Theorem vieta1lem2 20181

Description: Lemma for vieta1 20182: inductive step. Let z be a root of F. Then F = ( X p - z ) x. Q for some Q by the factor theorem, and Q is a degree- D polynomial, so by the induction hypothesis sum_ x e. ( `' Q " 0 ) x = -u (coeff ` Q ) ` ( D - 1 ) / (coeff ` Q ) ` D, so sum_ x e. R x = z - (coeff ` Q ) ` ( D - 1 ) / (coeff ` Q ) ` D. Now the coefficients of F are A ` ( D + 1 ) = (coeff ` Q ) ` D and A ` D = sum_ k e. ( 0 ... D ) (coeff ` X p - z ) ` k x. (coeff ` Q ) ` ( D - k ), which works out to -u z x. (coeff ` Q ) ` D + (coeff ` Q ) ` ( D - 1 ), so putting it all together we have sum_ x e. R x = -u A ` D / A ` ( D + 1 ) as we wanted to show. (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 ( X p o F - ( CC X. { z } ) ) )

Assertion

Ref	Expression
vieta1lem2	|- ( ph -> sum_ x e. R x = -u ( ( A ` ( N - 1 ) ) / ( A ` N ) ) )

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 vieta1lem2

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vieta1.5	. . . . 5 |- ( ph -> ( # ` R ) = N )
2		vieta1lem.7	. . . . . . 7 |- ( ph -> ( D + 1 ) = N )
3		vieta1lem.6	. . . . . . . 8 |- ( ph -> D e. NN )
4	3	peano2nnd 9973	. . . . . . 7 |- ( ph -> ( D + 1 ) e. NN )
5	2, 4	eqeltrrd 2479	. . . . . 6 |- ( ph -> N e. NN )
6	5	nnne0d 10000	. . . . 5 |- ( ph -> N =/= 0 )
7	1, 6	eqnetrd 2585	. . . 4 |- ( ph -> ( # ` R ) =/= 0 )
8		vieta1.4	. . . . . . . 8 |- ( ph -> F e. (Poly ` S ) )
9		vieta1.2	. . . . . . . . . 10 |- N = (deg ` F )
10	9, 6	syl5eqner 2592	. . . . . . . . 9 |- ( ph -> (deg ` F ) =/= 0 )
11		fveq2 5687	. . . . . . . . . . 11 |- ( F = 0 p -> (deg ` F ) = (deg ` 0 p ) )
12		dgr0 20133	. . . . . . . . . . 11 |- (deg ` 0 p ) = 0
13	11, 12	syl6eq 2452	. . . . . . . . . 10 |- ( F = 0 p -> (deg ` F ) = 0 )
14	13	necon3i 2606	. . . . . . . . 9 |- ( (deg ` F ) =/= 0 -> F =/= 0 p )
15	10, 14	syl 16	. . . . . . . 8 |- ( ph -> F =/= 0 p )
16		vieta1.3	. . . . . . . . 9 |- R = ( `' F " { 0 } )
17	16	fta1 20178	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ F =/= 0 p ) -> ( R e. Fin /\ ( # ` R ) <_ (deg ` F ) ) )
18	8, 15, 17	syl2anc 643	. . . . . . 7 |- ( ph -> ( R e. Fin /\ ( # ` R ) <_ (deg ` F ) ) )
19	18	simpld 446	. . . . . 6 |- ( ph -> R e. Fin )
20		hasheq0 11599	. . . . . 6 |- ( R e. Fin -> ( ( # ` R ) = 0 <-> R = (/) ) )
21	19, 20	syl 16	. . . . 5 |- ( ph -> ( ( # ` R ) = 0 <-> R = (/) ) )
22	21	necon3bid 2602	. . . 4 |- ( ph -> ( ( # ` R ) =/= 0 <-> R =/= (/) ) )
23	7, 22	mpbid 202	. . 3 |- ( ph -> R =/= (/) )
24		n0 3597	. . 3 |- ( R =/= (/) <-> E. z z e. R )
25	23, 24	sylib 189	. 2 |- ( ph -> E. z z e. R )
26		incom 3493	. . . . 5 |- ( { z } i^i ( `' Q " { 0 } ) ) = ( ( `' Q " { 0 } ) i^i { z } )
27		vieta1.1	. . . . . . . . . . 11 |- A = (coeff ` F )
28		vieta1lem.8	. . . . . . . . . . 11 |- ( 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 ) ) ) ) )
29		vieta1lem.9	. . . . . . . . . . 11 |- Q = ( F quot ( X p o F - ( CC X. { z } ) ) )
30	27, 9, 16, 8, 1, 3, 2, 28, 29	vieta1lem1 20180	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> ( Q e. (Poly ` CC ) /\ D = (deg ` Q ) ) )
31	30	simprd 450	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> D = (deg ` Q ) )
32	30	simpld 446	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> Q e. (Poly ` CC ) )
33		dgrcl 20105	. . . . . . . . . . 11 |- ( Q e. (Poly ` CC ) -> (deg ` Q ) e. NN0 )
34	32, 33	syl 16	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> (deg ` Q ) e. NN0 )
35	34	nn0red 10231	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> (deg ` Q ) e. RR )
36	31, 35	eqeltrd 2478	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> D e. RR )
37	36	ltp1d 9897	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> D < ( D + 1 ) )
38	36, 37	gtned 9164	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( D + 1 ) =/= D )
39		snssi 3902	. . . . . . . . . . 11 |- ( z e. ( `' Q " { 0 } ) -> { z } C_ ( `' Q " { 0 } ) )
40		ssequn1 3477	. . . . . . . . . . 11 |- ( { z } C_ ( `' Q " { 0 } ) <-> ( { z } u. ( `' Q " { 0 } ) ) = ( `' Q " { 0 } ) )
41	39, 40	sylib 189	. . . . . . . . . 10 |- ( z e. ( `' Q " { 0 } ) -> ( { z } u. ( `' Q " { 0 } ) ) = ( `' Q " { 0 } ) )
42	41	fveq2d 5691	. . . . . . . . 9 |- ( z e. ( `' Q " { 0 } ) -> ( # ` ( { z } u. ( `' Q " { 0 } ) ) ) = ( # ` ( `' Q " { 0 } ) ) )
43	8	adantr 452	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. R ) -> F e. (Poly ` S ) )
44		cnvimass 5183	. . . . . . . . . . . . . . . . . . . . 21 |- ( `' F " { 0 } ) C_ dom F
45	16, 44	eqsstri 3338	. . . . . . . . . . . . . . . . . . . 20 |- R C_ dom F
46		plyf 20070	. . . . . . . . . . . . . . . . . . . . 21 |- ( F e. (Poly ` S ) -> F : CC --> CC )
47		fdm 5554	. . . . . . . . . . . . . . . . . . . . 21 |- ( F : CC --> CC -> dom F = CC )
48	8, 46, 47	3syl 19	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> dom F = CC )
49	45, 48	syl5sseq 3356	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> R C_ CC )
50	49	sselda 3308	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. R ) -> z e. CC )
51	16	eleq2i 2468	. . . . . . . . . . . . . . . . . . . 20 |- ( z e. R <-> z e. ( `' F " { 0 } ) )
52	8, 46	syl 16	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> F : CC --> CC )
53		ffn 5550	. . . . . . . . . . . . . . . . . . . . 21 |- ( F : CC --> CC -> F Fn CC )
54		fniniseg 5810	. . . . . . . . . . . . . . . . . . . . 21 |- ( F Fn CC -> ( z e. ( `' F " { 0 } ) <-> ( z e. CC /\ ( F ` z ) = 0 ) ) )
55	52, 53, 54	3syl 19	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( z e. ( `' F " { 0 } ) <-> ( z e. CC /\ ( F ` z ) = 0 ) ) )
56	51, 55	syl5bb 249	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( z e. R <-> ( z e. CC /\ ( F ` z ) = 0 ) ) )
57	56	simplbda 608	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. R ) -> ( F ` z ) = 0 )
58		eqid 2404	. . . . . . . . . . . . . . . . . . 19 |- ( X p o F - ( CC X. { z } ) ) = ( X p o F - ( CC X. { z } ) )
59	58	facth 20176	. . . . . . . . . . . . . . . . . 18 |- ( ( F e. (Poly ` S ) /\ z e. CC /\ ( F ` z ) = 0 ) -> F = ( ( X p o F - ( CC X. { z } ) ) o F x. ( F quot ( X p o F - ( CC X. { z } ) ) ) ) )
60	43, 50, 57, 59	syl3anc 1184	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. R ) -> F = ( ( X p o F - ( CC X. { z } ) ) o F x. ( F quot ( X p o F - ( CC X. { z } ) ) ) ) )
