Theorem elaa2lem 40450

Description: Elementhood in the set of nonzero algebraic numbers. ' Only if ' part of elaa2 40451. (Contributed by Glauco Siliprandi, 5-Apr-2020.) (Revised by AV, 1-Oct-2020.)

Hypotheses

Ref	Expression
elaa2lem.a	|- ( ph -> A e. AA )
elaa2lem.an0	|- ( ph -> A =/= 0 )
elaa2lem.g	|- ( ph -> G e. (Poly ` ZZ ) )
elaa2lem.gn0	|- ( ph -> G =/= 0p )
elaa2lem.ga	|- ( ph -> ( G ` A ) = 0 )
elaa2lem.m	|- M = inf ( { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } , RR , < )
elaa2lem.i	|- I = ( k e. NN0 |-> ( (coeff ` G ) ` ( k + M ) ) )
elaa2lem.f	|- F = ( z e. CC |-> sum_ k e. ( 0 ... ( (deg ` G ) - M ) ) ( ( I ` k ) x. ( z ^ k ) ) )

Assertion

Ref	Expression
elaa2lem	|- ( 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 elaa2lem

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elaa2lem.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 11385	. . . . 5 |- ZZ C_ CC
4	3	a1i 11	. . . 4 |- ( ph -> ZZ C_ CC )
5		elaa2lem.g	. . . . . . . . 9 |- ( ph -> G e. (Poly ` ZZ ) )
6		dgrcl 23989	. . . . . . . . 9 |- ( G e. (Poly ` ZZ ) -> (deg ` G ) e. NN0 )
7	5, 6	syl 17	. . . . . . . 8 |- ( ph -> (deg ` G ) e. NN0 )
8	7	nn0zd 11480	. . . . . . 7 |- ( ph -> (deg ` G ) e. ZZ )
9		elaa2lem.m	. . . . . . . . 9 |- M = inf ( { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } , RR , < )
10		ssrab2 3687	. . . . . . . . . 10 |- { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } C_ NN0
11		nn0uz 11722	. . . . . . . . . . . . 13 |- NN0 = ( ZZ>= ` 0 )
12	10, 11	sseqtri 3637	. . . . . . . . . . . 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		elaa2lem.gn0	. . . . . . . . . . . . . . . . 17 |- ( ph -> G =/= 0p )
15	14	neneqd 2799	. . . . . . . . . . . . . . . 16 |- ( ph -> -. G = 0p )
16		eqid 2622	. . . . . . . . . . . . . . . . . 18 |- (deg ` G ) = (deg ` G )
17		eqid 2622	. . . . . . . . . . . . . . . . . 18 |- (coeff ` G ) = (coeff ` G )
18	16, 17	dgreq0 24021	. . . . . . . . . . . . . . . . 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 314	. . . . . . . . . . . . . . 15 |- ( ph -> -. ( (coeff ` G ) ` (deg ` G ) ) = 0 )
21	20	neqned 2801	. . . . . . . . . . . . . 14 |- ( ph -> ( (coeff ` G ) ` (deg ` G ) ) =/= 0 )
22	7, 21	jca 554	. . . . . . . . . . . . 13 |- ( ph -> ( (deg ` G ) e. NN0 /\ ( (coeff ` G ) ` (deg ` G ) ) =/= 0 ) )
23		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( n = (deg ` G ) -> ( (coeff ` G ) ` n ) = ( (coeff ` G ) ` (deg ` G ) ) )
24	23	neeq1d 2853	. . . . . . . . . . . . . 14 |- ( n = (deg ` G ) -> ( ( (coeff ` G ) ` n ) =/= 0 <-> ( (coeff ` G ) ` (deg ` G ) ) =/= 0 ) )
25	24	elrab 3363	. . . . . . . . . . . . 13 |- ( (deg ` G ) e. { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } <-> ( (deg ` G ) e. NN0 /\ ( (coeff ` G ) ` (deg ` G ) ) =/= 0 ) )
26	22, 25	sylibr 224	. . . . . . . . . . . 12 |- ( ph -> (deg ` G ) e. { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } )
27		ne0i 3921	. . . . . . . . . . . 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		infssuzcl 11772	. . . . . . . . . . 11 |- ( ( { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } C_ ( ZZ>= ` 0 ) /\ { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } =/= (/) ) -> inf ( { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } , RR , < ) e. { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } )
30	13, 28, 29	syl2anc 693	. . . . . . . . . 10 |- ( ph -> inf ( { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } , RR , < ) e. { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } )
