Theorem vieta1lem2 24066

Description: Lemma for vieta1 24067: 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 11037	. . . . . . 7 |- ( ph -> ( D + 1 ) e. NN )
5	2, 4	eqeltrrd 2702	. . . . . 6 |- ( ph -> N e. NN )
6	5	nnne0d 11065	. . . . 5 |- ( ph -> N =/= 0 )
7	1, 6	eqnetrd 2861	. . . 4 |- ( ph -> ( # ` R ) =/= 0 )
8		vieta1.4	. . . . . . . 8 |- ( ph -> F e. (Poly ` S ) )
9		vieta1.2	. . . . . . . . . 10 |- N = (deg ` F )
10	9, 6	syl5eqner 2869	. . . . . . . . 9 |- ( ph -> (deg ` F ) =/= 0 )
11		fveq2 6191	. . . . . . . . . . 11 |- ( F = 0p -> (deg ` F ) = (deg ` 0p ) )
12		dgr0 24018	. . . . . . . . . . 11 |- (deg ` 0p ) = 0
13	11, 12	syl6eq 2672	. . . . . . . . . 10 |- ( F = 0p -> (deg ` F ) = 0 )
14	13	necon3i 2826	. . . . . . . . 9 |- ( (deg ` F ) =/= 0 -> F =/= 0p )
15	10, 14	syl 17	. . . . . . . 8 |- ( ph -> F =/= 0p )
16		vieta1.3	. . . . . . . . 9 |- R = ( `' F " { 0 } )
17	16	fta1 24063	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ F =/= 0p ) -> ( R e. Fin /\ ( # ` R ) <_ (deg ` F ) ) )
18	8, 15, 17	syl2anc 693	. . . . . . 7 |- ( ph -> ( R e. Fin /\ ( # ` R ) <_ (deg ` F ) ) )
19	18	simpld 475	. . . . . 6 |- ( ph -> R e. Fin )
20		hasheq0 13154	. . . . . 6 |- ( R e. Fin -> ( ( # ` R ) = 0 <-> R = (/) ) )
21	19, 20	syl 17	. . . . 5 |- ( ph -> ( ( # ` R ) = 0 <-> R = (/) ) )
22	21	necon3bid 2838	. . . 4 |- ( ph -> ( ( # ` R ) =/= 0 <-> R =/= (/) ) )
23	7, 22	mpbid 222	. . 3 |- ( ph -> R =/= (/) )
24		n0 3931	. . 3 |- ( R =/= (/) <-> E. z z e. R )
25	23, 24	sylib 208	. 2 |- ( ph -> E. z z e. R )
26		incom 3805	. . . . 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 24065	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> ( Q e. (Poly ` CC ) /\ D = (deg ` Q ) ) )
31	30	simprd 479	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> D = (deg ` Q ) )
32	30	simpld 475	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> Q e. (Poly ` CC ) )
33		dgrcl 23989	. . . . . . . . . . 11 |- ( Q e. (Poly ` CC ) -> (deg ` Q ) e. NN0 )
34	32, 33	syl 17	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> (deg ` Q ) e. NN0 )
35	34	nn0red 11352	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> (deg ` Q ) e. RR )
36	31, 35	eqeltrd 2701	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> D e. RR )
37	36	ltp1d 10954	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> D < ( D + 1 ) )
38	36, 37	gtned 10172	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( D + 1 ) =/= D )
39		snssi 4339	. . . . . . . . . . 11 |- ( z e. ( `' Q " { 0 } ) -> { z } C_ ( `' Q " { 0 } ) )
40		ssequn1 3783	. . . . . . . . . . 11 |- ( { z } C_ ( `' Q " { 0 } ) <-> ( { z } u. ( `' Q " { 0 } ) ) = ( `' Q " { 0 } ) )
41	39, 40	sylib 208	. . . . . . . . . 10 |- ( z e. ( `' Q " { 0 } ) -> ( { z } u. ( `' Q " { 0 } ) ) = ( `' Q " { 0 } ) )
42	41	fveq2d 6195	. . . . . . . . 9 |- ( z e. ( `' Q " { 0 } ) -> ( # ` ( { z } u. ( `' Q " { 0 } ) ) ) = ( # ` ( `' Q " { 0 } ) ) )
43	8	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. R ) -> F e. (Poly ` S ) )
44		cnvimass 5485	. . . . . . . . . . . . . . . . . . . . 21 |- ( `' F " { 0 } ) C_ dom F
45	16, 44	eqsstri 3635	. . . . . . . . . . . . . . . . . . . 20 |- R C_ dom F
46		plyf 23954	. . . . . . . . . . . . . . . . . . . . 21 |- ( F e. (Poly ` S ) -> F : CC --> CC )
47		fdm 6051	. . . . . . . . . . . . . . . . . . . . 21 |- ( F : CC --> CC -> dom F = CC )
48	8, 46, 47	3syl 18	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> dom F = CC )
49	45, 48	syl5sseq 3653	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> R C_ CC )
50	49	sselda 3603	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. R ) -> z e. CC )
51	16	eleq2i 2693	. . . . . . . . . . . . . . . . . . . 20 |- ( z e. R <-> z e. ( `' F " { 0 } ) )
52		ffn 6045	. . . . . . . . . . . . . . . . . . . . 21 |- ( F : CC --> CC -> F Fn CC )
53		fniniseg 6338	. . . . . . . . . . . . . . . . . . . . 21 |- ( F Fn CC -> ( z e. ( `' F " { 0 } ) <-> ( z e. CC /\ ( F ` z ) = 0 ) ) )
54	8, 46, 52, 53	4syl 19	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( z e. ( `' F " { 0 } ) <-> ( z e. CC /\ ( F ` z ) = 0 ) ) )
55	51, 54	syl5bb 272	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( z e. R <-> ( z e. CC /\ ( F ` z ) = 0 ) ) )
56	55	simplbda 654	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. R ) -> ( F ` z ) = 0 )
57		eqid 2622	. . . . . . . . . . . . . . . . . . 19 |- ( Xp oF - ( CC X. { z } ) ) = ( Xp oF - ( CC X. { z } ) )
58	57	facth 24061	. . . . . . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. R ) -> F = ( ( Xp oF - ( CC X. { z } ) ) oF x. ( F quot ( Xp oF - ( CC X. { z } ) ) ) ) )