61	29	oveq2i 6051	. . . . . . . . . . . . . . . . 17 |- ( ( X p o F - ( CC X. { z } ) ) o F x. Q ) = ( ( X p o F - ( CC X. { z } ) ) o F x. ( F quot ( X p o F - ( CC X. { z } ) ) ) )
62	60, 61	syl6eqr 2454	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ z e. R ) -> F = ( ( X p o F - ( CC X. { z } ) ) o F x. Q ) )
63	62	cnveqd 5007	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> `' F = `' ( ( X p o F - ( CC X. { z } ) ) o F x. Q ) )
64	63	imaeq1d 5161	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( `' F " { 0 } ) = ( `' ( ( X p o F - ( CC X. { z } ) ) o F x. Q ) " { 0 } ) )
65	16, 64	syl5eq 2448	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> R = ( `' ( ( X p o F - ( CC X. { z } ) ) o F x. Q ) " { 0 } ) )
66		cnex 9027	. . . . . . . . . . . . . . 15 |- CC e. _V
67	66	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> CC e. _V )
68	58	plyremlem 20174	. . . . . . . . . . . . . . . . 17 |- ( z e. CC -> ( ( X p o F - ( CC X. { z } ) ) e. (Poly ` CC ) /\ (deg ` ( X p o F - ( CC X. { z } ) ) ) = 1 /\ ( `' ( X p o F - ( CC X. { z } ) ) " { 0 } ) = { z } ) )
69	50, 68	syl 16	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ z e. R ) -> ( ( X p o F - ( CC X. { z } ) ) e. (Poly ` CC ) /\ (deg ` ( X p o F - ( CC X. { z } ) ) ) = 1 /\ ( `' ( X p o F - ( CC X. { z } ) ) " { 0 } ) = { z } ) )
70	69	simp1d 969	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> ( X p o F - ( CC X. { z } ) ) e. (Poly ` CC ) )
71		plyf 20070	. . . . . . . . . . . . . . 15 |- ( ( X p o F - ( CC X. { z } ) ) e. (Poly ` CC ) -> ( X p o F - ( CC X. { z } ) ) : CC --> CC )
72	70, 71	syl 16	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( X p o F - ( CC X. { z } ) ) : CC --> CC )
73		plyf 20070	. . . . . . . . . . . . . . 15 |- ( Q e. (Poly ` CC ) -> Q : CC --> CC )
74	32, 73	syl 16	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> Q : CC --> CC )
75		ofmulrt 20152	. . . . . . . . . . . . . 14 |- ( ( CC e. _V /\ ( X p o F - ( CC X. { z } ) ) : CC --> CC /\ Q : CC --> CC ) -> ( `' ( ( X p o F - ( CC X. { z } ) ) o F x. Q ) " { 0 } ) = ( ( `' ( X p o F - ( CC X. { z } ) ) " { 0 } ) u. ( `' Q " { 0 } ) ) )
76	67, 72, 74, 75	syl3anc 1184	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( `' ( ( X p o F - ( CC X. { z } ) ) o F x. Q ) " { 0 } ) = ( ( `' ( X p o F - ( CC X. { z } ) ) " { 0 } ) u. ( `' Q " { 0 } ) ) )
77	69	simp3d 971	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( `' ( X p o F - ( CC X. { z } ) ) " { 0 } ) = { z } )
78	77	uneq1d 3460	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( ( `' ( X p o F - ( CC X. { z } ) ) " { 0 } ) u. ( `' Q " { 0 } ) ) = ( { z } u. ( `' Q " { 0 } ) ) )
79	65, 76, 78	3eqtrd 2440	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> R = ( { z } u. ( `' Q " { 0 } ) ) )
80	79	fveq2d 5691	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> ( # ` R ) = ( # ` ( { z } u. ( `' Q " { 0 } ) ) ) )
81	1, 2	eqtr4d 2439	. . . . . . . . . . . 12 |- ( ph -> ( # ` R ) = ( D + 1 ) )
82	81	adantr 452	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> ( # ` R ) = ( D + 1 ) )
83	80, 82	eqtr3d 2438	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> ( # ` ( { z } u. ( `' Q " { 0 } ) ) ) = ( D + 1 ) )
84	15	adantr 452	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. R ) -> F =/= 0 p )
85	62, 84	eqnetrrd 2587	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. R ) -> ( ( X p o F - ( CC X. { z } ) ) o F x. Q ) =/= 0 p )
86		plymul0or 20151	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( X p o F - ( CC X. { z } ) ) e. (Poly ` CC ) /\ Q e. (Poly ` CC ) ) -> ( ( ( X p o F - ( CC X. { z } ) ) o F x. Q ) = 0 p <-> ( ( X p o F - ( CC X. { z } ) ) = 0 p \/ Q = 0 p ) ) )
87	70, 32, 86	syl2anc 643	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. R ) -> ( ( ( X p o F - ( CC X. { z } ) ) o F x. Q ) = 0 p <-> ( ( X p o F - ( CC X. { z } ) ) = 0 p \/ Q = 0 p ) ) )
88	87	necon3abid 2600	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. R ) -> ( ( ( X p o F - ( CC X. { z } ) ) o F x. Q ) =/= 0 p <-> -. ( ( X p o F - ( CC X. { z } ) ) = 0 p \/ Q = 0 p ) ) )
89	85, 88	mpbid 202	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ z e. R ) -> -. ( ( X p o F - ( CC X. { z } ) ) = 0 p \/ Q = 0 p ) )
90		neanior 2652	. . . . . . . . . . . . . . . 16 |- ( ( ( X p o F - ( CC X. { z } ) ) =/= 0 p /\ Q =/= 0 p ) <-> -. ( ( X p o F - ( CC X. { z } ) ) = 0 p \/ Q = 0 p ) )
91	89, 90	sylibr 204	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> ( ( X p o F - ( CC X. { z } ) ) =/= 0 p /\ Q =/= 0 p ) )
92	91	simprd 450	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> Q =/= 0 p )
93		eqid 2404	. . . . . . . . . . . . . . 15 |- ( `' Q " { 0 } ) = ( `' Q " { 0 } )
94	93	fta1 20178	. . . . . . . . . . . . . 14 |- ( ( Q e. (Poly ` CC ) /\ Q =/= 0 p ) -> ( ( `' Q " { 0 } ) e. Fin /\ ( # ` ( `' Q " { 0 } ) ) <_ (deg ` Q ) ) )
95	32, 92, 94	syl2anc 643	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( ( `' Q " { 0 } ) e. Fin /\ ( # ` ( `' Q " { 0 } ) ) <_ (deg ` Q ) ) )
96	95	simprd 450	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( # ` ( `' Q " { 0 } ) ) <_ (deg ` Q ) )
97	96, 31	breqtrrd 4198	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> ( # ` ( `' Q " { 0 } ) ) <_ D )
98		snfi 7146	. . . . . . . . . . . . . 14 |- { z } e. Fin
99	95	simpld 446	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( `' Q " { 0 } ) e. Fin )
100		hashun2 11612	. . . . . . . . . . . . . 14 |- ( ( { z } e. Fin /\ ( `' Q " { 0 } ) e. Fin ) -> ( # ` ( { z } u. ( `' Q " { 0 } ) ) ) <_ ( ( # ` { z } ) + ( # ` ( `' Q " { 0 } ) ) ) )
101	98, 99, 100	sylancr 645	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( # ` ( { z } u. ( `' Q " { 0 } ) ) ) <_ ( ( # ` { z } ) + ( # ` ( `' Q " { 0 } ) ) ) )
102		ax-1cn 9004	. . . . . . . . . . . . . . 15 |- 1 e. CC
103	3	nncnd 9972	. . . . . . . . . . . . . . . 16 |- ( ph -> D e. CC )
104	103	adantr 452	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> D e. CC )
105		addcom 9208	. . . . . . . . . . . . . . 15 |- ( ( 1 e. CC /\ D e. CC ) -> ( 1 + D ) = ( D + 1 ) )
106	102, 104, 105	sylancr 645	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( 1 + D ) = ( D + 1 ) )