31	10, 30	sseldi 3601	. . . . . . . . 9 |- ( ph -> inf ( { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } , RR , < ) e. NN0 )
32	9, 31	syl5eqel 2705	. . . . . . . 8 |- ( ph -> M e. NN0 )
33	32	nn0zd 11480	. . . . . . 7 |- ( ph -> M e. ZZ )
34	8, 33	zsubcld 11487	. . . . . 6 |- ( ph -> ( (deg ` G ) - M ) e. ZZ )
35	9	a1i 11	. . . . . . . 8 |- ( ph -> M = inf ( { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } , RR , < ) )
36		infssuzle 11771	. . . . . . . . 9 |- ( ( { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } C_ ( ZZ>= ` 0 ) /\ (deg ` G ) e. { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } ) -> inf ( { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } , RR , < ) <_ (deg ` G ) )
37	13, 26, 36	syl2anc 693	. . . . . . . 8 |- ( ph -> inf ( { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } , RR , < ) <_ (deg ` G ) )
38	35, 37	eqbrtrd 4675	. . . . . . 7 |- ( ph -> M <_ (deg ` G ) )
39	7	nn0red 11352	. . . . . . . 8 |- ( ph -> (deg ` G ) e. RR )
40	32	nn0red 11352	. . . . . . . 8 |- ( ph -> M e. RR )
41	39, 40	subge0d 10617	. . . . . . 7 |- ( ph -> ( 0 <_ ( (deg ` G ) - M ) <-> M <_ (deg ` G ) ) )
42	38, 41	mpbird 247	. . . . . 6 |- ( ph -> 0 <_ ( (deg ` G ) - M ) )
43	34, 42	jca 554	. . . . 5 |- ( ph -> ( ( (deg ` G ) - M ) e. ZZ /\ 0 <_ ( (deg ` G ) - M ) ) )
44		elnn0z 11390	. . . . 5 |- ( ( (deg ` G ) - M ) e. NN0 <-> ( ( (deg ` G ) - M ) e. ZZ /\ 0 <_ ( (deg ` G ) - M ) ) )
45	43, 44	sylibr 224	. . . 4 |- ( ph -> ( (deg ` G ) - M ) e. NN0 )
46		id 22	. . . . . . . . 9 |- ( G e. (Poly ` ZZ ) -> G e. (Poly ` ZZ ) )
47		0zd 11389	. . . . . . . . 9 |- ( G e. (Poly ` ZZ ) -> 0 e. ZZ )
48	17	coef2 23987	. . . . . . . . 9 |- ( ( G e. (Poly ` ZZ ) /\ 0 e. ZZ ) -> (coeff ` G ) : NN0 --> ZZ )
49	46, 47, 48	syl2anc 693	. . . . . . . 8 |- ( G e. (Poly ` ZZ ) -> (coeff ` G ) : NN0 --> ZZ )
50	5, 49	syl 17	. . . . . . 7 |- ( ph -> (coeff ` G ) : NN0 --> ZZ )
51	50	adantr 481	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> (coeff ` G ) : NN0 --> ZZ )
52		simpr 477	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> k e. NN0 )
53	32	adantr 481	. . . . . . 7 |- ( ( ph /\ k e. NN0 ) -> M e. NN0 )
54	52, 53	nn0addcld 11355	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> ( k + M ) e. NN0 )
55	51, 54	ffvelrnd 6360	. . . . 5 |- ( ( ph /\ k e. NN0 ) -> ( (coeff ` G ) ` ( k + M ) ) e. ZZ )
56		elaa2lem.i	. . . . 5 |- I = ( k e. NN0 |-> ( (coeff ` G ) ` ( k + M ) ) )
57	55, 56	fmptd 6385	. . . 4 |- ( ph -> I : NN0 --> ZZ )
58		elplyr 23957	. . . 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 1326	. . 3 |- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... ( (deg ` G ) - M ) ) ( ( I ` k ) x. ( z ^ k ) ) ) e. (Poly ` ZZ ) )
60	2, 59	eqeltrd 2701	. 2 |- ( ph -> F e. (Poly ` ZZ ) )
61		simpr 477	. . . . . . . . . 10 |- ( ( ( ph /\ k e. NN0 ) /\ k <_ ( (deg ` G ) - M ) ) -> k <_ ( (deg ` G ) - M ) )
62	61	iftrued 4094	. . . . . . . . 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 4095	. . . . . . . . . . 11 |- ( -. k <_ ( (deg ` G ) - M ) -> if ( k <_ ( (deg ` G ) - M ) , ( (coeff ` G ) ` ( k + M ) ) , 0 ) = 0 )
64	63	adantl 482	. . . . . . . . . 10 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> if ( k <_ ( (deg ` G ) - M ) , ( (coeff ` G ) ` ( k + M ) ) , 0 ) = 0 )
65		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> -. k <_ ( (deg ` G ) - M ) )
66	39	ad2antrr 762	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> (deg ` G ) e. RR )
67	40	ad2antrr 762	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> M e. RR )