60	29	oveq2i 6661	. . . . . . . . . . . . . . . . 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 2674	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ z e. R ) -> F = ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) )
62	61	cnveqd 5298	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> `' F = `' ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) )
63	62	imaeq1d 5465	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( `' F " { 0 } ) = ( `' ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) " { 0 } ) )
64	16, 63	syl5eq 2668	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> R = ( `' ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) " { 0 } ) )
65		cnex 10017	. . . . . . . . . . . . . . 15 |- CC e. _V
66	65	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> CC e. _V )
67	57	plyremlem 24059	. . . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . . 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 1073	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> ( Xp oF - ( CC X. { z } ) ) e. (Poly ` CC ) )
70		plyf 23954	. . . . . . . . . . . . . . 15 |- ( ( Xp oF - ( CC X. { z } ) ) e. (Poly ` CC ) -> ( Xp oF - ( CC X. { z } ) ) : CC --> CC )
71	69, 70	syl 17	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( Xp oF - ( CC X. { z } ) ) : CC --> CC )
72		plyf 23954	. . . . . . . . . . . . . . 15 |- ( Q e. (Poly ` CC ) -> Q : CC --> CC )
73	32, 72	syl 17	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> Q : CC --> CC )
74		ofmulrt 24037	. . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . 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 1075	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( `' ( Xp oF - ( CC X. { z } ) ) " { 0 } ) = { z } )
77	76	uneq1d 3766	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( ( `' ( Xp oF - ( CC X. { z } ) ) " { 0 } ) u. ( `' Q " { 0 } ) ) = ( { z } u. ( `' Q " { 0 } ) ) )
78	64, 75, 77	3eqtrd 2660	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> R = ( { z } u. ( `' Q " { 0 } ) ) )
79	78	fveq2d 6195	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> ( # ` R ) = ( # ` ( { z } u. ( `' Q " { 0 } ) ) ) )
80	1, 2	eqtr4d 2659	. . . . . . . . . . . 12 |- ( ph -> ( # ` R ) = ( D + 1 ) )
81	80	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> ( # ` R ) = ( D + 1 ) )
82	79, 81	eqtr3d 2658	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> ( # ` ( { z } u. ( `' Q " { 0 } ) ) ) = ( D + 1 ) )
83	15	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. R ) -> F =/= 0p )
84	61, 83	eqnetrrd 2862	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. R ) -> ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) =/= 0p )
85		plymul0or 24036	. . . . . . . . . . . . . . . . . . 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 693	. . . . . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . . . . 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 222	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ z e. R ) -> -. ( ( Xp oF - ( CC X. { z } ) ) = 0p \/ Q = 0p ) )
89		neanior 2886	. . . . . . . . . . . . . . . 16 |- ( ( ( Xp oF - ( CC X. { z } ) ) =/= 0p /\ Q =/= 0p ) <-> -. ( ( Xp oF - ( CC X. { z } ) ) = 0p \/ Q = 0p ) )
90	88, 89	sylibr 224	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> ( ( Xp oF - ( CC X. { z } ) ) =/= 0p /\ Q =/= 0p ) )
91	90	simprd 479	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> Q =/= 0p )
92		eqid 2622	. . . . . . . . . . . . . . 15 |- ( `' Q " { 0 } ) = ( `' Q " { 0 } )
93	92	fta1 24063	. . . . . . . . . . . . . 14 |- ( ( Q e. (Poly ` CC ) /\ Q =/= 0p ) -> ( ( `' Q " { 0 } ) e. Fin /\ ( # ` ( `' Q " { 0 } ) ) <_ (deg ` Q ) ) )
94	32, 91, 93	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( ( `' Q " { 0 } ) e. Fin /\ ( # ` ( `' Q " { 0 } ) ) <_ (deg ` Q ) ) )
95	94	simprd 479	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( # ` ( `' Q " { 0 } ) ) <_ (deg ` Q ) )
96	95, 31	breqtrrd 4681	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> ( # ` ( `' Q " { 0 } ) ) <_ D )
97		snfi 8038	. . . . . . . . . . . . . 14 |- { z } e. Fin
98	94	simpld 475	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( `' Q " { 0 } ) e. Fin )
99		hashun2 13172	. . . . . . . . . . . . . 14 |- ( ( { z } e. Fin /\ ( `' Q " { 0 } ) e. Fin ) -> ( # ` ( { z } u. ( `' Q " { 0 } ) ) ) <_ ( ( # ` { z } ) + ( # ` ( `' Q " { 0 } ) ) ) )
100	97, 98, 99	sylancr 695	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( # ` ( { z } u. ( `' Q " { 0 } ) ) ) <_ ( ( # ` { z } ) + ( # ` ( `' Q " { 0 } ) ) ) )
101		ax-1cn 9994	. . . . . . . . . . . . . . 15 |- 1 e. CC
102	3	nncnd 11036	. . . . . . . . . . . . . . . 16 |- ( ph -> D e. CC )
103	102	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> D e. CC )
