0 } )  u.  ( `' ( F quot  G ) " { 0 } ) ) )
69 ofmulrt 23247 . . . . . . . . . . . . . 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 1269 . . . . . . . . . . . . 13  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( `' ( G  oF  x.  ( F quot  G ) ) " {
0 } )  =  ( ( `' G " { 0 } )  u.  ( `' ( F quot  G ) " { 0 } ) ) )
7168, 70eleqtrrd 2534 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  A  e.  ( `' ( G  oF  x.  ( F quot  G ) ) " { 0 } ) )
72 fniniseg 6008 . . . . . . . . . . . . 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 17 . . . . . . . . . . . 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 214 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( A  e.  CC  /\  (
( G  oF  x.  ( F quot  G
) ) `  A
)  =  0 ) )
7574simprd 465 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (
( G  oF  x.  ( F quot  G
) ) `  A
)  =  0 )
7675adantr 467 . . . . . . . . 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 6545 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  /\  A  e.  CC )  ->  (
( F  oF  -  ( G  oF  x.  ( F quot  G ) ) ) `  A )  =  ( ( F `  A
)  -  0 ) )
7877anabss3 833 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (
( F  oF  -  ( G  oF  x.  ( F quot  G ) ) ) `  A )  =  ( ( F `  A
)  -  0 ) )
7945, 78syl5eq 2499 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( R `  A )  =  ( ( F `
 A )  - 
0 ) )
8046ffvelrnda 6027 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F `  A )  e.  CC )
8180subid1d 9980 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (
( F `  A
)  -  0 )  =  ( F `  A ) )
8279, 81eqtrd 2487 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( R `  A )  =  ( F `  A ) )
83 fvex 5880 . . . . . . 7  |-  ( R `
 0 )  e. 
_V
8483fvconst2 6125 . . . . . 6  |-  ( A  e.  CC  ->  (
( CC  X.  {
( R `  0
) } ) `  A )  =  ( R `  0 ) )
8584adantl 468 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  (
( CC  X.  {
( R `  0
) } ) `  A )  =  ( R `  0 ) )
8644, 82, 853eqtr3d 2495 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F `  A )  =  ( R ` 
0 ) )
8786sneqd 3982 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  { ( F `  A ) }  =  { ( R `  0 ) } )
8887xpeq2d 4861 . 2  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( CC  X.  { ( F `
 A ) } )  =  ( CC 
X.  { ( R `
 0 ) } ) )
8943, 88eqtr4d 2490 1  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  R  =  ( CC  X.  { ( F `  A ) } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889    =/= wne 2624   _Vcvv 3047    u. cun 3404    C_ wss 3406   {csn 3970   class class class wbr 4405    X. cxp 4835   `'ccnv 4836   "cima 4840    Fn wfn 5580   -->wf 5581   ` cfv 5585  (class class class)co 6295    oFcof 6534   CCcc 9542   0cc0 9544   1c1 9545    x. cmul 9549    < clt 9680    - cmin 9865   NN0cn0 10876   0pc0p 22639  Polycply 23150   Xpcidp 23151  degcdgr 23153   quot cquot 23255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622  ax-addf 9623
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-sup 7961  df-inf 7962  df-oi 8030  df-card 8378  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-rp 11310  df-fz 11792  df-fzo 11923  df-fl 12035  df-seq 12221  df-exp 12280  df-hash 12523  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-clim 13564  df-rlim 13565  df-sum 13765  df-0p 22640  df-ply 23154  df-idp 23155  df-coe 23156  df-dgr 23157  df-quot 23256
This theorem is referenced by:  facth  23271
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