68	66, 67	resubcld 10458	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> ( (deg ` G ) - M ) e. RR )
69		nn0re 11301	. . . . . . . . . . . . . . . . 17 |- ( k e. NN0 -> k e. RR )
70	69	ad2antlr 763	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> k e. RR )
71	68, 70	ltnled 10184	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> ( ( (deg ` G ) - M ) < k <-> -. k <_ ( (deg ` G ) - M ) ) )
72	65, 71	mpbird 247	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> ( (deg ` G ) - M ) < k )
73	66, 67, 70	ltsubaddd 10623	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> ( ( (deg ` G ) - M ) < k <-> (deg ` G ) < ( k + M ) ) )
74	72, 73	mpbid 222	. . . . . . . . . . . . 13 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> (deg ` G ) < ( k + M ) )
75		olc 399	. . . . . . . . . . . . 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 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> G e. (Poly ` ZZ ) )
78	54	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> ( k + M ) e. NN0 )
79	16, 17	dgrlt 24022	. . . . . . . . . . . . 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 693	. . . . . . . . . . . 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 222	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> ( (deg ` G ) <_ ( k + M ) /\ ( (coeff ` G ) ` ( k + M ) ) = 0 ) )
82	81	simprd 479	. . . . . . . . . 10 |- ( ( ( ph /\ k e. NN0 ) /\ -. k <_ ( (deg ` G ) - M ) ) -> ( (coeff ` G ) ` ( k + M ) ) = 0 )
83	64, 82	eqtr4d 2659	. . . . . . . . 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 832	. . . . . . . 8 |- ( ( ph /\ k e. NN0 ) -> if ( k <_ ( (deg ` G ) - M ) , ( (coeff ` G ) ` ( k + M ) ) , 0 ) = ( (coeff ` G ) ` ( k + M ) ) )
85	84	mpteq2dva 4744	. . . . . . 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 6057	. . . . . . . . . 10 |- ( ph -> (coeff ` G ) : NN0 --> CC )
87	86	adantr 481	. . . . . . . . 9 |- ( ( ph /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> (coeff ` G ) : NN0 --> CC )
88		elfznn0 12433	. . . . . . . . . . 11 |- ( k e. ( 0 ... ( (deg ` G ) - M ) ) -> k e. NN0 )
89	88	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> k e. NN0 )
90	32	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> M e. NN0 )
91	89, 90	nn0addcld 11355	. . . . . . . . 9 |- ( ( ph /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> ( k + M ) e. NN0 )
92	87, 91	ffvelrnd 6360	. . . . . . . 8 |- ( ( ph /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> ( (coeff ` G ) ` ( k + M ) ) e. CC )
93		eqidd 2623	. . . . . . . . . . 11 |- ( ( ph /\ z e. CC ) -> ( 0 ... ( (deg ` G ) - M ) ) = ( 0 ... ( (deg ` G ) - M ) ) )
94		simpl 473	. . . . . . . . . . . . . 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 6293	. . . . . . . . . . . . . 14 |- ( ( ph /\ k e. NN0 ) -> ( I ` k ) = ( (coeff ` G ) ` ( k + M ) ) )
97	94, 89, 96	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> ( I ` k ) = ( (coeff ` G ) ` ( k + M ) ) )
98	97	adantlr 751	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> ( I ` k ) = ( (coeff ` G ) ` ( k + M ) ) )
99	98	oveq1d 6665	. . . . . . . . . . 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 14438	. . . . . . . . . 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 4744	. . . . . . . . 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 2656	. . . . . . . 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 23998	. . . . . . 7 |- ( ph -> (coeff ` F ) = ( k e. NN0 |-> if ( k <_ ( (deg ` G ) - M ) , ( (coeff ` G ) ` ( k + M ) ) , 0 ) ) )
104	85, 103, 95	3eqtr4d 2666	. . . . . 6 |- ( ph -> (coeff ` F ) = I )
