Theorem vieta1lem2 21736

Description: Lemma for vieta1 21737: inductive step. Let z be a root of F. Then F = ( Xp - 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 ` Xp - 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 ( Xp oF - ( 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 10335	. . . . . . 7 |- ( ph -> ( D + 1 ) e. NN )
5	2, 4	eqeltrrd 2516	. . . . . 6 |- ( ph -> N e. NN )
6	5	nnne0d 10362	. . . . 5 |- ( ph -> N =/= 0 )
7	1, 6	eqnetrd 2624	. . . 4 |- ( ph -> ( # ` R ) =/= 0 )
8		vieta1.4	. . . . . . . 8 |- ( ph -> F e. (Poly ` S ) )
9		vieta1.2	. . . . . . . . . 10 |- N = (deg ` F )
10	9, 6	syl5eqner 2631	. . . . . . . . 9 |- ( ph -> (deg ` F ) =/= 0 )
11		fveq2 5688	. . . . . . . . . . 11 |- ( F = 0p -> (deg ` F ) = (deg ` 0p ) )
12		dgr0 21688	. . . . . . . . . . 11 |- (deg ` 0p ) = 0
13	11, 12	syl6eq 2489	. . . . . . . . . 10 |- ( F = 0p -> (deg ` F ) = 0 )
14	13	necon3i 2648	. . . . . . . . 9 |- ( (deg ` F ) =/= 0 -> F =/= 0p )
15	10, 14	syl 16	. . . . . . . 8 |- ( ph -> F =/= 0p )
16		vieta1.3	. . . . . . . . 9 |- R = ( `' F " { 0 } )
17	16	fta1 21733	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ F =/= 0p ) -> ( R e. Fin /\ ( # ` R ) <_ (deg ` F ) ) )
18	8, 15, 17	syl2anc 656	. . . . . . 7 |- ( ph -> ( R e. Fin /\ ( # ` R ) <_ (deg ` F ) ) )
19	18	simpld 456	. . . . . 6 |- ( ph -> R e. Fin )
20		hasheq0 12127	. . . . . 6 |- ( R e. Fin -> ( ( # ` R ) = 0 <-> R = (/) ) )
21	19, 20	syl 16	. . . . 5 |- ( ph -> ( ( # ` R ) = 0 <-> R = (/) ) )
22	21	necon3bid 2641	. . . 4 |- ( ph -> ( ( # ` R ) =/= 0 <-> R =/= (/) ) )
23	7, 22	mpbid 210	. . 3 |- ( ph -> R =/= (/) )
24		n0 3643	. . 3 |- ( R =/= (/) <-> E. z z e. R )
25	23, 24	sylib 196	. 2 |- ( ph -> E. z z e. R )
26		incom 3540	. . . . 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 ( Xp oF - ( CC X. { z } ) ) )
30	27, 9, 16, 8, 1, 3, 2, 28, 29	vieta1lem1 21735	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> ( Q e. (Poly ` CC ) /\ D = (deg ` Q ) ) )
31	30	simprd 460	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> D = (deg ` Q ) )
32	30	simpld 456	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> Q e. (Poly ` CC ) )
33		dgrcl 21660	. . . . . . . . . . 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 10633	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> (deg ` Q ) e. RR )
36	31, 35	eqeltrd 2515	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> D e. RR )
37	36	ltp1d 10259	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> D < ( D + 1 ) )
38	36, 37	gtned 9505	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( D + 1 ) =/= D )
39		snssi 4014	. . . . . . . . . . 11 |- ( z e. ( `' Q " { 0 } ) -> { z } C_ ( `' Q " { 0 } ) )
40		ssequn1 3523	. . . . . . . . . . 11 |- ( { z } C_ ( `' Q " { 0 } ) <-> ( { z } u. ( `' Q " { 0 } ) ) = ( `' Q " { 0 } ) )
41	39, 40	sylib 196	. . . . . . . . . 10 |- ( z e. ( `' Q " { 0 } ) -> ( { z } u. ( `' Q " { 0 } ) ) = ( `' Q " { 0 } ) )
42	41	fveq2d 5692	. . . . . . . . 9 |- ( z e. ( `' Q " { 0 } ) -> ( # ` ( { z } u. ( `' Q " { 0 } ) ) ) = ( # ` ( `' Q " { 0 } ) ) )
43	8	adantr 462	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. R ) -> F e. (Poly ` S ) )
44		cnvimass 5186	. . . . . . . . . . . . . . . . . . . . 21 |- ( `' F " { 0 } ) C_ dom F
45	16, 44	eqsstri 3383	. . . . . . . . . . . . . . . . . . . 20 |- R C_ dom F
46		plyf 21625	. . . . . . . . . . . . . . . . . . . . 21 |- ( F e. (Poly ` S ) -> F : CC --> CC )
47		fdm 5560	. . . . . . . . . . . . . . . . . . . . 21 |- ( F : CC --> CC -> dom F = CC )
48	8, 46, 47	3syl 20	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> dom F = CC )
49	45, 48	syl5sseq 3401	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> R C_ CC )
50	49	sselda 3353	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. R ) -> z e. CC )
51	16	eleq2i 2505	. . . . . . . . . . . . . . . . . . . 20 |- ( z e. R <-> z e. ( `' F " { 0 } ) )
52		ffn 5556	. . . . . . . . . . . . . . . . . . . . 21 |- ( F : CC --> CC -> F Fn CC )
53		fniniseg 5821	. . . . . . . . . . . . . . . . . . . . 21 |- ( F Fn CC -> ( z e. ( `' F " { 0 } ) <-> ( z e. CC /\ ( F ` z ) = 0 ) ) )
54	8, 46, 52, 53	4syl 21	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( z e. ( `' F " { 0 } ) <-> ( z e. CC /\ ( F ` z ) = 0 ) ) )
55	51, 54	syl5bb 257	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( z e. R <-> ( z e. CC /\ ( F ` z ) = 0 ) ) )
56	55	simplbda 621	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. R ) -> ( F ` z ) = 0 )
57		eqid 2441	. . . . . . . . . . . . . . . . . . 19 |- ( Xp oF - ( CC X. { z } ) ) = ( Xp oF - ( CC X. { z } ) )
58	57	facth 21731	. . . . . . . . . . . . . . . . . 18 |- ( ( 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 } ) ) ) ) )
59	43, 50, 56, 58	syl3anc 1213	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. R ) -> F = ( ( Xp oF - ( CC X. { z } ) ) oF x. ( F quot ( Xp oF - ( CC X. { z } ) ) ) ) )
60	29	oveq2i 6101	. . . . . . . . . . . . . . . . 17 |- ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) = ( ( Xp oF - ( CC X. { z } ) ) oF x. ( F quot ( Xp oF - ( CC X. { z } ) ) ) )
61	59, 60	syl6eqr 2491	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ z e. R ) -> F = ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) )
62	61	cnveqd 5011	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> `' F = `' ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) )
63	62	imaeq1d 5165	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( `' F " { 0 } ) = ( `' ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) " { 0 } ) )
64	16, 63	syl5eq 2485	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> R = ( `' ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) " { 0 } ) )
65		cnex 9359	. . . . . . . . . . . . . . 15 |- CC e. _V
66	65	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> CC e. _V )
