Theorem vieta1lem2 22457

Description: Lemma for vieta1 22458: 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 10552	. . . . . . 7 |- ( ph -> ( D + 1 ) e. NN )
5	2, 4	eqeltrrd 2556	. . . . . 6 |- ( ph -> N e. NN )
6	5	nnne0d 10579	. . . . 5 |- ( ph -> N =/= 0 )
7	1, 6	eqnetrd 2760	. . . 4 |- ( ph -> ( # ` R ) =/= 0 )
8		vieta1.4	. . . . . . . 8 |- ( ph -> F e. (Poly ` S ) )
9		vieta1.2	. . . . . . . . . 10 |- N = (deg ` F )
10	9, 6	syl5eqner 2768	. . . . . . . . 9 |- ( ph -> (deg ` F ) =/= 0 )
11		fveq2 5865	. . . . . . . . . . 11 |- ( F = 0p -> (deg ` F ) = (deg ` 0p ) )
12		dgr0 22409	. . . . . . . . . . 11 |- (deg ` 0p ) = 0
13	11, 12	syl6eq 2524	. . . . . . . . . 10 |- ( F = 0p -> (deg ` F ) = 0 )
14	13	necon3i 2707	. . . . . . . . 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 22454	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ F =/= 0p ) -> ( R e. Fin /\ ( # ` R ) <_ (deg ` F ) ) )
18	8, 15, 17	syl2anc 661	. . . . . . 7 |- ( ph -> ( R e. Fin /\ ( # ` R ) <_ (deg ` F ) ) )
19	18	simpld 459	. . . . . 6 |- ( ph -> R e. Fin )
20		hasheq0 12400	. . . . . 6 |- ( R e. Fin -> ( ( # ` R ) = 0 <-> R = (/) ) )
21	19, 20	syl 16	. . . . 5 |- ( ph -> ( ( # ` R ) = 0 <-> R = (/) ) )
22	21	necon3bid 2725	. . . 4 |- ( ph -> ( ( # ` R ) =/= 0 <-> R =/= (/) ) )
23	7, 22	mpbid 210	. . 3 |- ( ph -> R =/= (/) )
24		n0 3794	. . 3 |- ( R =/= (/) <-> E. z z e. R )
25	23, 24	sylib 196	. 2 |- ( ph -> E. z z e. R )
26		incom 3691	. . . . 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 22456	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> ( Q e. (Poly ` CC ) /\ D = (deg ` Q ) ) )
31	30	simprd 463	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> D = (deg ` Q ) )
32	30	simpld 459	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> Q e. (Poly ` CC ) )
33		dgrcl 22381	. . . . . . . . . . 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 10852	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> (deg ` Q ) e. RR )
36	31, 35	eqeltrd 2555	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> D e. RR )
37	36	ltp1d 10475	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> D < ( D + 1 ) )
38	36, 37	gtned 9718	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( D + 1 ) =/= D )
39		snssi 4171	. . . . . . . . . . 11 |- ( z e. ( `' Q " { 0 } ) -> { z } C_ ( `' Q " { 0 } ) )
40		ssequn1 3674	. . . . . . . . . . 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 5869	. . . . . . . . 9 |- ( z e. ( `' Q " { 0 } ) -> ( # ` ( { z } u. ( `' Q " { 0 } ) ) ) = ( # ` ( `' Q " { 0 } ) ) )
43	8	adantr 465	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. R ) -> F e. (Poly ` S ) )
44		cnvimass 5356	. . . . . . . . . . . . . . . . . . . . 21 |- ( `' F " { 0 } ) C_ dom F
45	16, 44	eqsstri 3534	. . . . . . . . . . . . . . . . . . . 20 |- R C_ dom F
46		plyf 22346	. . . . . . . . . . . . . . . . . . . . 21 |- ( F e. (Poly ` S ) -> F : CC --> CC )
47		fdm 5734	. . . . . . . . . . . . . . . . . . . . 21 |- ( F : CC --> CC -> dom F = CC )
48	8, 46, 47	3syl 20	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> dom F = CC )
49	45, 48	syl5sseq 3552	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> R C_ CC )
50	49	sselda 3504	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. R ) -> z e. CC )
51	16	eleq2i 2545	. . . . . . . . . . . . . . . . . . . 20 |- ( z e. R <-> z e. ( `' F " { 0 } ) )
52		ffn 5730	. . . . . . . . . . . . . . . . . . . . 21 |- ( F : CC --> CC -> F Fn CC )
53		fniniseg 6001	. . . . . . . . . . . . . . . . . . . . 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 624	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. R ) -> ( F ` z ) = 0 )
57		eqid 2467	. . . . . . . . . . . . . . . . . . 19 |- ( Xp oF - ( CC X. { z } ) ) = ( Xp oF - ( CC X. { z } ) )
58	57	facth 22452	. . . . . . . . . . . . . . . . . 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 1228	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. R ) -> F = ( ( Xp oF - ( CC X. { z } ) ) oF x. ( F quot ( Xp oF - ( CC X. { z } ) ) ) ) )
60	29	oveq2i 6294	. . . . . . . . . . . . . . . . 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 2526	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ z e. R ) -> F = ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) )
62	61	cnveqd 5177	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> `' F = `' ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) )