105	104	fveq1d 6193	. . . . 5 |- ( ph -> ( (coeff ` F ) ` 0 ) = ( I ` 0 ) )
106		oveq1 6657	. . . . . . . . 9 |- ( k = 0 -> ( k + M ) = ( 0 + M ) )
107	106	adantl 482	. . . . . . . 8 |- ( ( ph /\ k = 0 ) -> ( k + M ) = ( 0 + M ) )
108	3, 33	sseldi 3601	. . . . . . . . . 10 |- ( ph -> M e. CC )
109	108	addid2d 10237	. . . . . . . . 9 |- ( ph -> ( 0 + M ) = M )
110	109	adantr 481	. . . . . . . 8 |- ( ( ph /\ k = 0 ) -> ( 0 + M ) = M )
111	107, 110	eqtrd 2656	. . . . . . 7 |- ( ( ph /\ k = 0 ) -> ( k + M ) = M )
112	111	fveq2d 6195	. . . . . 6 |- ( ( ph /\ k = 0 ) -> ( (coeff ` G ) ` ( k + M ) ) = ( (coeff ` G ) ` M ) )
113		0nn0 11307	. . . . . . 7 |- 0 e. NN0
114	113	a1i 11	. . . . . 6 |- ( ph -> 0 e. NN0 )
115	50, 32	ffvelrnd 6360	. . . . . 6 |- ( ph -> ( (coeff ` G ) ` M ) e. ZZ )
116	95, 112, 114, 115	fvmptd 6288	. . . . 5 |- ( ph -> ( I ` 0 ) = ( (coeff ` G ) ` M ) )
117		eqidd 2623	. . . . 5 |- ( ph -> ( (coeff ` G ) ` M ) = ( (coeff ` G ) ` M ) )
118	105, 116, 117	3eqtrd 2660	. . . 4 |- ( ph -> ( (coeff ` F ) ` 0 ) = ( (coeff ` G ) ` M ) )
119	35, 30	eqeltrd 2701	. . . . . 6 |- ( ph -> M e. { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } )
120		fveq2 6191	. . . . . . . 8 |- ( n = M -> ( (coeff ` G ) ` n ) = ( (coeff ` G ) ` M ) )
121	120	neeq1d 2853	. . . . . . 7 |- ( n = M -> ( ( (coeff ` G ) ` n ) =/= 0 <-> ( (coeff ` G ) ` M ) =/= 0 ) )
122	121	elrab 3363	. . . . . 6 |- ( M e. { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } <-> ( M e. NN0 /\ ( (coeff ` G ) ` M ) =/= 0 ) )
123	119, 122	sylib 208	. . . . 5 |- ( ph -> ( M e. NN0 /\ ( (coeff ` G ) ` M ) =/= 0 ) )
124	123	simprd 479	. . . 4 |- ( ph -> ( (coeff ` G ) ` M ) =/= 0 )
125	118, 124	eqnetrd 2861	. . 3 |- ( ph -> ( (coeff ` F ) ` 0 ) =/= 0 )
126	5, 47	syl 17	. . . . . . 7 |- ( ph -> 0 e. ZZ )
127		aasscn 24073	. . . . . . . . . . 11 |- AA C_ CC
128		elaa2lem.a	. . . . . . . . . . 11 |- ( ph -> A e. AA )
129	127, 128	sseldi 3601	. . . . . . . . . 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 13008	. . . . . . . 8 |- ( ( ph /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> ( A ^ k ) e. CC )
132	92, 131	mulcld 10060	. . . . . . 7 |- ( ( ph /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> ( ( (coeff ` G ) ` ( k + M ) ) x. ( A ^ k ) ) e. CC )
133		oveq1 6657	. . . . . . . . 9 |- ( k = ( j - M ) -> ( k + M ) = ( ( j - M ) + M ) )
134	133	fveq2d 6195	. . . . . . . 8 |- ( k = ( j - M ) -> ( (coeff ` G ) ` ( k + M ) ) = ( (coeff ` G ) ` ( ( j - M ) + M ) ) )
135		oveq2 6658	. . . . . . . 8 |- ( k = ( j - M ) -> ( A ^ k ) = ( A ^ ( j - M ) ) )
136	134, 135	oveq12d 6668	. . . . . . 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 14512	. . . . . 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 3601	. . . . . . . . . 10 |- ( ph -> (deg ` G ) e. CC )
139	138, 108	npcand 10396	. . . . . . . . 9 |- ( ph -> ( ( (deg ` G ) - M ) + M ) = (deg ` G ) )
140	109, 139	oveq12d 6668	. . . . . . . 8 |- ( ph -> ( ( 0 + M ) ... ( ( (deg ` G ) - M ) + M ) ) = ( M ... (deg ` G ) ) )
141	140	sumeq1d 14431	. . . . . . 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 12342	. . . . . . . . . . . . . 14 |- ( j e. ( M ... (deg ` G ) ) -> j e. ZZ )
143	142	adantl 482	. . . . . . . . . . . . 13 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> j e. ZZ )
144	3, 143	sseldi 3601	. . . . . . . . . . . 12 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> j e. CC )
145	108	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> M e. CC )