107	83, 106	eqtr4d 2439	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( # ` ( { z } u. ( `' Q " { 0 } ) ) ) = ( 1 + D ) )
108		hashsng 11602	. . . . . . . . . . . . . . 15 |- ( z e. R -> ( # ` { z } ) = 1 )
109	108	adantl 453	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( # ` { z } ) = 1 )
110	109	oveq1d 6055	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( ( # ` { z } ) + ( # ` ( `' Q " { 0 } ) ) ) = ( 1 + ( # ` ( `' Q " { 0 } ) ) ) )
111	101, 107, 110	3brtr3d 4201	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( 1 + D ) <_ ( 1 + ( # ` ( `' Q " { 0 } ) ) ) )
112		hashcl 11594	. . . . . . . . . . . . . . 15 |- ( ( `' Q " { 0 } ) e. Fin -> ( # ` ( `' Q " { 0 } ) ) e. NN0 )
113	99, 112	syl 16	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( # ` ( `' Q " { 0 } ) ) e. NN0 )
114	113	nn0red 10231	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( # ` ( `' Q " { 0 } ) ) e. RR )
115		1re 9046	. . . . . . . . . . . . . 14 |- 1 e. RR
116	115	a1i 11	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> 1 e. RR )
117	36, 114, 116	leadd2d 9577	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( D <_ ( # ` ( `' Q " { 0 } ) ) <-> ( 1 + D ) <_ ( 1 + ( # ` ( `' Q " { 0 } ) ) ) ) )
118	111, 117	mpbird 224	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> D <_ ( # ` ( `' Q " { 0 } ) ) )
119	114, 36	letri3d 9171	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> ( ( # ` ( `' Q " { 0 } ) ) = D <-> ( ( # ` ( `' Q " { 0 } ) ) <_ D /\ D <_ ( # ` ( `' Q " { 0 } ) ) ) ) )
120	97, 118, 119	mpbir2and 889	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> ( # ` ( `' Q " { 0 } ) ) = D )
121	83, 120	eqeq12d 2418	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> ( ( # ` ( { z } u. ( `' Q " { 0 } ) ) ) = ( # ` ( `' Q " { 0 } ) ) <-> ( D + 1 ) = D ) )
122	42, 121	syl5ib 211	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> ( z e. ( `' Q " { 0 } ) -> ( D + 1 ) = D ) )
123	122	necon3ad 2603	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( ( D + 1 ) =/= D -> -. z e. ( `' Q " { 0 } ) ) )
124	38, 123	mpd 15	. . . . . 6 |- ( ( ph /\ z e. R ) -> -. z e. ( `' Q " { 0 } ) )
125		disjsn 3828	. . . . . 6 |- ( ( ( `' Q " { 0 } ) i^i { z } ) = (/) <-> -. z e. ( `' Q " { 0 } ) )
126	124, 125	sylibr 204	. . . . 5 |- ( ( ph /\ z e. R ) -> ( ( `' Q " { 0 } ) i^i { z } ) = (/) )
127	26, 126	syl5eq 2448	. . . 4 |- ( ( ph /\ z e. R ) -> ( { z } i^i ( `' Q " { 0 } ) ) = (/) )
128	19	adantr 452	. . . 4 |- ( ( ph /\ z e. R ) -> R e. Fin )
129	49	adantr 452	. . . . 5 |- ( ( ph /\ z e. R ) -> R C_ CC )
130	129	sselda 3308	. . . 4 |- ( ( ( ph /\ z e. R ) /\ x e. R ) -> x e. CC )
131	127, 79, 128, 130	fsumsplit 12488	. . 3 |- ( ( ph /\ z e. R ) -> sum_ x e. R x = ( sum_ x e. { z } x + sum_ x e. ( `' Q " { 0 } ) x ) )
132		id 20	. . . . . . 7 |- ( x = z -> x = z )
133	132	sumsn 12489	. . . . . 6 |- ( ( z e. CC /\ z e. CC ) -> sum_ x e. { z } x = z )
134	50, 50, 133	syl2anc 643	. . . . 5 |- ( ( ph /\ z e. R ) -> sum_ x e. { z } x = z )
135	50	negnegd 9358	. . . . 5 |- ( ( ph /\ z e. R ) -> -u -u z = z )
136	134, 135	eqtr4d 2439	. . . 4 |- ( ( ph /\ z e. R ) -> sum_ x e. { z } x = -u -u z )
137	120, 31	eqtrd 2436	. . . . . 6 |- ( ( ph /\ z e. R ) -> ( # ` ( `' Q " { 0 } ) ) = (deg ` Q ) )
138	28	adantr 452	. . . . . . 7 |- ( ( ph /\ z e. R ) -> 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 ) ) ) ) )
139		fveq2 5687	. . . . . . . . . . 11 |- ( f = Q -> (deg ` f ) = (deg ` Q ) )
140	139	eqeq2d 2415	. . . . . . . . . 10 |- ( f = Q -> ( D = (deg ` f ) <-> D = (deg ` Q ) ) )
141		cnveq 5005	. . . . . . . . . . . . 13 |- ( f = Q -> `' f = `' Q )
142	141	imaeq1d 5161	. . . . . . . . . . . 12 |- ( f = Q -> ( `' f " { 0 } ) = ( `' Q " { 0 } ) )
143	142	fveq2d 5691	. . . . . . . . . . 11 |- ( f = Q -> ( # ` ( `' f " { 0 } ) ) = ( # ` ( `' Q " { 0 } ) ) )
144	143, 139	eqeq12d 2418	. . . . . . . . . 10 |- ( f = Q -> ( ( # ` ( `' f " { 0 } ) ) = (deg ` f ) <-> ( # ` ( `' Q " { 0 } ) ) = (deg ` Q ) ) )
145	140, 144	anbi12d 692	. . . . . . . . 9 |- ( f = Q -> ( ( D = (deg ` f ) /\ ( # ` ( `' f " { 0 } ) ) = (deg ` f ) ) <-> ( D = (deg ` Q ) /\ ( # ` ( `' Q " { 0 } ) ) = (deg ` Q ) ) ) )
146	142	sumeq1d 12450	. . . . . . . . . 10 |- ( f = Q -> sum_ x e. ( `' f " { 0 } ) x = sum_ x e. ( `' Q " { 0 } ) x )
147		fveq2 5687	. . . . . . . . . . . . 13 |- ( f = Q -> (coeff ` f ) = (coeff ` Q ) )
148	139	oveq1d 6055	. . . . . . . . . . . . 13 |- ( f = Q -> ( (deg ` f ) - 1 ) = ( (deg ` Q ) - 1 ) )
149	147, 148	fveq12d 5693	. . . . . . . . . . . 12 |- ( f = Q -> ( (coeff ` f ) ` ( (deg ` f ) - 1 ) ) = ( (coeff ` Q ) ` ( (deg ` Q ) - 1 ) ) )
150	147, 139	fveq12d 5693	. . . . . . . . . . . 12 |- ( f = Q -> ( (coeff ` f ) ` (deg ` f ) ) = ( (coeff ` Q ) ` (deg ` Q ) ) )
151	149, 150	oveq12d 6058	. . . . . . . . . . 11 |- ( f = Q -> ( ( (coeff ` f ) ` ( (deg ` f ) - 1 ) ) / ( (coeff ` f ) ` (deg ` f ) ) ) = ( ( (coeff ` Q ) ` ( (deg ` Q ) - 1 ) ) / ( (coeff ` Q ) ` (deg ` Q ) ) ) )
152	151	negeqd 9256	. . . . . . . . . 10 |- ( f = Q -> -u ( ( (coeff ` f ) ` ( (deg ` f ) - 1 ) ) / ( (coeff ` f ) ` (deg ` f ) ) ) = -u ( ( (coeff ` Q ) ` ( (deg ` Q ) - 1 ) ) / ( (coeff ` Q ) ` (deg ` Q ) ) ) )
153	146, 152	eqeq12d 2418	. . . . . . . . 9 |- ( f = Q -> ( sum_ x e. ( `' f " { 0 } ) x = -u ( ( (coeff ` f ) ` ( (deg ` f ) - 1 ) ) / ( (coeff ` f ) ` (deg ` f ) ) ) <-> sum_ x e. ( `' Q " { 0 } ) x = -u ( ( (coeff ` Q ) ` ( (deg ` Q ) - 1 ) ) / ( (coeff ` Q ) ` (deg ` Q ) ) ) ) )
154	145, 153	imbi12d 312	. . . . . . . 8 |- ( f = Q -> ( ( ( D = (deg ` f ) /\ ( # ` ( `' f " { 0 } ) ) = (deg ` f ) ) -> sum_ x e. ( `' f " { 0 } ) x = -u ( ( (coeff ` f ) ` ( (deg ` f ) - 1 ) ) / ( (coeff ` f ) ` (deg ` f ) ) ) ) <-> ( ( D = (deg ` Q ) /\ ( # ` ( `' Q " { 0 } ) ) = (deg ` Q ) ) -> sum_ x e. ( `' Q " { 0 } ) x = -u ( ( (coeff ` Q ) ` ( (deg ` Q ) - 1 ) ) / ( (coeff ` Q ) ` (deg ` Q ) ) ) ) ) )
155	154	rspcv 3008	. . . . . . 7 |- ( Q e. (Poly ` CC ) -> ( 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 ) ) ) ) -> ( ( D = (deg ` Q ) /\ ( # ` ( `' Q " { 0 } ) ) = (deg ` Q ) ) -> sum_ x e. ( `' Q " { 0 } ) x = -u ( ( (coeff ` Q ) ` ( (deg ` Q ) - 1 ) ) / ( (coeff ` Q ) ` (deg ` Q ) ) ) ) ) )