104		addcom 10222	. . . . . . . . . . . . . . 15 |- ( ( 1 e. CC /\ D e. CC ) -> ( 1 + D ) = ( D + 1 ) )
105	101, 103, 104	sylancr 695	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( 1 + D ) = ( D + 1 ) )
106	82, 105	eqtr4d 2659	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( # ` ( { z } u. ( `' Q " { 0 } ) ) ) = ( 1 + D ) )
107		hashsng 13159	. . . . . . . . . . . . . . 15 |- ( z e. R -> ( # ` { z } ) = 1 )
108	107	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( # ` { z } ) = 1 )
109	108	oveq1d 6665	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( ( # ` { z } ) + ( # ` ( `' Q " { 0 } ) ) ) = ( 1 + ( # ` ( `' Q " { 0 } ) ) ) )
110	100, 106, 109	3brtr3d 4684	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( 1 + D ) <_ ( 1 + ( # ` ( `' Q " { 0 } ) ) ) )
111		hashcl 13147	. . . . . . . . . . . . . . 15 |- ( ( `' Q " { 0 } ) e. Fin -> ( # ` ( `' Q " { 0 } ) ) e. NN0 )
112	98, 111	syl 17	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( # ` ( `' Q " { 0 } ) ) e. NN0 )
113	112	nn0red 11352	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( # ` ( `' Q " { 0 } ) ) e. RR )
114		1red 10055	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> 1 e. RR )
115	36, 113, 114	leadd2d 10622	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( D <_ ( # ` ( `' Q " { 0 } ) ) <-> ( 1 + D ) <_ ( 1 + ( # ` ( `' Q " { 0 } ) ) ) ) )
116	110, 115	mpbird 247	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> D <_ ( # ` ( `' Q " { 0 } ) ) )
117	113, 36	letri3d 10179	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> ( ( # ` ( `' Q " { 0 } ) ) = D <-> ( ( # ` ( `' Q " { 0 } ) ) <_ D /\ D <_ ( # ` ( `' Q " { 0 } ) ) ) ) )
118	96, 116, 117	mpbir2and 957	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> ( # ` ( `' Q " { 0 } ) ) = D )
119	82, 118	eqeq12d 2637	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> ( ( # ` ( { z } u. ( `' Q " { 0 } ) ) ) = ( # ` ( `' Q " { 0 } ) ) <-> ( D + 1 ) = D ) )
120	42, 119	syl5ib 234	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> ( z e. ( `' Q " { 0 } ) -> ( D + 1 ) = D ) )
121	120	necon3ad 2807	. . . . . . 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 4246	. . . . . 6 |- ( ( ( `' Q " { 0 } ) i^i { z } ) = (/) <-> -. z e. ( `' Q " { 0 } ) )
124	122, 123	sylibr 224	. . . . 5 |- ( ( ph /\ z e. R ) -> ( ( `' Q " { 0 } ) i^i { z } ) = (/) )
125	26, 124	syl5eq 2668	. . . 4 |- ( ( ph /\ z e. R ) -> ( { z } i^i ( `' Q " { 0 } ) ) = (/) )
126	19	adantr 481	. . . 4 |- ( ( ph /\ z e. R ) -> R e. Fin )
127	49	adantr 481	. . . . 5 |- ( ( ph /\ z e. R ) -> R C_ CC )
128	127	sselda 3603	. . . 4 |- ( ( ( ph /\ z e. R ) /\ x e. R ) -> x e. CC )
129	125, 78, 126, 128	fsumsplit 14471	. . 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 14475	. . . . . 6 |- ( ( z e. CC /\ z e. CC ) -> sum_ x e. { z } x = z )
132	50, 50, 131	syl2anc 693	. . . . 5 |- ( ( ph /\ z e. R ) -> sum_ x e. { z } x = z )
133	50	negnegd 10383	. . . . 5 |- ( ( ph /\ z e. R ) -> -u -u z = z )
134	132, 133	eqtr4d 2659	. . . 4 |- ( ( ph /\ z e. R ) -> sum_ x e. { z } x = -u -u z )
135	118, 31	eqtrd 2656	. . . . . 6 |- ( ( ph /\ z e. R ) -> ( # ` ( `' Q " { 0 } ) ) = (deg ` Q ) )
136	28	adantr 481	. . . . . . 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 6191	. . . . . . . . . . 11 |- ( f = Q -> (deg ` f ) = (deg ` Q ) )
138	137	eqeq2d 2632	. . . . . . . . . 10 |- ( f = Q -> ( D = (deg ` f ) <-> D = (deg ` Q ) ) )
139		cnveq 5296	. . . . . . . . . . . . 13 |- ( f = Q -> `' f = `' Q )
140	139	imaeq1d 5465	. . . . . . . . . . . 12 |- ( f = Q -> ( `' f " { 0 } ) = ( `' Q " { 0 } ) )
141	140	fveq2d 6195	. . . . . . . . . . 11 |- ( f = Q -> ( # ` ( `' f " { 0 } ) ) = ( # ` ( `' Q " { 0 } ) ) )
142	141, 137	eqeq12d 2637	. . . . . . . . . 10 |- ( f = Q -> ( ( # ` ( `' f " { 0 } ) ) = (deg ` f ) <-> ( # ` ( `' Q " { 0 } ) ) = (deg ` Q ) ) )
143	138, 142	anbi12d 747	. . . . . . . . 9 |- ( f = Q -> ( ( D = (deg ` f ) /\ ( # ` ( `' f " { 0 } ) ) = (deg ` f ) ) <-> ( D = (deg ` Q ) /\ ( # ` ( `' Q " { 0 } ) ) = (deg ` Q ) ) ) )
144	140	sumeq1d 14431	. . . . . . . . . 10 |- ( f = Q -> sum_ x e. ( `' f " { 0 } ) x = sum_ x e. ( `' Q " { 0 } ) x )
145		fveq2 6191	. . . . . . . . . . . . 13 |- ( f = Q -> (coeff ` f ) = (coeff ` Q ) )
146	137	oveq1d 6665	. . . . . . . . . . . . 13 |- ( f = Q -> ( (deg ` f ) - 1 ) = ( (deg ` Q ) - 1 ) )
147	145, 146	fveq12d 6197	. . . . . . . . . . . 12 |- ( f = Q -> ( (coeff ` f ) ` ( (deg ` f ) - 1 ) ) = ( (coeff ` Q ) ` ( (deg ` Q ) - 1 ) ) )
148	145, 137	fveq12d 6197	. . . . . . . . . . . 12 |- ( f = Q -> ( (coeff ` f ) ` (deg ` f ) ) = ( (coeff ` Q ) ` (deg ` Q ) ) )