146	144, 145	npcand 10396	. . . . . . . . . . 11 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> ( ( j - M ) + M ) = j )
147	146	fveq2d 6195	. . . . . . . . . 10 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> ( (coeff ` G ) ` ( ( j - M ) + M ) ) = ( (coeff ` G ) ` j ) )
148	147	oveq1d 6665	. . . . . . . . 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 481	. . . . . . . . . . . 12 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> A e. CC )
150		elaa2lem.an0	. . . . . . . . . . . . 13 |- ( ph -> A =/= 0 )
151	150	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> A =/= 0 )
152	33	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> M e. ZZ )
153	149, 151, 152, 143	expsubd 13019	. . . . . . . . . . 11 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> ( A ^ ( j - M ) ) = ( ( A ^ j ) / ( A ^ M ) ) )
154	153	oveq2d 6666	. . . . . . . . . 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 481	. . . . . . . . . . . . 13 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> (coeff ` G ) : NN0 --> CC )
156		0red 10041	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> 0 e. RR )
157	40	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> M e. RR )
158	143	zred 11482	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> j e. RR )
159	32	nn0ge0d 11354	. . . . . . . . . . . . . . . . 17 |- ( ph -> 0 <_ M )
160	159	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> 0 <_ M )
161		elfzle1 12344	. . . . . . . . . . . . . . . . 17 |- ( j e. ( M ... (deg ` G ) ) -> M <_ j )
162	161	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> M <_ j )
163	156, 157, 158, 160, 162	letrd 10194	. . . . . . . . . . . . . . 15 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> 0 <_ j )
164	143, 163	jca 554	. . . . . . . . . . . . . 14 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> ( j e. ZZ /\ 0 <_ j ) )
165		elnn0z 11390	. . . . . . . . . . . . . 14 |- ( j e. NN0 <-> ( j e. ZZ /\ 0 <_ j ) )
166	164, 165	sylibr 224	. . . . . . . . . . . . 13 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> j e. NN0 )
167	155, 166	ffvelrnd 6360	. . . . . . . . . . . 12 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> ( (coeff ` G ) ` j ) e. CC )
168	149, 166	expcld 13008	. . . . . . . . . . . 12 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> ( A ^ j ) e. CC )
169	129, 32	expcld 13008	. . . . . . . . . . . . 13 |- ( ph -> ( A ^ M ) e. CC )
170	169	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> ( A ^ M ) e. CC )
171	149, 151, 152	expne0d 13014	. . . . . . . . . . . 12 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> ( A ^ M ) =/= 0 )
172	167, 168, 170, 171	divassd 10836	. . . . . . . . . . 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 2628	. . . . . . . . . 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 2657	. . . . . . . . 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 2659	. . . . . . . 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 14433	. . . . . . 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 2656	. . . . . 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 2711	. . . . . . . 8 |- ( ph -> M e. ( ZZ>= ` 0 ) )
179		fzss1 12380	. . . . . . . 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 10060	. . . . . . . 8 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) e. CC )
182	181, 170, 171	divcld 10801	. . . . . . 7 |- ( ( ph /\ j e. ( M ... (deg ` G ) ) ) -> ( ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) / ( A ^ M ) ) e. CC )
183	33	ad2antrr 762	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. j < M ) -> M e. ZZ )