63	62	imaeq1d 5335	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( `' F " { 0 } ) = ( `' ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) " { 0 } ) )
64	16, 63	syl5eq 2520	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> R = ( `' ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) " { 0 } ) )
65		cnex 9572	. . . . . . . . . . . . . . 15 |- CC e. _V
66	65	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> CC e. _V )
67	57	plyremlem 22450	. . . . . . . . . . . . . . . . 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 1008	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> ( Xp oF - ( CC X. { z } ) ) e. (Poly ` CC ) )
70		plyf 22346	. . . . . . . . . . . . . . 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 22346	. . . . . . . . . . . . . . 15 |- ( Q e. (Poly ` CC ) -> Q : CC --> CC )
73	32, 72	syl 16	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> Q : CC --> CC )
74		ofmulrt 22428	. . . . . . . . . . . . . 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 1228	. . . . . . . . . . . . 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 1010	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( `' ( Xp oF - ( CC X. { z } ) ) " { 0 } ) = { z } )
77	76	uneq1d 3657	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( ( `' ( Xp oF - ( CC X. { z } ) ) " { 0 } ) u. ( `' Q " { 0 } ) ) = ( { z } u. ( `' Q " { 0 } ) ) )
78	64, 75, 77	3eqtrd 2512	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> R = ( { z } u. ( `' Q " { 0 } ) ) )
79	78	fveq2d 5869	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> ( # ` R ) = ( # ` ( { z } u. ( `' Q " { 0 } ) ) ) )
80	1, 2	eqtr4d 2511	. . . . . . . . . . . 12 |- ( ph -> ( # ` R ) = ( D + 1 ) )
81	80	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> ( # ` R ) = ( D + 1 ) )
82	79, 81	eqtr3d 2510	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> ( # ` ( { z } u. ( `' Q " { 0 } ) ) ) = ( D + 1 ) )
83	15	adantr 465	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. R ) -> F =/= 0p )
84	61, 83	eqnetrrd 2761	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. R ) -> ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) =/= 0p )
85		plymul0or 22427	. . . . . . . . . . . . . . . . . . 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 661	. . . . . . . . . . . . . . . . . 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 2713	. . . . . . . . . . . . . . . . 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 2792	. . . . . . . . . . . . . . . 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 463	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> Q =/= 0p )
92		eqid 2467	. . . . . . . . . . . . . . 15 |- ( `' Q " { 0 } ) = ( `' Q " { 0 } )
93	92	fta1 22454	. . . . . . . . . . . . . 14 |- ( ( Q e. (Poly ` CC ) /\ Q =/= 0p ) -> ( ( `' Q " { 0 } ) e. Fin /\ ( # ` ( `' Q " { 0 } ) ) <_ (deg ` Q ) ) )
94	32, 91, 93	syl2anc 661	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( ( `' Q " { 0 } ) e. Fin /\ ( # ` ( `' Q " { 0 } ) ) <_ (deg ` Q ) ) )
95	94	simprd 463	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( # ` ( `' Q " { 0 } ) ) <_ (deg ` Q ) )
96	95, 31	breqtrrd 4473	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> ( # ` ( `' Q " { 0 } ) ) <_ D )
97		snfi 7596	. . . . . . . . . . . . . 14 |- { z } e. Fin
98	94	simpld 459	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( `' Q " { 0 } ) e. Fin )
99		hashun2 12418	. . . . . . . . . . . . . 14 |- ( ( { z } e. Fin /\ ( `' Q " { 0 } ) e. Fin ) -> ( # ` ( { z } u. ( `' Q " { 0 } ) ) ) <_ ( ( # ` { z } ) + ( # ` ( `' Q " { 0 } ) ) ) )
100	97, 98, 99	sylancr 663	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( # ` ( { z } u. ( `' Q " { 0 } ) ) ) <_ ( ( # ` { z } ) + ( # ` ( `' Q " { 0 } ) ) ) )
101		ax-1cn 9549	. . . . . . . . . . . . . . 15 |- 1 e. CC
102	3	nncnd 10551	. . . . . . . . . . . . . . . 16 |- ( ph -> D e. CC )
103	102	adantr 465	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> D e. CC )
104		addcom 9764	. . . . . . . . . . . . . . 15 |- ( ( 1 e. CC /\ D e. CC ) -> ( 1 + D ) = ( D + 1 ) )
105	101, 103, 104	sylancr 663	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( 1 + D ) = ( D + 1 ) )
106	82, 105	eqtr4d 2511	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( # ` ( { z } u. ( `' Q " { 0 } ) ) ) = ( 1 + D ) )
