Theorem aareccl 24081

Description: The reciprocal of an algebraic number is algebraic. (Contributed by Mario Carneiro, 24-Jul-2014.)

Assertion

Ref	Expression
aareccl	|- ( ( A e. AA /\ A =/= 0 ) -> ( 1 / A ) e. AA )

Proof of Theorem aareccl

Dummy variables f g k n z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elaa 24071	. . . 4 |- ( A e. AA <-> ( A e. CC /\ E. f e. ( (Poly ` ZZ ) \ { 0p } ) ( f ` A ) = 0 ) )
2	1	simprbi 480	. . 3 |- ( A e. AA -> E. f e. ( (Poly ` ZZ ) \ { 0p } ) ( f ` A ) = 0 )
3	2	adantr 481	. 2 |- ( ( A e. AA /\ A =/= 0 ) -> E. f e. ( (Poly ` ZZ ) \ { 0p } ) ( f ` A ) = 0 )
4		aacn 24072	. . . . 5 |- ( A e. AA -> A e. CC )
5		reccl 10692	. . . . 5 |- ( ( A e. CC /\ A =/= 0 ) -> ( 1 / A ) e. CC )
6	4, 5	sylan 488	. . . 4 |- ( ( A e. AA /\ A =/= 0 ) -> ( 1 / A ) e. CC )
7	6	adantr 481	. . 3 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> ( 1 / A ) e. CC )
8		zsscn 11385	. . . . . . 7 |- ZZ C_ CC
9	8	a1i 11	. . . . . 6 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> ZZ C_ CC )
10		simprl 794	. . . . . . . . 9 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> f e. ( (Poly ` ZZ ) \ { 0p } ) )
11		eldifsn 4317	. . . . . . . . 9 |- ( f e. ( (Poly ` ZZ ) \ { 0p } ) <-> ( f e. (Poly ` ZZ ) /\ f =/= 0p ) )
12	10, 11	sylib 208	. . . . . . . 8 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> ( f e. (Poly ` ZZ ) /\ f =/= 0p ) )
13	12	simpld 475	. . . . . . 7 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> f e. (Poly ` ZZ ) )
14		dgrcl 23989	. . . . . . 7 |- ( f e. (Poly ` ZZ ) -> (deg ` f ) e. NN0 )
15	13, 14	syl 17	. . . . . 6 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> (deg ` f ) e. NN0 )
16	13	adantr 481	. . . . . . . 8 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> f e. (Poly ` ZZ ) )
17		0z 11388	. . . . . . . 8 |- 0 e. ZZ
18		eqid 2622	. . . . . . . . 9 |- (coeff ` f ) = (coeff ` f )
19	18	coef2 23987	. . . . . . . 8 |- ( ( f e. (Poly ` ZZ ) /\ 0 e. ZZ ) -> (coeff ` f ) : NN0 --> ZZ )
20	16, 17, 19	sylancl 694	. . . . . . 7 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> (coeff ` f ) : NN0 --> ZZ )
21		fznn0sub 12373	. . . . . . . 8 |- ( k e. ( 0 ... (deg ` f ) ) -> ( (deg ` f ) - k ) e. NN0 )
22	21	adantl 482	. . . . . . 7 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> ( (deg ` f ) - k ) e. NN0 )
23	20, 22	ffvelrnd 6360	. . . . . 6 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> ( (coeff ` f ) ` ( (deg ` f ) - k ) ) e. ZZ )
24	9, 15, 23	elplyd 23958	. . . . 5 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) e. (Poly ` ZZ ) )
25		0cn 10032	. . . . . 6 |- 0 e. CC
26		eqid 2622	. . . . . . . . . 10 |- (coeff ` ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) ) = (coeff ` ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) )
27	26	coefv0 24004	. . . . . . . . 9 |- ( ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) e. (Poly ` ZZ ) -> ( ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) ` 0 ) = ( (coeff ` ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) ) ` 0 ) )
28	24, 27	syl 17	. . . . . . . 8 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> ( ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) ` 0 ) = ( (coeff ` ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) ) ` 0 ) )
29	23	zcnd 11483	. . . . . . . . . 10 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> ( (coeff ` f ) ` ( (deg ` f ) - k ) ) e. CC )
