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Theorem facth 19688
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 2285 . . . . 5  |-  ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )  =  ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )
31, 2plyrem 19687 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC )  ->  ( F  o F  -  ( G  o F  x.  ( F quot  G ) ) )  =  ( CC  X.  { ( F `  A ) } ) )
433adant3 975 . . 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 957 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( F `  A )  =  0 )
65sneqd 3655 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  { ( F `  A ) }  =  { 0 } )
76xpeq2d 4715 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( CC  X.  { ( F `
 A ) } )  =  ( CC 
X.  { 0 } ) )
84, 7eqtrd 2317 . 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 8820 . . . 4  |-  CC  e.  _V
109a1i 10 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  CC  e.  _V )
11 simp1 955 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  F  e.  (Poly `  S )
)
12 plyf 19582 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
1311, 12syl 15 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  F : CC --> CC )
141plyremlem 19686 . . . . . . 7  |-  ( A  e.  CC  ->  ( G  e.  (Poly `  CC )  /\  (deg `  G
)  =  1  /\  ( `' G " { 0 } )  =  { A }
) )
15143ad2ant2 977 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( G  e.  (Poly `  CC )  /\  (deg `  G
)  =  1  /\  ( `' G " { 0 } )  =  { A }
) )
1615simp1d 967 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  G  e.  (Poly `  CC )
)
17 plyssc 19584 . . . . . . 7  |-  (Poly `  S )  C_  (Poly `  CC )
1817, 11sseldi 3180 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  F  e.  (Poly `  CC )
)
1915simp2d 968 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  (deg `  G )  =  1 )
20 ax-1ne0 8808 . . . . . . . . 9  |-  1  =/=  0
2120a1i 10 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  1  =/=  0 )
2219, 21eqnetrd 2466 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  (deg `  G )  =/=  0
)
23 fveq2 5527 . . . . . . . . 9  |-  ( G  =  0 p  -> 
(deg `  G )  =  (deg `  0 p
) )
24 dgr0 19645 . . . . . . . . 9  |-  (deg ` 
0 p )  =  0
2523, 24syl6eq 2333 . . . . . . . 8  |-  ( G  =  0 p  -> 
(deg `  G )  =  0 )
2625necon3i 2487 . . . . . . 7  |-  ( (deg
`  G )  =/=  0  ->  G  =/=  0 p )
2722, 26syl 15 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  G  =/=  0 p )
28 quotcl2 19684 . . . . . 6  |-  ( ( F  e.  (Poly `  CC )  /\  G  e.  (Poly `  CC )  /\  G  =/=  0 p )  ->  ( F quot  G )  e.  (Poly `  CC ) )
2918, 16, 27, 28syl3anc 1182 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( F quot  G )  e.  (Poly `  CC ) )
30 plymulcl 19605 . . . . 5  |-  ( ( G  e.  (Poly `  CC )  /\  ( F quot  G )  e.  (Poly `  CC ) )  -> 
( G  o F  x.  ( F quot  G
) )  e.  (Poly `  CC ) )
3116, 29, 30syl2anc 642 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( G  o F  x.  ( F quot  G ) )  e.  (Poly `  CC )
)
32 plyf 19582 . . . 4  |-  ( ( G  o F  x.  ( F quot  G )
)  e.  (Poly `  CC )  ->  ( G  o F  x.  ( F quot  G ) ) : CC --> CC )
3331, 32syl 15 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  A  e.  CC  /\  ( F `
 A )  =  0 )  ->  ( G  o F  x.  ( F quot  G ) ) : CC --> CC )
34 ofsubeq0 9745 . . 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 1182 . 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 201 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 176    /\ w3a 934    = wceq 1625    e. wcel 1686    =/= wne 2448   _Vcvv 2790   {csn 3642    X. cxp 4689   `'ccnv 4690   "cima 4694   -->wf 5253   ` cfv 5257  (class class class)co 5860    o Fcof 6078   CCcc 8737   0cc0 8739   1c1 8740    x. cmul 8744    - cmin 9039   0 pc0p 19026  Polycply 19568   X pcidp 19569  degcdgr 19571   quot cquot 19672
This theorem is referenced by:  fta1lem  19689  vieta1lem1  19692  vieta1lem2  19693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817  ax-addf 8818
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-er 6662  df-map 6776  df-pm 6777  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-sup 7196  df-oi 7227  df-card 7574  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-n0 9968  df-z 10027  df-uz 10233  df-rp 10357  df-fz 10785  df-fzo 10873  df-fl 10927  df-seq 11049  df-exp 11107  df-hash 11340  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-clim 11964  df-rlim 11965  df-sum 12161  df-0p 19027  df-ply 19572  df-idp 19573  df-coe 19574  df-dgr 19575  df-quot 19673
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