107		hashsng 12405	. . . . . . . . . . . . . . 15 |- ( z e. R -> ( # ` { z } ) = 1 )
108	107	adantl 466	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( # ` { z } ) = 1 )
109	108	oveq1d 6298	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( ( # ` { z } ) + ( # ` ( `' Q " { 0 } ) ) ) = ( 1 + ( # ` ( `' Q " { 0 } ) ) ) )
110	100, 106, 109	3brtr3d 4476	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( 1 + D ) <_ ( 1 + ( # ` ( `' Q " { 0 } ) ) ) )
111		hashcl 12395	. . . . . . . . . . . . . . 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 10852	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( # ` ( `' Q " { 0 } ) ) e. RR )
114		1red 9610	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> 1 e. RR )
115	36, 113, 114	leadd2d 10146	. . . . . . . . . . . 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 9725	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> ( ( # ` ( `' Q " { 0 } ) ) = D <-> ( ( # ` ( `' Q " { 0 } ) ) <_ D /\ D <_ ( # ` ( `' Q " { 0 } ) ) ) ) )
118	96, 116, 117	mpbir2and 920	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> ( # ` ( `' Q " { 0 } ) ) = D )
119	82, 118	eqeq12d 2489	. . . . . . . . 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 2677	. . . . . . 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 4088	. . . . . 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 2520	. . . 4 |- ( ( ph /\ z e. R ) -> ( { z } i^i ( `' Q " { 0 } ) ) = (/) )
126	19	adantr 465	. . . 4 |- ( ( ph /\ z e. R ) -> R e. Fin )
127	49	adantr 465	. . . . 5 |- ( ( ph /\ z e. R ) -> R C_ CC )
128	127	sselda 3504	. . . 4 |- ( ( ( ph /\ z e. R ) /\ x e. R ) -> x e. CC )
129	125, 78, 126, 128	fsumsplit 13524	. . 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 13525	. . . . . 6 |- ( ( z e. CC /\ z e. CC ) -> sum_ x e. { z } x = z )
132	50, 50, 131	syl2anc 661	. . . . 5 |- ( ( ph /\ z e. R ) -> sum_ x e. { z } x = z )
133	50	negnegd 9920	. . . . 5 |- ( ( ph /\ z e. R ) -> -u -u z = z )
134	132, 133	eqtr4d 2511	. . . 4 |- ( ( ph /\ z e. R ) -> sum_ x e. { z } x = -u -u z )
135	118, 31	eqtrd 2508	. . . . . 6 |- ( ( ph /\ z e. R ) -> ( # ` ( `' Q " { 0 } ) ) = (deg ` Q ) )
136	28	adantr 465	. . . . . . 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 5865	. . . . . . . . . . 11 |- ( f = Q -> (deg ` f ) = (deg ` Q ) )
138	137	eqeq2d 2481	. . . . . . . . . 10 |- ( f = Q -> ( D = (deg ` f ) <-> D = (deg ` Q ) ) )
139		cnveq 5175	. . . . . . . . . . . . 13 |- ( f = Q -> `' f = `' Q )
140	139	imaeq1d 5335	. . . . . . . . . . . 12 |- ( f = Q -> ( `' f " { 0 } ) = ( `' Q " { 0 } ) )
141	140	fveq2d 5869	. . . . . . . . . . 11 |- ( f = Q -> ( # ` ( `' f " { 0 } ) ) = ( # ` ( `' Q " { 0 } ) ) )
142	141, 137	eqeq12d 2489	. . . . . . . . . 10 |- ( f = Q -> ( ( # ` ( `' f " { 0 } ) ) = (deg ` f ) <-> ( # ` ( `' Q " { 0 } ) ) = (deg ` Q ) ) )
143	138, 142	anbi12d 710	. . . . . . . . 9 |- ( f = Q -> ( ( D = (deg ` f ) /\ ( # ` ( `' f " { 0 } ) ) = (deg ` f ) ) <-> ( D = (deg ` Q ) /\ ( # ` ( `' Q " { 0 } ) ) = (deg ` Q ) ) ) )
144	140	sumeq1d 13485	. . . . . . . . . 10 |- ( f = Q -> sum_ x e. ( `' f " { 0 } ) x = sum_ x e. ( `' Q " { 0 } ) x )
145		fveq2 5865	. . . . . . . . . . . . 13 |- ( f = Q -> (coeff ` f ) = (coeff ` Q ) )
146	137	oveq1d 6298	. . . . . . . . . . . . 13 |- ( f = Q -> ( (deg ` f ) - 1 ) = ( (deg ` Q ) - 1 ) )
147	145, 146	fveq12d 5871	. . . . . . . . . . . 12 |- ( f = Q -> ( (coeff ` f ) ` ( (deg ` f ) - 1 ) ) = ( (coeff ` Q ) ` ( (deg ` Q ) - 1 ) ) )
148	145, 137	fveq12d 5871	. . . . . . . . . . . 12 |- ( f = Q -> ( (coeff ` f ) ` (deg ` f ) ) = ( (coeff ` Q ) ` (deg ` Q ) ) )
149	147, 148	oveq12d 6301	. . . . . . . . . . 11 |- ( f = Q -> ( ( (coeff ` f ) ` ( (deg ` f ) - 1 ) ) / ( (coeff ` f ) ` (deg ` f ) ) ) = ( ( (coeff ` Q ) ` ( (deg ` Q ) - 1 ) ) / ( (coeff ` Q ) ` (deg ` Q ) ) ) )
150	149	negeqd 9813	. . . . . . . . . 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 2489	. . . . . . . . 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 3210	. . . . . . 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 679	. . . . 5 |- ( ( ph /\ z e. R ) -> sum_ x e. ( `' Q " { 0 } ) x = -u ( ( (coeff ` Q ) ` ( (deg ` Q ) - 1 ) ) / ( (coeff ` Q ) ` (deg ` Q ) ) ) )
156	31	oveq1d 6298	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> ( D - 1 ) = ( (deg ` Q ) - 1 ) )