30		eqidd 2623	. . . . . . . . . 10 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) )
31	24, 15, 29, 30	coeeq2 23998	. . . . . . . . 9 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> (coeff ` ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) ) = ( k e. NN0 |-> if ( k <_ (deg ` f ) , ( (coeff ` f ) ` ( (deg ` f ) - k ) ) , 0 ) ) )
32	31	fveq1d 6193	. . . . . . . 8 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> ( (coeff ` ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) ) ` 0 ) = ( ( k e. NN0 |-> if ( k <_ (deg ` f ) , ( (coeff ` f ) ` ( (deg ` f ) - k ) ) , 0 ) ) ` 0 ) )
33		0nn0 11307	. . . . . . . . . 10 |- 0 e. NN0
34		breq1 4656	. . . . . . . . . . . 12 |- ( k = 0 -> ( k <_ (deg ` f ) <-> 0 <_ (deg ` f ) ) )
35		oveq2 6658	. . . . . . . . . . . . 13 |- ( k = 0 -> ( (deg ` f ) - k ) = ( (deg ` f ) - 0 ) )
36	35	fveq2d 6195	. . . . . . . . . . . 12 |- ( k = 0 -> ( (coeff ` f ) ` ( (deg ` f ) - k ) ) = ( (coeff ` f ) ` ( (deg ` f ) - 0 ) ) )
37	34, 36	ifbieq1d 4109	. . . . . . . . . . 11 |- ( k = 0 -> if ( k <_ (deg ` f ) , ( (coeff ` f ) ` ( (deg ` f ) - k ) ) , 0 ) = if ( 0 <_ (deg ` f ) , ( (coeff ` f ) ` ( (deg ` f ) - 0 ) ) , 0 ) )
38		eqid 2622	. . . . . . . . . . 11 |- ( k e. NN0 |-> if ( k <_ (deg ` f ) , ( (coeff ` f ) ` ( (deg ` f ) - k ) ) , 0 ) ) = ( k e. NN0 |-> if ( k <_ (deg ` f ) , ( (coeff ` f ) ` ( (deg ` f ) - k ) ) , 0 ) )
39		fvex 6201	. . . . . . . . . . . 12 |- ( (coeff ` f ) ` ( (deg ` f ) - 0 ) ) e. _V
40		c0ex 10034	. . . . . . . . . . . 12 |- 0 e. _V
41	39, 40	ifex 4156	. . . . . . . . . . 11 |- if ( 0 <_ (deg ` f ) , ( (coeff ` f ) ` ( (deg ` f ) - 0 ) ) , 0 ) e. _V
42	37, 38, 41	fvmpt 6282	. . . . . . . . . 10 |- ( 0 e. NN0 -> ( ( k e. NN0 |-> if ( k <_ (deg ` f ) , ( (coeff ` f ) ` ( (deg ` f ) - k ) ) , 0 ) ) ` 0 ) = if ( 0 <_ (deg ` f ) , ( (coeff ` f ) ` ( (deg ` f ) - 0 ) ) , 0 ) )
43	33, 42	ax-mp 5	. . . . . . . . 9 |- ( ( k e. NN0 |-> if ( k <_ (deg ` f ) , ( (coeff ` f ) ` ( (deg ` f ) - k ) ) , 0 ) ) ` 0 ) = if ( 0 <_ (deg ` f ) , ( (coeff ` f ) ` ( (deg ` f ) - 0 ) ) , 0 )
44	15	nn0ge0d 11354	. . . . . . . . . . 11 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> 0 <_ (deg ` f ) )
45	44	iftrued 4094	. . . . . . . . . 10 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> if ( 0 <_ (deg ` f ) , ( (coeff ` f ) ` ( (deg ` f ) - 0 ) ) , 0 ) = ( (coeff ` f ) ` ( (deg ` f ) - 0 ) ) )
46	15	nn0cnd 11353	. . . . . . . . . . . 12 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> (deg ` f ) e. CC )
47	46	subid1d 10381	. . . . . . . . . . 11 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> ( (deg ` f ) - 0 ) = (deg ` f ) )
48	47	fveq2d 6195	. . . . . . . . . 10 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> ( (coeff ` f ) ` ( (deg ` f ) - 0 ) ) = ( (coeff ` f ) ` (deg ` f ) ) )
49	45, 48	eqtrd 2656	. . . . . . . . 9 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> if ( 0 <_ (deg ` f ) , ( (coeff ` f ) ` ( (deg ` f ) - 0 ) ) , 0 ) = ( (coeff ` f ) ` (deg ` f ) ) )
50	43, 49	syl5eq 2668	. . . . . . . 8 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> ( ( k e. NN0 |-> if ( k <_ (deg ` f ) , ( (coeff ` f ) ` ( (deg ` f ) - k ) ) , 0 ) ) ` 0 ) = ( (coeff ` f ) ` (deg ` f ) ) )
51	28, 32, 50	3eqtrd 2660	. . . . . . 7 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> ( ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) ` 0 ) = ( (coeff ` f ) ` (deg ` f ) ) )
52	12	simprd 479	. . . . . . . 8 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> f =/= 0p )
