Theorem plyrem 24060

Description: The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 15260). 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

Step	Hyp	Ref	Expression
1		plyssc 23956	. . . . . . . 8 |- (Poly ` S ) C_ (Poly ` CC )
2		simpl 473	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> F e. (Poly ` S ) )
3	1, 2	sseldi 3601	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> F e. (Poly ` CC ) )
4		plyrem.1	. . . . . . . . . 10 |- G = ( Xp oF - ( CC X. { A } ) )
5	4	plyremlem 24059	. . . . . . . . 9 |- ( A e. CC -> ( G e. (Poly ` CC ) /\ (deg ` G ) = 1 /\ ( `' G " { 0 } ) = { A } ) )
6	5	adantl 482	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> ( G e. (Poly ` CC ) /\ (deg ` G ) = 1 /\ ( `' G " { 0 } ) = { A } ) )
7	6	simp1d 1073	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> G e. (Poly ` CC ) )
8	6	simp2d 1074	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> (deg ` G ) = 1 )
9		ax-1ne0 10005	. . . . . . . . . 10 |- 1 =/= 0
10	9	a1i 11	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> 1 =/= 0 )
11	8, 10	eqnetrd 2861	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> (deg ` G ) =/= 0 )
12		fveq2 6191	. . . . . . . . . 10 |- ( G = 0p -> (deg ` G ) = (deg ` 0p ) )
13		dgr0 24018	. . . . . . . . . 10 |- (deg ` 0p ) = 0
14	12, 13	syl6eq 2672	. . . . . . . . 9 |- ( G = 0p -> (deg ` G ) = 0 )
15	14	necon3i 2826	. . . . . . . 8 |- ( (deg ` G ) =/= 0 -> G =/= 0p )
16	11, 15	syl 17	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> G =/= 0p )
17		plyrem.2	. . . . . . . 8 |- R = ( F oF - ( G oF x. ( F quot G ) ) )
18	17	quotdgr 24058	. . . . . . 7 |- ( ( F e. (Poly ` CC ) /\ G e. (Poly ` CC ) /\ G =/= 0p ) -> ( R = 0p \/ (deg ` R ) < (deg ` G ) ) )
19	3, 7, 16, 18	syl3anc 1326	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> ( R = 0p \/ (deg ` R ) < (deg ` G ) ) )
20		0lt1 10550	. . . . . . . 8 |- 0 < 1
21	20, 8	syl5breqr 4691	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> 0 < (deg ` G ) )
22		fveq2 6191	. . . . . . . . 9 |- ( R = 0p -> (deg ` R ) = (deg ` 0p ) )
23	22, 13	syl6eq 2672	. . . . . . . 8 |- ( R = 0p -> (deg ` R ) = 0 )
24	23	breq1d 4663	. . . . . . 7 |- ( R = 0p -> ( (deg ` R ) < (deg ` G ) <-> 0 < (deg ` G ) ) )
25	21, 24	syl5ibrcom 237	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> ( R = 0p -> (deg ` R ) < (deg ` G ) ) )
26		pm2.62 425	. . . . . 6 |- ( ( R = 0p \/ (deg ` R ) < (deg ` G ) ) -> ( ( R = 0p -> (deg ` R ) < (deg ` G ) ) -> (deg ` R ) < (deg ` G ) ) )
27	19, 25, 26	sylc 65	. . . . 5 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> (deg ` R ) < (deg ` G ) )
28	27, 8	breqtrd 4679	. . . 4 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> (deg ` R ) < 1 )
29		quotcl2 24057	. . . . . . . . . 10 |- ( ( F e. (Poly ` CC ) /\ G e. (Poly ` CC ) /\ G =/= 0p ) -> ( F quot G ) e. (Poly ` CC ) )
30	3, 7, 16, 29	syl3anc 1326	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> ( F quot G ) e. (Poly ` CC ) )
31		plymulcl 23977	. . . . . . . . 9 |- ( ( G e. (Poly ` CC ) /\ ( F quot G ) e. (Poly ` CC ) ) -> ( G oF x. ( F quot G ) ) e. (Poly ` CC ) )
32	7, 30, 31	syl2anc 693	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> ( G oF x. ( F quot G ) ) e. (Poly ` CC ) )
33		plysubcl 23978	. . . . . . . 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 ) )
