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Theorem facth 20225
Description: The factor theorem. If a polynomial  F has a root at  A, then  G  =  x  -  A is a factor of  F (and the other factor is  F quot  G). (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypothesis
Ref Expression
facth.1  |-  G  =  ( X p  o F  -  ( CC  X.  { A } ) )
Assertion
Ref Expression
facth  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  F  =  ( G  o F  x.  ( F quot  G ) ) )

Proof of Theorem facth
StepHypRef Expression
1 facth.1 . . . . 5  |-  G  =  ( X p  o F  -  ( CC  X.  { A } ) )
2 eqid 2438 . . . . 5  |-  ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )  =  ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )
31, 2plyrem 20224 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )  =  ( CC  X.  { ( F `  A ) } ) )
433adant3 978 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )  =  ( CC  X.  { ( F `  A ) } ) )
5 simp3 960 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( F `  A )  =  0 )
65sneqd 3829 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  { ( F `  A ) }  =  { 0 } )
76xpeq2d 4904 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( CC  X.  { ( F `
 A ) } )  =  ( CC 
X.  { 0 } ) )
84, 7eqtrd 2470 . 2  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )  =  ( CC  X.  { 0 } ) )
9 cnex 9073 . . . 4  |-  CC  e.  _V
109a1i 11 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  CC  e.  _V )
11 simp1 958 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  F  e.  (Poly `  S )
)
12 plyf 20119 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
1311, 12syl 16 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  F : CC --> CC )
141plyremlem 20223 . . . . . . 7  |-  ( A  e.  CC  ->  ( G  e.  (Poly `  CC )  /\  (deg `  G
)  =  1  /\  ( `' G " { 0 } )  =  { A }
) )
15143ad2ant2 980 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( G  e.  (Poly `  CC )  /\  (deg `  G
)  =  1  /\  ( `' G " { 0 } )  =  { A }
) )
1615simp1d 970 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  G  e.  (Poly `  CC )
)
17 plyssc 20121 . . . . . . 7  |-  (Poly `  S )  C_  (Poly `  CC )
1817, 11sseldi 3348 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  F  e.  (Poly `  CC )
)
1915simp2d 971 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  (deg `  G )  =  1 )
20 ax-1ne0 9061 . . . . . . . . 9  |-  1  =/=  0
2120a1i 11 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  1  =/=  0 )
2219, 21eqnetrd 2621 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  (deg `  G )  =/=  0
)
23 fveq2 5730 . . . . . . . . 9  |-  ( G  =  0 p  -> 
(deg `  G )  =  (deg `  0 p
) )
24 dgr0 20182 . . . . . . . . 9  |-  (deg ` 
0 p )  =  0
2523, 24syl6eq 2486 . . . . . . . 8  |-  ( G  =  0 p  -> 
(deg `  G )  =  0 )
2625necon3i 2645 . . . . . . 7  |-  ( (deg
`  G )  =/=  0  ->  G  =/=  0 p )
2722, 26syl 16 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  G  =/=  0 p )
28 quotcl2 20221 . . . . . 6  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )  /\  G  =/=  0 p )  ->  ( F quot  G )  e.  (Poly `  CC ) )
2918, 16, 27, 28syl3anc 1185 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( F quot  G )  e.  (Poly `  CC ) )
30 plymulcl 20142 . . . . 5  |-  ( ( G  e.  (Poly `  CC )  /\  ( F quot  G )  e.  (Poly `  CC ) )  -> 
( G  o F  x.  ( F quot  G
) )  e.  (Poly `  CC ) )
3116, 29, 30syl2anc 644 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( G  o F  x.  ( F quot  G ) )  e.  (Poly `  CC )
)
32 plyf 20119 . . . 4  |-  ( ( G  o F  x.  ( F quot  G )
)  e.  (Poly `  CC )  ->  ( G  o F  x.  ( F quot  G ) ) : CC --> CC )
3331, 32syl 16 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( G  o F  x.  ( F quot  G ) ) : CC --> CC )
34 ofsubeq0 9999 . . 3  |-  ( ( CC  e.  _V  /\  F : CC --> CC  /\  ( G  o F  x.  ( F quot  G ) ) : CC --> CC )  ->  ( ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )  =  ( CC  X.  { 0 } )  <-> 
F  =  ( G  o F  x.  ( F quot  G ) ) ) )
3510, 13, 33, 34syl3anc 1185 . 2  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  (
( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )  =  ( CC  X.  {
0 } )  <->  F  =  ( G  o F  x.  ( F quot  G ) ) ) )
368, 35mpbid 203 1  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  F  =  ( G  o F  x.  ( F quot  G ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   _Vcvv 2958   {csn 3816    X. cxp 4878   `'ccnv 4879   "cima 4883   -->wf 5452   ` cfv 5456  (class class class)co 6083    o Fcof 6305   CCcc 8990   0cc0 8992   1c1 8993    x. cmul 8997    - cmin 9293   0 pc0p 19563  Polycply 20105   X pcidp 20106  degcdgr 20108   quot cquot 20209
This theorem is referenced by:  fta1lem  20226  vieta1lem1  20229  vieta1lem2  20230
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-fz 11046  df-fzo 11138  df-fl 11204  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-rlim 12285  df-sum 12482  df-0p 19564  df-ply 20109  df-idp 20110  df-coe 20111  df-dgr 20112  df-quot 20210
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