53		eqid 2622	. . . . . . . . . . 11 |- (deg ` f ) = (deg ` f )
54	53, 18	dgreq0 24021	. . . . . . . . . 10 |- ( f e. (Poly ` ZZ ) -> ( f = 0p <-> ( (coeff ` f ) ` (deg ` f ) ) = 0 ) )
55	13, 54	syl 17	. . . . . . . . 9 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> ( f = 0p <-> ( (coeff ` f ) ` (deg ` f ) ) = 0 ) )
56	55	necon3bid 2838	. . . . . . . 8 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> ( f =/= 0p <-> ( (coeff ` f ) ` (deg ` f ) ) =/= 0 ) )
57	52, 56	mpbid 222	. . . . . . 7 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> ( (coeff ` f ) ` (deg ` f ) ) =/= 0 )
58	51, 57	eqnetrd 2861	. . . . . 6 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> ( ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) ` 0 ) =/= 0 )
59		ne0p 23963	. . . . . 6 |- ( ( 0 e. CC /\ ( ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) ` 0 ) =/= 0 ) -> ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) =/= 0p )
60	25, 58, 59	sylancr 695	. . . . 5 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) =/= 0p )
61		eldifsn 4317	. . . . 5 |- ( ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) e. ( (Poly ` ZZ ) \ { 0p } ) <-> ( ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) e. (Poly ` ZZ ) /\ ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) =/= 0p ) )
62	24, 60, 61	sylanbrc 698	. . . 4 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) e. ( (Poly ` ZZ ) \ { 0p } ) )
63		oveq1 6657	. . . . . . . . 9 |- ( z = ( 1 / A ) -> ( z ^ k ) = ( ( 1 / A ) ^ k ) )
64	63	oveq2d 6666	. . . . . . . 8 |- ( z = ( 1 / A ) -> ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) = ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( ( 1 / A ) ^ k ) ) )
65	64	sumeq2sdv 14435	. . . . . . 7 |- ( z = ( 1 / A ) -> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) = sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( ( 1 / A ) ^ k ) ) )
66		eqid 2622	. . . . . . 7 |- ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) )
67		sumex 14418	. . . . . . 7 |- sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( ( 1 / A ) ^ k ) ) e. _V
68	65, 66, 67	fvmpt 6282	. . . . . 6 |- ( ( 1 / A ) e. CC -> ( ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) ` ( 1 / A ) ) = sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( ( 1 / A ) ^ k ) ) )
69	7, 68	syl 17	. . . . 5 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> ( ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) ` ( 1 / A ) ) = sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( ( 1 / A ) ^ k ) ) )
70	18	coef3 23988	. . . . . . . . . . 11 |- ( f e. (Poly ` ZZ ) -> (coeff ` f ) : NN0 --> CC )
71	13, 70	syl 17	. . . . . . . . . 10 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> (coeff ` f ) : NN0 --> CC )
72		elfznn0 12433	. . . . . . . . . 10 |- ( n e. ( 0 ... (deg ` f ) ) -> n e. NN0 )
73		ffvelrn 6357	. . . . . . . . . 10 |- ( ( (coeff ` f ) : NN0 --> CC /\ n e. NN0 ) -> ( (coeff ` f ) ` n ) e. CC )
74	71, 72, 73	syl2an 494	. . . . . . . . 9 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ n e. ( 0 ... (deg ` f ) ) ) -> ( (coeff ` f ) ` n ) e. CC )
75	4	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> A e. CC )
76		expcl 12878	. . . . . . . . . 10 |- ( ( A e. CC /\ n e. NN0 ) -> ( A ^ n ) e. CC )
77	75, 72, 76	syl2an 494	. . . . . . . . 9 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ n e. ( 0 ... (deg ` f ) ) ) -> ( A ^ n ) e. CC )