149	147, 148	oveq12d 6668	. . . . . . . . . . 11 |- ( f = Q -> ( ( (coeff ` f ) ` ( (deg ` f ) - 1 ) ) / ( (coeff ` f ) ` (deg ` f ) ) ) = ( ( (coeff ` Q ) ` ( (deg ` Q ) - 1 ) ) / ( (coeff ` Q ) ` (deg ` Q ) ) ) )
150	149	negeqd 10275	. . . . . . . . . 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 2637	. . . . . . . . 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 334	. . . . . . . 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 3305	. . . . . . 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 65	. . . . . 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 715	. . . . 5 |- ( ( ph /\ z e. R ) -> sum_ x e. ( `' Q " { 0 } ) x = -u ( ( (coeff ` Q ) ` ( (deg ` Q ) - 1 ) ) / ( (coeff ` Q ) ` (deg ` Q ) ) ) )
156	31	oveq1d 6665	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> ( D - 1 ) = ( (deg ` Q ) - 1 ) )
157	156	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( (coeff ` Q ) ` ( D - 1 ) ) = ( (coeff ` Q ) ` ( (deg ` Q ) - 1 ) ) )
158	61	fveq2d 6195	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> (coeff ` F ) = (coeff ` ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) ) )
159	27, 158	syl5eq 2668	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> A = (coeff ` ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) ) )
160	61	fveq2d 6195	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> (deg ` F ) = (deg ` ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) ) )
161	68	simp2d 1074	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> (deg ` ( Xp oF - ( CC X. { z } ) ) ) = 1 )
162		ax-1ne0 10005	. . . . . . . . . . . . . . 15 |- 1 =/= 0
163	162	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> 1 =/= 0 )
164	161, 163	eqnetrd 2861	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> (deg ` ( Xp oF - ( CC X. { z } ) ) ) =/= 0 )
165		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( ( Xp oF - ( CC X. { z } ) ) = 0p -> (deg ` ( Xp oF - ( CC X. { z } ) ) ) = (deg ` 0p ) )
166	165, 12	syl6eq 2672	. . . . . . . . . . . . . 14 |- ( ( Xp oF - ( CC X. { z } ) ) = 0p -> (deg ` ( Xp oF - ( CC X. { z } ) ) ) = 0 )
167	166	necon3i 2826	. . . . . . . . . . . . 13 |- ( (deg ` ( Xp oF - ( CC X. { z } ) ) ) =/= 0 -> ( Xp oF - ( CC X. { z } ) ) =/= 0p )
168	164, 167	syl 17	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( Xp oF - ( CC X. { z } ) ) =/= 0p )
169		eqid 2622	. . . . . . . . . . . . 13 |- (deg ` ( Xp oF - ( CC X. { z } ) ) ) = (deg ` ( Xp oF - ( CC X. { z } ) ) )
170		eqid 2622	. . . . . . . . . . . . 13 |- (deg ` Q ) = (deg ` Q )
171	169, 170	dgrmul 24026	. . . . . . . . . . . 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 1327	. . . . . . . . . . 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 2656	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> (deg ` F ) = ( (deg ` ( Xp oF - ( CC X. { z } ) ) ) + (deg ` Q ) ) )
174	9, 173	syl5eq 2668	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> N = ( (deg ` ( Xp oF - ( CC X. { z } ) ) ) + (deg ` Q ) ) )
175	159, 174	fveq12d 6197	. . . . . . . 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 2622	. . . . . . . . . 10 |- (coeff ` ( Xp oF - ( CC X. { z } ) ) ) = (coeff ` ( Xp oF - ( CC X. { z } ) ) )
177		eqid 2622	. . . . . . . . . 10 |- (coeff ` Q ) = (coeff ` Q )
178	176, 177, 169, 170	coemulhi 24010	. . . . . . . . 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 693	. . . . . . . 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 6195	. . . . . . . . . . 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 3624	. . . . . . . . . . . . . . 15 |- CC C_ CC
182		plyid 23965	. . . . . . . . . . . . . . 15 |- ( ( CC C_ CC /\ 1 e. CC ) -> Xp e. (Poly ` CC ) )
183	181, 101, 182	mp2an 708	. . . . . . . . . . . . . 14 |- Xp e. (Poly ` CC )
184		plyconst 23962	. . . . . . . . . . . . . . 15 |- ( ( CC C_ CC /\ z e. CC ) -> ( CC X. { z } ) e. (Poly ` CC ) )
185	181, 50, 184	sylancr 695	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( CC X. { z } ) e. (Poly ` CC ) )
186		eqid 2622	. . . . . . . . . . . . . . 15 |- (coeff ` Xp ) = (coeff ` Xp )
187		eqid 2622	. . . . . . . . . . . . . . 15 |- (coeff ` ( CC X. { z } ) ) = (coeff ` ( CC X. { z } ) )
188	186, 187	coesub 24013	. . . . . . . . . . . . . 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 695	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> (coeff ` ( Xp oF - ( CC X. { z } ) ) ) = ( (coeff ` Xp ) oF - (coeff ` ( CC X. { z } ) ) ) )
190	189	fveq1d 6193	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 1 ) = ( ( (coeff ` Xp ) oF - (coeff ` ( CC X. { z } ) ) ) ` 1 ) )
191		1nn0 11308	. . . . . . . . . . . . . 14 |- 1 e. NN0
192	186	coef3 23988	. . . . . . . . . . . . . . . . 17 |- ( Xp e. (Poly ` CC ) -> (coeff ` Xp ) : NN0 --> CC )
