Theorem vieta1lem1 24065

Description: Lemma for vieta1 24067. (Contributed by Mario Carneiro, 28-Jul-2014.)

Hypotheses

Ref	Expression
vieta1.1	|- A = (coeff ` F )
vieta1.2	|- N = (deg ` F )
vieta1.3	|- R = ( `' F " { 0 } )
vieta1.4	|- ( ph -> F e. (Poly ` S ) )
vieta1.5	|- ( ph -> ( # ` R ) = N )
vieta1lem.6	|- ( ph -> D e. NN )
vieta1lem.7	|- ( ph -> ( D + 1 ) = N )
vieta1lem.8	|- ( ph -> A. f e. (Poly ` CC ) ( ( D = (deg ` f ) /\ ( # ` ( `' f " { 0 } ) ) = (deg ` f ) ) -> sum_ x e. ( `' f " { 0 } ) x = -u ( ( (coeff ` f ) ` ( (deg ` f ) - 1 ) ) / ( (coeff ` f ) ` (deg ` f ) ) ) ) )
vieta1lem.9	|- Q = ( F quot ( Xp oF - ( CC X. { z } ) ) )

Assertion

Ref	Expression
vieta1lem1	|- ( ( ph /\ z e. R ) -> ( Q e. (Poly ` CC ) /\ D = (deg ` Q ) ) )

Distinct variable groups:   D, f   f, F   z, f, N   x, f, Q   R, f   x, z, R   A, f, z   ph, x, z

Allowed substitution hints:   ph( f)   A( x)   D( x, z)   Q( z)   S( x, z, f)   F( x, z)   N( x)