78	74, 77	mulcld 10060	. . . . . . . 8 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ n e. ( 0 ... (deg ` f ) ) ) -> ( ( (coeff ` f ) ` n ) x. ( A ^ n ) ) e. CC )
79	75, 15	expcld 13008	. . . . . . . . 9 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> ( A ^ (deg ` f ) ) e. CC )
80	79	adantr 481	. . . . . . . 8 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ n e. ( 0 ... (deg ` f ) ) ) -> ( A ^ (deg ` f ) ) e. CC )
81		simplr 792	. . . . . . . . . 10 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> A =/= 0 )
82	15	nn0zd 11480	. . . . . . . . . 10 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> (deg ` f ) e. ZZ )
83	75, 81, 82	expne0d 13014	. . . . . . . . 9 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> ( A ^ (deg ` f ) ) =/= 0 )
84	83	adantr 481	. . . . . . . 8 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ n e. ( 0 ... (deg ` f ) ) ) -> ( A ^ (deg ` f ) ) =/= 0 )
85	78, 80, 84	divcld 10801	. . . . . . 7 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ n e. ( 0 ... (deg ` f ) ) ) -> ( ( ( (coeff ` f ) ` n ) x. ( A ^ n ) ) / ( A ^ (deg ` f ) ) ) e. CC )
86		fveq2 6191	. . . . . . . . 9 |- ( n = ( ( 0 + (deg ` f ) ) - k ) -> ( (coeff ` f ) ` n ) = ( (coeff ` f ) ` ( ( 0 + (deg ` f ) ) - k ) ) )
87		oveq2 6658	. . . . . . . . 9 |- ( n = ( ( 0 + (deg ` f ) ) - k ) -> ( A ^ n ) = ( A ^ ( ( 0 + (deg ` f ) ) - k ) ) )
88	86, 87	oveq12d 6668	. . . . . . . 8 |- ( n = ( ( 0 + (deg ` f ) ) - k ) -> ( ( (coeff ` f ) ` n ) x. ( A ^ n ) ) = ( ( (coeff ` f ) ` ( ( 0 + (deg ` f ) ) - k ) ) x. ( A ^ ( ( 0 + (deg ` f ) ) - k ) ) ) )
89	88	oveq1d 6665	. . . . . . 7 |- ( n = ( ( 0 + (deg ` f ) ) - k ) -> ( ( ( (coeff ` f ) ` n ) x. ( A ^ n ) ) / ( A ^ (deg ` f ) ) ) = ( ( ( (coeff ` f ) ` ( ( 0 + (deg ` f ) ) - k ) ) x. ( A ^ ( ( 0 + (deg ` f ) ) - k ) ) ) / ( A ^ (deg ` f ) ) ) )
90	85, 89	fsumrev2 14514	. . . . . 6 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> sum_ n e. ( 0 ... (deg ` f ) ) ( ( ( (coeff ` f ) ` n ) x. ( A ^ n ) ) / ( A ^ (deg ` f ) ) ) = sum_ k e. ( 0 ... (deg ` f ) ) ( ( ( (coeff ` f ) ` ( ( 0 + (deg ` f ) ) - k ) ) x. ( A ^ ( ( 0 + (deg ` f ) ) - k ) ) ) / ( A ^ (deg ` f ) ) ) )
91	46	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> (deg ` f ) e. CC )
92	91	addid2d 10237	. . . . . . . . . . . 12 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> ( 0 + (deg ` f ) ) = (deg ` f ) )
93	92	oveq1d 6665	. . . . . . . . . . 11 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> ( ( 0 + (deg ` f ) ) - k ) = ( (deg ` f ) - k ) )
94	93	fveq2d 6195	. . . . . . . . . 10 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> ( (coeff ` f ) ` ( ( 0 + (deg ` f ) ) - k ) ) = ( (coeff ` f ) ` ( (deg ` f ) - k ) ) )
95	93	oveq2d 6666	. . . . . . . . . . 11 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> ( A ^ ( ( 0 + (deg ` f ) ) - k ) ) = ( A ^ ( (deg ` f ) - k ) ) )
96	75	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> A e. CC )
97	81	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> A =/= 0 )
98		elfznn0 12433	. . . . . . . . . . . . . 14 |- ( k e. ( 0 ... (deg ` f ) ) -> k e. NN0 )
99	98	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> k e. NN0 )
100	99	nn0zd 11480	. . . . . . . . . . . 12 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> k e. ZZ )
101	82	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> (deg ` f ) e. ZZ )
102	96, 97, 100, 101	expsubd 13019	. . . . . . . . . . 11 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> ( A ^ ( (deg ` f ) - k ) ) = ( ( A ^ (deg ` f ) ) / ( A ^ k ) ) )