193		ffn 6045	. . . . . . . . . . . . . . . . 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 23988	. . . . . . . . . . . . . . . 16 |- ( ( CC X. { z } ) e. (Poly ` CC ) -> (coeff ` ( CC X. { z } ) ) : NN0 --> CC )
197		ffn 6045	. . . . . . . . . . . . . . . 16 |- ( (coeff ` ( CC X. { z } ) ) : NN0 --> CC -> (coeff ` ( CC X. { z } ) ) Fn NN0 )
198	185, 196, 197	3syl 18	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> (coeff ` ( CC X. { z } ) ) Fn NN0 )
199		nn0ex 11298	. . . . . . . . . . . . . . . 16 |- NN0 e. _V
200	199	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> NN0 e. _V )
201		inidm 3822	. . . . . . . . . . . . . . 15 |- ( NN0 i^i NN0 ) = NN0
202		coeidp 24019	. . . . . . . . . . . . . . . . 17 |- ( 1 e. NN0 -> ( (coeff ` Xp ) ` 1 ) = if ( 1 = 1 , 1 , 0 ) )
203	202	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> ( (coeff ` Xp ) ` 1 ) = if ( 1 = 1 , 1 , 0 ) )
204		eqid 2622	. . . . . . . . . . . . . . . . 17 |- 1 = 1
205	204	iftruei 4093	. . . . . . . . . . . . . . . 16 |- if ( 1 = 1 , 1 , 0 ) = 1
206	203, 205	syl6eq 2672	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> ( (coeff ` Xp ) ` 1 ) = 1 )
207		0lt1 10550	. . . . . . . . . . . . . . . . . 18 |- 0 < 1
208		0re 10040	. . . . . . . . . . . . . . . . . . 19 |- 0 e. RR
209		1re 10039	. . . . . . . . . . . . . . . . . . 19 |- 1 e. RR
210	208, 209	ltnlei 10158	. . . . . . . . . . . . . . . . . 18 |- ( 0 < 1 <-> -. 1 <_ 0 )
211	207, 210	mpbi 220	. . . . . . . . . . . . . . . . 17 |- -. 1 <_ 0
212	50	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> z e. CC )
213		0dgr 24001	. . . . . . . . . . . . . . . . . . 19 |- ( z e. CC -> (deg ` ( CC X. { z } ) ) = 0 )
214	212, 213	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> (deg ` ( CC X. { z } ) ) = 0 )
215	214	breq2d 4665	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> ( 1 <_ (deg ` ( CC X. { z } ) ) <-> 1 <_ 0 ) )
216	211, 215	mtbiri 317	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> -. 1 <_ (deg ` ( CC X. { z } ) ) )
217		eqid 2622	. . . . . . . . . . . . . . . . . . . 20 |- (deg ` ( CC X. { z } ) ) = (deg ` ( CC X. { z } ) )
218	187, 217	dgrub 23990	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( CC X. { z } ) e. (Poly ` CC ) /\ 1 e. NN0 /\ ( (coeff ` ( CC X. { z } ) ) ` 1 ) =/= 0 ) -> 1 <_ (deg ` ( CC X. { z } ) ) )
219	218	3expia 1267	. . . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> ( ( (coeff ` ( CC X. { z } ) ) ` 1 ) =/= 0 -> 1 <_ (deg ` ( CC X. { z } ) ) ) )
221	220	necon1bd 2812	. . . . . . . . . . . . . . . 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 6906	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> ( ( (coeff ` Xp ) oF - (coeff ` ( CC X. { z } ) ) ) ` 1 ) = ( 1 - 0 ) )
224	191, 223	mpan2 707	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` Xp ) oF - (coeff ` ( CC X. { z } ) ) ) ` 1 ) = ( 1 - 0 ) )
225		1m0e1 11131	. . . . . . . . . . . . 13 |- ( 1 - 0 ) = 1
226	224, 225	syl6eq 2672	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` Xp ) oF - (coeff ` ( CC X. { z } ) ) ) ` 1 ) = 1 )
227	190, 226	eqtrd 2656	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 1 ) = 1 )
228	180, 227	eqtrd 2656	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` (deg ` ( Xp oF - ( CC X. { z } ) ) ) ) = 1 )
229	228	oveq1d 6665	. . . . . . . . 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 23988	. . . . . . . . . . . 12 |- ( Q e. (Poly ` CC ) -> (coeff ` Q ) : NN0 --> CC )
231	32, 230	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> (coeff ` Q ) : NN0 --> CC )
232	231, 34	ffvelrnd 6360	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> ( (coeff ` Q ) ` (deg ` Q ) ) e. CC )
233	232	mulid2d 10058	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> ( 1 x. ( (coeff ` Q ) ` (deg ` Q ) ) ) = ( (coeff ` Q ) ` (deg ` Q ) ) )
234	229, 233	eqtrd 2656	. . . . . . . 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 2660	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( A ` N ) = ( (coeff ` Q ) ` (deg ` Q ) ) )
236	157, 235	oveq12d 6668	. . . . . 6 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) = ( ( (coeff ` Q ) ` ( (deg ` Q ) - 1 ) ) / ( (coeff ` Q ) ` (deg ` Q ) ) ) )
237	236	negeqd 10275	. . . . 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 2659	. . . 4 |- ( ( ph /\ z e. R ) -> sum_ x e. ( `' Q " { 0 } ) x = -u ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) )
239	134, 238	oveq12d 6668	. . 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 10379	. . . . 5 |- ( ( ph /\ z e. R ) -> -u z e. CC )
241		nnm1nn0 11334	. . . . . . . . 9 |- ( D e. NN -> ( D - 1 ) e. NN0 )
242	3, 241	syl 17	. . . . . . . 8 |- ( ph -> ( D - 1 ) e. NN0 )
243	242	adantr 481	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( D - 1 ) e. NN0 )
