Theorem elaa2lemOLD 38210

Description: Elementhood in the set of nonzero algebraic numbers. ' Only if ' part of elaa2 38211. (Contributed by Glauco Siliprandi, 5-Apr-2020.) Obsolete version of elaa2lem 38209 as of 1-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
elaa2lemOLD.a	|- ( ph -> A e. AA )
elaa2lemOLD.an0	|- ( ph -> A =/= 0 )
elaa2lemOLD.g	|- ( ph -> G e. (Poly ` ZZ ) )
elaa2lemOLD.gn0	|- ( ph -> G =/= 0p )
elaa2lemOLD.ga	|- ( ph -> ( G ` A ) = 0 )
elaa2lemOLD.m	|- M = sup ( { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } , RR , `' < )
elaa2lemOLD.i	|- I = ( k e. NN0 |-> ( (coeff ` G ) ` ( k + M ) ) )
elaa2lemOLD.f	|- F = ( z e. CC |-> sum_ k e. ( 0 ... ( (deg ` G ) - M ) ) ( ( I ` k ) x. ( z ^ k ) ) )

Assertion

Ref	Expression
elaa2lemOLD	|- ( ph -> E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) )

Distinct variable groups:   A, f   A, k, z   f, F   k, G   n, G   z, G   k, I, z   k, M   n, M   z, M   ph, k, z

Allowed substitution hints:   ph( f, n)   A( n)   F( z, k, n)   G( f)   I( f, n)   M( f)

Proof of Theorem elaa2lemOLD

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elaa2lemOLD.f	. . . 4 |- F = ( z e. CC |-> sum_ k e. ( 0 ... ( (deg ` G ) - M ) ) ( ( I ` k ) x. ( z ^ k ) ) )
2	1	a1i 11	. . 3 |- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... ( (deg ` G ) - M ) ) ( ( I ` k ) x. ( z ^ k ) ) ) )
3		zsscn 10969	. . . . 5 |- ZZ C_ CC
4	3	a1i 11	. . . 4 |- ( ph -> ZZ C_ CC )
5		elaa2lemOLD.g	. . . . . . . . 9 |- ( ph -> G e. (Poly ` ZZ ) )
6		dgrcl 23266	. . . . . . . . 9 |- ( G e. (Poly ` ZZ ) -> (deg ` G ) e. NN0 )
7	5, 6	syl 17	. . . . . . . 8 |- ( ph -> (deg ` G ) e. NN0 )
8	7	nn0zd 11061	. . . . . . 7 |- ( ph -> (deg ` G ) e. ZZ )
9		elaa2lemOLD.m	. . . . . . . . 9 |- M = sup ( { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } , RR , `' < )
10		ssrab2 3500	. . . . . . . . . 10 |- { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } C_ NN0
11		nn0uz 11217	. . . . . . . . . . . . 13 |- NN0 = ( ZZ>= ` 0 )
12	10, 11	sseqtri 3450	. . . . . . . . . . . 12 |- { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } C_ ( ZZ>= ` 0 )
13	12	a1i 11	. . . . . . . . . . 11 |- ( ph -> { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } C_ ( ZZ>= ` 0 ) )
14		elaa2lemOLD.gn0	. . . . . . . . . . . . . . . . 17 |- ( ph -> G =/= 0p )
15	14	neneqd 2648	. . . . . . . . . . . . . . . 16 |- ( ph -> -. G = 0p )
16		eqid 2471	. . . . . . . . . . . . . . . . . 18 |- (deg ` G ) = (deg ` G )
17		eqid 2471	. . . . . . . . . . . . . . . . . 18 |- (coeff ` G ) = (coeff ` G )
18	16, 17	dgreq0 23298	. . . . . . . . . . . . . . . . 17 |- ( G e. (Poly ` ZZ ) -> ( G = 0p <-> ( (coeff ` G ) ` (deg ` G ) ) = 0 ) )
19	5, 18	syl 17	. . . . . . . . . . . . . . . 16 |- ( ph -> ( G = 0p <-> ( (coeff ` G ) ` (deg ` G ) ) = 0 ) )
20	15, 19	mtbid 307	. . . . . . . . . . . . . . 15 |- ( ph -> -. ( (coeff ` G ) ` (deg ` G ) ) = 0 )
21	20	neqned 2650	. . . . . . . . . . . . . 14 |- ( ph -> ( (coeff ` G ) ` (deg ` G ) ) =/= 0 )
22	7, 21	jca 541	. . . . . . . . . . . . 13 |- ( ph -> ( (deg ` G ) e. NN0 /\ ( (coeff ` G ) ` (deg ` G ) ) =/= 0 ) )
23		fveq2 5879	. . . . . . . . . . . . . . 15 |- ( n = (deg ` G ) -> ( (coeff ` G ) ` n ) = ( (coeff ` G ) ` (deg ` G ) ) )
24	23	neeq1d 2702	. . . . . . . . . . . . . 14 |- ( n = (deg ` G ) -> ( ( (coeff ` G ) ` n ) =/= 0 <-> ( (coeff ` G ) ` (deg ` G ) ) =/= 0 ) )
25	24	elrab 3184	. . . . . . . . . . . . 13 |- ( (deg ` G ) e. { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } <-> ( (deg ` G ) e. NN0 /\ ( (coeff ` G ) ` (deg ` G ) ) =/= 0 ) )
26	22, 25	sylibr 217	. . . . . . . . . . . 12 |- ( ph -> (deg ` G ) e. { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } )
27		ne0i 3728	. . . . . . . . . . . 12 |- ( (deg ` G ) e. { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } -> { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } =/= (/) )
28	26, 27	syl 17	. . . . . . . . . . 11 |- ( ph -> { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } =/= (/) )
29		infmssuzclOLD 11270	. . . . . . . . . . 11 |- ( ( { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } C_ ( ZZ>= ` 0 ) /\ { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } =/= (/) ) -> sup ( { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } , RR , `' < ) e. { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } )
30	13, 28, 29	syl2anc 673	. . . . . . . . . 10 |- ( ph -> sup ( { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } , RR , `' < ) e. { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } )
31	10, 30	sseldi 3416	. . . . . . . . 9 |- ( ph -> sup ( { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } , RR , `' < ) e. NN0 )
32	9, 31	syl5eqel 2553	. . . . . . . 8 |- ( ph -> M e. NN0 )
33	32	nn0zd 11061	. . . . . . 7 |- ( ph -> M e. ZZ )
34	8, 33	zsubcld 11068	. . . . . 6 |- ( ph -> ( (deg ` G ) - M ) e. ZZ )
35	9	a1i 11	. . . . . . . 8 |- ( ph -> M = sup ( { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } , RR , `' < ) )
36		infmssuzleOLD 11269	. . . . . . . . 9 |- ( ( { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } C_ ( ZZ>= ` 0 ) /\ (deg ` G ) e. { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } ) -> sup ( { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } , RR , `' < ) <_ (deg ` G ) )
37	13, 26, 36	syl2anc 673	. . . . . . . 8 |- ( ph -> sup ( { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } , RR , `' < ) <_ (deg ` G ) )
38	35, 37	eqbrtrd 4416	. . . . . . 7 |- ( ph -> M <_ (deg ` G ) )
39	7	nn0red 10950	. . . . . . . 8 |- ( ph -> (deg ` G ) e. RR )
40	32	nn0red 10950	. . . . . . . 8 |- ( ph -> M e. RR )
41	39, 40	subge0d 10224	. . . . . . 7 |- ( ph -> ( 0 <_ ( (deg ` G ) - M ) <-> M <_ (deg ` G ) ) )
42	38, 41	mpbird 240	. . . . . 6 |- ( ph -> 0 <_ ( (deg ` G ) - M ) )
43	34, 42	jca 541	. . . . 5 |- ( ph -> ( ( (deg ` G ) - M ) e. ZZ /\ 0 <_ ( (deg ` G ) - M ) ) )
44		elnn0z 10974	. . . . 5 |- ( ( (deg ` G ) - M ) e. NN0 <-> ( ( (deg ` G ) - M ) e. ZZ /\ 0 <_ ( (deg ` G ) - M ) ) )
45	43, 44	sylibr 217	. . . 4 |- ( ph -> ( (deg ` G ) - M ) e. NN0 )
46		id 22	. . . . . . . . 9 |- ( G e. (Poly ` ZZ ) -> G e. (Poly ` ZZ ) )
47		0zd 10973	. . . . . . . . 9 |- ( G e. (Poly ` ZZ ) -> 0 e. ZZ )
48	17	coef2 23264	. . . . . . . . 9 |- ( ( G e. (Poly ` ZZ ) /\ 0 e. ZZ ) -> (coeff ` G ) : NN0 --> ZZ )
49	46, 47, 48	syl2anc 673	. . . . . . . 8 |- ( G e. (Poly ` ZZ ) -> (coeff ` G ) : NN0 --> ZZ )
50	5, 49	syl 17	. . . . . . 7 |- ( ph -> (coeff ` G ) : NN0 --> ZZ )
51	50	adantr 472	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> (coeff ` G ) : NN0 --> ZZ )
52		simpr 468	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> k e. NN0 )
53	32	adantr 472	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> M e. NN0 )
54	52, 53	nn0addcld 10953	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> ( k + M ) e. NN0 )
55	51, 54	ffvelrnd 6038	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> ( (coeff ` G ) ` ( k + M ) ) e. ZZ )
56		elaa2lemOLD.i	. . . . 5 |- I = ( k e. NN0 |-> ( (coeff ` G ) ` ( k + M ) ) )
57	55, 56	fmptd 6061	. . . 4 |- ( ph -> I : NN0 --> ZZ )
58		elplyr 23234	. . . 4 |- ( ( ZZ C_ CC /\ ( (deg ` G ) - M ) e. NN0 /\ I : NN0 --> ZZ ) -> ( z e. CC |-> sum_ k e. ( 0 ... ( (deg ` G ) - M ) ) ( ( I ` k ) x. ( z ^ k ) ) ) e. (Poly ` ZZ ) )
59	4, 45, 57, 58	syl3anc 1292	. . 3 |- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... ( (deg ` G ) - M ) ) ( ( I ` k ) x. ( z ^ k ) ) ) e. (Poly ` ZZ ) )
60	2, 59	eqeltrd 2549	. 2 |- ( ph -> F e. (Poly ` ZZ ) )
61		simpr 468	. . . . . . . . . 10 |- ( ( ( ph /\ k e. NN0 ) /\ k <_ ( (deg ` G ) - M ) ) -> k <_ ( (deg ` G ) - M ) )
62	61	iftrued 3880	. . . . . . . . 9 |- ( ( ( ph /\ k e. NN0 ) /\ k <_ ( (deg ` G ) - M ) ) -> if ( k <_ ( (deg ` G ) - M ) , ( (coeff ` G ) ` ( k + M ) ) , 0 ) = ( (coeff ` G ) ` ( k + M ) ) )
63		iffalse 3881	. . . . . . . . . . 11 |- ( -. k <_ ( (deg ` G ) - M ) -> if ( k <_ ( (deg ` G ) - M ) , ( (coeff ` G ) ` ( k + M ) ) , 0 ) = 0 )
64	63	adantl 473	. . . . . . . . . 10 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> if ( k <_ ( (deg ` G ) - M ) , ( (coeff ` G ) ` ( k + M ) ) , 0 ) = 0 )
65		simpr 468	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> -. k <_ ( (deg ` G ) - M ) )
66	39	ad2antrr 740	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> (deg ` G ) e. RR )
67	40	ad2antrr 740	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> M e. RR )
68	66, 67	resubcld 10068	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> ( (deg ` G ) - M ) e. RR )
69		nn0re 10902	. . . . . . . . . . . . . . . . 17 |- ( k e. NN0 -> k e. RR )
70	69	ad2antlr 741	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> k e. RR )
71	68, 70	ltnled 9799	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> ( ( (deg ` G ) - M ) < k <-> -. k <_ ( (deg ` G ) - M ) ) )
72	65, 71	mpbird 240	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> ( (deg ` G ) - M ) < k )