67	57	plyremlem 21729	. . . . . . . . . . . . . . . . 17 |- ( 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 } ) )
68	50, 67	syl 16	. . . . . . . . . . . . . . . 16 |- ( ( 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 } ) )
69	68	simp1d 995	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> ( Xp oF - ( CC X. { z } ) ) e. (Poly ` CC ) )
70		plyf 21625	. . . . . . . . . . . . . . 15 |- ( ( Xp oF - ( CC X. { z } ) ) e. (Poly ` CC ) -> ( Xp oF - ( CC X. { z } ) ) : CC --> CC )
71	69, 70	syl 16	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( Xp oF - ( CC X. { z } ) ) : CC --> CC )
72		plyf 21625	. . . . . . . . . . . . . . 15 |- ( Q e. (Poly ` CC ) -> Q : CC --> CC )
73	32, 72	syl 16	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> Q : CC --> CC )
74		ofmulrt 21707	. . . . . . . . . . . . . 14 |- ( ( CC e. _V /\ ( Xp oF - ( CC X. { z } ) ) : CC --> CC /\ Q : CC --> CC ) -> ( `' ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) " { 0 } ) = ( ( `' ( Xp oF - ( CC X. { z } ) ) " { 0 } ) u. ( `' Q " { 0 } ) ) )
75	66, 71, 73, 74	syl3anc 1213	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( `' ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) " { 0 } ) = ( ( `' ( Xp oF - ( CC X. { z } ) ) " { 0 } ) u. ( `' Q " { 0 } ) ) )
76	68	simp3d 997	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( `' ( Xp oF - ( CC X. { z } ) ) " { 0 } ) = { z } )
77	76	uneq1d 3506	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( ( `' ( Xp oF - ( CC X. { z } ) ) " { 0 } ) u. ( `' Q " { 0 } ) ) = ( { z } u. ( `' Q " { 0 } ) ) )
78	64, 75, 77	3eqtrd 2477	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> R = ( { z } u. ( `' Q " { 0 } ) ) )
79	78	fveq2d 5692	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> ( # ` R ) = ( # ` ( { z } u. ( `' Q " { 0 } ) ) ) )
80	1, 2	eqtr4d 2476	. . . . . . . . . . . 12 |- ( ph -> ( # ` R ) = ( D + 1 ) )
81	80	adantr 462	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> ( # ` R ) = ( D + 1 ) )
82	79, 81	eqtr3d 2475	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> ( # ` ( { z } u. ( `' Q " { 0 } ) ) ) = ( D + 1 ) )
83	15	adantr 462	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. R ) -> F =/= 0p )
84	61, 83	eqnetrrd 2626	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. R ) -> ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) =/= 0p )
85		plymul0or 21706	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( 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 ) ) )
86	69, 32, 85	syl2anc 656	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. R ) -> ( ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) = 0p <-> ( ( Xp oF - ( CC X. { z } ) ) = 0p \/ Q = 0p ) ) )
87	86	necon3abid 2639	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. R ) -> ( ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) =/= 0p <-> -. ( ( Xp oF - ( CC X. { z } ) ) = 0p \/ Q = 0p ) ) )
88	84, 87	mpbid 210	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ z e. R ) -> -. ( ( Xp oF - ( CC X. { z } ) ) = 0p \/ Q = 0p ) )
89		neanior 2695	. . . . . . . . . . . . . . . 16 |- ( ( ( Xp oF - ( CC X. { z } ) ) =/= 0p /\ Q =/= 0p ) <-> -. ( ( Xp oF - ( CC X. { z } ) ) = 0p \/ Q = 0p ) )
90	88, 89	sylibr 212	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> ( ( Xp oF - ( CC X. { z } ) ) =/= 0p /\ Q =/= 0p ) )
91	90	simprd 460	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> Q =/= 0p )
92		eqid 2441	. . . . . . . . . . . . . . 15 |- ( `' Q " { 0 } ) = ( `' Q " { 0 } )
93	92	fta1 21733	. . . . . . . . . . . . . 14 |- ( ( Q e. (Poly ` CC ) /\ Q =/= 0p ) -> ( ( `' Q " { 0 } ) e. Fin /\ ( # ` ( `' Q " { 0 } ) ) <_ (deg ` Q ) ) )
94	32, 91, 93	syl2anc 656	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( ( `' Q " { 0 } ) e. Fin /\ ( # ` ( `' Q " { 0 } ) ) <_ (deg ` Q ) ) )
95	94	simprd 460	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( # ` ( `' Q " { 0 } ) ) <_ (deg ` Q ) )
96	95, 31	breqtrrd 4315	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> ( # ` ( `' Q " { 0 } ) ) <_ D )
97		snfi 7386	. . . . . . . . . . . . . 14 |- { z } e. Fin
98	94	simpld 456	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( `' Q " { 0 } ) e. Fin )
99		hashun2 12142	. . . . . . . . . . . . . 14 |- ( ( { z } e. Fin /\ ( `' Q " { 0 } ) e. Fin ) -> ( # ` ( { z } u. ( `' Q " { 0 } ) ) ) <_ ( ( # ` { z } ) + ( # ` ( `' Q " { 0 } ) ) ) )
100	97, 98, 99	sylancr 658	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( # ` ( { z } u. ( `' Q " { 0 } ) ) ) <_ ( ( # ` { z } ) + ( # ` ( `' Q " { 0 } ) ) ) )
101		ax-1cn 9336	. . . . . . . . . . . . . . 15 |- 1 e. CC
102	3	nncnd 10334	. . . . . . . . . . . . . . . 16 |- ( ph -> D e. CC )
103	102	adantr 462	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> D e. CC )
104		addcom 9551	. . . . . . . . . . . . . . 15 |- ( ( 1 e. CC /\ D e. CC ) -> ( 1 + D ) = ( D + 1 ) )
105	101, 103, 104	sylancr 658	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( 1 + D ) = ( D + 1 ) )
106	82, 105	eqtr4d 2476	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( # ` ( { z } u. ( `' Q " { 0 } ) ) ) = ( 1 + D ) )
107		hashsng 12132	. . . . . . . . . . . . . . 15 |- ( z e. R -> ( # ` { z } ) = 1 )
108	107	adantl 463	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( # ` { z } ) = 1 )
109	108	oveq1d 6105	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( ( # ` { z } ) + ( # ` ( `' Q " { 0 } ) ) ) = ( 1 + ( # ` ( `' Q " { 0 } ) ) ) )
110	100, 106, 109	3brtr3d 4318	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( 1 + D ) <_ ( 1 + ( # ` ( `' Q " { 0 } ) ) ) )
111		hashcl 12122	. . . . . . . . . . . . . . 15 |- ( ( `' Q " { 0 } ) e. Fin -> ( # ` ( `' Q " { 0 } ) ) e. NN0 )
112	98, 111	syl 16	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( # ` ( `' Q " { 0 } ) ) e. NN0 )