103	95, 102	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> ( A ^ ( ( 0 + (deg ` f ) ) - k ) ) = ( ( A ^ (deg ` f ) ) / ( A ^ k ) ) )
104	94, 103	oveq12d 6668	. . . . . . . . 9 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> ( ( (coeff ` f ) ` ( ( 0 + (deg ` f ) ) - k ) ) x. ( A ^ ( ( 0 + (deg ` f ) ) - k ) ) ) = ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( ( A ^ (deg ` f ) ) / ( A ^ k ) ) ) )
105	104	oveq1d 6665	. . . . . . . 8 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> ( ( ( (coeff ` f ) ` ( ( 0 + (deg ` f ) ) - k ) ) x. ( A ^ ( ( 0 + (deg ` f ) ) - k ) ) ) / ( A ^ (deg ` f ) ) ) = ( ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( ( A ^ (deg ` f ) ) / ( A ^ k ) ) ) / ( A ^ (deg ` f ) ) ) )
106	79	adantr 481	. . . . . . . . . 10 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> ( A ^ (deg ` f ) ) e. CC )
107		expcl 12878	. . . . . . . . . . 11 |- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ k ) e. CC )
108	75, 98, 107	syl2an 494	. . . . . . . . . 10 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> ( A ^ k ) e. CC )
109	96, 97, 100	expne0d 13014	. . . . . . . . . 10 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> ( A ^ k ) =/= 0 )
110	106, 108, 109	divcld 10801	. . . . . . . . 9 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> ( ( A ^ (deg ` f ) ) / ( A ^ k ) ) e. CC )
111	83	adantr 481	. . . . . . . . 9 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> ( A ^ (deg ` f ) ) =/= 0 )
112	29, 110, 106, 111	divassd 10836	. . . . . . . 8 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> ( ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( ( A ^ (deg ` f ) ) / ( A ^ k ) ) ) / ( A ^ (deg ` f ) ) ) = ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( ( ( A ^ (deg ` f ) ) / ( A ^ k ) ) / ( A ^ (deg ` f ) ) ) ) )
113	106, 111	dividd 10799	. . . . . . . . . . 11 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> ( ( A ^ (deg ` f ) ) / ( A ^ (deg ` f ) ) ) = 1 )
114	113	oveq1d 6665	. . . . . . . . . 10 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> ( ( ( A ^ (deg ` f ) ) / ( A ^ (deg ` f ) ) ) / ( A ^ k ) ) = ( 1 / ( A ^ k ) ) )
115	106, 108, 106, 109, 111	divdiv32d 10826	. . . . . . . . . 10 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> ( ( ( A ^ (deg ` f ) ) / ( A ^ k ) ) / ( A ^ (deg ` f ) ) ) = ( ( ( A ^ (deg ` f ) ) / ( A ^ (deg ` f ) ) ) / ( A ^ k ) ) )
116	96, 97, 100	exprecd 13016	. . . . . . . . . 10 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> ( ( 1 / A ) ^ k ) = ( 1 / ( A ^ k ) ) )
117	114, 115, 116	3eqtr4d 2666	. . . . . . . . 9 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> ( ( ( A ^ (deg ` f ) ) / ( A ^ k ) ) / ( A ^ (deg ` f ) ) ) = ( ( 1 / A ) ^ k ) )
118	117	oveq2d 6666	. . . . . . . 8 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( ( ( A ^ (deg ` f ) ) / ( A ^ k ) ) / ( A ^ (deg ` f ) ) ) ) = ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( ( 1 / A ) ^ k ) ) )
119	105, 112, 118	3eqtrd 2660	. . . . . . 7 |- ( ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) /\ k e. ( 0 ... (deg ` f ) ) ) -> ( ( ( (coeff ` f ) ` ( ( 0 + (deg ` f ) ) - k ) ) x. ( A ^ ( ( 0 + (deg ` f ) ) - k ) ) ) / ( A ^ (deg ` f ) ) ) = ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( ( 1 / A ) ^ k ) ) )