157	156	fveq2d 5869	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( (coeff ` Q ) ` ( D - 1 ) ) = ( (coeff ` Q ) ` ( (deg ` Q ) - 1 ) ) )
158	61	fveq2d 5869	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> (coeff ` F ) = (coeff ` ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) ) )
159	27, 158	syl5eq 2520	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> A = (coeff ` ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) ) )
160	61	fveq2d 5869	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> (deg ` F ) = (deg ` ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) ) )
161	68	simp2d 1009	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> (deg ` ( Xp oF - ( CC X. { z } ) ) ) = 1 )
162		ax-1ne0 9560	. . . . . . . . . . . . . . 15 |- 1 =/= 0
163	162	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> 1 =/= 0 )
164	161, 163	eqnetrd 2760	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> (deg ` ( Xp oF - ( CC X. { z } ) ) ) =/= 0 )
165		fveq2 5865	. . . . . . . . . . . . . . 15 |- ( ( Xp oF - ( CC X. { z } ) ) = 0p -> (deg ` ( Xp oF - ( CC X. { z } ) ) ) = (deg ` 0p ) )
166	165, 12	syl6eq 2524	. . . . . . . . . . . . . 14 |- ( ( Xp oF - ( CC X. { z } ) ) = 0p -> (deg ` ( Xp oF - ( CC X. { z } ) ) ) = 0 )
167	166	necon3i 2707	. . . . . . . . . . . . 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 2467	. . . . . . . . . . . . 13 |- (deg ` ( Xp oF - ( CC X. { z } ) ) ) = (deg ` ( Xp oF - ( CC X. { z } ) ) )
170		eqid 2467	. . . . . . . . . . . . 13 |- (deg ` Q ) = (deg ` Q )
171	169, 170	dgrmul 22417	. . . . . . . . . . . 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 1229	. . . . . . . . . . 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 2508	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> (deg ` F ) = ( (deg ` ( Xp oF - ( CC X. { z } ) ) ) + (deg ` Q ) ) )
174	9, 173	syl5eq 2520	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> N = ( (deg ` ( Xp oF - ( CC X. { z } ) ) ) + (deg ` Q ) ) )
175	159, 174	fveq12d 5871	. . . . . . . 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 2467	. . . . . . . . . 10 |- (coeff ` ( Xp oF - ( CC X. { z } ) ) ) = (coeff ` ( Xp oF - ( CC X. { z } ) ) )
177		eqid 2467	. . . . . . . . . 10 |- (coeff ` Q ) = (coeff ` Q )
178	176, 177, 169, 170	coemulhi 22401	. . . . . . . . 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 661	. . . . . . . 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 5869	. . . . . . . . . . 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 3523	. . . . . . . . . . . . . . 15 |- CC C_ CC
182		plyid 22357	. . . . . . . . . . . . . . 15 |- ( ( CC C_ CC /\ 1 e. CC ) -> Xp e. (Poly ` CC ) )
183	181, 101, 182	mp2an 672	. . . . . . . . . . . . . 14 |- Xp e. (Poly ` CC )
184		plyconst 22354	. . . . . . . . . . . . . . 15 |- ( ( CC C_ CC /\ z e. CC ) -> ( CC X. { z } ) e. (Poly ` CC ) )
185	181, 50, 184	sylancr 663	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( CC X. { z } ) e. (Poly ` CC ) )
186		eqid 2467	. . . . . . . . . . . . . . 15 |- (coeff ` Xp ) = (coeff ` Xp )
187		eqid 2467	. . . . . . . . . . . . . . 15 |- (coeff ` ( CC X. { z } ) ) = (coeff ` ( CC X. { z } ) )
188	186, 187	coesub 22404	. . . . . . . . . . . . . 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 663	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> (coeff ` ( Xp oF - ( CC X. { z } ) ) ) = ( (coeff ` Xp ) oF - (coeff ` ( CC X. { z } ) ) ) )
190	189	fveq1d 5867	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 1 ) = ( ( (coeff ` Xp ) oF - (coeff ` ( CC X. { z } ) ) ) ` 1 ) )
191		1nn0 10810	. . . . . . . . . . . . . 14 |- 1 e. NN0
192	186	coef3 22380	. . . . . . . . . . . . . . . . 17 |- ( Xp e. (Poly ` CC ) -> (coeff ` Xp ) : NN0 --> CC )
193		ffn 5730	. . . . . . . . . . . . . . . . 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 22380	. . . . . . . . . . . . . . . 16 |- ( ( CC X. { z } ) e. (Poly ` CC ) -> (coeff ` ( CC X. { z } ) ) : NN0 --> CC )
197		ffn 5730	. . . . . . . . . . . . . . . 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 10800	. . . . . . . . . . . . . . . 16 |- NN0 e. _V
200	199	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> NN0 e. _V )
201		inidm 3707	. . . . . . . . . . . . . . 15 |- ( NN0 i^i NN0 ) = NN0
202		coeidp 22410	. . . . . . . . . . . . . . . . 17 |- ( 1 e. NN0 -> ( (coeff ` Xp ) ` 1 ) = if ( 1 = 1 , 1 , 0 ) )