73	66, 67, 70	ltsubaddd 10230	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> ( ( (deg ` G ) - M ) < k <-> (deg ` G ) < ( k + M ) ) )
74	72, 73	mpbid 215	. . . . . . . . . . . . 13 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> (deg ` G ) < ( k + M ) )
75		olc 391	. . . . . . . . . . . . 13 |- ( (deg ` G ) < ( k + M ) -> ( G = 0p \/ (deg ` G ) < ( k + M ) ) )
76	74, 75	syl 17	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> ( G = 0p \/ (deg ` G ) < ( k + M ) ) )
77	5	ad2antrr 740	. . . . . . . . . . . . 13 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> G e. (Poly ` ZZ ) )
78	54	adantr 472	. . . . . . . . . . . . 13 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> ( k + M ) e. NN0 )
79	16, 17	dgrlt 23299	. . . . . . . . . . . . 13 |- ( ( G e. (Poly ` ZZ ) /\ ( k + M ) e. NN0 ) -> ( ( G = 0p \/ (deg ` G ) < ( k + M ) ) <-> ( (deg ` G ) <_ ( k + M ) /\ ( (coeff ` G ) ` ( k + M ) ) = 0 ) ) )
80	77, 78, 79	syl2anc 673	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> ( ( G = 0p \/ (deg ` G ) < ( k + M ) ) <-> ( (deg ` G ) <_ ( k + M ) /\ ( (coeff ` G ) ` ( k + M ) ) = 0 ) ) )
81	76, 80	mpbid 215	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> ( (deg ` G ) <_ ( k + M ) /\ ( (coeff ` G ) ` ( k + M ) ) = 0 ) )
82	81	simprd 470	. . . . . . . . . 10 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> ( (coeff ` G ) ` ( k + M ) ) = 0 )
83	64, 82	eqtr4d 2508	. . . . . . . . 9 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> if ( k <_ ( (deg ` G ) - M ) , ( (coeff ` G ) ` ( k + M ) ) , 0 ) = ( (coeff ` G ) ` ( k + M ) ) )
84	62, 83	pm2.61dan 808	. . . . . . . 8 |- ( ( ph /\ k e. NN0 ) -> if ( k <_ ( (deg ` G ) - M ) , ( (coeff ` G ) ` ( k + M ) ) , 0 ) = ( (coeff ` G ) ` ( k + M ) ) )
85	84	mpteq2dva 4482	. . . . . . 7 |- ( ph -> ( k e. NN0 |-> if ( k <_ ( (deg ` G ) - M ) , ( (coeff ` G ) ` ( k + M ) ) , 0 ) ) = ( k e. NN0 |-> ( (coeff ` G ) ` ( k + M ) ) ) )
86	50, 4	fssd 5750	. . . . . . . . . 10 |- ( ph -> (coeff ` G ) : NN0 --> CC )
87	86	adantr 472	. . . . . . . . 9 |- ( ( ph /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> (coeff ` G ) : NN0 --> CC )
88		elfznn0 11913	. . . . . . . . . . 11 |- ( k e. ( 0 ... ( (deg ` G ) - M ) ) -> k e. NN0 )
89	88	adantl 473	. . . . . . . . . 10 |- ( ( ph /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> k e. NN0 )
90	32	adantr 472	. . . . . . . . . 10 |- ( ( ph /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> M e. NN0 )
91	89, 90	nn0addcld 10953	. . . . . . . . 9 |- ( ( ph /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> ( k + M ) e. NN0 )
92	87, 91	ffvelrnd 6038	. . . . . . . 8 |- ( ( ph /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> ( (coeff ` G ) ` ( k + M ) ) e. CC )
93		eqidd 2472	. . . . . . . . . . 11 |- ( ( ph /\ z e. CC ) -> ( 0 ... ( (deg ` G ) - M ) ) = ( 0 ... ( (deg ` G ) - M ) ) )
94		simpl 464	. . . . . . . . . . . . . 14 |- ( ( ph /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> ph )
95	56	a1i 11	. . . . . . . . . . . . . . 15 |- ( ph -> I = ( k e. NN0 |-> ( (coeff ` G ) ` ( k + M ) ) ) )
96	95, 55	fvmpt2d 5974	. . . . . . . . . . . . . 14 |- ( ( ph /\ k e. NN0 ) -> ( I ` k ) = ( (coeff ` G ) ` ( k + M ) ) )
97	94, 89, 96	syl2anc 673	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> ( I ` k ) = ( (coeff ` G ) ` ( k + M ) ) )
98	97	adantlr 729	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> ( I ` k ) = ( (coeff ` G ) ` ( k + M ) ) )
99	98	oveq1d 6323	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> ( ( I ` k ) x. ( z ^ k ) ) = ( ( (coeff ` G ) ` ( k + M ) ) x. ( z ^ k ) ) )
100	93, 99	sumeq12rdv 13850	. . . . . . . . . 10 |- ( ( ph /\ z e. CC ) -> sum_ k e. ( 0 ... ( (deg ` G ) - M ) ) ( ( I ` k ) x. ( z ^ k ) ) = sum_ k e. ( 0 ... ( (deg ` G ) - M ) ) ( ( (coeff ` G ) ` ( k + M ) ) x. ( z ^ k ) ) )
101	100	mpteq2dva 4482	. . . . . . . . 9 |- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... ( (deg ` G ) - M ) ) ( ( I ` k ) x. ( z ^ k ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... ( (deg ` G ) - M ) ) ( ( (coeff ` G ) ` ( k + M ) ) x. ( z ^ k ) ) ) )
102	2, 101	eqtrd 2505	. . . . . . . 8 |- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... ( (deg ` G ) - M ) ) ( ( (coeff ` G ) ` ( k + M ) ) x. ( z ^ k ) ) ) )
103	60, 45, 92, 102	coeeq2 23275	. . . . . . 7 |- ( ph -> (coeff ` F ) = ( k e. NN0 |-> if ( k <_ ( (deg ` G ) - M ) , ( (coeff ` G ) ` ( k + M ) ) , 0 ) ) )
104	85, 103, 95	3eqtr4d 2515	. . . . . 6 |- ( ph -> (coeff ` F ) = I )