113	112	nn0red 10633	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( # ` ( `' Q " { 0 } ) ) e. RR )
114		1red 9397	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> 1 e. RR )
115	36, 113, 114	leadd2d 9930	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( D <_ ( # ` ( `' Q " { 0 } ) ) <-> ( 1 + D ) <_ ( 1 + ( # ` ( `' Q " { 0 } ) ) ) ) )
116	110, 115	mpbird 232	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> D <_ ( # ` ( `' Q " { 0 } ) ) )
117	113, 36	letri3d 9512	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> ( ( # ` ( `' Q " { 0 } ) ) = D <-> ( ( # ` ( `' Q " { 0 } ) ) <_ D /\ D <_ ( # ` ( `' Q " { 0 } ) ) ) ) )
118	96, 116, 117	mpbir2and 908	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> ( # ` ( `' Q " { 0 } ) ) = D )
119	82, 118	eqeq12d 2455	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> ( ( # ` ( { z } u. ( `' Q " { 0 } ) ) ) = ( # ` ( `' Q " { 0 } ) ) <-> ( D + 1 ) = D ) )
120	42, 119	syl5ib 219	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> ( z e. ( `' Q " { 0 } ) -> ( D + 1 ) = D ) )
121	120	necon3ad 2642	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( ( D + 1 ) =/= D -> -. z e. ( `' Q " { 0 } ) ) )
122	38, 121	mpd 15	. . . . . 6 |- ( ( ph /\ z e. R ) -> -. z e. ( `' Q " { 0 } ) )
123		disjsn 3933	. . . . . 6 |- ( ( ( `' Q " { 0 } ) i^i { z } ) = (/) <-> -. z e. ( `' Q " { 0 } ) )
124	122, 123	sylibr 212	. . . . 5 |- ( ( ph /\ z e. R ) -> ( ( `' Q " { 0 } ) i^i { z } ) = (/) )
125	26, 124	syl5eq 2485	. . . 4 |- ( ( ph /\ z e. R ) -> ( { z } i^i ( `' Q " { 0 } ) ) = (/) )
126	19	adantr 462	. . . 4 |- ( ( ph /\ z e. R ) -> R e. Fin )
127	49	adantr 462	. . . . 5 |- ( ( ph /\ z e. R ) -> R C_ CC )
128	127	sselda 3353	. . . 4 |- ( ( ( ph /\ z e. R ) /\ x e. R ) -> x e. CC )
129	125, 78, 126, 128	fsumsplit 13212	. . 3 |- ( ( ph /\ z e. R ) -> sum_ x e. R x = ( sum_ x e. { z } x + sum_ x e. ( `' Q " { 0 } ) x ) )
130		id 22	. . . . . . 7 |- ( x = z -> x = z )
131	130	sumsn 13213	. . . . . 6 |- ( ( z e. CC /\ z e. CC ) -> sum_ x e. { z } x = z )
132	50, 50, 131	syl2anc 656	. . . . 5 |- ( ( ph /\ z e. R ) -> sum_ x e. { z } x = z )
133	50	negnegd 9706	. . . . 5 |- ( ( ph /\ z e. R ) -> -u -u z = z )
134	132, 133	eqtr4d 2476	. . . 4 |- ( ( ph /\ z e. R ) -> sum_ x e. { z } x = -u -u z )
135	118, 31	eqtrd 2473	. . . . . 6 |- ( ( ph /\ z e. R ) -> ( # ` ( `' Q " { 0 } ) ) = (deg ` Q ) )
136	28	adantr 462	. . . . . . 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 ) ) ) ) )
137		fveq2 5688	. . . . . . . . . . 11 |- ( f = Q -> (deg ` f ) = (deg ` Q ) )
138	137	eqeq2d 2452	. . . . . . . . . 10 |- ( f = Q -> ( D = (deg ` f ) <-> D = (deg ` Q ) ) )
139		cnveq 5009	. . . . . . . . . . . . 13 |- ( f = Q -> `' f = `' Q )
140	139	imaeq1d 5165	. . . . . . . . . . . 12 |- ( f = Q -> ( `' f " { 0 } ) = ( `' Q " { 0 } ) )
141	140	fveq2d 5692	. . . . . . . . . . 11 |- ( f = Q -> ( # ` ( `' f " { 0 } ) ) = ( # ` ( `' Q " { 0 } ) ) )
142	141, 137	eqeq12d 2455	. . . . . . . . . 10 |- ( f = Q -> ( ( # ` ( `' f " { 0 } ) ) = (deg ` f ) <-> ( # ` ( `' Q " { 0 } ) ) = (deg ` Q ) ) )
143	138, 142	anbi12d 705	. . . . . . . . 9 |- ( f = Q -> ( ( D = (deg ` f ) /\ ( # ` ( `' f " { 0 } ) ) = (deg ` f ) ) <-> ( D = (deg ` Q ) /\ ( # ` ( `' Q " { 0 } ) ) = (deg ` Q ) ) ) )
144	140	sumeq1d 13174	. . . . . . . . . 10 |- ( f = Q -> sum_ x e. ( `' f " { 0 } ) x = sum_ x e. ( `' Q " { 0 } ) x )
145		fveq2 5688	. . . . . . . . . . . . 13 |- ( f = Q -> (coeff ` f ) = (coeff ` Q ) )
146	137	oveq1d 6105	. . . . . . . . . . . . 13 |- ( f = Q -> ( (deg ` f ) - 1 ) = ( (deg ` Q ) - 1 ) )
147	145, 146	fveq12d 5694	. . . . . . . . . . . 12 |- ( f = Q -> ( (coeff ` f ) ` ( (deg ` f ) - 1 ) ) = ( (coeff ` Q ) ` ( (deg ` Q ) - 1 ) ) )
148	145, 137	fveq12d 5694	. . . . . . . . . . . 12 |- ( f = Q -> ( (coeff ` f ) ` (deg ` f ) ) = ( (coeff ` Q ) ` (deg ` Q ) ) )
149	147, 148	oveq12d 6108	. . . . . . . . . . 11 |- ( f = Q -> ( ( (coeff ` f ) ` ( (deg ` f ) - 1 ) ) / ( (coeff ` f ) ` (deg ` f ) ) ) = ( ( (coeff ` Q ) ` ( (deg ` Q ) - 1 ) ) / ( (coeff ` Q ) ` (deg ` Q ) ) ) )
150	149	negeqd 9600	. . . . . . . . . 10 |- ( f = Q -> -u ( ( (coeff ` f ) ` ( (deg ` f ) - 1 ) ) / ( (coeff ` f ) ` (deg ` f ) ) ) = -u ( ( (coeff ` Q ) ` ( (deg ` Q ) - 1 ) ) / ( (coeff ` Q ) ` (deg ` Q ) ) ) )
151	144, 150	eqeq12d 2455	. . . . . . . . 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 ) ) ) ) )
152	143, 151	imbi12d 320	. . . . . . . 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 ) ) ) ) ) )
153	152	rspcv 3066	. . . . . . 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 ) ) ) ) ) )
154	32, 136, 153	sylc 60	. . . . . 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 ) ) ) ) )
155	31, 135, 154	mp2and 674	. . . . 5 |- ( ( ph /\ z e. R ) -> sum_ x e. ( `' Q " { 0 } ) x = -u ( ( (coeff ` Q ) ` ( (deg ` Q ) - 1 ) ) / ( (coeff ` Q ) ` (deg ` Q ) ) ) )
156	31	oveq1d 6105	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> ( D - 1 ) = ( (deg ` Q ) - 1 ) )
157	156	fveq2d 5692	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( (coeff ` Q ) ` ( D - 1 ) ) = ( (coeff ` Q ) ` ( (deg ` Q ) - 1 ) ) )
158	61	fveq2d 5692	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> (coeff ` F ) = (coeff ` ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) ) )
159	27, 158	syl5eq 2485	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> A = (coeff ` ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) ) )
160	61	fveq2d 5692	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> (deg ` F ) = (deg ` ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) ) )
161	68	simp2d 996	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> (deg ` ( Xp oF - ( CC X. { z } ) ) ) = 1 )
162		ax-1ne0 9347	. . . . . . . . . . . . . . 15 |- 1 =/= 0
163	162	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> 1 =/= 0 )