156	32, 138, 155	sylc 58	. . . . . 6 |- ( ( ph /\ z e. R ) -> ( ( D = (deg ` Q ) /\ ( # ` ( `' Q " { 0 } ) ) = (deg ` Q ) ) -> sum_ x e. ( `' Q " { 0 } ) x = -u ( ( (coeff ` Q ) ` ( (deg ` Q ) - 1 ) ) / ( (coeff ` Q ) ` (deg ` Q ) ) ) ) )
157	31, 137, 156	mp2and 661	. . . . 5 |- ( ( ph /\ z e. R ) -> sum_ x e. ( `' Q " { 0 } ) x = -u ( ( (coeff ` Q ) ` ( (deg ` Q ) - 1 ) ) / ( (coeff ` Q ) ` (deg ` Q ) ) ) )
158	31	oveq1d 6055	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> ( D - 1 ) = ( (deg ` Q ) - 1 ) )
159	158	fveq2d 5691	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( (coeff ` Q ) ` ( D - 1 ) ) = ( (coeff ` Q ) ` ( (deg ` Q ) - 1 ) ) )
160	62	fveq2d 5691	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> (coeff ` F ) = (coeff ` ( ( X p o F - ( CC X. { z } ) ) o F x. Q ) ) )
161	27, 160	syl5eq 2448	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> A = (coeff ` ( ( X p o F - ( CC X. { z } ) ) o F x. Q ) ) )
162	62	fveq2d 5691	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> (deg ` F ) = (deg ` ( ( X p o F - ( CC X. { z } ) ) o F x. Q ) ) )
163	69	simp2d 970	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> (deg ` ( X p o F - ( CC X. { z } ) ) ) = 1 )
164		ax-1ne0 9015	. . . . . . . . . . . . . . 15 |- 1 =/= 0
165	164	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> 1 =/= 0 )
166	163, 165	eqnetrd 2585	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> (deg ` ( X p o F - ( CC X. { z } ) ) ) =/= 0 )
167		fveq2 5687	. . . . . . . . . . . . . . 15 |- ( ( X p o F - ( CC X. { z } ) ) = 0 p -> (deg ` ( X p o F - ( CC X. { z } ) ) ) = (deg ` 0 p ) )
168	167, 12	syl6eq 2452	. . . . . . . . . . . . . 14 |- ( ( X p o F - ( CC X. { z } ) ) = 0 p -> (deg ` ( X p o F - ( CC X. { z } ) ) ) = 0 )
169	168	necon3i 2606	. . . . . . . . . . . . 13 |- ( (deg ` ( X p o F - ( CC X. { z } ) ) ) =/= 0 -> ( X p o F - ( CC X. { z } ) ) =/= 0 p )
170	166, 169	syl 16	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( X p o F - ( CC X. { z } ) ) =/= 0 p )
171		eqid 2404	. . . . . . . . . . . . 13 |- (deg ` ( X p o F - ( CC X. { z } ) ) ) = (deg ` ( X p o F - ( CC X. { z } ) ) )
172		eqid 2404	. . . . . . . . . . . . 13 |- (deg ` Q ) = (deg ` Q )
173	171, 172	dgrmul 20141	. . . . . . . . . . . 12 |- ( ( ( ( X p o F - ( CC X. { z } ) ) e. (Poly ` CC ) /\ ( X p o F - ( CC X. { z } ) ) =/= 0 p ) /\ ( Q e. (Poly ` CC ) /\ Q =/= 0 p ) ) -> (deg ` ( ( X p o F - ( CC X. { z } ) ) o F x. Q ) ) = ( (deg ` ( X p o F - ( CC X. { z } ) ) ) + (deg ` Q ) ) )
174	70, 170, 32, 92, 173	syl22anc 1185	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> (deg ` ( ( X p o F - ( CC X. { z } ) ) o F x. Q ) ) = ( (deg ` ( X p o F - ( CC X. { z } ) ) ) + (deg ` Q ) ) )
175	162, 174	eqtrd 2436	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> (deg ` F ) = ( (deg ` ( X p o F - ( CC X. { z } ) ) ) + (deg ` Q ) ) )
176	9, 175	syl5eq 2448	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> N = ( (deg ` ( X p o F - ( CC X. { z } ) ) ) + (deg ` Q ) ) )
177	161, 176	fveq12d 5693	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> ( A ` N ) = ( (coeff ` ( ( X p o F - ( CC X. { z } ) ) o F x. Q ) ) ` ( (deg ` ( X p o F - ( CC X. { z } ) ) ) + (deg ` Q ) ) ) )
178		eqid 2404	. . . . . . . . . 10 |- (coeff ` ( X p o F - ( CC X. { z } ) ) ) = (coeff ` ( X p o F - ( CC X. { z } ) ) )
179		eqid 2404	. . . . . . . . . 10 |- (coeff ` Q ) = (coeff ` Q )
180	178, 179, 171, 172	coemulhi 20125	. . . . . . . . 9 |- ( ( ( X p o F - ( CC X. { z } ) ) e. (Poly ` CC ) /\ Q e. (Poly ` CC ) ) -> ( (coeff ` ( ( X p o F - ( CC X. { z } ) ) o F x. Q ) ) ` ( (deg ` ( X p o F - ( CC X. { z } ) ) ) + (deg ` Q ) ) ) = ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` (deg ` ( X p o F - ( CC X. { z } ) ) ) ) x. ( (coeff ` Q ) ` (deg ` Q ) ) ) )
181	70, 32, 180	syl2anc 643	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( ( X p o F - ( CC X. { z } ) ) o F x. Q ) ) ` ( (deg ` ( X p o F - ( CC X. { z } ) ) ) + (deg ` Q ) ) ) = ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` (deg ` ( X p o F - ( CC X. { z } ) ) ) ) x. ( (coeff ` Q ) ` (deg ` Q ) ) ) )
182	163	fveq2d 5691	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` (deg ` ( X p o F - ( CC X. { z } ) ) ) ) = ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` 1 ) )
183		ssid 3327	. . . . . . . . . . . . . . 15 |- CC C_ CC
184		plyid 20081	. . . . . . . . . . . . . . 15 |- ( ( CC C_ CC /\ 1 e. CC ) -> X p e. (Poly ` CC ) )
185	183, 102, 184	mp2an 654	. . . . . . . . . . . . . 14 |- X p e. (Poly ` CC )
186		plyconst 20078	. . . . . . . . . . . . . . 15 |- ( ( CC C_ CC /\ z e. CC ) -> ( CC X. { z } ) e. (Poly ` CC ) )
187	183, 50, 186	sylancr 645	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( CC X. { z } ) e. (Poly ` CC ) )
188		eqid 2404	. . . . . . . . . . . . . . 15 |- (coeff ` X p ) = (coeff ` X p )
189		eqid 2404	. . . . . . . . . . . . . . 15 |- (coeff ` ( CC X. { z } ) ) = (coeff ` ( CC X. { z } ) )
190	188, 189	coesub 20128	. . . . . . . . . . . . . 14 |- ( ( X p e. (Poly ` CC ) /\ ( CC X. { z } ) e. (Poly ` CC ) ) -> (coeff ` ( X p o F - ( CC X. { z } ) ) ) = ( (coeff ` X p ) o F - (coeff ` ( CC X. { z } ) ) ) )
191	185, 187, 190	sylancr 645	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> (coeff ` ( X p o F - ( CC X. { z } ) ) ) = ( (coeff ` X p ) o F - (coeff ` ( CC X. { z } ) ) ) )
192	191	fveq1d 5689	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` 1 ) = ( ( (coeff ` X p ) o F - (coeff ` ( CC X. { z } ) ) ) ` 1 ) )
193		1nn0 10193	. . . . . . . . . . . . . 14 |- 1 e. NN0
194	188	coef3 20104	. . . . . . . . . . . . . . . . 17 |- ( X p e. (Poly ` CC ) -> (coeff ` X p ) : NN0 --> CC )
195		ffn 5550	. . . . . . . . . . . . . . . . 17 |- ( (coeff ` X p ) : NN0 --> CC -> (coeff ` X p ) Fn NN0 )
196	185, 194, 195	mp2b 10	. . . . . . . . . . . . . . . 16 |- (coeff ` X p ) Fn NN0
197	196	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> (coeff ` X p ) Fn NN0 )
198	189	coef3 20104	. . . . . . . . . . . . . . . 16 |- ( ( CC X. { z } ) e. (Poly ` CC ) -> (coeff ` ( CC X. { z } ) ) : NN0 --> CC )
199		ffn 5550	. . . . . . . . . . . . . . . 16 |- ( (coeff ` ( CC X. { z } ) ) : NN0 --> CC -> (coeff ` ( CC X. { z } ) ) Fn NN0 )
200	187, 198, 199	3syl 19	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> (coeff ` ( CC X. { z } ) ) Fn NN0 )