105	104	fveq1d 5881	. . . . 5 |- ( ph -> ( (coeff ` F ) ` 0 ) = ( I ` 0 ) )
106		oveq1 6315	. . . . . . . . 9 |- ( k = 0 -> ( k + M ) = ( 0 + M ) )
107	106	adantl 473	. . . . . . . 8 |- ( ( ph /\ k = 0 ) -> ( k + M ) = ( 0 + M ) )
108	3, 33	sseldi 3416	. . . . . . . . . 10 |- ( ph -> M e. CC )
109	108	addid2d 9852	. . . . . . . . 9 |- ( ph -> ( 0 + M ) = M )
110	109	adantr 472	. . . . . . . 8 |- ( ( ph /\ k = 0 ) -> ( 0 + M ) = M )
111	107, 110	eqtrd 2505	. . . . . . 7 |- ( ( ph /\ k = 0 ) -> ( k + M ) = M )
112	111	fveq2d 5883	. . . . . 6 |- ( ( ph /\ k = 0 ) -> ( (coeff ` G ) ` ( k + M ) ) = ( (coeff ` G ) ` M ) )
113		0nn0 10908	. . . . . . 7 |- 0 e. NN0
114	113	a1i 11	. . . . . 6 |- ( ph -> 0 e. NN0 )
115	50, 32	ffvelrnd 6038	. . . . . 6 |- ( ph -> ( (coeff ` G ) ` M ) e. ZZ )
116	95, 112, 114, 115	fvmptd 5969	. . . . 5 |- ( ph -> ( I ` 0 ) = ( (coeff ` G ) ` M ) )
117		eqidd 2472	. . . . 5 |- ( ph -> ( (coeff ` G ) ` M ) = ( (coeff ` G ) ` M ) )
118	105, 116, 117	3eqtrd 2509	. . . 4 |- ( ph -> ( (coeff ` F ) ` 0 ) = ( (coeff ` G ) ` M ) )
119	35, 30	eqeltrd 2549	. . . . . 6 |- ( ph -> M e. { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } )
120		fveq2 5879	. . . . . . . 8 |- ( n = M -> ( (coeff ` G ) ` n ) = ( (coeff ` G ) ` M ) )
121	120	neeq1d 2702	. . . . . . 7 |- ( n = M -> ( ( (coeff ` G ) ` n ) =/= 0 <-> ( (coeff ` G ) ` M ) =/= 0 ) )
122	121	elrab 3184	. . . . . 6 |- ( M e. { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } <-> ( M e. NN0 /\ ( (coeff ` G ) ` M ) =/= 0 ) )
123	119, 122	sylib 201	. . . . 5 |- ( ph -> ( M e. NN0 /\ ( (coeff ` G ) ` M ) =/= 0 ) )
124	123	simprd 470	. . . 4 |- ( ph -> ( (coeff ` G ) ` M ) =/= 0 )
125	118, 124	eqnetrd 2710	. . 3 |- ( ph -> ( (coeff ` F ) ` 0 ) =/= 0 )
126	5, 47	syl 17	. . . . . . 7 |- ( ph -> 0 e. ZZ )
127		aasscn 23350	. . . . . . . . . . 11 |- AA C_ CC
128		elaa2lemOLD.a	. . . . . . . . . . 11 |- ( ph -> A e. AA )
129	127, 128	sseldi 3416	. . . . . . . . . 10 |- ( ph -> A e. CC )
130	94, 129	syl 17	. . . . . . . . 9 |- ( ( ph /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> A e. CC )
131	130, 89	expcld 12454	. . . . . . . 8 |- ( ( ph /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> ( A ^ k ) e. CC )
132	92, 131	mulcld 9681	. . . . . . 7 |- ( ( ph /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> ( ( (coeff ` G ) ` ( k + M ) ) x. ( A ^ k ) ) e. CC )
133		oveq1 6315	. . . . . . . . 9 |- ( k = ( j - M ) -> ( k + M ) = ( ( j - M ) + M ) )
134	133	fveq2d 5883	. . . . . . . 8 |- ( k = ( j - M ) -> ( (coeff ` G ) ` ( k + M ) ) = ( (coeff ` G ) ` ( ( j - M ) + M ) ) )
135		oveq2 6316	. . . . . . . 8 |- ( k = ( j - M ) -> ( A ^ k ) = ( A ^ ( j - M ) ) )
136	134, 135	oveq12d 6326	. . . . . . 7 |- ( k = ( j - M ) -> ( ( (coeff ` G ) ` ( k + M ) ) x. ( A ^ k ) ) = ( ( (coeff ` G ) ` ( ( j - M ) + M ) ) x. ( A ^ ( j - M ) ) ) )
137	33, 126, 34, 132, 136	fsumshft 13918	. . . . . 6 |- ( ph -> sum_ k e. ( 0 ... ( (deg ` G ) - M ) ) ( ( (coeff ` G ) ` ( k + M ) ) x. ( A ^ k ) ) = sum_ j e. ( ( 0 + M ) ... ( ( (deg ` G ) - M ) + M ) ) ( ( (coeff ` G ) ` ( ( j - M ) + M ) ) x. ( A ^ ( j - M ) ) ) )
138	3, 8	sseldi 3416	. . . . . . . . . 10 |- ( ph -> (deg ` G ) e. CC )
139	138, 108	npcand 10009	. . . . . . . . 9 |- ( ph -> ( ( (deg ` G ) - M ) + M ) = (deg ` G ) )
140	109, 139	oveq12d 6326	. . . . . . . 8 |- ( ph -> ( ( 0 + M ) ... ( ( (deg ` G ) - M ) + M ) ) = ( M ... (deg ` G ) ) )
141	140	sumeq1d 13844	. . . . . . 7 |- ( ph -> sum_ j e. ( ( 0 + M ) ... ( ( (deg ` G ) - M ) + M ) ) ( ( (coeff ` G ) ` ( ( j - M ) + M ) ) x. ( A ^ ( j - M ) ) ) = sum_ j e. ( M ... (deg ` G ) ) ( ( (coeff ` G ) ` ( ( j - M ) + M ) ) x. ( A ^ ( j - M ) ) ) )
142		elfzelz 11826	. . . . . . . . . . . . . 14 |- ( j e. ( M ... (deg ` G ) ) -> j e. ZZ )
143	142	adantl 473	. . . . . . . . . . . . 13 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> j e. ZZ )
144	3, 143	sseldi 3416	. . . . . . . . . . . 12 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> j e. CC )
145	108	adantr 472	. . . . . . . . . . . 12 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> M e. CC )
146	144, 145	npcand 10009	. . . . . . . . . . 11 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> ( ( j - M ) + M ) = j )
147	146	fveq2d 5883	. . . . . . . . . 10 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> ( (coeff ` G ) ` ( ( j - M ) + M ) ) = ( (coeff ` G ) ` j ) )