164	161, 163	eqnetrd 2624	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> (deg ` ( Xp oF - ( CC X. { z } ) ) ) =/= 0 )
165		fveq2 5688	. . . . . . . . . . . . . . 15 |- ( ( Xp oF - ( CC X. { z } ) ) = 0p -> (deg ` ( Xp oF - ( CC X. { z } ) ) ) = (deg ` 0p ) )
166	165, 12	syl6eq 2489	. . . . . . . . . . . . . 14 |- ( ( Xp oF - ( CC X. { z } ) ) = 0p -> (deg ` ( Xp oF - ( CC X. { z } ) ) ) = 0 )
167	166	necon3i 2648	. . . . . . . . . . . . 13 |- ( (deg ` ( Xp oF - ( CC X. { z } ) ) ) =/= 0 -> ( Xp oF - ( CC X. { z } ) ) =/= 0p )
168	164, 167	syl 16	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( Xp oF - ( CC X. { z } ) ) =/= 0p )
169		eqid 2441	. . . . . . . . . . . . 13 |- (deg ` ( Xp oF - ( CC X. { z } ) ) ) = (deg ` ( Xp oF - ( CC X. { z } ) ) )
170		eqid 2441	. . . . . . . . . . . . 13 |- (deg ` Q ) = (deg ` Q )
171	169, 170	dgrmul 21696	. . . . . . . . . . . 12 |- ( ( ( ( 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 ) ) )
172	69, 168, 32, 91, 171	syl22anc 1214	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> (deg ` ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) ) = ( (deg ` ( Xp oF - ( CC X. { z } ) ) ) + (deg ` Q ) ) )
173	160, 172	eqtrd 2473	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> (deg ` F ) = ( (deg ` ( Xp oF - ( CC X. { z } ) ) ) + (deg ` Q ) ) )
174	9, 173	syl5eq 2485	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> N = ( (deg ` ( Xp oF - ( CC X. { z } ) ) ) + (deg ` Q ) ) )
175	159, 174	fveq12d 5694	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> ( A ` N ) = ( (coeff ` ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) ) ` ( (deg ` ( Xp oF - ( CC X. { z } ) ) ) + (deg ` Q ) ) ) )
176		eqid 2441	. . . . . . . . . 10 |- (coeff ` ( Xp oF - ( CC X. { z } ) ) ) = (coeff ` ( Xp oF - ( CC X. { z } ) ) )
177		eqid 2441	. . . . . . . . . 10 |- (coeff ` Q ) = (coeff ` Q )
178	176, 177, 169, 170	coemulhi 21680	. . . . . . . . 9 |- ( ( ( Xp oF - ( CC X. { z } ) ) e. (Poly ` CC ) /\ Q e. (Poly ` CC ) ) -> ( (coeff ` ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) ) ` ( (deg ` ( Xp oF - ( CC X. { z } ) ) ) + (deg ` Q ) ) ) = ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` (deg ` ( Xp oF - ( CC X. { z } ) ) ) ) x. ( (coeff ` Q ) ` (deg ` Q ) ) ) )
179	69, 32, 178	syl2anc 656	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) ) ` ( (deg ` ( Xp oF - ( CC X. { z } ) ) ) + (deg ` Q ) ) ) = ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` (deg ` ( Xp oF - ( CC X. { z } ) ) ) ) x. ( (coeff ` Q ) ` (deg ` Q ) ) ) )
180	161	fveq2d 5692	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` (deg ` ( Xp oF - ( CC X. { z } ) ) ) ) = ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 1 ) )
181		ssid 3372	. . . . . . . . . . . . . . 15 |- CC C_ CC
182		plyid 21636	. . . . . . . . . . . . . . 15 |- ( ( CC C_ CC /\ 1 e. CC ) -> Xp e. (Poly ` CC ) )
183	181, 101, 182	mp2an 667	. . . . . . . . . . . . . 14 |- Xp e. (Poly ` CC )
184		plyconst 21633	. . . . . . . . . . . . . . 15 |- ( ( CC C_ CC /\ z e. CC ) -> ( CC X. { z } ) e. (Poly ` CC ) )
185	181, 50, 184	sylancr 658	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( CC X. { z } ) e. (Poly ` CC ) )
186		eqid 2441	. . . . . . . . . . . . . . 15 |- (coeff ` Xp ) = (coeff ` Xp )
187		eqid 2441	. . . . . . . . . . . . . . 15 |- (coeff ` ( CC X. { z } ) ) = (coeff ` ( CC X. { z } ) )
188	186, 187	coesub 21683	. . . . . . . . . . . . . 14 |- ( ( Xp e. (Poly ` CC ) /\ ( CC X. { z } ) e. (Poly ` CC ) ) -> (coeff ` ( Xp oF - ( CC X. { z } ) ) ) = ( (coeff ` Xp ) oF - (coeff ` ( CC X. { z } ) ) ) )
189	183, 185, 188	sylancr 658	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> (coeff ` ( Xp oF - ( CC X. { z } ) ) ) = ( (coeff ` Xp ) oF - (coeff ` ( CC X. { z } ) ) ) )
190	189	fveq1d 5690	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 1 ) = ( ( (coeff ` Xp ) oF - (coeff ` ( CC X. { z } ) ) ) ` 1 ) )
191		1nn0 10591	. . . . . . . . . . . . . 14 |- 1 e. NN0
192	186	coef3 21659	. . . . . . . . . . . . . . . . 17 |- ( Xp e. (Poly ` CC ) -> (coeff ` Xp ) : NN0 --> CC )
193		ffn 5556	. . . . . . . . . . . . . . . . 17 |- ( (coeff ` Xp ) : NN0 --> CC -> (coeff ` Xp ) Fn NN0 )
194	183, 192, 193	mp2b 10	. . . . . . . . . . . . . . . 16 |- (coeff ` Xp ) Fn NN0
195	194	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> (coeff ` Xp ) Fn NN0 )
196	187	coef3 21659	. . . . . . . . . . . . . . . 16 |- ( ( CC X. { z } ) e. (Poly ` CC ) -> (coeff ` ( CC X. { z } ) ) : NN0 --> CC )
197		ffn 5556	. . . . . . . . . . . . . . . 16 |- ( (coeff ` ( CC X. { z } ) ) : NN0 --> CC -> (coeff ` ( CC X. { z } ) ) Fn NN0 )
198	185, 196, 197	3syl 20	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> (coeff ` ( CC X. { z } ) ) Fn NN0 )
199		nn0ex 10581	. . . . . . . . . . . . . . . 16 |- NN0 e. _V
200	199	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> NN0 e. _V )
201		inidm 3556	. . . . . . . . . . . . . . 15 |- ( NN0 i^i NN0 ) = NN0
202		coeidp 21689	. . . . . . . . . . . . . . . . 17 |- ( 1 e. NN0 -> ( (coeff ` Xp ) ` 1 ) = if ( 1 = 1 , 1 , 0 ) )
203	202	adantl 463	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> ( (coeff ` Xp ) ` 1 ) = if ( 1 = 1 , 1 , 0 ) )
204		eqid 2441	. . . . . . . . . . . . . . . . 17 |- 1 = 1
205	204	iftruei 3795	. . . . . . . . . . . . . . . 16 |- if ( 1 = 1 , 1 , 0 ) = 1
206	203, 205	syl6eq 2489	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> ( (coeff ` Xp ) ` 1 ) = 1 )
207		0lt1 9858	. . . . . . . . . . . . . . . . . 18 |- 0 < 1
208		0re 9382	. . . . . . . . . . . . . . . . . . 19 |- 0 e. RR
209		1re 9381	. . . . . . . . . . . . . . . . . . 19 |- 1 e. RR
210	208, 209	ltnlei 9491	. . . . . . . . . . . . . . . . . 18 |- ( 0 < 1 <-> -. 1 <_ 0 )
211	207, 210	mpbi 208	. . . . . . . . . . . . . . . . 17 |- -. 1 <_ 0