201		nn0ex 10183	. . . . . . . . . . . . . . . 16 |- NN0 e. _V
202	201	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> NN0 e. _V )
203		inidm 3510	. . . . . . . . . . . . . . 15 |- ( NN0 i^i NN0 ) = NN0
204		coeidp 20134	. . . . . . . . . . . . . . . . 17 |- ( 1 e. NN0 -> ( (coeff ` X p ) ` 1 ) = if ( 1 = 1 , 1 , 0 ) )
205	204	adantl 453	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> ( (coeff ` X p ) ` 1 ) = if ( 1 = 1 , 1 , 0 ) )
206		eqid 2404	. . . . . . . . . . . . . . . . 17 |- 1 = 1
207		iftrue 3705	. . . . . . . . . . . . . . . . 17 |- ( 1 = 1 -> if ( 1 = 1 , 1 , 0 ) = 1 )
208	206, 207	ax-mp 8	. . . . . . . . . . . . . . . 16 |- if ( 1 = 1 , 1 , 0 ) = 1
209	205, 208	syl6eq 2452	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> ( (coeff ` X p ) ` 1 ) = 1 )
210		0lt1 9506	. . . . . . . . . . . . . . . . . 18 |- 0 < 1
211		0re 9047	. . . . . . . . . . . . . . . . . . 19 |- 0 e. RR
212	211, 115	ltnlei 9150	. . . . . . . . . . . . . . . . . 18 |- ( 0 < 1 <-> -. 1 <_ 0 )
213	210, 212	mpbi 200	. . . . . . . . . . . . . . . . 17 |- -. 1 <_ 0
214	50	adantr 452	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> z e. CC )
215		0dgr 20117	. . . . . . . . . . . . . . . . . . 19 |- ( z e. CC -> (deg ` ( CC X. { z } ) ) = 0 )
216	214, 215	syl 16	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> (deg ` ( CC X. { z } ) ) = 0 )
217	216	breq2d 4184	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> ( 1 <_ (deg ` ( CC X. { z } ) ) <-> 1 <_ 0 ) )
218	213, 217	mtbiri 295	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> -. 1 <_ (deg ` ( CC X. { z } ) ) )
219		eqid 2404	. . . . . . . . . . . . . . . . . . . 20 |- (deg ` ( CC X. { z } ) ) = (deg ` ( CC X. { z } ) )
220	189, 219	dgrub 20106	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( CC X. { z } ) e. (Poly ` CC ) /\ 1 e. NN0 /\ ( (coeff ` ( CC X. { z } ) ) ` 1 ) =/= 0 ) -> 1 <_ (deg ` ( CC X. { z } ) ) )
221	220	3expia 1155	. . . . . . . . . . . . . . . . . 18 |- ( ( ( CC X. { z } ) e. (Poly ` CC ) /\ 1 e. NN0 ) -> ( ( (coeff ` ( CC X. { z } ) ) ` 1 ) =/= 0 -> 1 <_ (deg ` ( CC X. { z } ) ) ) )
222	187, 221	sylan 458	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> ( ( (coeff ` ( CC X. { z } ) ) ` 1 ) =/= 0 -> 1 <_ (deg ` ( CC X. { z } ) ) ) )
223	222	necon1bd 2635	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> ( -. 1 <_ (deg ` ( CC X. { z } ) ) -> ( (coeff ` ( CC X. { z } ) ) ` 1 ) = 0 ) )
224	218, 223	mpd 15	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> ( (coeff ` ( CC X. { z } ) ) ` 1 ) = 0 )
225	197, 200, 202, 202, 203, 209, 224	ofval 6273	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> ( ( (coeff ` X p ) o F - (coeff ` ( CC X. { z } ) ) ) ` 1 ) = ( 1 - 0 ) )
226	193, 225	mpan2 653	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` X p ) o F - (coeff ` ( CC X. { z } ) ) ) ` 1 ) = ( 1 - 0 ) )
227	102	subid1i 9328	. . . . . . . . . . . . 13 |- ( 1 - 0 ) = 1
228	226, 227	syl6eq 2452	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` X p ) o F - (coeff ` ( CC X. { z } ) ) ) ` 1 ) = 1 )
229	192, 228	eqtrd 2436	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` 1 ) = 1 )
230	182, 229	eqtrd 2436	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` (deg ` ( X p o F - ( CC X. { z } ) ) ) ) = 1 )
231	230	oveq1d 6055	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` (deg ` ( X p o F - ( CC X. { z } ) ) ) ) x. ( (coeff ` Q ) ` (deg ` Q ) ) ) = ( 1 x. ( (coeff ` Q ) ` (deg ` Q ) ) ) )
232	179	coef3 20104	. . . . . . . . . . . 12 |- ( Q e. (Poly ` CC ) -> (coeff ` Q ) : NN0 --> CC )
233	32, 232	syl 16	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> (coeff ` Q ) : NN0 --> CC )
234	233, 34	ffvelrnd 5830	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> ( (coeff ` Q ) ` (deg ` Q ) ) e. CC )
235	234	mulid2d 9062	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> ( 1 x. ( (coeff ` Q ) ` (deg ` Q ) ) ) = ( (coeff ` Q ) ` (deg ` Q ) ) )
236	231, 235	eqtrd 2436	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` (deg ` ( X p o F - ( CC X. { z } ) ) ) ) x. ( (coeff ` Q ) ` (deg ` Q ) ) ) = ( (coeff ` Q ) ` (deg ` Q ) ) )
237	177, 181, 236	3eqtrd 2440	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( A ` N ) = ( (coeff ` Q ) ` (deg ` Q ) ) )
238	159, 237	oveq12d 6058	. . . . . 6 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) = ( ( (coeff ` Q ) ` ( (deg ` Q ) - 1 ) ) / ( (coeff ` Q ) ` (deg ` Q ) ) ) )
239	238	negeqd 9256	. . . . 5 |- ( ( ph /\ z e. R ) -> -u ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) = -u ( ( (coeff ` Q ) ` ( (deg ` Q ) - 1 ) ) / ( (coeff ` Q ) ` (deg ` Q ) ) ) )
240	157, 239	eqtr4d 2439	. . . 4 |- ( ( ph /\ z e. R ) -> sum_ x e. ( `' Q " { 0 } ) x = -u ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) )
241	136, 240	oveq12d 6058	. . 3 |- ( ( ph /\ z e. R ) -> ( sum_ x e. { z } x + sum_ x e. ( `' Q " { 0 } ) x ) = ( -u -u z + -u ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) ) )
242	50	negcld 9354	. . . . 5 |- ( ( ph /\ z e. R ) -> -u z e. CC )
243		nnm1nn0 10217	. . . . . . . . 9 |- ( D e. NN -> ( D - 1 ) e. NN0 )
244	3, 243	syl 16	. . . . . . . 8 |- ( ph -> ( D - 1 ) e. NN0 )
245	244	adantr 452	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( D - 1 ) e. NN0 )
246	233, 245	ffvelrnd 5830	. . . . . 6 |- ( ( ph /\ z e. R ) -> ( (coeff ` Q ) ` ( D - 1 ) ) e. CC )
247	237, 234	eqeltrd 2478	. . . . . 6 |- ( ( ph /\ z e. R ) -> ( A ` N ) e. CC )
248	9, 27	dgreq0 20136	. . . . . . . . 9 |- ( F e. (Poly ` S ) -> ( F = 0 p <-> ( A ` N ) = 0 ) )
249	43, 248	syl 16	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> ( F = 0 p <-> ( A ` N ) = 0 ) )
250	249	necon3bid 2602	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( F =/= 0 p <-> ( A ` N ) =/= 0 ) )
251	84, 250	mpbid 202	. . . . . 6 |- ( ( ph /\ z e. R ) -> ( A ` N ) =/= 0 )
252	246, 247, 251	divcld 9746	. . . . 5 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) e. CC )
253	242, 252	negdid 9380	. . . 4 |- ( ( ph /\ z e. R ) -> -u ( -u z + ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) ) = ( -u -u z + -u ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) ) )
254	242, 247	mulcld 9064	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( -u z x. ( A ` N ) ) e. CC )