184	8	ad2antrr 762	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. j < M ) -> (deg ` G ) e. ZZ )
185		eldifi 3732	. . . . . . . . . . . . . . . . . . 19 |- ( j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) -> j e. ( 0 ... (deg ` G ) ) )
186		elfznn0 12433	. . . . . . . . . . . . . . . . . . . 20 |- ( j e. ( 0 ... (deg ` G ) ) -> j e. NN0 )
187	186	nn0zd 11480	. . . . . . . . . . . . . . . . . . 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 763	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. j < M ) -> j e. ZZ )
190	183, 184, 189	3jca 1242	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. j < M ) -> ( M e. ZZ /\ (deg ` G ) e. ZZ /\ j e. ZZ ) )
191		simpr 477	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. j < M ) -> -. j < M )
192	40	ad2antrr 762	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. j < M ) -> M e. RR )
193	189	zred 11482	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. j < M ) -> j e. RR )
194	192, 193	lenltd 10183	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. j < M ) -> ( M <_ j <-> -. j < M ) )
195	191, 194	mpbird 247	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. j < M ) -> M <_ j )
196		elfzle2 12345	. . . . . . . . . . . . . . . . . 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 763	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. j < M ) -> j <_ (deg ` G ) )
199	190, 195, 198	jca32 558	. . . . . . . . . . . . . . 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 12333	. . . . . . . . . . . . . . 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 224	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. j < M ) -> j e. ( M ... (deg ` G ) ) )
202		eldifn 3733	. . . . . . . . . . . . . . 15 |- ( j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) -> -. j e. ( M ... (deg ` G ) ) )
203	202	ad2antlr 763	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. j < M ) -> -. j e. ( M ... (deg ` G ) ) )
204	201, 203	condan 835	. . . . . . . . . . . . 13 |- ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) -> j < M )
205	204	adantr 481	. . . . . . . . . . . 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 = inf ( { 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 481	. . . . . . . . . . . . . . . . . 18 |- ( ( j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) /\ -. ( (coeff ` G ) ` j ) = 0 ) -> j e. NN0 )
210		neqne 2802	. . . . . . . . . . . . . . . . . . 19 |- ( -. ( (coeff ` G ) ` j ) = 0 -> ( (coeff ` G ) ` j ) =/= 0 )
211	210	adantl 482	. . . . . . . . . . . . . . . . . 18 |- ( ( j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) /\ -. ( (coeff ` G ) ` j ) = 0 ) -> ( (coeff ` G ) ` j ) =/= 0 )
212	209, 211	jca 554	. . . . . . . . . . . . . . . . 17 |- ( ( j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) /\ -. ( (coeff ` G ) ` j ) = 0 ) -> ( j e. NN0 /\ ( (coeff ` G ) ` j ) =/= 0 ) )
213		fveq2 6191	. . . . . . . . . . . . . . . . . . 19 |- ( n = j -> ( (coeff ` G ) ` n ) = ( (coeff ` G ) ` j ) )
214	213	neeq1d 2853	. . . . . . . . . . . . . . . . . 18 |- ( n = j -> ( ( (coeff ` G ) ` n ) =/= 0 <-> ( (coeff ` G ) ` j ) =/= 0 ) )
215	214	elrab 3363	. . . . . . . . . . . . . . . . 17 |- ( j e. { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } <-> ( j e. NN0 /\ ( (coeff ` G ) ` j ) =/= 0 ) )
216	212, 215	sylibr 224	. . . . . . . . . . . . . . . 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 750	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. ( (coeff ` G ) ` j ) = 0 ) -> j e. { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } )
218		infssuzle 11771	. . . . . . . . . . . . . . 15 |- ( ( { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } C_ ( ZZ>= ` 0 ) /\ j e. { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } ) -> inf ( { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } , RR , < ) <_ j )
219	207, 217, 218	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. ( (coeff ` G ) ` j ) = 0 ) -> inf ( { n e. NN0 | ( (coeff ` G ) ` n ) =/= 0 } , RR , < ) <_ j )
220	206, 219	eqbrtrd 4675	. . . . . . . . . . . . 13 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. ( (coeff ` G ) ` j ) = 0 ) -> M <_ j )
221	40	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. ( (coeff ` G ) ` j ) = 0 ) -> M e. RR )
222	188	zred 11482	. . . . . . . . . . . . . . 15 |- ( j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) -> j e. RR )
223	222	ad2antlr 763	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. ( (coeff ` G ) ` j ) = 0 ) -> j e. RR )
224	221, 223	lenltd 10183	. . . . . . . . . . . . 13 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. ( (coeff ` G ) ` j ) = 0 ) -> ( M <_ j <-> -. j < M ) )
225	220, 224	mpbid 222	. . . . . . . . . . . 12 |- ( ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) /\ -. ( (coeff ` G ) ` j ) = 0 ) -> -. j < M )
226	205, 225	condan 835	. . . . . . . . . . 11 |- ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) -> ( (coeff ` G ) ` j ) = 0 )
227	226	oveq1d 6665	. . . . . . . . . 10 |- ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) -> ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) = ( 0 x. ( A ^ j ) ) )
228	129	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) -> A e. CC )
229	208	adantl 482	. . . . . . . . . . . 12 |- ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) -> j e. NN0 )
230	228, 229	expcld 13008	. . . . . . . . . . 11 |- ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) -> ( A ^ j ) e. CC )
231	230	mul02d 10234	. . . . . . . . . 10 |- ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) -> ( 0 x. ( A ^ j ) ) = 0 )
232	227, 231	eqtrd 2656	. . . . . . . . 9 |- ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) -> ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) = 0 )
233	232	oveq1d 6665	. . . . . . . 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 13014	. . . . . . . . . 10 |- ( ph -> ( A ^ M ) =/= 0 )
235	169, 234	div0d 10800	. . . . . . . . 9 |- ( ph -> ( 0 / ( A ^ M ) ) = 0 )
236	235	adantr 481	. . . . . . . 8 |- ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) -> ( 0 / ( A ^ M ) ) = 0 )
237	233, 236	eqtrd 2656	. . . . . . 7 |- ( ( ph /\ j e. ( ( 0 ... (deg ` G ) ) \ ( M ... (deg ` G ) ) ) ) -> ( ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) / ( A ^ M ) ) = 0 )
238		fzfid 12772	. . . . . . 7 |- ( ph -> ( 0 ... (deg ` G ) ) e. Fin )
239	180, 182, 237, 238	fsumss 14456	. . . . . 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 2660	. . . . 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 487	. . . . . . . . . 10 |- ( ( ph /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> ( (coeff ` G ) ` ( k + M ) ) e. ZZ )
242	56	fvmpt2 6291	. . . . . . . . . 10 |- ( ( k e. NN0 /\ ( (coeff ` G ) ` ( k + M ) ) e. ZZ ) -> ( I ` k ) = ( (coeff ` G ) ` ( k + M ) ) )
243	89, 241, 242	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> ( I ` k ) = ( (coeff ` G ) ` ( k + M ) ) )
244	243	adantlr 751	. . . . . . . 8 |- ( ( ( ph /\ z = A ) /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> ( I ` k ) = ( (coeff ` G ) ` ( k + M ) ) )