244	231, 243	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ z e. R ) -> ( (coeff ` Q ) ` ( D - 1 ) ) e. CC )
245	235, 232	eqeltrd 2701	. . . . . 6 |- ( ( ph /\ z e. R ) -> ( A ` N ) e. CC )
246	9, 27	dgreq0 24021	. . . . . . . . 9 |- ( F e. (Poly ` S ) -> ( F = 0p <-> ( A ` N ) = 0 ) )
247	43, 246	syl 17	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> ( F = 0p <-> ( A ` N ) = 0 ) )
248	247	necon3bid 2838	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( F =/= 0p <-> ( A ` N ) =/= 0 ) )
249	83, 248	mpbid 222	. . . . . 6 |- ( ( ph /\ z e. R ) -> ( A ` N ) =/= 0 )
250	244, 245, 249	divcld 10801	. . . . 5 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) e. CC )
251	240, 250	negdid 10405	. . . 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 10060	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( -u z x. ( A ` N ) ) e. CC )
253	252, 244, 245, 249	divdird 10839	. . . . . 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 11334	. . . . . . . . . . 11 |- ( N e. NN -> ( N - 1 ) e. NN0 )
255	5, 254	syl 17	. . . . . . . . . 10 |- ( ph -> ( N - 1 ) e. NN0 )
256	255	adantr 481	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> ( N - 1 ) e. NN0 )
257	176, 177	coemul 24008	. . . . . . . . 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 1326	. . . . . . . 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 6193	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> ( A ` ( N - 1 ) ) = ( (coeff ` ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) ) ` ( N - 1 ) ) )
260		1e0p1 11552	. . . . . . . . . . . 12 |- 1 = ( 0 + 1 )
261	260	oveq2i 6661	. . . . . . . . . . 11 |- ( 0 ... 1 ) = ( 0 ... ( 0 + 1 ) )
262	261	sumeq1i 14428	. . . . . . . . . 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 11307	. . . . . . . . . . . . 13 |- 0 e. NN0
264		nn0uz 11722	. . . . . . . . . . . . 13 |- NN0 = ( ZZ>= ` 0 )
265	263, 264	eleqtri 2699	. . . . . . . . . . . 12 |- 0 e. ( ZZ>= ` 0 )
266	265	a1i 11	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> 0 e. ( ZZ>= ` 0 ) )
267	261	eleq2i 2693	. . . . . . . . . . . 12 |- ( k e. ( 0 ... 1 ) <-> k e. ( 0 ... ( 0 + 1 ) ) )
268	176	coef3 23988	. . . . . . . . . . . . . . 15 |- ( ( Xp oF - ( CC X. { z } ) ) e. (Poly ` CC ) -> (coeff ` ( Xp oF - ( CC X. { z } ) ) ) : NN0 --> CC )
269	69, 268	syl 17	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> (coeff ` ( Xp oF - ( CC X. { z } ) ) ) : NN0 --> CC )
270		elfznn0 12433	. . . . . . . . . . . . . 14 |- ( k e. ( 0 ... 1 ) -> k e. NN0 )
271		ffvelrn 6357	. . . . . . . . . . . . . 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 494	. . . . . . . . . . . . 13 |- ( ( ( ph /\ z e. R ) /\ k e. ( 0 ... 1 ) ) -> ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) e. CC )
273	2	oveq1d 6665	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( ( D + 1 ) - 1 ) = ( N - 1 ) )
274		pncan 10287	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( D e. CC /\ 1 e. CC ) -> ( ( D + 1 ) - 1 ) = D )
275	102, 101, 274	sylancl 694	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( ( D + 1 ) - 1 ) = D )
276	273, 275	eqtr3d 2658	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( N - 1 ) = D )
277	276	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. R ) -> ( N - 1 ) = D )
278	3	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. R ) -> D e. NN )
279	277, 278	eqeltrd 2701	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. R ) -> ( N - 1 ) e. NN )
280		nnuz 11723	. . . . . . . . . . . . . . . . 17 |- NN = ( ZZ>= ` 1 )
281	279, 280	syl6eleq 2711	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ z e. R ) -> ( N - 1 ) e. ( ZZ>= ` 1 ) )
282		fzss2 12381	. . . . . . . . . . . . . . . 16 |- ( ( N - 1 ) e. ( ZZ>= ` 1 ) -> ( 0 ... 1 ) C_ ( 0 ... ( N - 1 ) ) )
283	281, 282	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> ( 0 ... 1 ) C_ ( 0 ... ( N - 1 ) ) )
284	283	sselda 3603	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ z e. R ) /\ k e. ( 0 ... 1 ) ) -> k e. ( 0 ... ( N - 1 ) ) )
285		fznn0sub 12373	. . . . . . . . . . . . . . 15 |- ( k e. ( 0 ... ( N - 1 ) ) -> ( ( N - 1 ) - k ) e. NN0 )
286		ffvelrn 6357	. . . . . . . . . . . . . . 15 |- ( ( (coeff ` Q ) : NN0 --> CC /\ ( ( N - 1 ) - k ) e. NN0 ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) e. CC )
287	231, 285, 286	syl2an 494	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ z e. R ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) e. CC )
288	284, 287	syldan 487	. . . . . . . . . . . . 13 |- ( ( ( ph /\ z e. R ) /\ k e. ( 0 ... 1 ) ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) e. CC )