203	202	adantl 466	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> ( (coeff ` Xp ) ` 1 ) = if ( 1 = 1 , 1 , 0 ) )
204		eqid 2467	. . . . . . . . . . . . . . . . 17 |- 1 = 1
205	204	iftruei 3946	. . . . . . . . . . . . . . . 16 |- if ( 1 = 1 , 1 , 0 ) = 1
206	203, 205	syl6eq 2524	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> ( (coeff ` Xp ) ` 1 ) = 1 )
207		0lt1 10074	. . . . . . . . . . . . . . . . . 18 |- 0 < 1
208		0re 9595	. . . . . . . . . . . . . . . . . . 19 |- 0 e. RR
209		1re 9594	. . . . . . . . . . . . . . . . . . 19 |- 1 e. RR
210	208, 209	ltnlei 9704	. . . . . . . . . . . . . . . . . 18 |- ( 0 < 1 <-> -. 1 <_ 0 )
211	207, 210	mpbi 208	. . . . . . . . . . . . . . . . 17 |- -. 1 <_ 0
212	50	adantr 465	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> z e. CC )
213		0dgr 22393	. . . . . . . . . . . . . . . . . . 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 4459	. . . . . . . . . . . . . . . . 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 2467	. . . . . . . . . . . . . . . . . . . 20 |- (deg ` ( CC X. { z } ) ) = (deg ` ( CC X. { z } ) )
218	187, 217	dgrub 22382	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( CC X. { z } ) e. (Poly ` CC ) /\ 1 e. NN0 /\ ( (coeff ` ( CC X. { z } ) ) ` 1 ) =/= 0 ) -> 1 <_ (deg ` ( CC X. { z } ) ) )
219	218	3expia 1198	. . . . . . . . . . . . . . . . . 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 471	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> ( ( (coeff ` ( CC X. { z } ) ) ` 1 ) =/= 0 -> 1 <_ (deg ` ( CC X. { z } ) ) ) )
221	220	necon1bd 2685	. . . . . . . . . . . . . . . 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 6532	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ z e. R ) /\ 1 e. NN0 ) -> ( ( (coeff ` Xp ) oF - (coeff ` ( CC X. { z } ) ) ) ` 1 ) = ( 1 - 0 ) )
224	191, 223	mpan2 671	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` Xp ) oF - (coeff ` ( CC X. { z } ) ) ) ` 1 ) = ( 1 - 0 ) )
225		1m0e1 10645	. . . . . . . . . . . . 13 |- ( 1 - 0 ) = 1
226	224, 225	syl6eq 2524	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` Xp ) oF - (coeff ` ( CC X. { z } ) ) ) ` 1 ) = 1 )
227	190, 226	eqtrd 2508	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 1 ) = 1 )
228	180, 227	eqtrd 2508	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` (deg ` ( Xp oF - ( CC X. { z } ) ) ) ) = 1 )
229	228	oveq1d 6298	. . . . . . . . 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 22380	. . . . . . . . . . . 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 6021	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> ( (coeff ` Q ) ` (deg ` Q ) ) e. CC )
233	232	mulid2d 9613	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> ( 1 x. ( (coeff ` Q ) ` (deg ` Q ) ) ) = ( (coeff ` Q ) ` (deg ` Q ) ) )
234	229, 233	eqtrd 2508	. . . . . . . 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 2512	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( A ` N ) = ( (coeff ` Q ) ` (deg ` Q ) ) )
236	157, 235	oveq12d 6301	. . . . . 6 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) = ( ( (coeff ` Q ) ` ( (deg ` Q ) - 1 ) ) / ( (coeff ` Q ) ` (deg ` Q ) ) ) )
237	236	negeqd 9813	. . . . 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 2511	. . . 4 |- ( ( ph /\ z e. R ) -> sum_ x e. ( `' Q " { 0 } ) x = -u ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) )
239	134, 238	oveq12d 6301	. . 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 9916	. . . . 5 |- ( ( ph /\ z e. R ) -> -u z e. CC )
241		nnm1nn0 10836	. . . . . . . . 9 |- ( D e. NN -> ( D - 1 ) e. NN0 )
242	3, 241	syl 16	. . . . . . . 8 |- ( ph -> ( D - 1 ) e. NN0 )
243	242	adantr 465	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( D - 1 ) e. NN0 )
244	231, 243	ffvelrnd 6021	. . . . . 6 |- ( ( ph /\ z e. R ) -> ( (coeff ` Q ) ` ( D - 1 ) ) e. CC )
245	235, 232	eqeltrd 2555	. . . . . 6 |- ( ( ph /\ z e. R ) -> ( A ` N ) e. CC )
246	9, 27	dgreq0 22412	. . . . . . . . 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 2725	. . . . . . 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 10319	. . . . 5 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) e. CC )
251	240, 250	negdid 9942	. . . 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 9615	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( -u z x. ( A ` N ) ) e. CC )