245		oveq1 6657	. . . . . . . . 9 |- ( z = A -> ( z ^ k ) = ( A ^ k ) )
246	245	ad2antlr 763	. . . . . . . 8 |- ( ( ( ph /\ z = A ) /\ k e. ( 0 ... ( (deg ` G ) - M ) ) ) -> ( z ^ k ) = ( A ^ k ) )
247	244, 246	oveq12d 6668	. . . . . . 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 14433	. . . . . 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 12772	. . . . . . 7 |- ( ph -> ( 0 ... ( (deg ` G ) - M ) ) e. Fin )
250	249, 132	fsumcl 14464	. . . . . 6 |- ( ph -> sum_ k e. ( 0 ... ( (deg ` G ) - M ) ) ( ( (coeff ` G ) ` ( k + M ) ) x. ( A ^ k ) ) e. CC )
251	2, 248, 129, 250	fvmptd 6288	. . . . 5 |- ( ph -> ( F ` A ) = sum_ k e. ( 0 ... ( (deg ` G ) - M ) ) ( ( (coeff ` G ) ` ( k + M ) ) x. ( A ^ k ) ) )
252	17, 16	coeid2 23995	. . . . . . . 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 693	. . . . . . 7 |- ( ph -> ( G ` A ) = sum_ j e. ( 0 ... (deg ` G ) ) ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) )
254	253	oveq1d 6665	. . . . . 6 |- ( ph -> ( ( G ` A ) / ( A ^ M ) ) = ( sum_ j e. ( 0 ... (deg ` G ) ) ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) / ( A ^ M ) ) )
255	86	adantr 481	. . . . . . . . 9 |- ( ( ph /\ j e. ( 0 ... (deg ` G ) ) ) -> (coeff ` G ) : NN0 --> CC )
256	186	adantl 482	. . . . . . . . 9 |- ( ( ph /\ j e. ( 0 ... (deg ` G ) ) ) -> j e. NN0 )
257	255, 256	ffvelrnd 6360	. . . . . . . 8 |- ( ( ph /\ j e. ( 0 ... (deg ` G ) ) ) -> ( (coeff ` G ) ` j ) e. CC )
258	129	adantr 481	. . . . . . . . 9 |- ( ( ph /\ j e. ( 0 ... (deg ` G ) ) ) -> A e. CC )
259	258, 256	expcld 13008	. . . . . . . 8 |- ( ( ph /\ j e. ( 0 ... (deg ` G ) ) ) -> ( A ^ j ) e. CC )
260	257, 259	mulcld 10060	. . . . . . 7 |- ( ( ph /\ j e. ( 0 ... (deg ` G ) ) ) -> ( ( (coeff ` G ) ` j ) x. ( A ^ j ) ) e. CC )
261	238, 169, 260, 234	fsumdivc 14518	. . . . . 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 2656	. . . . 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 2666	. . . 4 |- ( ph -> ( F ` A ) = ( ( G ` A ) / ( A ^ M ) ) )
264		elaa2lem.ga	. . . . 5 |- ( ph -> ( G ` A ) = 0 )
265	264	oveq1d 6665	. . . 4 |- ( ph -> ( ( G ` A ) / ( A ^ M ) ) = ( 0 / ( A ^ M ) ) )
266	263, 265, 235	3eqtrd 2660	. . 3 |- ( ph -> ( F ` A ) = 0 )
267	125, 266	jca 554	. 2 |- ( ph -> ( ( (coeff ` F ) ` 0 ) =/= 0 /\ ( F ` A ) = 0 ) )
268		fveq2 6191	. . . . . 6 |- ( f = F -> (coeff ` f ) = (coeff ` F ) )
269	268	fveq1d 6193	. . . . 5 |- ( f = F -> ( (coeff ` f ) ` 0 ) = ( (coeff ` F ) ` 0 ) )
270	269	neeq1d 2853	. . . 4 |- ( f = F -> ( ( (coeff ` f ) ` 0 ) =/= 0 <-> ( (coeff ` F ) ` 0 ) =/= 0 ) )
271		fveq1 6190	. . . . 5 |- ( f = F -> ( f ` A ) = ( F ` A ) )
272	271	eqeq1d 2624	. . . 4 |- ( f = F -> ( ( f ` A ) = 0 <-> ( F ` A ) = 0 ) )
273	270, 272	anbi12d 747	. . 3 |- ( f = F -> ( ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) <-> ( ( (coeff ` F ) ` 0 ) =/= 0 /\ ( F ` A ) = 0 ) ) )
274	273	rspcev 3309	. 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 693	1 |- ( ph -> E. f e. (Poly ` ZZ ) ( ( (coeff ` f ) ` 0 ) =/= 0 /\ ( f ` A ) = 0 ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   \ cdif 3571   C_ wss 3574   (/)c0 3915   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650  infcinf 8347   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   ^cexp 12860   sum_csu 14416   0pc0p 23436  Polycply 23940  coeffccoe 23942  degcdgr 23943   AAcaa 24069

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-0p 23437  df-ply 23944  df-coe 23946  df-dgr 23947  df-aa 24070

This theorem is referenced by:  elaa2  40451