148	147	oveq1d 6323	. . . . . . . . 9 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> ( ( (coeff ` G ) ` ( ( j - M ) + M ) ) x. ( A ^ ( j - M ) ) ) = ( ( (coeff ` G ) ` j ) x. ( A ^ ( j - M ) ) ) )
149	129	adantr 472	. . . . . . . . . . . 12 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> A e. CC )
150		elaa2lemOLD.an0	. . . . . . . . . . . . 13 |- ( ph -> A =/= 0 )
151	150	adantr 472	. . . . . . . . . . . 12 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> A =/= 0 )
152	33	adantr 472	. . . . . . . . . . . 12 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> M e. ZZ )
153	149, 151, 152, 143	expsubd 12465	. . . . . . . . . . 11 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> ( A ^ ( j - M ) ) = ( ( A ^ j ) / ( A ^ M ) ) )
154	153	oveq2d 6324	. . . . . . . . . 10 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> ( ( (coeff ` G ) ` j ) x. ( A ^ ( j - M ) ) ) = ( ( (coeff ` G ) ` j ) x. ( ( A ^ j ) / ( A ^ M ) ) ) )
155	86	adantr 472	. . . . . . . . . . . . 13 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> (coeff ` G ) : NN0 --> CC )
156		0red 9662	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> 0 e. RR )
157	40	adantr 472	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> M e. RR )
158	143	zred 11063	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> j e. RR )
159	32	nn0ge0d 10952	. . . . . . . . . . . . . . . . 17 |- ( ph -> 0 <_ M )
160	159	adantr 472	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> 0 <_ M )
161		elfzle1 11828	. . . . . . . . . . . . . . . . 17 |- ( j e. ( M ... (deg ` G ) ) -> M <_ j )
162	161	adantl 473	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> M <_ j )
163	156, 157, 158, 160, 162	letrd 9809	. . . . . . . . . . . . . . 15 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> 0 <_ j )
164	143, 163	jca 541	. . . . . . . . . . . . . 14 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> ( j e. ZZ /\ 0 <_ j ) )
165		elnn0z 10974	. . . . . . . . . . . . . 14 |- ( j e. NN0 <-> ( j e. ZZ /\ 0 <_ j ) )
166	164, 165	sylibr 217	. . . . . . . . . . . . 13 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> j e. NN0 )
167	155, 166	ffvelrnd 6038	. . . . . . . . . . . 12 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> ( (coeff ` G ) ` j ) e. CC )
168	149, 166	expcld 12454	. . . . . . . . . . . 12 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> ( A ^ j ) e. CC )
169	129, 32	expcld 12454	. . . . . . . . . . . . 13 |- ( ph -> ( A ^ M ) e. CC )
170	169	adantr 472	. . . . . . . . . . . 12 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> ( A ^ M ) e. CC )
171	149, 151, 152	expne0d 12460	. . . . . . . . . . . 12 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> ( A ^ M ) =/= 0 )
172	167, 168, 170, 171	divassd 10440	. . . . . . . . . . 11 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> ( ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) / ( A ^ M ) ) = ( ( (coeff ` G ) ` j ) x. ( ( A ^ j ) / ( A ^ M ) ) ) )
173	172	eqcomd 2477	. . . . . . . . . 10 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> ( ( (coeff ` G ) ` j ) x. ( ( A ^ j ) / ( A ^ M ) ) ) = ( ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) / ( A ^ M ) ) )
174	154, 173	eqtr2d 2506	. . . . . . . . 9 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> ( ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) / ( A ^ M ) ) = ( ( (coeff ` G ) ` j ) x. ( A ^ ( j - M ) ) ) )
175	148, 174	eqtr4d 2508	. . . . . . . 8 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> ( ( (coeff ` G ) ` ( ( j - M ) + M ) ) x. ( A ^ ( j - M ) ) ) = ( ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) / ( A ^ M ) ) )
176	175	sumeq2dv 13846	. . . . . . 7 |- ( ph -> sum_ j e. ( M ... (deg ` G ) ) ( ( (coeff ` G ) ` ( ( j - M ) + M ) ) x. ( A ^ ( j - M ) ) ) = sum_ j e. ( M ... (deg ` G ) ) ( ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) / ( A ^ M ) ) )
177	141, 176	eqtrd 2505	. . . . . 6 |- ( ph -> sum_ j e. ( ( 0 + M ) ... ( ( (deg ` G ) - M ) + M ) ) ( ( (coeff ` G ) ` ( ( j - M ) + M ) ) x. ( A ^ ( j - M ) ) ) = sum_ j e. ( M ... (deg ` G ) ) ( ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) / ( A ^ M ) ) )
178	32, 11	syl6eleq 2559	. . . . . . . 8 |- ( ph -> M e. ( ZZ>= ` 0 ) )
179		fzss1 11863	. . . . . . . 8 |- ( M e. ( ZZ>= ` 0 ) -> ( M ... (deg ` G ) ) C_ ( 0 ... (deg ` G ) ) )
180	178, 179	syl 17	. . . . . . 7 |- ( ph -> ( M ... (deg ` G ) ) C_ ( 0 ... (deg ` G ) ) )
181	167, 168	mulcld 9681	. . . . . . . 8 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) e. CC )