Proof of Theorem vieta1lem1

Step	Hyp	Ref	Expression
1		vieta1lem.9	. . 3 |- Q = ( F quot ( Xp oF - ( CC X. { z } ) ) )
2		plyssc 23956	. . . . 5 |- (Poly ` S ) C_ (Poly ` CC )
3		vieta1.4	. . . . . 6 |- ( ph -> F e. (Poly ` S ) )
4	3	adantr 481	. . . . 5 |- ( ( ph /\ z e. R ) -> F e. (Poly ` S ) )
5	2, 4	sseldi 3601	. . . 4 |- ( ( ph /\ z e. R ) -> F e. (Poly ` CC ) )
6		vieta1.3	. . . . . . . . 9 |- R = ( `' F " { 0 } )
7		cnvimass 5485	. . . . . . . . 9 |- ( `' F " { 0 } ) C_ dom F
8	6, 7	eqsstri 3635	. . . . . . . 8 |- R C_ dom F
9		plyf 23954	. . . . . . . . . 10 |- ( F e. (Poly ` S ) -> F : CC --> CC )
10	3, 9	syl 17	. . . . . . . . 9 |- ( ph -> F : CC --> CC )
11		fdm 6051	. . . . . . . . 9 |- ( F : CC --> CC -> dom F = CC )
12	10, 11	syl 17	. . . . . . . 8 |- ( ph -> dom F = CC )
13	8, 12	syl5sseq 3653	. . . . . . 7 |- ( ph -> R C_ CC )
14	13	sselda 3603	. . . . . 6 |- ( ( ph /\ z e. R ) -> z e. CC )
15		eqid 2622	. . . . . . 7 |- ( Xp oF - ( CC X. { z } ) ) = ( Xp oF - ( CC X. { z } ) )
16	15	plyremlem 24059	. . . . . 6 |- ( z e. CC -> ( ( Xp oF - ( CC X. { z } ) ) e. (Poly ` CC ) /\ (deg ` ( Xp oF - ( CC X. { z } ) ) ) = 1 /\ ( `' ( Xp oF - ( CC X. { z } ) ) " { 0 } ) = { z } ) )
17	14, 16	syl 17	. . . . 5 |- ( ( ph /\ z e. R ) -> ( ( Xp oF - ( CC X. { z } ) ) e. (Poly ` CC ) /\ (deg ` ( Xp oF - ( CC X. { z } ) ) ) = 1 /\ ( `' ( Xp oF - ( CC X. { z } ) ) " { 0 } ) = { z } ) )
18	17	simp1d 1073	. . . 4 |- ( ( ph /\ z e. R ) -> ( Xp oF - ( CC X. { z } ) ) e. (Poly ` CC ) )
19	17	simp2d 1074	. . . . . 6 |- ( ( ph /\ z e. R ) -> (deg ` ( Xp oF - ( CC X. { z } ) ) ) = 1 )
20		ax-1ne0 10005	. . . . . . 7 |- 1 =/= 0
21	20	a1i 11	. . . . . 6 |- ( ( ph /\ z e. R ) -> 1 =/= 0 )
22	19, 21	eqnetrd 2861	. . . . 5 |- ( ( ph /\ z e. R ) -> (deg ` ( Xp oF - ( CC X. { z } ) ) ) =/= 0 )
23		fveq2 6191	. . . . . . 7 |- ( ( Xp oF - ( CC X. { z } ) ) = 0p -> (deg ` ( Xp oF - ( CC X. { z } ) ) ) = (deg ` 0p ) )
24		dgr0 24018	. . . . . . 7 |- (deg ` 0p ) = 0
25	23, 24	syl6eq 2672	. . . . . 6 |- ( ( Xp oF - ( CC X. { z } ) ) = 0p -> (deg ` ( Xp oF - ( CC X. { z } ) ) ) = 0 )
26	25	necon3i 2826	. . . . 5 |- ( (deg ` ( Xp oF - ( CC X. { z } ) ) ) =/= 0 -> ( Xp oF - ( CC X. { z } ) ) =/= 0p )
27	22, 26	syl 17	. . . 4 |- ( ( ph /\ z e. R ) -> ( Xp oF - ( CC X. { z } ) ) =/= 0p )
28		quotcl2 24057	. . . 4 |- ( ( F e. (Poly ` CC ) /\ ( Xp oF - ( CC X. { z } ) ) e. (Poly ` CC ) /\ ( Xp oF - ( CC X. { z } ) ) =/= 0p ) -> ( F quot ( Xp oF - ( CC X. { z } ) ) ) e. (Poly ` CC ) )
29	5, 18, 27, 28	syl3anc 1326	. . 3 |- ( ( ph /\ z e. R ) -> ( F quot ( Xp oF - ( CC X. { z } ) ) ) e. (Poly ` CC ) )
30	1, 29	syl5eqel 2705	. 2 |- ( ( ph /\ z e. R ) -> Q e. (Poly ` CC ) )
31		ax-1cn 9994	. . . 4 |- 1 e. CC
32	31	a1i 11	. . 3 |- ( ( ph /\ z e. R ) -> 1 e. CC )
33		vieta1lem.6	. . . . 5 |- ( ph -> D e. NN )
34	33	nncnd 11036	. . . 4 |- ( ph -> D e. CC )
35	34	adantr 481	. . 3 |- ( ( ph /\ z e. R ) -> D e. CC )
36		dgrcl 23989	. . . . 5 |- ( Q e. (Poly ` CC ) -> (deg ` Q ) e. NN0 )
37	30, 36	syl 17	. . . 4 |- ( ( ph /\ z e. R ) -> (deg ` Q ) e. NN0 )
38	37	nn0cnd 11353	. . 3 |- ( ( ph /\ z e. R ) -> (deg ` Q ) e. CC )
39		addcom 10222	. . . . 5 |- ( ( 1 e. CC /\ D e. CC ) -> ( 1 + D ) = ( D + 1 ) )
40	31, 35, 39	sylancr 695	. . . 4 |- ( ( ph /\ z e. R ) -> ( 1 + D ) = ( D + 1 ) )
41		vieta1lem.7	. . . . . . 7 |- ( ph -> ( D + 1 ) = N )
42		vieta1.2	. . . . . . 7 |- N = (deg ` F )
43	41, 42	syl6eq 2672	. . . . . 6 |- ( ph -> ( D + 1 ) = (deg ` F ) )
44	43	adantr 481	. . . . 5 |- ( ( ph /\ z e. R ) -> ( D + 1 ) = (deg ` F ) )
45	6	eleq2i 2693	. . . . . . . . . 10 |- ( z e. R <-> z e. ( `' F " { 0 } ) )
46		ffn 6045	. . . . . . . . . . . 12 |- ( F : CC --> CC -> F Fn CC )
47	10, 46	syl 17	. . . . . . . . . . 11 |- ( ph -> F Fn CC )
48		fniniseg 6338	. . . . . . . . . . 11 |- ( F Fn CC -> ( z e. ( `' F " { 0 } ) <-> ( z e. CC /\ ( F ` z ) = 0 ) ) )
49	47, 48	syl 17	. . . . . . . . . 10 |- ( ph -> ( z e. ( `' F " { 0 } ) <-> ( z e. CC /\ ( F ` z ) = 0 ) ) )
50	45, 49	syl5bb 272	. . . . . . . . 9 |- ( ph -> ( z e. R <-> ( z e. CC /\ ( F ` z ) = 0 ) ) )
51	50	simplbda 654	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> ( F ` z ) = 0 )