255	254, 246, 247, 251	divdird 9784	. . . . . 6 |- ( ( ph /\ z e. R ) -> ( ( ( -u z x. ( A ` N ) ) + ( (coeff ` Q ) ` ( D - 1 ) ) ) / ( A ` N ) ) = ( ( ( -u z x. ( A ` N ) ) / ( A ` N ) ) + ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) ) )
256		nnm1nn0 10217	. . . . . . . . . . 11 |- ( N e. NN -> ( N - 1 ) e. NN0 )
257	5, 256	syl 16	. . . . . . . . . 10 |- ( ph -> ( N - 1 ) e. NN0 )
258	257	adantr 452	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> ( N - 1 ) e. NN0 )
259	178, 179	coemul 20123	. . . . . . . . 9 |- ( ( ( X p o F - ( CC X. { z } ) ) e. (Poly ` CC ) /\ Q e. (Poly ` CC ) /\ ( N - 1 ) e. NN0 ) -> ( (coeff ` ( ( X p o F - ( CC X. { z } ) ) o F x. Q ) ) ` ( N - 1 ) ) = sum_ k e. ( 0 ... ( N - 1 ) ) ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) )
260	70, 32, 258, 259	syl3anc 1184	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( ( X p o F - ( CC X. { z } ) ) o F x. Q ) ) ` ( N - 1 ) ) = sum_ k e. ( 0 ... ( N - 1 ) ) ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) )
261	161	fveq1d 5689	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> ( A ` ( N - 1 ) ) = ( (coeff ` ( ( X p o F - ( CC X. { z } ) ) o F x. Q ) ) ` ( N - 1 ) ) )
262		1e0p1 10366	. . . . . . . . . . . 12 |- 1 = ( 0 + 1 )
263	262	oveq2i 6051	. . . . . . . . . . 11 |- ( 0 ... 1 ) = ( 0 ... ( 0 + 1 ) )
264	263	sumeq1i 12447	. . . . . . . . . 10 |- sum_ k e. ( 0 ... 1 ) ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) = sum_ k e. ( 0 ... ( 0 + 1 ) ) ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) )
265		0nn0 10192	. . . . . . . . . . . . 13 |- 0 e. NN0
266		nn0uz 10476	. . . . . . . . . . . . 13 |- NN0 = ( ZZ>= ` 0 )
267	265, 266	eleqtri 2476	. . . . . . . . . . . 12 |- 0 e. ( ZZ>= ` 0 )
268	267	a1i 11	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> 0 e. ( ZZ>= ` 0 ) )
269	263	eleq2i 2468	. . . . . . . . . . . 12 |- ( k e. ( 0 ... 1 ) <-> k e. ( 0 ... ( 0 + 1 ) ) )
270	178	coef3 20104	. . . . . . . . . . . . . . 15 |- ( ( X p o F - ( CC X. { z } ) ) e. (Poly ` CC ) -> (coeff ` ( X p o F - ( CC X. { z } ) ) ) : NN0 --> CC )
271	70, 270	syl 16	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> (coeff ` ( X p o F - ( CC X. { z } ) ) ) : NN0 --> CC )
272		elfznn0 11039	. . . . . . . . . . . . . 14 |- ( k e. ( 0 ... 1 ) -> k e. NN0 )
273		ffvelrn 5827	. . . . . . . . . . . . . 14 |- ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) : NN0 --> CC /\ k e. NN0 ) -> ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) e. CC )
274	271, 272, 273	syl2an 464	. . . . . . . . . . . . 13 |- ( ( ( ph /\ z e. R ) /\ k e. ( 0 ... 1 ) ) -> ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) e. CC )
275	2	oveq1d 6055	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( ( D + 1 ) - 1 ) = ( N - 1 ) )
276		pncan 9267	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( D e. CC /\ 1 e. CC ) -> ( ( D + 1 ) - 1 ) = D )
277	103, 102, 276	sylancl 644	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( ( D + 1 ) - 1 ) = D )
278	275, 277	eqtr3d 2438	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( N - 1 ) = D )
279	278	adantr 452	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. R ) -> ( N - 1 ) = D )
280	3	adantr 452	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. R ) -> D e. NN )
281	279, 280	eqeltrd 2478	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. R ) -> ( N - 1 ) e. NN )
282		nnuz 10477	. . . . . . . . . . . . . . . . 17 |- NN = ( ZZ>= ` 1 )
283	281, 282	syl6eleq 2494	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ z e. R ) -> ( N - 1 ) e. ( ZZ>= ` 1 ) )
284		fzss2 11048	. . . . . . . . . . . . . . . 16 |- ( ( N - 1 ) e. ( ZZ>= ` 1 ) -> ( 0 ... 1 ) C_ ( 0 ... ( N - 1 ) ) )
285	283, 284	syl 16	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> ( 0 ... 1 ) C_ ( 0 ... ( N - 1 ) ) )
286	285	sselda 3308	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ z e. R ) /\ k e. ( 0 ... 1 ) ) -> k e. ( 0 ... ( N - 1 ) ) )
287		fznn0sub 11041	. . . . . . . . . . . . . . 15 |- ( k e. ( 0 ... ( N - 1 ) ) -> ( ( N - 1 ) - k ) e. NN0 )
288		ffvelrn 5827	. . . . . . . . . . . . . . 15 |- ( ( (coeff ` Q ) : NN0 --> CC /\ ( ( N - 1 ) - k ) e. NN0 ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) e. CC )
289	233, 287, 288	syl2an 464	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ z e. R ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) e. CC )
290	286, 289	syldan 457	. . . . . . . . . . . . 13 |- ( ( ( ph /\ z e. R ) /\ k e. ( 0 ... 1 ) ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) e. CC )
291	274, 290	mulcld 9064	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. R ) /\ k e. ( 0 ... 1 ) ) -> ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) e. CC )
292	269, 291	sylan2br 463	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. R ) /\ k e. ( 0 ... ( 0 + 1 ) ) ) -> ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) e. CC )
293		id 20	. . . . . . . . . . . . . 14 |- ( k = ( 0 + 1 ) -> k = ( 0 + 1 ) )
294	293, 262	syl6eqr 2454	. . . . . . . . . . . . 13 |- ( k = ( 0 + 1 ) -> k = 1 )
295	294	fveq2d 5691	. . . . . . . . . . . 12 |- ( k = ( 0 + 1 ) -> ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) = ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` 1 ) )
296	294	oveq2d 6056	. . . . . . . . . . . . 13 |- ( k = ( 0 + 1 ) -> ( ( N - 1 ) - k ) = ( ( N - 1 ) - 1 ) )
297	296	fveq2d 5691	. . . . . . . . . . . 12 |- ( k = ( 0 + 1 ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) = ( (coeff ` Q ) ` ( ( N - 1 ) - 1 ) ) )
298	295, 297	oveq12d 6058	. . . . . . . . . . 11 |- ( k = ( 0 + 1 ) -> ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) = ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` 1 ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - 1 ) ) ) )
299	268, 292, 298	fsump1 12495	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> sum_ k e. ( 0 ... ( 0 + 1 ) ) ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) = ( sum_ k e. ( 0 ... 0 ) ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) + ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` 1 ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - 1 ) ) ) ) )