182	181, 170, 171	divcld 10405	. . . . . . 7 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> ( ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) / ( A ^ M ) ) e. CC )
183	33	ad2antrr 740	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. j < M ) -> M e. ZZ )
184	8	ad2antrr 740	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. j < M ) -> (deg ` G ) e. ZZ )
185		eldifi 3544	. . . . . . . . . . . . . . . . . . 19 |- ( j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) -> j e. ( 0 ... (deg ` G ) ) )
186		elfznn0 11913	. . . . . . . . . . . . . . . . . . . 20 |- ( j e. ( 0 ... (deg ` G ) ) -> j e. NN0 )
187	186	nn0zd 11061	. . . . . . . . . . . . . . . . . . 19 |- ( j e. ( 0 ... (deg ` G ) ) -> j e. ZZ )
188	185, 187	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) -> j e. ZZ )
189	188	ad2antlr 741	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. j < M ) -> j e. ZZ )
190	183, 184, 189	3jca 1210	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. j < M ) -> ( M e. ZZ /\ (deg ` G ) e. ZZ /\ j e. ZZ ) )
191		simpr 468	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. j < M ) -> -. j < M )
192	40	ad2antrr 740	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. j < M ) -> M e. RR )
193	189	zred 11063	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. j < M ) -> j e. RR )
194	192, 193	lenltd 9798	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. j < M ) -> ( M <_ j <-> -. j < M ) )
195	191, 194	mpbird 240	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. j < M ) -> M <_ j )
196		elfzle2 11829	. . . . . . . . . . . . . . . . . 18 |- ( j e. ( 0 ... (deg ` G ) ) -> j <_ (deg ` G ) )
197	185, 196	syl 17	. . . . . . . . . . . . . . . . 17 |- ( j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) -> j <_ (deg ` G ) )
198	197	ad2antlr 741	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. j < M ) -> j <_ (deg ` G ) )
199	190, 195, 198	jca32 544	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. j < M ) -> ( ( M e. ZZ /\ (deg ` G ) e. ZZ /\ j e. ZZ ) /\ ( M <_ j /\ j <_ (deg ` G ) ) ) )
200		elfz2 11817	. . . . . . . . . . . . . . 15 |- ( j e. ( M ... (deg ` G ) ) <-> ( ( M e. ZZ /\ (deg ` G ) e. ZZ /\ j e. ZZ ) /\ ( M <_ j /\ j <_ (deg ` G ) ) ) )
201	199, 200	sylibr 217	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. j < M ) -> j e. ( M ... (deg ` G ) ) )
202		eldifn 3545	. . . . . . . . . . . . . . 15 |- ( j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) -> -. j e. ( M ... (deg ` G ) ) )
203	202	ad2antlr 741	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. j < M ) -> -. j e. ( M ... (deg ` G ) ) )
204	201, 203	condan 811	. . . . . . . . . . . . 13 |- ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) -> j < M )
205	204	adantr 472	. . . . . . . . . . . 12 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. ( (coeff ` G ) ` j ) = 0 ) -> j < M )
206	9	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. ( (coeff ` G ) ` j ) = 0 ) -> M = sup ( { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } , RR , `' < ) )
207	12	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. ( (coeff ` G ) ` j ) = 0 ) -> { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } C_ ( ZZ>= ` 0 ) )
208	185, 186	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) -> j e. NN0 )
209	208	adantr 472	. . . . . . . . . . . . . . . . . 18 |- ( ( j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) /\ -. ( (coeff ` G ) ` j ) = 0 ) -> j e. NN0 )
210		neqne 2651	. . . . . . . . . . . . . . . . . . 19 |- ( -. ( (coeff ` G ) ` j ) = 0 -> ( (coeff ` G ) ` j ) =/= 0 )
211	210	adantl 473	. . . . . . . . . . . . . . . . . 18 |- ( ( j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) /\ -. ( (coeff ` G ) ` j ) = 0 ) -> ( (coeff ` G ) ` j ) =/= 0 )
212	209, 211	jca 541	. . . . . . . . . . . . . . . . 17 |- ( ( j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) /\ -. ( (coeff ` G ) ` j ) = 0 ) -> ( j e. NN0 /\ ( (coeff ` G ) ` j ) =/= 0 ) )
213		fveq2 5879	. . . . . . . . . . . . . . . . . . 19 |- ( n = j -> ( (coeff ` G ) ` n ) = ( (coeff ` G ) ` j ) )
214	213	neeq1d 2702	. . . . . . . . . . . . . . . . . 18 |- ( n = j -> ( ( (coeff ` G ) ` n ) =/= 0 <-> ( (coeff ` G ) ` j ) =/= 0 ) )
215	214	elrab 3184	. . . . . . . . . . . . . . . . 17 |- ( j e. { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } <-> ( j e. NN0 /\ ( (coeff ` G ) ` j ) =/= 0 ) )