289	272, 288	mulcld 10060	. . . . . . . . . . . 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 493	. . . . . . . . . . 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 2674	. . . . . . . . . . . . 13 |- ( k = ( 0 + 1 ) -> k = 1 )
293	292	fveq2d 6195	. . . . . . . . . . . 12 |- ( k = ( 0 + 1 ) -> ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) = ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 1 ) )
294	292	oveq2d 6666	. . . . . . . . . . . . 13 |- ( k = ( 0 + 1 ) -> ( ( N - 1 ) - k ) = ( ( N - 1 ) - 1 ) )
295	294	fveq2d 6195	. . . . . . . . . . . 12 |- ( k = ( 0 + 1 ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) = ( (coeff ` Q ) ` ( ( N - 1 ) - 1 ) ) )
296	293, 295	oveq12d 6668	. . . . . . . . . . 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 14487	. . . . . . . . . 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 2668	. . . . . . . . 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 3733	. . . . . . . . . . . . . 14 |- ( k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) -> -. k e. ( 0 ... 1 ) )
300	299	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> -. k e. ( 0 ... 1 ) )
301		eldifi 3732	. . . . . . . . . . . . . . . . 17 |- ( k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) -> k e. ( 0 ... ( N - 1 ) ) )
302		elfznn0 12433	. . . . . . . . . . . . . . . . 17 |- ( k e. ( 0 ... ( N - 1 ) ) -> k e. NN0 )
303	301, 302	syl 17	. . . . . . . . . . . . . . . 16 |- ( k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) -> k e. NN0 )
304	176, 169	dgrub 23990	. . . . . . . . . . . . . . . . 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 1267	. . . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . . 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 12338	. . . . . . . . . . . . . . . . . . 19 |- ( k e. ( 0 ... ( N - 1 ) ) -> k e. ( ZZ>= ` 0 ) )
308	301, 307	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) -> k e. ( ZZ>= ` 0 ) )
309	308	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> k e. ( ZZ>= ` 0 ) )
310		1z 11407	. . . . . . . . . . . . . . . . 17 |- 1 e. ZZ
311		elfz5 12334	. . . . . . . . . . . . . . . . 17 |- ( ( k e. ( ZZ>= ` 0 ) /\ 1 e. ZZ ) -> ( k e. ( 0 ... 1 ) <-> k <_ 1 ) )
312	309, 310, 311	sylancl 694	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> ( k e. ( 0 ... 1 ) <-> k <_ 1 ) )
313	161	breq2d 4665	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. R ) -> ( k <_ (deg ` ( Xp oF - ( CC X. { z } ) ) ) <-> k <_ 1 ) )
314	313	adantr 481	. . . . . . . . . . . . . . . 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 271	. . . . . . . . . . . . . . 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 249	. . . . . . . . . . . . . 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 2812	. . . . . . . . . . . . 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 6665	. . . . . . . . . . 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 491	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) e. CC )
321	320	mul02d 10234	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> ( 0 x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) = 0 )
322	319, 321	eqtrd 2656	. . . . . . . . . 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 12772	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> ( 0 ... ( N - 1 ) ) e. Fin )
324	283, 289, 322, 323	fsumss 14456	. . . . . . . . 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 11388	. . . . . . . . . . . 12 |- 0 e. ZZ
326	189	fveq1d 6193	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 0 ) = ( ( (coeff ` Xp ) oF - (coeff ` ( CC X. { z } ) ) ) ` 0 ) )
327		coeidp 24019	. . . . . . . . . . . . . . . . . . . 20 |- ( 0 e. NN0 -> ( (coeff ` Xp ) ` 0 ) = if ( 0 = 1 , 1 , 0 ) )
328	162	nesymi 2851	. . . . . . . . . . . . . . . . . . . . 21 |- -. 0 = 1
329	328	iffalsei 4096	. . . . . . . . . . . . . . . . . . . 20 |- if ( 0 = 1 , 1 , 0 ) = 0
330	327, 329	syl6eq 2672	. . . . . . . . . . . . . . . . . . 19 |- ( 0 e. NN0 -> ( (coeff ` Xp ) ` 0 ) = 0 )
331	330	adantl 482	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ z e. R ) /\ 0 e. NN0 ) -> ( (coeff ` Xp ) ` 0 ) = 0 )
332		0cn 10032	. . . . . . . . . . . . . . . . . . . . 21 |- 0 e. CC
333		vex 3203	. . . . . . . . . . . . . . . . . . . . . 22 |- z e. _V
334	333	fvconst2 6469	. . . . . . . . . . . . . . . . . . . . 21 |- ( 0 e. CC -> ( ( CC X. { z } ) ` 0 ) = z )
335	332, 334	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 |- ( ( CC X. { z } ) ` 0 ) = z
336	187	coefv0 24004	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( CC X. { z } ) e. (Poly ` CC ) -> ( ( CC X. { z } ) ` 0 ) = ( (coeff ` ( CC X. { z } ) ) ` 0 ) )
337	185, 336	syl 17	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ z e. R ) -> ( ( CC X. { z } ) ` 0 ) = ( (coeff ` ( CC X. { z } ) ) ` 0 ) )