253	252, 244, 245, 249	divdird 10357	. . . . . 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 10836	. . . . . . . . . . 11 |- ( N e. NN -> ( N - 1 ) e. NN0 )
255	5, 254	syl 16	. . . . . . . . . 10 |- ( ph -> ( N - 1 ) e. NN0 )
256	255	adantr 465	. . . . . . . . 9 |- ( ( ph /\ z e. R ) -> ( N - 1 ) e. NN0 )
257	176, 177	coemul 22399	. . . . . . . . 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 1228	. . . . . . . 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 5867	. . . . . . . 8 |- ( ( ph /\ z e. R ) -> ( A ` ( N - 1 ) ) = ( (coeff ` ( ( Xp oF - ( CC X. { z } ) ) oF x. Q ) ) ` ( N - 1 ) ) )
260		1e0p1 11003	. . . . . . . . . . . 12 |- 1 = ( 0 + 1 )
261	260	oveq2i 6294	. . . . . . . . . . 11 |- ( 0 ... 1 ) = ( 0 ... ( 0 + 1 ) )
262	261	sumeq1i 13482	. . . . . . . . . 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 10809	. . . . . . . . . . . . 13 |- 0 e. NN0
264		nn0uz 11115	. . . . . . . . . . . . 13 |- NN0 = ( ZZ>= ` 0 )
265	263, 264	eleqtri 2553	. . . . . . . . . . . 12 |- 0 e. ( ZZ>= ` 0 )
266	265	a1i 11	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> 0 e. ( ZZ>= ` 0 ) )
267	261	eleq2i 2545	. . . . . . . . . . . 12 |- ( k e. ( 0 ... 1 ) <-> k e. ( 0 ... ( 0 + 1 ) ) )
268	176	coef3 22380	. . . . . . . . . . . . . . 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 11769	. . . . . . . . . . . . . 14 |- ( k e. ( 0 ... 1 ) -> k e. NN0 )
271		ffvelrn 6018	. . . . . . . . . . . . . 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 477	. . . . . . . . . . . . 13 |- ( ( ( ph /\ z e. R ) /\ k e. ( 0 ... 1 ) ) -> ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) e. CC )
273	2	oveq1d 6298	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( ( D + 1 ) - 1 ) = ( N - 1 ) )
274		pncan 9825	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( D e. CC /\ 1 e. CC ) -> ( ( D + 1 ) - 1 ) = D )
275	102, 101, 274	sylancl 662	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( ( D + 1 ) - 1 ) = D )
276	273, 275	eqtr3d 2510	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( N - 1 ) = D )
277	276	adantr 465	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. R ) -> ( N - 1 ) = D )
278	3	adantr 465	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. R ) -> D e. NN )
279	277, 278	eqeltrd 2555	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. R ) -> ( N - 1 ) e. NN )
280		nnuz 11116	. . . . . . . . . . . . . . . . 17 |- NN = ( ZZ>= ` 1 )
281	279, 280	syl6eleq 2565	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ z e. R ) -> ( N - 1 ) e. ( ZZ>= ` 1 ) )
282		fzss2 11722	. . . . . . . . . . . . . . . 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 3504	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ z e. R ) /\ k e. ( 0 ... 1 ) ) -> k e. ( 0 ... ( N - 1 ) ) )
285		fznn0sub 11715	. . . . . . . . . . . . . . 15 |- ( k e. ( 0 ... ( N - 1 ) ) -> ( ( N - 1 ) - k ) e. NN0 )
286		ffvelrn 6018	. . . . . . . . . . . . . . 15 |- ( ( (coeff ` Q ) : NN0 --> CC /\ ( ( N - 1 ) - k ) e. NN0 ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) e. CC )
287	231, 285, 286	syl2an 477	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ z e. R ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) e. CC )
288	284, 287	syldan 470	. . . . . . . . . . . . 13 |- ( ( ( ph /\ z e. R ) /\ k e. ( 0 ... 1 ) ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) e. CC )
289	272, 288	mulcld 9615	. . . . . . . . . . . 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 476	. . . . . . . . . . 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 2526	. . . . . . . . . . . . 13 |- ( k = ( 0 + 1 ) -> k = 1 )
293	292	fveq2d 5869	. . . . . . . . . . . 12 |- ( k = ( 0 + 1 ) -> ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) = ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 1 ) )
294	292	oveq2d 6299	. . . . . . . . . . . . 13 |- ( k = ( 0 + 1 ) -> ( ( N - 1 ) - k ) = ( ( N - 1 ) - 1 ) )
295	294	fveq2d 5869	. . . . . . . . . . . 12 |- ( k = ( 0 + 1 ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) = ( (coeff ` Q ) ` ( ( N - 1 ) - 1 ) ) )
296	293, 295	oveq12d 6301	. . . . . . . . . . 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 13533	. . . . . . . . . 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 2520	. . . . . . . . 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 3627	. . . . . . . . . . . . . 14 |- ( k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) -> -. k e. ( 0 ... 1 ) )