212	50	adantr 462	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> z e. CC )
213		0dgr 21672	. . . . . . . . . . . . . . . . . . 19 |- ( z e. CC -> (deg ` ( CC X. { z } ) ) = 0 )
214	212, 213	syl 16	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> (deg ` ( CC X. { z } ) ) = 0 )
215	214	breq2d 4301	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> ( 1 <_ (deg ` ( CC X. { z } ) ) <-> 1 <_ 0 ) )
216	211, 215	mtbiri 303	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> -. 1 <_ (deg ` ( CC X. { z } ) ) )
217		eqid 2441	. . . . . . . . . . . . . . . . . . . 20 |- (deg ` ( CC X. { z } ) ) = (deg ` ( CC X. { z } ) )
218	187, 217	dgrub 21661	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( CC X. { z } ) e. (Poly ` CC ) /\ 1 e. NN0 /\ ( (coeff ` ( CC X. { z } ) ) ` 1 ) =/= 0 ) -> 1 <_ (deg ` ( CC X. { z } ) ) )
219	218	3expia 1184	. . . . . . . . . . . . . . . . . 18 |- ( ( ( CC X. { z } ) e. (Poly ` CC ) /\ 1 e. NN0 ) -> ( ( (coeff ` ( CC X. { z } ) ) ` 1 ) =/= 0 -> 1 <_ (deg ` ( CC X. { z } ) ) ) )
220	185, 219	sylan 468	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> ( ( (coeff ` ( CC X. { z } ) ) ` 1 ) =/= 0 -> 1 <_ (deg ` ( CC X. { z } ) ) ) )
221	220	necon1bd 2677	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> ( -. 1 <_ (deg ` ( CC X. { z } ) ) -> ( (coeff ` ( CC X. { z } ) ) ` 1 ) = 0 ) )
222	216, 221	mpd 15	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> ( (coeff ` ( CC X. { z } ) ) ` 1 ) = 0 )
223	195, 198, 200, 200, 201, 206, 222	ofval 6328	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> ( ( (coeff ` Xp ) oF - (coeff ` ( CC X. { z } ) ) ) ` 1 ) = ( 1 - 0 ) )
224	191, 223	mpan2 666	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` Xp ) oF - (coeff ` ( CC X. { z } ) ) ) ` 1 ) = ( 1 - 0 ) )
225		1m0e1 10428	. . . . . . . . . . . . 13 |- ( 1 - 0 ) = 1
226	224, 225	syl6eq 2489	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` Xp ) oF - (coeff ` ( CC X. { z } ) ) ) ` 1 ) = 1 )
227	190, 226	eqtrd 2473	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 1 ) = 1 )
228	180, 227	eqtrd 2473	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` (deg ` ( Xp oF - ( CC X. { z } ) ) ) ) = 1 )
229	228	oveq1d 6105	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` (deg ` ( Xp oF - ( CC X. { z } ) ) ) ) x. ( (coeff ` Q ) ` (deg ` Q ) ) ) = ( 1 x. ( (coeff ` Q ) ` (deg ` Q ) ) ) )
230	177	coef3 21659	. . . . . . . . . . . 12 |- ( Q e. (Poly ` CC ) -> (coeff ` Q ) : NN0 --> CC )
231	32, 230	syl 16	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> (coeff ` Q ) : NN0 --> CC )
232	231, 34	ffvelrnd 5841	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> ( (coeff ` Q ) ` (deg ` Q ) ) e. CC )
233	232	mulid2d 9400	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> ( 1 x. ( (coeff ` Q ) ` (deg ` Q ) ) ) = ( (coeff ` Q ) ` (deg ` Q ) ) )
234	229, 233	eqtrd 2473	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` (deg ` ( Xp oF - ( CC X. { z } ) ) ) ) x. ( (coeff ` Q ) ` (deg ` Q ) ) ) = ( (coeff ` Q ) ` (deg ` Q ) ) )
235	175, 179, 234	3eqtrd 2477	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( A ` N ) = ( (coeff ` Q ) ` (deg ` Q ) ) )
236	157, 235	oveq12d 6108	. . . . . 6 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) = ( ( (coeff ` Q ) ` ( (deg ` Q ) - 1 ) ) / ( (coeff ` Q ) ` (deg ` Q ) ) ) )
237	236	negeqd 9600	. . . . 5 |- ( ( ph /\ z e. R ) -> -u ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) = -u ( ( (coeff ` Q ) ` ( (deg ` Q ) - 1 ) ) / ( (coeff ` Q ) ` (deg ` Q ) ) ) )
238	155, 237	eqtr4d 2476	. . . 4 |- ( ( ph /\ z e. R ) -> sum_ x e. ( `' Q " { 0 } ) x = -u ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) )
239	134, 238	oveq12d 6108	. . 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 ) ) ) )
240	50	negcld 9702	. . . . 5 |- ( ( ph /\ z e. R ) -> -u z e. CC )
241		nnm1nn0 10617	. . . . . . . . 9 |- ( D e. NN -> ( D - 1 ) e. NN0 )
242	3, 241	syl 16	. . . . . . . 8 |- ( ph -> ( D - 1 ) e. NN0 )
243	242	adantr 462	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( D - 1 ) e. NN0 )
244	231, 243	ffvelrnd 5841	. . . . . 6 |- ( ( ph /\ z e. R ) -> ( (coeff ` Q ) ` ( D - 1 ) ) e. CC )
245	235, 232	eqeltrd 2515	. . . . . 6 |- ( ( ph /\ z e. R ) -> ( A ` N ) e. CC )
246	9, 27	dgreq0 21691	. . . . . . . . 9 |- ( F e. (Poly ` S ) -> ( F = 0p <-> ( A ` N ) = 0 ) )
247	43, 246	syl 16	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> ( F = 0p <-> ( A ` N ) = 0 ) )
248	247	necon3bid 2641	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( F =/= 0p <-> ( A ` N ) =/= 0 ) )
249	83, 248	mpbid 210	. . . . . 6 |- ( ( ph /\ z e. R ) -> ( A ` N ) =/= 0 )
250	244, 245, 249	divcld 10103	. . . . 5 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) e. CC )
251	240, 250	negdid 9728	. . . 4 |- ( ( ph /\ z e. R ) -> -u ( -u z + ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) ) = ( -u -u z + -u ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) ) )
252	240, 245	mulcld 9402	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( -u z x. ( A ` N ) ) e. CC )
253	252, 244, 245, 249	divdird 10141	. . . . . 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 ) ) ) )
254		nnm1nn0 10617	. . . . . . . . . . 11 |- ( N e. NN -> ( N - 1 ) e. NN0 )
255	5, 254	syl 16	. . . . . . . . . 10 |- ( ph -> ( N - 1 ) e. NN0 )
256	255	adantr 462	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> ( N - 1 ) e. NN0 )
257	176, 177	coemul 21678	. . . . . . . . 9 |- ( ( ( Xp oF - ( CC X. { z } ) ) e. (Poly ` CC ) /\ Q e. (Poly ` CC ) /\ ( N - 1 ) e. NN0 ) -> ( (coeff ` ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) ) ` ( N - 1 ) ) = sum_ k e. ( 0 ... ( N - 1 ) ) ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) )