52	15	facth 24061	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ z e. CC /\ ( F ` z ) = 0 ) -> F = ( ( Xp oF - ( CC X. { z } ) ) oF x. ( F quot ( Xp oF - ( CC X. { z } ) ) ) ) )
53	4, 14, 51, 52	syl3anc 1326	. . . . . . 7 |- ( ( ph /\ z e. R ) -> F = ( ( Xp oF - ( CC X. { z } ) ) oF x. ( F quot ( Xp oF - ( CC X. { z } ) ) ) ) )
54	1	oveq2i 6661	. . . . . . 7 |- ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) = ( ( Xp oF - ( CC X. { z } ) ) oF x. ( F quot ( Xp oF - ( CC X. { z } ) ) ) )
55	53, 54	syl6eqr 2674	. . . . . 6 |- ( ( ph /\ z e. R ) -> F = ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) )
56	55	fveq2d 6195	. . . . 5 |- ( ( ph /\ z e. R ) -> (deg ` F ) = (deg ` ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) ) )
57	33	peano2nnd 11037	. . . . . . . . . . . . . . 15 |- ( ph -> ( D + 1 ) e. NN )
58	41, 57	eqeltrrd 2702	. . . . . . . . . . . . . 14 |- ( ph -> N e. NN )
59	58	nnne0d 11065	. . . . . . . . . . . . 13 |- ( ph -> N =/= 0 )
60	42, 59	syl5eqner 2869	. . . . . . . . . . . 12 |- ( ph -> (deg ` F ) =/= 0 )
61		fveq2 6191	. . . . . . . . . . . . . 14 |- ( F = 0p -> (deg ` F ) = (deg ` 0p ) )
62	61, 24	syl6eq 2672	. . . . . . . . . . . . 13 |- ( F = 0p -> (deg ` F ) = 0 )
63	62	necon3i 2826	. . . . . . . . . . . 12 |- ( (deg ` F ) =/= 0 -> F =/= 0p )
64	60, 63	syl 17	. . . . . . . . . . 11 |- ( ph -> F =/= 0p )
65	64	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> F =/= 0p )
66	55, 65	eqnetrrd 2862	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) =/= 0p )
67		plymul0or 24036	. . . . . . . . . . 11 |- ( ( ( Xp oF - ( CC X. { z } ) ) e. (Poly ` CC ) /\ Q e. (Poly ` CC ) ) -> ( ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) = 0p <-> ( ( Xp oF - ( CC X. { z } ) ) = 0p \/ Q = 0p ) ) )
68	18, 30, 67	syl2anc 693	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> ( ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) = 0p <-> ( ( Xp oF - ( CC X. { z } ) ) = 0p \/ Q = 0p ) ) )
69	68	necon3abid 2830	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> ( ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) =/= 0p <-> -. ( ( Xp oF - ( CC X. { z } ) ) = 0p \/ Q = 0p ) ) )
70	66, 69	mpbid 222	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> -. ( ( Xp oF - ( CC X. { z } ) ) = 0p \/ Q = 0p ) )
71		neanior 2886	. . . . . . . 8 |- ( ( ( Xp oF - ( CC X. { z } ) ) =/= 0p /\ Q =/= 0p ) <-> -. ( ( Xp oF - ( CC X. { z } ) ) = 0p \/ Q = 0p ) )
72	70, 71	sylibr 224	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( ( Xp oF - ( CC X. { z } ) ) =/= 0p /\ Q =/= 0p ) )
73	72	simprd 479	. . . . . 6 |- ( ( ph /\ z e. R ) -> Q =/= 0p )
74		eqid 2622	. . . . . . 7 |- (deg ` ( Xp oF - ( CC X. { z } ) ) ) = (deg ` ( Xp oF - ( CC X. { z } ) ) )
75		eqid 2622	. . . . . . 7 |- (deg ` Q ) = (deg ` Q )
76	74, 75	dgrmul 24026	. . . . . 6 |- ( ( ( ( Xp oF - ( CC X. { z } ) ) e. (Poly ` CC ) /\ ( Xp oF - ( CC X. { z } ) ) =/= 0p ) /\ ( Q e. (Poly ` CC ) /\ Q =/= 0p ) ) -> (deg ` ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) ) = ( (deg ` ( Xp oF - ( CC X. { z } ) ) ) + (deg ` Q ) ) )
77	18, 27, 30, 73, 76	syl22anc 1327	. . . . 5 |- ( ( ph /\ z e. R ) -> (deg ` ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) ) = ( (deg ` ( Xp oF - ( CC X. { z } ) ) ) + (deg ` Q ) ) )
78	44, 56, 77	3eqtrd 2660	. . . 4 |- ( ( ph /\ z e. R ) -> ( D + 1 ) = ( (deg ` ( Xp oF - ( CC X. { z } ) ) ) + (deg ` Q ) ) )
79	19	oveq1d 6665	. . . 4 |- ( ( ph /\ z e. R ) -> ( (deg ` ( Xp oF - ( CC X. { z } ) ) ) + (deg ` Q ) ) = ( 1 + (deg ` Q ) ) )
80	40, 78, 79	3eqtrd 2660	. . 3 |- ( ( ph /\ z e. R ) -> ( 1 + D ) = ( 1 + (deg ` Q ) ) )
81	32, 35, 38, 80	addcanad 10241	. 2 |- ( ( ph /\ z e. R ) -> D = (deg ` Q ) )
82	30, 81	jca 554	1 |- ( ( ph /\ z e. R ) -> ( Q e. (Poly ` CC ) /\ D = (deg ` Q ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {csn 4177   X. cxp 5112   `'ccnv 5113   dom cdm 5114   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   NN0cn0 11292   #chash 13117   sum_csu 14416   0pc0p 23436  Polycply 23940   Xpcidp 23941  coeffccoe 23942  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:  vieta1lem2  24066