300	264, 299	syl5eq 2448	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> sum_ k e. ( 0 ... 1 ) ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) = ( sum_ k e. ( 0 ... 0 ) ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) + ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` 1 ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - 1 ) ) ) ) )
301		eldifn 3430	. . . . . . . . . . . . . 14 |- ( k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) -> -. k e. ( 0 ... 1 ) )
302	301	adantl 453	. . . . . . . . . . . . 13 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> -. k e. ( 0 ... 1 ) )
303		eldifi 3429	. . . . . . . . . . . . . . . . 17 |- ( k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) -> k e. ( 0 ... ( N - 1 ) ) )
304		elfznn0 11039	. . . . . . . . . . . . . . . . 17 |- ( k e. ( 0 ... ( N - 1 ) ) -> k e. NN0 )
305	303, 304	syl 16	. . . . . . . . . . . . . . . 16 |- ( k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) -> k e. NN0 )
306	178, 171	dgrub 20106	. . . . . . . . . . . . . . . . 17 |- ( ( ( X p o F - ( CC X. { z } ) ) e. (Poly ` CC ) /\ k e. NN0 /\ ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) =/= 0 ) -> k <_ (deg ` ( X p o F - ( CC X. { z } ) ) ) )
307	306	3expia 1155	. . . . . . . . . . . . . . . 16 |- ( ( ( X p o F - ( CC X. { z } ) ) e. (Poly ` CC ) /\ k e. NN0 ) -> ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) =/= 0 -> k <_ (deg ` ( X p o F - ( CC X. { z } ) ) ) ) )
308	70, 305, 307	syl2an 464	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) =/= 0 -> k <_ (deg ` ( X p o F - ( CC X. { z } ) ) ) ) )
309		elfzuz 11011	. . . . . . . . . . . . . . . . . . 19 |- ( k e. ( 0 ... ( N - 1 ) ) -> k e. ( ZZ>= ` 0 ) )
310	303, 309	syl 16	. . . . . . . . . . . . . . . . . 18 |- ( k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) -> k e. ( ZZ>= ` 0 ) )
311	310	adantl 453	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> k e. ( ZZ>= ` 0 ) )
312		1z 10267	. . . . . . . . . . . . . . . . 17 |- 1 e. ZZ
313		elfz5 11007	. . . . . . . . . . . . . . . . 17 |- ( ( k e. ( ZZ>= ` 0 ) /\ 1 e. ZZ ) -> ( k e. ( 0 ... 1 ) <-> k <_ 1 ) )
314	311, 312, 313	sylancl 644	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> ( k e. ( 0 ... 1 ) <-> k <_ 1 ) )
315	163	breq2d 4184	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. R ) -> ( k <_ (deg ` ( X p o F - ( CC X. { z } ) ) ) <-> k <_ 1 ) )
316	315	adantr 452	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> ( k <_ (deg ` ( X p o F - ( CC X. { z } ) ) ) <-> k <_ 1 ) )
317	314, 316	bitr4d 248	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> ( k e. ( 0 ... 1 ) <-> k <_ (deg ` ( X p o F - ( CC X. { z } ) ) ) ) )
318	308, 317	sylibrd 226	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) =/= 0 -> k e. ( 0 ... 1 ) ) )
319	318	necon1bd 2635	. . . . . . . . . . . . 13 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> ( -. k e. ( 0 ... 1 ) -> ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) = 0 ) )
320	302, 319	mpd 15	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) = 0 )
321	320	oveq1d 6055	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) = ( 0 x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) )
322	303, 289	sylan2 461	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) e. CC )
323	322	mul02d 9220	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> ( 0 x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) = 0 )
324	321, 323	eqtrd 2436	. . . . . . . . . 10 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) = 0 )
325		fzfid 11267	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> ( 0 ... ( N - 1 ) ) e. Fin )
326	285, 291, 324, 325	fsumss 12474	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> sum_ k e. ( 0 ... 1 ) ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) = sum_ k e. ( 0 ... ( N - 1 ) ) ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) )
327		0z 10249	. . . . . . . . . . . 12 |- 0 e. ZZ
328	191	fveq1d 5689	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` 0 ) = ( ( (coeff ` X p ) o F - (coeff ` ( CC X. { z } ) ) ) ` 0 ) )
329		coeidp 20134	. . . . . . . . . . . . . . . . . . . 20 |- ( 0 e. NN0 -> ( (coeff ` X p ) ` 0 ) = if ( 0 = 1 , 1 , 0 ) )
330	164	necomi 2649	. . . . . . . . . . . . . . . . . . . . . 22 |- 0 =/= 1
331		df-ne 2569	. . . . . . . . . . . . . . . . . . . . . 22 |- ( 0 =/= 1 <-> -. 0 = 1 )
332	330, 331	mpbi 200	. . . . . . . . . . . . . . . . . . . . 21 |- -. 0 = 1
333		iffalse 3706	. . . . . . . . . . . . . . . . . . . . 21 |- ( -. 0 = 1 -> if ( 0 = 1 , 1 , 0 ) = 0 )
334	332, 333	ax-mp 8	. . . . . . . . . . . . . . . . . . . 20 |- if ( 0 = 1 , 1 , 0 ) = 0
335	329, 334	syl6eq 2452	. . . . . . . . . . . . . . . . . . 19 |- ( 0 e. NN0 -> ( (coeff ` X p ) ` 0 ) = 0 )
336	335	adantl 453	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ z e. R ) /\ 0 e. NN0 ) -> ( (coeff ` X p ) ` 0 ) = 0 )
337		0cn 9040	. . . . . . . . . . . . . . . . . . . . 21 |- 0 e. CC
338		vex 2919	. . . . . . . . . . . . . . . . . . . . . 22 |- z e. _V
339	338	fvconst2 5906	. . . . . . . . . . . . . . . . . . . . 21 |- ( 0 e. CC -> ( ( CC X. { z } ) ` 0 ) = z )
340	337, 339	ax-mp 8	. . . . . . . . . . . . . . . . . . . 20 |- ( ( CC X. { z } ) ` 0 ) = z
341	189	coefv0 20119	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( CC X. { z } ) e. (Poly ` CC ) -> ( ( CC X. { z } ) ` 0 ) = ( (coeff ` ( CC X. { z } ) ) ` 0 ) )
342	187, 341	syl 16	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ z e. R ) -> ( ( CC X. { z } ) ` 0 ) = ( (coeff ` ( CC X. { z } ) ) ` 0 ) )
343	340, 342	syl5reqr 2451	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( CC X. { z } ) ) ` 0 ) = z )
344	343	adantr 452	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ z e. R ) /\ 0 e. NN0 ) -> ( (coeff ` ( CC X. { z } ) ) ` 0 ) = z )
345	197, 200, 202, 202, 203, 336, 344	ofval 6273	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ z e. R ) /\ 0 e. NN0 ) -> ( ( (coeff ` X p ) o F - (coeff ` ( CC X. { z } ) ) ) ` 0 ) = ( 0 - z ) )
346	265, 345	mpan2 653	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` X p ) o F - (coeff ` ( CC X. { z } ) ) ) ` 0 ) = ( 0 - z ) )
347		df-neg 9250	. . . . . . . . . . . . . . . 16 |- -u z = ( 0 - z )
348	346, 347	syl6eqr 2454	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` X p ) o F - (coeff ` ( CC X. { z } ) ) ) ` 0 ) = -u z )