258	69, 32, 256, 257	syl3anc 1213	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) ) ` ( N - 1 ) ) = sum_ k e. ( 0 ... ( N - 1 ) ) ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) )
259	159	fveq1d 5690	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> ( A ` ( N - 1 ) ) = ( (coeff ` ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) ) ` ( N - 1 ) ) )
260		1e0p1 10779	. . . . . . . . . . . 12 |- 1 = ( 0 + 1 )
261	260	oveq2i 6101	. . . . . . . . . . 11 |- ( 0 ... 1 ) = ( 0 ... ( 0 + 1 ) )
262	261	sumeq1i 13171	. . . . . . . . . 10 |- sum_ k e. ( 0 ... 1 ) ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) = sum_ k e. ( 0 ... ( 0 + 1 ) ) ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) )
263		0nn0 10590	. . . . . . . . . . . . 13 |- 0 e. NN0
264		nn0uz 10891	. . . . . . . . . . . . 13 |- NN0 = ( ZZ>= ` 0 )
265	263, 264	eleqtri 2513	. . . . . . . . . . . 12 |- 0 e. ( ZZ>= ` 0 )
266	265	a1i 11	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> 0 e. ( ZZ>= ` 0 ) )
267	261	eleq2i 2505	. . . . . . . . . . . 12 |- ( k e. ( 0 ... 1 ) <-> k e. ( 0 ... ( 0 + 1 ) ) )
268	176	coef3 21659	. . . . . . . . . . . . . . 15 |- ( ( Xp oF - ( CC X. { z } ) ) e. (Poly ` CC ) -> (coeff ` ( Xp oF - ( CC X. { z } ) ) ) : NN0 --> CC )
269	69, 268	syl 16	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> (coeff ` ( Xp oF - ( CC X. { z } ) ) ) : NN0 --> CC )
270		elfznn0 11477	. . . . . . . . . . . . . 14 |- ( k e. ( 0 ... 1 ) -> k e. NN0 )
271		ffvelrn 5838	. . . . . . . . . . . . . 14 |- ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) : NN0 --> CC /\ k e. NN0 ) -> ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) e. CC )
272	269, 270, 271	syl2an 474	. . . . . . . . . . . . 13 |- ( ( ( ph /\ z e. R ) /\ k e. ( 0 ... 1 ) ) -> ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) e. CC )
273	2	oveq1d 6105	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( ( D + 1 ) - 1 ) = ( N - 1 ) )
274		pncan 9612	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( D e. CC /\ 1 e. CC ) -> ( ( D + 1 ) - 1 ) = D )
275	102, 101, 274	sylancl 657	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( ( D + 1 ) - 1 ) = D )
276	273, 275	eqtr3d 2475	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( N - 1 ) = D )
277	276	adantr 462	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. R ) -> ( N - 1 ) = D )
278	3	adantr 462	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. R ) -> D e. NN )
279	277, 278	eqeltrd 2515	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. R ) -> ( N - 1 ) e. NN )
280		nnuz 10892	. . . . . . . . . . . . . . . . 17 |- NN = ( ZZ>= ` 1 )
281	279, 280	syl6eleq 2531	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ z e. R ) -> ( N - 1 ) e. ( ZZ>= ` 1 ) )
282		fzss2 11494	. . . . . . . . . . . . . . . 16 |- ( ( N - 1 ) e. ( ZZ>= ` 1 ) -> ( 0 ... 1 ) C_ ( 0 ... ( N - 1 ) ) )
283	281, 282	syl 16	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> ( 0 ... 1 ) C_ ( 0 ... ( N - 1 ) ) )
284	283	sselda 3353	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ z e. R ) /\ k e. ( 0 ... 1 ) ) -> k e. ( 0 ... ( N - 1 ) ) )
285		fznn0sub 11483	. . . . . . . . . . . . . . 15 |- ( k e. ( 0 ... ( N - 1 ) ) -> ( ( N - 1 ) - k ) e. NN0 )
286		ffvelrn 5838	. . . . . . . . . . . . . . 15 |- ( ( (coeff ` Q ) : NN0 --> CC /\ ( ( N - 1 ) - k ) e. NN0 ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) e. CC )
287	231, 285, 286	syl2an 474	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ z e. R ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) e. CC )
288	284, 287	syldan 467	. . . . . . . . . . . . 13 |- ( ( ( ph /\ z e. R ) /\ k e. ( 0 ... 1 ) ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) e. CC )
289	272, 288	mulcld 9402	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. R ) /\ k e. ( 0 ... 1 ) ) -> ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) e. CC )
290	267, 289	sylan2br 473	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. R ) /\ k e. ( 0 ... ( 0 + 1 ) ) ) -> ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) e. CC )
291		id 22	. . . . . . . . . . . . . 14 |- ( k = ( 0 + 1 ) -> k = ( 0 + 1 ) )
292	291, 260	syl6eqr 2491	. . . . . . . . . . . . 13 |- ( k = ( 0 + 1 ) -> k = 1 )
293	292	fveq2d 5692	. . . . . . . . . . . 12 |- ( k = ( 0 + 1 ) -> ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) = ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 1 ) )
294	292	oveq2d 6106	. . . . . . . . . . . . 13 |- ( k = ( 0 + 1 ) -> ( ( N - 1 ) - k ) = ( ( N - 1 ) - 1 ) )
295	294	fveq2d 5692	. . . . . . . . . . . 12 |- ( k = ( 0 + 1 ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) = ( (coeff ` Q ) ` ( ( N - 1 ) - 1 ) ) )
296	293, 295	oveq12d 6108	. . . . . . . . . . 11 |- ( k = ( 0 + 1 ) -> ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) = ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 1 ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - 1 ) ) ) )
297	266, 290, 296	fsump1 13219	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> sum_ k e. ( 0 ... ( 0 + 1 ) ) ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) = ( sum_ k e. ( 0 ... 0 ) ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) + ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 1 ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - 1 ) ) ) ) )
298	262, 297	syl5eq 2485	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> sum_ k e. ( 0 ... 1 ) ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) = ( sum_ k e. ( 0 ... 0 ) ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) + ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 1 ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - 1 ) ) ) ) )
299		eldifn 3476	. . . . . . . . . . . . . 14 |- ( k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) -> -. k e. ( 0 ... 1 ) )
300	299	adantl 463	. . . . . . . . . . . . 13 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> -. k e. ( 0 ... 1 ) )