34	3, 32, 33	syl2anc 693	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> ( F oF - ( G oF x. ( F quot G ) ) ) e. (Poly ` CC ) )
35	17, 34	syl5eqel 2705	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> R e. (Poly ` CC ) )
36		dgrcl 23989	. . . . . 6 |- ( R e. (Poly ` CC ) -> (deg ` R ) e. NN0 )
37	35, 36	syl 17	. . . . 5 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> (deg ` R ) e. NN0 )
38		nn0lt10b 11439	. . . . 5 |- ( (deg ` R ) e. NN0 -> ( (deg ` R ) < 1 <-> (deg ` R ) = 0 ) )
39	37, 38	syl 17	. . . 4 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> ( (deg ` R ) < 1 <-> (deg ` R ) = 0 ) )
40	28, 39	mpbid 222	. . 3 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> (deg ` R ) = 0 )
41		0dgrb 24002	. . . 4 |- ( R e. (Poly ` CC ) -> ( (deg ` R ) = 0 <-> R = ( CC X. { ( R ` 0 ) } ) ) )
42	35, 41	syl 17	. . 3 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> ( (deg ` R ) = 0 <-> R = ( CC X. { ( R ` 0 ) } ) ) )
43	40, 42	mpbid 222	. 2 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> R = ( CC X. { ( R ` 0 ) } ) )
44	43	fveq1d 6193	. . . . 5 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> ( R ` A ) = ( ( CC X. { ( R ` 0 ) } ) ` A ) )
45	17	fveq1i 6192	. . . . . . 7 |- ( R ` A ) = ( ( F oF - ( G oF x. ( F quot G ) ) ) ` A )
46		plyf 23954	. . . . . . . . . . 11 |- ( F e. (Poly ` S ) -> F : CC --> CC )
47	46	adantr 481	. . . . . . . . . 10 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> F : CC --> CC )
48		ffn 6045	. . . . . . . . . 10 |- ( F : CC --> CC -> F Fn CC )
49	47, 48	syl 17	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> F Fn CC )
50		plyf 23954	. . . . . . . . . . . 12 |- ( G e. (Poly ` CC ) -> G : CC --> CC )
51	7, 50	syl 17	. . . . . . . . . . 11 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> G : CC --> CC )
52		ffn 6045	. . . . . . . . . . 11 |- ( G : CC --> CC -> G Fn CC )
53	51, 52	syl 17	. . . . . . . . . 10 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> G Fn CC )
54		plyf 23954	. . . . . . . . . . . 12 |- ( ( F quot G ) e. (Poly ` CC ) -> ( F quot G ) : CC --> CC )
55	30, 54	syl 17	. . . . . . . . . . 11 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> ( F quot G ) : CC --> CC )
56		ffn 6045	. . . . . . . . . . 11 |- ( ( F quot G ) : CC --> CC -> ( F quot G ) Fn CC )
57	55, 56	syl 17	. . . . . . . . . 10 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> ( F quot G ) Fn CC )
58		cnex 10017	. . . . . . . . . . 11 |- CC e. _V
59	58	a1i 11	. . . . . . . . . 10 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> CC e. _V )
60		inidm 3822	. . . . . . . . . 10 |- ( CC i^i CC ) = CC
61	53, 57, 59, 59, 60	offn 6908	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> ( G oF x. ( F quot G ) ) Fn CC )
62		eqidd 2623	. . . . . . . . 9 |- ( ( ( F e. (Poly ` S ) /\ A e. CC ) /\ A e. CC ) -> ( F ` A ) = ( F ` A ) )
63	6	simp3d 1075	. . . . . . . . . . . . . . 15 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> ( `' G " { 0 } ) = { A } )
64		ssun1 3776	. . . . . . . . . . . . . . 15 |- ( `' G " { 0 } ) C_ ( ( `' G " { 0 } ) u. ( `' ( F quot G ) " { 0 } ) )
65	63, 64	syl6eqssr 3656	. . . . . . . . . . . . . 14 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> { A } C_ ( ( `' G " { 0 } ) u. ( `' ( F quot G ) " { 0 } ) ) )