349	328, 348	eqtrd 2436	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` 0 ) = -u z )
350	279	oveq1d 6055	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. R ) -> ( ( N - 1 ) - 0 ) = ( D - 0 ) )
351	104	subid1d 9356	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. R ) -> ( D - 0 ) = D )
352	350, 351, 31	3eqtrd 2440	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ z e. R ) -> ( ( N - 1 ) - 0 ) = (deg ` Q ) )
353	352	fveq2d 5691	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - 0 ) ) = ( (coeff ` Q ) ` (deg ` Q ) ) )
354	353, 237	eqtr4d 2439	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - 0 ) ) = ( A ` N ) )
355	349, 354	oveq12d 6058	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` 0 ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - 0 ) ) ) = ( -u z x. ( A ` N ) ) )
356	355, 254	eqeltrd 2478	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` 0 ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - 0 ) ) ) e. CC )
357		fveq2 5687	. . . . . . . . . . . . . 14 |- ( k = 0 -> ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) = ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` 0 ) )
358		oveq2 6048	. . . . . . . . . . . . . . 15 |- ( k = 0 -> ( ( N - 1 ) - k ) = ( ( N - 1 ) - 0 ) )
359	358	fveq2d 5691	. . . . . . . . . . . . . 14 |- ( k = 0 -> ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) = ( (coeff ` Q ) ` ( ( N - 1 ) - 0 ) ) )
360	357, 359	oveq12d 6058	. . . . . . . . . . . . 13 |- ( k = 0 -> ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) = ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` 0 ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - 0 ) ) ) )
361	360	fsum1 12490	. . . . . . . . . . . 12 |- ( ( 0 e. ZZ /\ ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` 0 ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - 0 ) ) ) e. CC ) -> sum_ k e. ( 0 ... 0 ) ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) = ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` 0 ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - 0 ) ) ) )
362	327, 356, 361	sylancr 645	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> sum_ k e. ( 0 ... 0 ) ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) = ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` 0 ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - 0 ) ) ) )
363	362, 355	eqtrd 2436	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> sum_ k e. ( 0 ... 0 ) ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) = ( -u z x. ( A ` N ) ) )
364	279	oveq1d 6055	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( ( N - 1 ) - 1 ) = ( D - 1 ) )
365	364	fveq2d 5691	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - 1 ) ) = ( (coeff ` Q ) ` ( D - 1 ) ) )
366	229, 365	oveq12d 6058	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` 1 ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - 1 ) ) ) = ( 1 x. ( (coeff ` Q ) ` ( D - 1 ) ) ) )
367	246	mulid2d 9062	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> ( 1 x. ( (coeff ` Q ) ` ( D - 1 ) ) ) = ( (coeff ` Q ) ` ( D - 1 ) ) )
368	366, 367	eqtrd 2436	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` 1 ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - 1 ) ) ) = ( (coeff ` Q ) ` ( D - 1 ) ) )
369	363, 368	oveq12d 6058	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> ( sum_ k e. ( 0 ... 0 ) ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) + ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` 1 ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - 1 ) ) ) ) = ( ( -u z x. ( A ` N ) ) + ( (coeff ` Q ) ` ( D - 1 ) ) ) )
370	300, 326, 369	3eqtr3rd 2445	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> ( ( -u z x. ( A ` N ) ) + ( (coeff ` Q ) ` ( D - 1 ) ) ) = sum_ k e. ( 0 ... ( N - 1 ) ) ( ( (coeff ` ( X p o F - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) )
371	260, 261, 370	3eqtr4rd 2447	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( ( -u z x. ( A ` N ) ) + ( (coeff ` Q ) ` ( D - 1 ) ) ) = ( A ` ( N - 1 ) ) )
372	371	oveq1d 6055	. . . . . 6 |- ( ( ph /\ z e. R ) -> ( ( ( -u z x. ( A ` N ) ) + ( (coeff ` Q ) ` ( D - 1 ) ) ) / ( A ` N ) ) = ( ( A ` ( N - 1 ) ) / ( A ` N ) ) )
373	242, 247, 251	divcan4d 9752	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( ( -u z x. ( A ` N ) ) / ( A ` N ) ) = -u z )
374	373	oveq1d 6055	. . . . . 6 |- ( ( ph /\ z e. R ) -> ( ( ( -u z x. ( A ` N ) ) / ( A ` N ) ) + ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) ) = ( -u z + ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) ) )
375	255, 372, 374	3eqtr3rd 2445	. . . . 5 |- ( ( ph /\ z e. R ) -> ( -u z + ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) ) = ( ( A ` ( N - 1 ) ) / ( A ` N ) ) )
376	375	negeqd 9256	. . . 4 |- ( ( ph /\ z e. R ) -> -u ( -u z + ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) ) = -u ( ( A ` ( N - 1 ) ) / ( A ` N ) ) )
377	253, 376	eqtr3d 2438	. . 3 |- ( ( ph /\ z e. R ) -> ( -u -u z + -u ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) ) = -u ( ( A ` ( N - 1 ) ) / ( A ` N ) ) )
378	131, 241, 377	3eqtrd 2440	. 2 |- ( ( ph /\ z e. R ) -> sum_ x e. R x = -u ( ( A ` ( N - 1 ) ) / ( A ` N ) ) )
379	25, 378	exlimddv 1645	1 |- ( ph -> sum_ x e. R x = -u ( ( A ` ( N - 1 ) ) / ( A ` N ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 177   \/ wo 358   /\ wa 359   /\ w3a 936   E.wex 1547   = wceq 1649   e. wcel 1721   =/= wne 2567   A.wral 2666   _Vcvv 2916   \ cdif 3277   u. cun 3278   i^i cin 3279   C_ wss 3280   (/)c0 3588   ifcif 3699   {csn 3774   class class class wbr 4172   X. cxp 4835   `'ccnv 4836   dom cdm 4837   "cima 4840   Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   o Fcof 6262   Fincfn 7068   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947   + caddc 8949   x. cmul 8951   < clt 9076   <_ cle 9077   - cmin 9247   -ucneg 9248   / cdiv 9633   NNcn 9956   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999   #chash 11573   sum_csu 12434   0 pc0p 19514  Polycply 20056   X pcidp 20057  coeffccoe 20058  degcdgr 20059   quot cquot 20160

This theorem is referenced by:  vieta1  20182

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435  df-0p 19515  df-ply 20060  df-idp 20061  df-coe 20062  df-dgr 20063  df-quot 20161