301		eldifi 3475	. . . . . . . . . . . . . . . . 17 |- ( k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) -> k e. ( 0 ... ( N - 1 ) ) )
302		elfznn0 11477	. . . . . . . . . . . . . . . . 17 |- ( k e. ( 0 ... ( N - 1 ) ) -> k e. NN0 )
303	301, 302	syl 16	. . . . . . . . . . . . . . . 16 |- ( k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) -> k e. NN0 )
304	176, 169	dgrub 21661	. . . . . . . . . . . . . . . . 17 |- ( ( ( Xp oF - ( CC X. { z } ) ) e. (Poly ` CC ) /\ k e. NN0 /\ ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) =/= 0 ) -> k <_ (deg ` ( Xp oF - ( CC X. { z } ) ) ) )
305	304	3expia 1184	. . . . . . . . . . . . . . . 16 |- ( ( ( Xp oF - ( CC X. { z } ) ) e. (Poly ` CC ) /\ k e. NN0 ) -> ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) =/= 0 -> k <_ (deg ` ( Xp oF - ( CC X. { z } ) ) ) ) )
306	69, 303, 305	syl2an 474	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) =/= 0 -> k <_ (deg ` ( Xp oF - ( CC X. { z } ) ) ) ) )
307		elfzuz 11445	. . . . . . . . . . . . . . . . . . 19 |- ( k e. ( 0 ... ( N - 1 ) ) -> k e. ( ZZ>= ` 0 ) )
308	301, 307	syl 16	. . . . . . . . . . . . . . . . . 18 |- ( k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) -> k e. ( ZZ>= ` 0 ) )
309	308	adantl 463	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> k e. ( ZZ>= ` 0 ) )
310		1z 10672	. . . . . . . . . . . . . . . . 17 |- 1 e. ZZ
311		elfz5 11441	. . . . . . . . . . . . . . . . 17 |- ( ( k e. ( ZZ>= ` 0 ) /\ 1 e. ZZ ) -> ( k e. ( 0 ... 1 ) <-> k <_ 1 ) )
312	309, 310, 311	sylancl 657	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> ( k e. ( 0 ... 1 ) <-> k <_ 1 ) )
313	161	breq2d 4301	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. R ) -> ( k <_ (deg ` ( Xp oF - ( CC X. { z } ) ) ) <-> k <_ 1 ) )
314	313	adantr 462	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> ( k <_ (deg ` ( Xp oF - ( CC X. { z } ) ) ) <-> k <_ 1 ) )
315	312, 314	bitr4d 256	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> ( k e. ( 0 ... 1 ) <-> k <_ (deg ` ( Xp oF - ( CC X. { z } ) ) ) ) )
316	306, 315	sylibrd 234	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) =/= 0 -> k e. ( 0 ... 1 ) ) )
317	316	necon1bd 2677	. . . . . . . . . . . . 13 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> ( -. k e. ( 0 ... 1 ) -> ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) = 0 ) )
318	300, 317	mpd 15	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) = 0 )
319	318	oveq1d 6105	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) = ( 0 x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) )
320	301, 287	sylan2 471	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) e. CC )
321	320	mul02d 9563	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> ( 0 x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) = 0 )
322	319, 321	eqtrd 2473	. . . . . . . . . 10 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) = 0 )
323		fzfid 11791	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> ( 0 ... ( N - 1 ) ) e. Fin )
324	283, 289, 322, 323	fsumss 13198	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> sum_ k e. ( 0 ... 1 ) ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) = sum_ k e. ( 0 ... ( N - 1 ) ) ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) )
325		0z 10653	. . . . . . . . . . . 12 |- 0 e. ZZ
326	189	fveq1d 5690	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 0 ) = ( ( (coeff ` Xp ) oF - (coeff ` ( CC X. { z } ) ) ) ` 0 ) )
327		coeidp 21689	. . . . . . . . . . . . . . . . . . . 20 |- ( 0 e. NN0 -> ( (coeff ` Xp ) ` 0 ) = if ( 0 = 1 , 1 , 0 ) )
328	162	nesymi 2646	. . . . . . . . . . . . . . . . . . . . 21 |- -. 0 = 1
329	328	iffalsei 3797	. . . . . . . . . . . . . . . . . . . 20 |- if ( 0 = 1 , 1 , 0 ) = 0
330	327, 329	syl6eq 2489	. . . . . . . . . . . . . . . . . . 19 |- ( 0 e. NN0 -> ( (coeff ` Xp ) ` 0 ) = 0 )
331	330	adantl 463	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ z e. R ) /\ 0 e. NN0 ) -> ( (coeff ` Xp ) ` 0 ) = 0 )
332		0cn 9374	. . . . . . . . . . . . . . . . . . . . 21 |- 0 e. CC
333		vex 2973	. . . . . . . . . . . . . . . . . . . . . 22 |- z e. _V
334	333	fvconst2 5930	. . . . . . . . . . . . . . . . . . . . 21 |- ( 0 e. CC -> ( ( CC X. { z } ) ` 0 ) = z )
335	332, 334	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 |- ( ( CC X. { z } ) ` 0 ) = z
336	187	coefv0 21674	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( CC X. { z } ) e. (Poly ` CC ) -> ( ( CC X. { z } ) ` 0 ) = ( (coeff ` ( CC X. { z } ) ) ` 0 ) )
337	185, 336	syl 16	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ z e. R ) -> ( ( CC X. { z } ) ` 0 ) = ( (coeff ` ( CC X. { z } ) ) ` 0 ) )
338	335, 337	syl5reqr 2488	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( CC X. { z } ) ) ` 0 ) = z )
339	338	adantr 462	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ z e. R ) /\ 0 e. NN0 ) -> ( (coeff ` ( CC X. { z } ) ) ` 0 ) = z )
340	195, 198, 200, 200, 201, 331, 339	ofval 6328	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ z e. R ) /\ 0 e. NN0 ) -> ( ( (coeff ` Xp ) oF - (coeff ` ( CC X. { z } ) ) ) ` 0 ) = ( 0 - z ) )
341	263, 340	mpan2 666	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` Xp ) oF - (coeff ` ( CC X. { z } ) ) ) ` 0 ) = ( 0 - z ) )
342		df-neg 9594	. . . . . . . . . . . . . . . 16 |- -u z = ( 0 - z )
343	341, 342	syl6eqr 2491	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` Xp ) oF - (coeff ` ( CC X. { z } ) ) ) ` 0 ) = -u z )
344	326, 343	eqtrd 2473	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 0 ) = -u z )
345	277	oveq1d 6105	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. R ) -> ( ( N - 1 ) - 0 ) = ( D - 0 ) )
346	103	subid1d 9704	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. R ) -> ( D - 0 ) = D )