66		snssg 4327	. . . . . . . . . . . . . . 15 |- ( A e. CC -> ( A e. ( ( `' G " { 0 } ) u. ( `' ( F quot G ) " { 0 } ) ) <-> { A } C_ ( ( `' G " { 0 } ) u. ( `' ( F quot G ) " { 0 } ) ) ) )
67	66	adantl 482	. . . . . . . . . . . . . 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 } ) ) ) )
68	65, 67	mpbird 247	. . . . . . . . . . . . 13 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> A e. ( ( `' G " { 0 } ) u. ( `' ( F quot G ) " { 0 } ) ) )
69		ofmulrt 24037	. . . . . . . . . . . . . 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 } ) ) )
70	59, 51, 55, 69	syl3anc 1326	. . . . . . . . . . . . 13 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> ( `' ( G oF x. ( F quot G ) ) " { 0 } ) = ( ( `' G " { 0 } ) u. ( `' ( F quot G ) " { 0 } ) ) )
71	68, 70	eleqtrrd 2704	. . . . . . . . . . . 12 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> A e. ( `' ( G oF x. ( F quot G ) ) " { 0 } ) )
72		fniniseg 6338	. . . . . . . . . . . . 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 ) ) )
73	61, 72	syl 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 ) ) )
74	71, 73	mpbid 222	. . . . . . . . . . 11 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> ( A e. CC /\ ( ( G oF x. ( F quot G ) ) ` A ) = 0 ) )
75	74	simprd 479	. . . . . . . . . 10 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> ( ( G oF x. ( F quot G ) ) ` A ) = 0 )
76	75	adantr 481	. . . . . . . . 9 |- ( ( ( F e. (Poly ` S ) /\ A e. CC ) /\ A e. CC ) -> ( ( G oF x. ( F quot G ) ) ` A ) = 0 )
77	49, 61, 59, 59, 60, 62, 76	ofval 6906	. . . . . . . 8 |- ( ( ( F e. (Poly ` S ) /\ A e. CC ) /\ A e. CC ) -> ( ( F oF - ( G oF x. ( F quot G ) ) ) ` A ) = ( ( F ` A ) - 0 ) )
78	77	anabss3 864	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> ( ( F oF - ( G oF x. ( F quot G ) ) ) ` A ) = ( ( F ` A ) - 0 ) )
79	45, 78	syl5eq 2668	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> ( R ` A ) = ( ( F ` A ) - 0 ) )
80	46	ffvelrnda 6359	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> ( F ` A ) e. CC )
81	80	subid1d 10381	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> ( ( F ` A ) - 0 ) = ( F ` A ) )
82	79, 81	eqtrd 2656	. . . . 5 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> ( R ` A ) = ( F ` A ) )
83		fvex 6201	. . . . . . 7 |- ( R ` 0 ) e. _V
84	83	fvconst2 6469	. . . . . 6 |- ( A e. CC -> ( ( CC X. { ( R ` 0 ) } ) ` A ) = ( R ` 0 ) )
85	84	adantl 482	. . . . 5 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> ( ( CC X. { ( R ` 0 ) } ) ` A ) = ( R ` 0 ) )
86	44, 82, 85	3eqtr3d 2664	. . . 4 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> ( F ` A ) = ( R ` 0 ) )
87	86	sneqd 4189	. . 3 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> { ( F ` A ) } = { ( R ` 0 ) } )
88	87	xpeq2d 5139	. 2 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> ( CC X. { ( F ` A ) } ) = ( CC X. { ( R ` 0 ) } ) )
89	43, 88	eqtr4d 2659	1 |- ( ( F e. (Poly ` S ) /\ A e. CC ) -> R = ( CC X. { ( F ` A ) } ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   u. cun 3572   C_ wss 3574   {csn 4177   class class class wbr 4653   X. cxp 5112   `'ccnv 5113   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   0cc0 9936   1c1 9937   x. cmul 9941   < clt 10074   - cmin 10266   NN0cn0 11292   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:  facth  24061