300	299	adantl 466	. . . . . . . . . . . . 13 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> -. k e. ( 0 ... 1 ) )
301		eldifi 3626	. . . . . . . . . . . . . . . . 17 |- ( k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) -> k e. ( 0 ... ( N - 1 ) ) )
302		elfznn0 11769	. . . . . . . . . . . . . . . . 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 22382	. . . . . . . . . . . . . . . . 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 1198	. . . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . . 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 11683	. . . . . . . . . . . . . . . . . . 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 466	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> k e. ( ZZ>= ` 0 ) )
310		1z 10893	. . . . . . . . . . . . . . . . 17 |- 1 e. ZZ
311		elfz5 11679	. . . . . . . . . . . . . . . . 17 |- ( ( k e. ( ZZ>= ` 0 ) /\ 1 e. ZZ ) -> ( k e. ( 0 ... 1 ) <-> k <_ 1 ) )
312	309, 310, 311	sylancl 662	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> ( k e. ( 0 ... 1 ) <-> k <_ 1 ) )
313	161	breq2d 4459	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. R ) -> ( k <_ (deg ` ( Xp oF - ( CC X. { z } ) ) ) <-> k <_ 1 ) )
314	313	adantr 465	. . . . . . . . . . . . . . . 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 2685	. . . . . . . . . . . . 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 6298	. . . . . . . . . . 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 474	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) e. CC )
321	320	mul02d 9776	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. R ) /\ k e. ( ( 0 ... ( N - 1 ) ) \ ( 0 ... 1 ) ) ) -> ( 0 x. ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) ) = 0 )
322	319, 321	eqtrd 2508	. . . . . . . . . 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 12050	. . . . . . . . . 10 |- ( ( ph /\ z e. R ) -> ( 0 ... ( N - 1 ) ) e. Fin )
324	283, 289, 322, 323	fsumss 13509	. . . . . . . . 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 10874	. . . . . . . . . . . 12 |- 0 e. ZZ
326	189	fveq1d 5867	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 0 ) = ( ( (coeff ` Xp ) oF - (coeff ` ( CC X. { z } ) ) ) ` 0 ) )
327		coeidp 22410	. . . . . . . . . . . . . . . . . . . 20 |- ( 0 e. NN0 -> ( (coeff ` Xp ) ` 0 ) = if ( 0 = 1 , 1 , 0 ) )
328	162	nesymi 2740	. . . . . . . . . . . . . . . . . . . . 21 |- -. 0 = 1
329	328	iffalsei 3949	. . . . . . . . . . . . . . . . . . . 20 |- if ( 0 = 1 , 1 , 0 ) = 0
330	327, 329	syl6eq 2524	. . . . . . . . . . . . . . . . . . 19 |- ( 0 e. NN0 -> ( (coeff ` Xp ) ` 0 ) = 0 )
331	330	adantl 466	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ z e. R ) /\ 0 e. NN0 ) -> ( (coeff ` Xp ) ` 0 ) = 0 )
332		0cn 9587	. . . . . . . . . . . . . . . . . . . . 21 |- 0 e. CC
333		vex 3116	. . . . . . . . . . . . . . . . . . . . . 22 |- z e. _V
334	333	fvconst2 6115	. . . . . . . . . . . . . . . . . . . . 21 |- ( 0 e. CC -> ( ( CC X. { z } ) ` 0 ) = z )
335	332, 334	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 |- ( ( CC X. { z } ) ` 0 ) = z
336	187	coefv0 22395	. . . . . . . . . . . . . . . . . . . . 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 2523	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( CC X. { z } ) ) ` 0 ) = z )
339	338	adantr 465	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ z e. R ) /\ 0 e. NN0 ) -> ( (coeff ` ( CC X. { z } ) ) ` 0 ) = z )
340	195, 198, 200, 200, 201, 331, 339	ofval 6532	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ z e. R ) /\ 0 e. NN0 ) -> ( ( (coeff ` Xp ) oF - (coeff ` ( CC X. { z } ) ) ) ` 0 ) = ( 0 - z ) )
341	263, 340	mpan2 671	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` Xp ) oF - (coeff ` ( CC X. { z } ) ) ) ` 0 ) = ( 0 - z ) )
342		df-neg 9807	. . . . . . . . . . . . . . . 16 |- -u z = ( 0 - z )
343	341, 342	syl6eqr 2526	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` Xp ) oF - (coeff ` ( CC X. { z } ) ) ) ` 0 ) = -u z )
344	326, 343	eqtrd 2508	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 0 ) = -u z )
345	277	oveq1d 6298	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. R ) -> ( ( N - 1 ) - 0 ) = ( D - 0 ) )
346	103	subid1d 9918	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. R ) -> ( D - 0 ) = D )
347	345, 346, 31	3eqtrd 2512	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ z e. R ) -> ( ( N - 1 ) - 0 ) = (deg ` Q ) )