216	212, 215	sylibr 217	. . . . . . . . . . . . . . . 16 |- ( ( j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) /\ -. ( (coeff ` G ) ` j ) = 0 ) -> j e. { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } )
217	216	adantll 728	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. ( (coeff ` G ) ` j ) = 0 ) -> j e. { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } )
218		infmssuzleOLD 11269	. . . . . . . . . . . . . . 15 |- ( ( { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } C_ ( ZZ>= ` 0 ) /\ j e. { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } ) -> sup ( { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } , RR , `' < ) <_ j )
219	207, 217, 218	syl2anc 673	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. ( (coeff ` G ) ` j ) = 0 ) -> sup ( { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } , RR , `' < ) <_ j )
220	206, 219	eqbrtrd 4416	. . . . . . . . . . . . 13 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. ( (coeff ` G ) ` j ) = 0 ) -> M <_ j )
221	40	ad2antrr 740	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. ( (coeff ` G ) ` j ) = 0 ) -> M e. RR )
222	188	zred 11063	. . . . . . . . . . . . . . 15 |- ( j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) -> j e. RR )
223	222	ad2antlr 741	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. ( (coeff ` G ) ` j ) = 0 ) -> j e. RR )
224	221, 223	lenltd 9798	. . . . . . . . . . . . 13 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. ( (coeff ` G ) ` j ) = 0 ) -> ( M <_ j <-> -. j < M ) )
225	220, 224	mpbid 215	. . . . . . . . . . . 12 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. ( (coeff ` G ) ` j ) = 0 ) -> -. j < M )
226	205, 225	condan 811	. . . . . . . . . . 11 |- ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) -> ( (coeff ` G ) ` j ) = 0 )
227	226	oveq1d 6323	. . . . . . . . . 10 |- ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) -> ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) = ( 0 x. ( A ^ j ) ) )
228	129	adantr 472	. . . . . . . . . . . 12 |- ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) -> A e. CC )
229	208	adantl 473	. . . . . . . . . . . 12 |- ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) -> j e. NN0 )
230	228, 229	expcld 12454	. . . . . . . . . . 11 |- ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) -> ( A ^ j ) e. CC )
231	230	mul02d 9849	. . . . . . . . . 10 |- ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) -> ( 0 x. ( A ^ j ) ) = 0 )
232	227, 231	eqtrd 2505	. . . . . . . . 9 |- ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) -> ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) = 0 )
233	232	oveq1d 6323	. . . . . . . 8 |- ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) -> ( ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) / ( A ^ M ) ) = ( 0 / ( A ^ M ) ) )
234	129, 150, 33	expne0d 12460	. . . . . . . . . 10 |- ( ph -> ( A ^ M ) =/= 0 )
235	169, 234	div0d 10404	. . . . . . . . 9 |- ( ph -> ( 0 / ( A ^ M ) ) = 0 )
236	235	adantr 472	. . . . . . . 8 |- ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) -> ( 0 / ( A ^ M ) ) = 0 )
237	233, 236	eqtrd 2505	. . . . . . 7 |- ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) -> ( ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) / ( A ^ M ) ) = 0 )
238		fzfid 12224	. . . . . . 7 |- ( ph -> ( 0 ... (deg ` G ) ) e. Fin )
239	180, 182, 237, 238	fsumss 13868	. . . . . 6 |- ( ph -> sum_ j e. ( M ... (deg ` G ) ) ( ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) / ( A ^ M ) ) = sum_ j e. ( 0 ... (deg ` G ) ) ( ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) / ( A ^ M ) ) )
240	137, 177, 239	3eqtrd 2509	. . . . 5 |- ( ph -> sum_ k e. ( 0 ... ( (deg ` G ) - M ) ) ( ( (coeff ` G ) ` ( k + M ) ) x. ( A ^ k ) ) = sum_ j e. ( 0 ... (deg ` G ) ) ( ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) / ( A ^ M ) ) )
241	89, 55	syldan 478	. . . . . . . . . 10 |- ( ( ph /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> ( (coeff ` G ) ` ( k + M ) ) e. ZZ )
242	56	fvmpt2 5972	. . . . . . . . . 10 |- ( ( k e. NN0 /\ ( (coeff ` G ) ` ( k + M ) ) e. ZZ ) -> ( I ` k ) = ( (coeff ` G ) ` ( k + M ) ) )
243	89, 241, 242	syl2anc 673	. . . . . . . . 9 |- ( ( ph /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> ( I ` k ) = ( (coeff ` G ) ` ( k + M ) ) )
244	243	adantlr 729	. . . . . . . 8 |- ( ( ( ph /\ z = A ) /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> ( I ` k ) = ( (coeff ` G ) ` ( k + M ) ) )
245		oveq1 6315	. . . . . . . . 9 |- ( z = A -> ( z ^ k ) = ( A ^ k ) )
