Theorem facth 24061

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). This is part of Metamath 100 proof #89. (Contributed by Mario Carneiro, 26-Jul-2014.)

Hypothesis

Ref	Expression
facth.1	|- G = ( Xp oF - ( CC X. { A } ) )

Assertion

Ref	Expression
facth	|- ( ( F e. (Poly ` S ) /\ A e. CC /\ ( F ` A ) = 0 ) -> F = ( G oF x. ( F quot G ) ) )

Proof of Theorem facth

Step	Hyp	Ref	Expression
1		facth.1	. . . . 5 |- G = ( Xp oF - ( CC X. { A } ) )
2		eqid 2622	. . . . 5 |- ( F oF - ( G oF x. ( F quot G ) ) ) = ( F oF - ( G oF x. ( F quot G ) ) )
3	1, 2	plyrem 24060	. . . 4 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> ( F oF - ( G oF x. ( F quot G ) ) ) = ( CC X. { ( F ` A ) } ) )
4	3	3adant3 1081	. . 3 |- ( ( F e. (Poly ` S ) /\ A e. CC /\ ( F ` A ) = 0 ) -> ( F oF - ( G oF x. ( F quot G ) ) ) = ( CC X. { ( F ` A ) } ) )
5		simp3 1063	. . . . 5 |- ( ( F e. (Poly ` S ) /\ A e. CC /\ ( F ` A ) = 0 ) -> ( F ` A ) = 0 )
6	5	sneqd 4189	. . . 4 |- ( ( F e. (Poly ` S ) /\ A e. CC /\ ( F ` A ) = 0 ) -> { ( F ` A ) } = { 0 } )
7	6	xpeq2d 5139	. . 3 |- ( ( F e. (Poly ` S ) /\ A e. CC /\ ( F ` A ) = 0 ) -> ( CC X. { ( F ` A ) } ) = ( CC X. { 0 } ) )
8	4, 7	eqtrd 2656	. 2 |- ( ( F e. (Poly ` S ) /\ A e. CC /\ ( F ` A ) = 0 ) -> ( F oF - ( G oF x. ( F quot G ) ) ) = ( CC X. { 0 } ) )
9		cnex 10017	. . . 4 |- CC e. _V
10	9	a1i 11	. . 3 |- ( ( F e. (Poly ` S ) /\ A e. CC /\ ( F ` A ) = 0 ) -> CC e. _V )
11		simp1 1061	. . . 4 |- ( ( F e. (Poly ` S ) /\ A e. CC /\ ( F ` A ) = 0 ) -> F e. (Poly ` S ) )
12		plyf 23954	. . . 4 |- ( F e. (Poly ` S ) -> F : CC --> CC )
13	11, 12	syl 17	. . 3 |- ( ( F e. (Poly ` S ) /\ A e. CC /\ ( F ` A ) = 0 ) -> F : CC --> CC )
14	1	plyremlem 24059	. . . . . . 7 |- ( A e. CC -> ( G e. (Poly ` CC ) /\ (deg ` G ) = 1 /\ ( `' G " { 0 } ) = { A } ) )
15	14	3ad2ant2 1083	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ A e. CC /\ ( F ` A ) = 0 ) -> ( G e. (Poly ` CC ) /\ (deg ` G ) = 1 /\ ( `' G " { 0 } ) = { A } ) )
16	15	simp1d 1073	. . . . 5 |- ( ( F e. (Poly ` S ) /\ A e. CC /\ ( F ` A ) = 0 ) -> G e. (Poly ` CC ) )
17		plyssc 23956	. . . . . . 7 |- (Poly ` S ) C_ (Poly ` CC )
18	17, 11	sseldi 3601	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ A e. CC /\ ( F ` A ) = 0 ) -> F e. (Poly ` CC ) )
19	15	simp2d 1074	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ A e. CC /\ ( F ` A ) = 0 ) -> (deg ` G ) = 1 )
20		ax-1ne0 10005	. . . . . . . . 9 |- 1 =/= 0
21	20	a1i 11	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ A e. CC /\ ( F ` A ) = 0 ) -> 1 =/= 0 )
22	19, 21	eqnetrd 2861	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ A e. CC /\ ( F ` A ) = 0 ) -> (deg ` G ) =/= 0 )
23		fveq2 6191	. . . . . . . . 9 |- ( G = 0p -> (deg ` G ) = (deg ` 0p ) )
24		dgr0 24018	. . . . . . . . 9 |- (deg ` 0p ) = 0
25	23, 24	syl6eq 2672	. . . . . . . 8 |- ( G = 0p -> (deg ` G ) = 0 )
26	25	necon3i 2826	. . . . . . 7 |- ( (deg ` G ) =/= 0 -> G =/= 0p )
27	22, 26	syl 17	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ A e. CC /\ ( F ` A ) = 0 ) -> G =/= 0p )
28		quotcl2 24057	. . . . . 6 |- ( ( F e. (Poly ` CC ) /\ G e. (Poly ` CC ) /\ G =/= 0p ) -> ( F quot G ) e. (Poly ` CC ) )
29	18, 16, 27, 28	syl3anc 1326	. . . . 5 |- ( ( F e. (Poly ` S ) /\ A e. CC /\ ( F ` A ) = 0 ) -> ( F quot G ) e. (Poly ` CC ) )
30		plymulcl 23977	. . . . 5 |- ( ( G e. (Poly ` CC ) /\ ( F quot G ) e. (Poly ` CC ) ) -> ( G oF x. ( F quot G ) ) e. (Poly ` CC ) )
31	16, 29, 30	syl2anc 693	. . . 4 |- ( ( F e. (Poly ` S ) /\ A e. CC /\ ( F ` A ) = 0 ) -> ( G oF x. ( F quot G ) ) e. (Poly ` CC ) )
32		plyf 23954	. . . 4 |- ( ( G oF x. ( F quot G ) ) e. (Poly ` CC ) -> ( G oF x. ( F quot G ) ) : CC --> CC )
33	31, 32	syl 17	. . 3 |- ( ( F e. (Poly ` S ) /\ A e. CC /\ ( F ` A ) = 0 ) -> ( G oF x. ( F quot G ) ) : CC --> CC )
34		ofsubeq0 11017	. . 3 |- ( ( CC e. _V /\ F : CC --> CC /\ ( G oF x. ( F quot G ) ) : CC --> CC ) -> ( ( F oF - ( G oF x. ( F quot G ) ) ) = ( CC X. { 0 } ) <-> F = ( G oF x. ( F quot G ) ) ) )
35	10, 13, 33, 34	syl3anc 1326	. 2 |- ( ( F e. (Poly ` S ) /\ A e. CC /\ ( F ` A ) = 0 ) -> ( ( F oF - ( G oF x. ( F quot G ) ) ) = ( CC X. { 0 } ) <-> F = ( G oF x. ( F quot G ) ) ) )
36	8, 35	mpbid 222	1 |- ( ( F e. (Poly ` S ) /\ A e. CC /\ ( F ` A ) = 0 ) -> F = ( G oF x. ( F quot G ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   {csn 4177   X. cxp 5112   `'ccnv 5113   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   0cc0 9936   1c1 9937   x. cmul 9941   - cmin 10266   0pc0p 23436  Polycply 23940   Xpcidp 23941  degcdgr 23943   quot cquot 24045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-0p 23437  df-ply 23944  df-idp 23945  df-coe 23946  df-dgr 23947  df-quot 24046

This theorem is referenced by:  fta1lem  24062  vieta1lem1  24065  vieta1lem2  24066