348	347	fveq2d 5869	. . . . . . . . . . . . . . 15 |- ( ( ph /\ z e. R ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - 0 ) ) = ( (coeff ` Q ) ` (deg ` Q ) ) )
349	348, 235	eqtr4d 2511	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. R ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - 0 ) ) = ( A ` N ) )
350	344, 349	oveq12d 6301	. . . . . . . . . . . . 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 2555	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 0 ) x. ( (coeff ` Q ) ` ( ( N - 1 ) - 0 ) ) ) e. CC )
352		fveq2 5865	. . . . . . . . . . . . . 14 |- ( k = 0 -> ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` k ) = ( (coeff ` ( Xp oF - ( CC X. { z } ) ) ) ` 0 ) )
353		oveq2 6291	. . . . . . . . . . . . . . 15 |- ( k = 0 -> ( ( N - 1 ) - k ) = ( ( N - 1 ) - 0 ) )
354	353	fveq2d 5869	. . . . . . . . . . . . . 14 |- ( k = 0 -> ( (coeff ` Q ) ` ( ( N - 1 ) - k ) ) = ( (coeff ` Q ) ` ( ( N - 1 ) - 0 ) ) )
355	352, 354	oveq12d 6301	. . . . . . . . . . . . 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 13526	. . . . . . . . . . . 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 663	. . . . . . . . . . 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 2508	. . . . . . . . . 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 6298	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. R ) -> ( ( N - 1 ) - 1 ) = ( D - 1 ) )
360	359	fveq2d 5869	. . . . . . . . . . . 12 |- ( ( ph /\ z e. R ) -> ( (coeff ` Q ) ` ( ( N - 1 ) - 1 ) ) = ( (coeff ` Q ) ` ( D - 1 ) ) )
361	227, 360	oveq12d 6301	. . . . . . . . . . 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 9613	. . . . . . . . . . 11 |- ( ( ph /\ z e. R ) -> ( 1 x. ( (coeff ` Q ) ` ( D - 1 ) ) ) = ( (coeff ` Q ) ` ( D - 1 ) ) )
363	361, 362	eqtrd 2508	. . . . . . . . . 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 6301	. . . . . . . . 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 2517	. . . . . . . 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 2519	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( ( -u z x. ( A ` N ) ) + ( (coeff ` Q ) ` ( D - 1 ) ) ) = ( A ` ( N - 1 ) ) )
367	366	oveq1d 6298	. . . . . 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 10325	. . . . . . 7 |- ( ( ph /\ z e. R ) -> ( ( -u z x. ( A ` N ) ) / ( A ` N ) ) = -u z )
369	368	oveq1d 6298	. . . . . 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 2517	. . . . 5 |- ( ( ph /\ z e. R ) -> ( -u z + ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) ) = ( ( A ` ( N - 1 ) ) / ( A ` N ) ) )
371	370	negeqd 9813	. . . 4 |- ( ( ph /\ z e. R ) -> -u ( -u z + ( ( (coeff ` Q ) ` ( D - 1 ) ) / ( A ` N ) ) ) = -u ( ( A ` ( N - 1 ) ) / ( A ` N ) ) )
372	251, 371	eqtr3d 2510	. . 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 2512	. 2 |- ( ( ph /\ z e. R ) -> sum_ x e. R x = -u ( ( A ` ( N - 1 ) ) / ( A ` N ) ) )
374	25, 373	exlimddv 1702	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 973   = wceq 1379   E.wex 1596   e. wcel 1767   =/= wne 2662   A.wral 2814   _Vcvv 3113   \ cdif 3473   u. cun 3474   i^i cin 3475   C_ wss 3476   (/)c0 3785   ifcif 3939   {csn 4027   class class class wbr 4447   X. cxp 4997   `'ccnv 4998   dom cdm 4999   "cima 5002   Fn wfn 5582   -->wf 5583   ` cfv 5587  (class class class)co 6283   oFcof 6521   Fincfn 7516   CCcc 9489   RRcr 9490   0cc0 9491   1c1 9492   + caddc 9494   x. cmul 9496   < clt 9627   <_ cle 9628   - cmin 9804   -ucneg 9805   / cdiv 10205   NNcn 10535   NN0cn0 10794   ZZcz 10863   ZZ>=cuz 11081   ...cfz 11671   #chash 12372   sum_csu 13470   0pc0p 21827  Polycply 22332   Xpcidp 22333  coeffccoe 22334  degcdgr 22335   quot cquot 22436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569  ax-addf 9570

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-of 6523  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7900  df-oi 7934  df-card 8319  df-cda 8547  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-n0 10795  df-z 10864  df-uz 11082  df-rp 11220  df-fz 11672  df-fzo 11792  df-fl 11896  df-seq 12075  df-exp 12134  df-hash 12373  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-clim 13273  df-rlim 13274  df-sum 13471  df-0p 21828  df-ply 22336  df-idp 22337  df-coe 22338  df-dgr 22339  df-quot 22437

This theorem is referenced by:  vieta1  22458