347	345, 346, 31	3eqtrd 2477	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ z e. R ) -> ( ( N - 1 ) - 0 ) = (deg ` Q ) )
348	347	fveq2d 5692	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - 0 ) ) = ( (coeff ` Q ) ` (deg ` Q ) ) )
349	348, 235	eqtr4d 2476	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - 0 ) ) = ( A ` N ) )
350	344, 349	oveq12d 6108	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 0 ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - 0 ) ) ) = ( -u z x. ( A ` N ) ) )
351	350, 252	eqeltrd 2515	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 0 ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - 0 ) ) ) e. CC )
352		fveq2 5688	. . . . . . . . . . . . . 14 |- ( k = 0 -> ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) = ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 0 ) )
353		oveq2 6098	. . . . . . . . . . . . . . 15 |- ( k = 0 -> ( ( N - 1 ) - k ) = ( ( N - 1 ) - 0 ) )
354	353	fveq2d 5692	. . . . . . . . . . . . . 14 |- ( k = 0 -> ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) = ( (coeff ` Q ) ` ( ( N - 1 ) - 0 ) ) )
355	352, 354	oveq12d 6108	. . . . . . . . . . . . 13 |- ( k = 0 -> ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) = ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 0 ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - 0 ) ) ) )
356	355	fsum1 13214	. . . . . . . . . . . 12 |- ( ( 0 e. ZZ /\ ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 0 ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - 0 ) ) ) e. CC ) -> sum_ k e. ( 0 ... 0 ) ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) = ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 0 ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - 0 ) ) ) )
357	325, 351, 356	sylancr 658	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> sum_ k e. ( 0 ... 0 ) ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) = ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 0 ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - 0 ) ) ) )
358	357, 350	eqtrd 2473	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> sum_ k e. ( 0 ... 0 ) ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) = ( -u z x. ( A ` N ) ) )
359	277	oveq1d 6105	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( ( N - 1 ) - 1 ) = ( D - 1 ) )
360	359	fveq2d 5692	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - 1 ) ) = ( (coeff ` Q ) ` ( D - 1 ) ) )
361	227, 360	oveq12d 6108	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 1 ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - 1 ) ) ) = ( 1 x. ( (coeff ` Q ) ` ( D - 1 ) ) ) )
362	244	mulid2d 9400	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> ( 1 x. ( (coeff ` Q ) ` ( D - 1 ) ) ) = ( (coeff ` Q ) ` ( D - 1 ) ) )
363	361, 362	eqtrd 2473	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 1 ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - 1 ) ) ) = ( (coeff ` Q ) ` ( D - 1 ) ) )
364	358, 363	oveq12d 6108	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> ( sum_ k e. ( 0 ... 0 ) ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) + ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 1 ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - 1 ) ) ) ) = ( ( -u z x. ( A ` N ) ) + ( (coeff ` Q ) ` ( D - 1 ) ) ) )
365	298, 324, 364	3eqtr3rd 2482	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> ( ( -u z x. ( A ` N ) ) + ( (coeff ` Q ) ` ( D - 1 ) ) ) = sum_ k e. ( 0 ... ( N - 1 ) ) ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) )
366	258, 259, 365	3eqtr4rd 2484	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( ( -u z x. ( A ` N ) ) + ( (coeff ` Q ) ` ( D - 1 ) ) ) = ( A ` ( N - 1 ) ) )
367	366	oveq1d 6105	. . . . . 6 |- ( ( ph /\ z e. R ) -> ( ( ( -u z x. ( A ` N ) ) + ( (coeff ` Q ) ` ( D - 1 ) ) ) / ( A ` N ) ) = ( ( A ` ( N - 1 ) ) / ( A ` N ) ) )
368	240, 245, 249	divcan4d 10109	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( ( -u z x. ( A ` N ) ) / ( A ` N ) ) = -u z )
369	368	oveq1d 6105	. . . . . 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 ) ) ) )
370	253, 367, 369	3eqtr3rd 2482	. . . . 5 |- ( ( ph /\ z e. R ) -> ( -u z + ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) ) = ( ( A ` ( N - 1 ) ) / ( A ` N ) ) )
371	370	negeqd 9600	. . . 4 |- ( ( ph /\ z e. R ) -> -u ( -u z + ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) ) = -u ( ( A ` ( N - 1 ) ) / ( A ` N ) ) )
372	251, 371	eqtr3d 2475	. . 3 |- ( ( ph /\ z e. R ) -> ( -u -u z + -u ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) ) = -u ( ( A ` ( N - 1 ) ) / ( A ` N ) ) )
373	129, 239, 372	3eqtrd 2477	. 2 |- ( ( ph /\ z e. R ) -> sum_ x e. R x = -u ( ( A ` ( N - 1 ) ) / ( A ` N ) ) )
374	25, 373	exlimddv 1697	1 |- ( ph -> sum_ x e. R x = -u ( ( A ` ( N - 1 ) ) / ( A ` N ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 960   = wceq 1364   E.wex 1591   e. wcel 1761   =/= wne 2604   A.wral 2713   _Vcvv 2970   \ cdif 3322   u. cun 3323   i^i cin 3324   C_ wss 3325   (/)c0 3634   ifcif 3788   {csn 3874   class class class wbr 4289   X. cxp 4834   `'ccnv 4835   dom cdm 4836   "cima 4839   Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   oFcof 6317   Fincfn 7306   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279   + caddc 9281   x. cmul 9283   < clt 9414   <_ cle 9415   - cmin 9591   -ucneg 9592   / cdiv 9989   NNcn 10318   NN0cn0 10575   ZZcz 10642   ZZ>=cuz 10857   ...cfz 11433   #chash 12099   sum_csu 13159   0pc0p 21106  Polycply 21611   Xpcidp 21612  coeffccoe 21613  degcdgr 21614   quot cquot 21715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fz 11434  df-fzo 11545  df-fl 11638  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-rlim 12963  df-sum 13160  df-0p 21107  df-ply 21615  df-idp 21616  df-coe 21617  df-dgr 21618  df-quot 21716

This theorem is referenced by:  vieta1  21737