338	335, 337	syl5reqr 2671	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( CC X. { z } ) ) ` 0 ) = z )
339	338	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ z e. R ) /\ 0 e. NN0 ) -> ( (coeff ` ( CC X. { z } ) ) ` 0 ) = z )
340	195, 198, 200, 200, 201, 331, 339	ofval 6906	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ z e. R ) /\ 0 e. NN0 ) -> ( ( (coeff ` Xp ) oF - (coeff ` ( CC X. { z } ) ) ) ` 0 ) = ( 0 - z ) )
341	263, 340	mpan2 707	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` Xp ) oF - (coeff ` ( CC X. { z } ) ) ) ` 0 ) = ( 0 - z ) )
342		df-neg 10269	. . . . . . . . . . . . . . . 16 |- -u z = ( 0 - z )
343	341, 342	syl6eqr 2674	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` Xp ) oF - (coeff ` ( CC X. { z } ) ) ) ` 0 ) = -u z )
344	326, 343	eqtrd 2656	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 0 ) = -u z )
345	277	oveq1d 6665	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. R ) -> ( ( N - 1 ) - 0 ) = ( D - 0 ) )
346	103	subid1d 10381	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. R ) -> ( D - 0 ) = D )
347	345, 346, 31	3eqtrd 2660	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ z e. R ) -> ( ( N - 1 ) - 0 ) = (deg ` Q ) )
348	347	fveq2d 6195	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - 0 ) ) = ( (coeff ` Q ) ` (deg ` Q ) ) )
349	348, 235	eqtr4d 2659	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - 0 ) ) = ( A ` N ) )
350	344, 349	oveq12d 6668	. . . . . . . . . . . . 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 2701	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 0 ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - 0 ) ) ) e. CC )
352		fveq2 6191	. . . . . . . . . . . . . 14 |- ( k = 0 -> ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) = ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 0 ) )
353		oveq2 6658	. . . . . . . . . . . . . . 15 |- ( k = 0 -> ( ( N - 1 ) - k ) = ( ( N - 1 ) - 0 ) )
354	353	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( k = 0 -> ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) = ( (coeff ` Q ) ` ( ( N - 1 ) - 0 ) ) )
355	352, 354	oveq12d 6668	. . . . . . . . . . . . 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 14476	. . . . . . . . . . . 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 695	. . . . . . . . . . 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 2656	. . . . . . . . . 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 6665	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( ( N - 1 ) - 1 ) = ( D - 1 ) )
360	359	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - 1 ) ) = ( (coeff ` Q ) ` ( D - 1 ) ) )
361	227, 360	oveq12d 6668	. . . . . . . . . . 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 10058	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> ( 1 x. ( (coeff ` Q ) ` ( D - 1 ) ) ) = ( (coeff ` Q ) ` ( D - 1 ) ) )
363	361, 362	eqtrd 2656	. . . . . . . . . 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 6668	. . . . . . . . 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 2665	. . . . . . . 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 2667	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( ( -u z x. ( A ` N ) ) + ( (coeff ` Q ) ` ( D - 1 ) ) ) = ( A ` ( N - 1 ) ) )
367	366	oveq1d 6665	. . . . . 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 10807	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( ( -u z x. ( A ` N ) ) / ( A ` N ) ) = -u z )
369	368	oveq1d 6665	. . . . . 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 2665	. . . . 5 |- ( ( ph /\ z e. R ) -> ( -u z + ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) ) = ( ( A ` ( N - 1 ) ) / ( A ` N ) ) )
371	370	negeqd 10275	. . . 4 |- ( ( ph /\ z e. R ) -> -u ( -u z + ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) ) = -u ( ( A ` ( N - 1 ) ) / ( A ` N ) ) )
372	251, 371	eqtr3d 2658	. . 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 2660	. 2 |- ( ( ph /\ z e. R ) -> sum_ x e. R x = -u ( ( A ` ( N - 1 ) ) / ( A ` N ) ) )
374	25, 373	exlimddv 1863	1 |- ( ph -> sum_ x e. R x = -u ( ( A ` ( N - 1 ) ) / ( A ` N ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   ifcif 4086   {csn 4177   class class class wbr 4653   X. cxp 5112   `'ccnv 5113   dom cdm 5114   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   #chash 13117   sum_csu 14416   0pc0p 23436  Polycply 23940   Xpcidp 23941  coeffccoe 23942  degcdgr 23943   quot cquot 24045

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-0p 23437  df-ply 23944  df-idp 23945  df-coe 23946  df-dgr 23947  df-quot 24046

This theorem is referenced by:  vieta1  24067