246	245	ad2antlr 741	. . . . . . . 8 |- ( ( ( ph /\ z = A ) /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> ( z ^ k ) = ( A ^ k ) )
247	244, 246	oveq12d 6326	. . . . . . 7 |- ( ( ( ph /\ z = A ) /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> ( ( I ` k ) x. ( z ^ k ) ) = ( ( (coeff ` G ) ` ( k + M ) ) x. ( A ^ k ) ) )
248	247	sumeq2dv 13846	. . . . . 6 |- ( ( ph /\ z = A ) -> sum_ k e. ( 0 ... ( (deg ` G ) - M ) ) ( ( I ` k ) x. ( z ^ k ) ) = sum_ k e. ( 0 ... ( (deg ` G ) - M ) ) ( ( (coeff ` G ) ` ( k + M ) ) x. ( A ^ k ) ) )
249		fzfid 12224	. . . . . . 7 |- ( ph -> ( 0 ... ( (deg ` G ) - M ) ) e. Fin )
250	249, 132	fsumcl 13876	. . . . . 6 |- ( ph -> sum_ k e. ( 0 ... ( (deg ` G ) - M ) ) ( ( (coeff ` G ) ` ( k + M ) ) x. ( A ^ k ) ) e. CC )
251	2, 248, 129, 250	fvmptd 5969	. . . . 5 |- ( ph -> ( F ` A ) = sum_ k e. ( 0 ... ( (deg ` G ) - M ) ) ( ( (coeff ` G ) ` ( k + M ) ) x. ( A ^ k ) ) )
252	17, 16	coeid2 23272	. . . . . . . 8 |- ( ( G e. (Poly ` ZZ ) /\ A e. CC ) -> ( G ` A ) = sum_ j e. ( 0 ... (deg ` G ) ) ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) )
253	5, 129, 252	syl2anc 673	. . . . . . 7 |- ( ph -> ( G ` A ) = sum_ j e. ( 0 ... (deg ` G ) ) ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) )
254	253	oveq1d 6323	. . . . . 6 |- ( ph -> ( ( G ` A ) / ( A ^ M ) ) = ( sum_ j e. ( 0 ... (deg ` G ) ) ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) / ( A ^ M ) ) )
255	86	adantr 472	. . . . . . . . 9 |- ( ( ph /\ j e. ( 0 ... (deg ` G ) ) ) -> (coeff ` G ) : NN0 --> CC )
256	186	adantl 473	. . . . . . . . 9 |- ( ( ph /\ j e. ( 0 ... (deg ` G ) ) ) -> j e. NN0 )
257	255, 256	ffvelrnd 6038	. . . . . . . 8 |- ( ( ph /\ j e. ( 0 ... (deg ` G ) ) ) -> ( (coeff ` G ) ` j ) e. CC )
258	129	adantr 472	. . . . . . . . 9 |- ( ( ph /\ j e. ( 0 ... (deg ` G ) ) ) -> A e. CC )
259	258, 256	expcld 12454	. . . . . . . 8 |- ( ( ph /\ j e. ( 0 ... (deg ` G ) ) ) -> ( A ^ j ) e. CC )
260	257, 259	mulcld 9681	. . . . . . 7 |- ( ( ph /\ j e. ( 0 ... (deg ` G ) ) ) -> ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) e. CC )
261	238, 169, 260, 234	fsumdivc 13924	. . . . . 6 |- ( ph -> ( sum_ j e. ( 0 ... (deg ` G ) ) ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) / ( A ^ M ) ) = sum_ j e. ( 0 ... (deg ` G ) ) ( ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) / ( A ^ M ) ) )
262	254, 261	eqtrd 2505	. . . . 5 |- ( ph -> ( ( G ` A ) / ( A ^ M ) ) = sum_ j e. ( 0 ... (deg ` G ) ) ( ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) / ( A ^ M ) ) )
263	240, 251, 262	3eqtr4d 2515	. . . 4 |- ( ph -> ( F ` A ) = ( ( G ` A ) / ( A ^ M ) ) )
264		elaa2lemOLD.ga	. . . . 5 |- ( ph -> ( G ` A ) = 0 )
265	264	oveq1d 6323	. . . 4 |- ( ph -> ( ( G ` A ) / ( A ^ M ) ) = ( 0 / ( A ^ M ) ) )
266	263, 265, 235	3eqtrd 2509	. . 3 |- ( ph -> ( F ` A ) = 0 )
267	125, 266	jca 541	. 2 |- ( ph -> ( ( (coeff ` F ) ` 0 ) =/= 0 /\ ( F ` A ) = 0 ) )
268		fveq2 5879	. . . . . 6 |- ( f = F -> (coeff ` f ) = (coeff ` F ) )
269	268	fveq1d 5881	. . . . 5 |- ( f = F -> ( (coeff ` f ) ` 0 ) = ( (coeff ` F ) ` 0 ) )
270	269	neeq1d 2702	. . . 4 |- ( f = F -> ( ( (coeff ` f ) ` 0 ) =/= 0 <-> ( (coeff ` F ) ` 0 ) =/= 0 ) )
271		fveq1 5878	. . . . 5 |- ( f = F -> ( f ` A ) = ( F ` A ) )
272	271	eqeq1d 2473	. . . 4 |- ( f = F -> ( ( f ` A ) = 0 <-> ( F ` A ) = 0 ) )
273	270, 272	anbi12d 725	. . 3 |- ( f = F -> ( ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) <-> ( ( (coeff ` F ) ` 0 ) =/= 0 /\ ( F ` A ) = 0 ) ) )
274	273	rspcev 3136	. 2 |- ( ( F e. (Poly ` ZZ ) /\ ( ( (coeff ` F ) ` 0 ) =/= 0 /\ ( F ` A ) = 0 ) ) -> E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) )
275	60, 267, 274	syl2anc 673	1 |- ( ph -> E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   \/ wo 375   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   E.wrex 2757   {crab 2760   \ cdif 3387   C_ wss 3390   (/)c0 3722   ifcif 3872   class class class wbr 4395   |-> cmpt 4454   `'ccnv 4838   -->wf 5585   ` cfv 5589  (class class class)co 6308   supcsup 7972   CCcc 9555   RRcr 9556   0cc0 9557   + caddc 9560   x. cmul 9562   < clt 9693   <_ cle 9694   - cmin 9880   / cdiv 10291   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810   ^cexp 12310   sum_csu 13829   0pc0p 22706  Polycply 23217  coeffccoe 23219  degcdgr 23220   AAcaa 23346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-rlim 13630  df-sum 13830  df-0p 22707  df-ply 23221  df-coe 23223  df-dgr 23224  df-aa 23347

This theorem is referenced by: (None)