120	119	sumeq2dv 14433	. . . . . 6 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> sum_ k e. ( 0 ... (deg ` f ) ) ( ( ( (coeff ` f ) ` ( ( 0 + (deg ` f ) ) - k ) ) x. ( A ^ ( ( 0 + (deg ` f ) ) - k ) ) ) / ( A ^ (deg ` f ) ) ) = sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( ( 1 / A ) ^ k ) ) )
121	90, 120	eqtrd 2656	. . . . 5 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> sum_ n e. ( 0 ... (deg ` f ) ) ( ( ( (coeff ` f ) ` n ) x. ( A ^ n ) ) / ( A ^ (deg ` f ) ) ) = sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( ( 1 / A ) ^ k ) ) )
122	18, 53	coeid2 23995	. . . . . . . . 9 |- ( ( f e. (Poly ` ZZ ) /\ A e. CC ) -> ( f ` A ) = sum_ n e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` n ) x. ( A ^ n ) ) )
123	13, 75, 122	syl2anc 693	. . . . . . . 8 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> ( f ` A ) = sum_ n e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` n ) x. ( A ^ n ) ) )
124		simprr 796	. . . . . . . 8 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> ( f ` A ) = 0 )
125	123, 124	eqtr3d 2658	. . . . . . 7 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> sum_ n e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` n ) x. ( A ^ n ) ) = 0 )
126	125	oveq1d 6665	. . . . . 6 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> ( sum_ n e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` n ) x. ( A ^ n ) ) / ( A ^ (deg ` f ) ) ) = ( 0 / ( A ^ (deg ` f ) ) ) )
127		fzfid 12772	. . . . . . 7 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> ( 0 ... (deg ` f ) ) e. Fin )
128	127, 79, 78, 83	fsumdivc 14518	. . . . . 6 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> ( sum_ n e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` n ) x. ( A ^ n ) ) / ( A ^ (deg ` f ) ) ) = sum_ n e. ( 0 ... (deg ` f ) ) ( ( ( (coeff ` f ) ` n ) x. ( A ^ n ) ) / ( A ^ (deg ` f ) ) ) )
129	79, 83	div0d 10800	. . . . . 6 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> ( 0 / ( A ^ (deg ` f ) ) ) = 0 )
130	126, 128, 129	3eqtr3d 2664	. . . . 5 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> sum_ n e. ( 0 ... (deg ` f ) ) ( ( ( (coeff ` f ) ` n ) x. ( A ^ n ) ) / ( A ^ (deg ` f ) ) ) = 0 )
131	69, 121, 130	3eqtr2d 2662	. . . 4 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> ( ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) ` ( 1 / A ) ) = 0 )
132		fveq1 6190	. . . . . 6 |- ( g = ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) -> ( g ` ( 1 / A ) ) = ( ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) ` ( 1 / A ) ) )
133	132	eqeq1d 2624	. . . . 5 |- ( g = ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) -> ( ( g ` ( 1 / A ) ) = 0 <-> ( ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) ` ( 1 / A ) ) = 0 ) )
134	133	rspcev 3309	. . . 4 |- ( ( ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( ( z e. CC |-> sum_ k e. ( 0 ... (deg ` f ) ) ( ( (coeff ` f ) ` ( (deg ` f ) - k ) ) x. ( z ^ k ) ) ) ` ( 1 / A ) ) = 0 ) -> E. g e. ( (Poly ` ZZ ) \ { 0p } ) ( g ` ( 1 / A ) ) = 0 )
135	62, 131, 134	syl2anc 693	. . 3 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> E. g e. ( (Poly ` ZZ ) \ { 0p } ) ( g ` ( 1 / A ) ) = 0 )
136		elaa 24071	. . 3 |- ( ( 1 / A ) e. AA <-> ( ( 1 / A ) e. CC /\ E. g e. ( (Poly ` ZZ ) \ { 0p } ) ( g ` ( 1 / A ) ) = 0 ) )
137	7, 135, 136	sylanbrc 698	. 2 |- ( ( ( A e. AA /\ A =/= 0 ) /\ ( f e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( f ` A ) = 0 ) ) -> ( 1 / A ) e. AA )
138	3, 137	rexlimddv 3035	1 |- ( ( A e. AA /\ A =/= 0 ) -> ( 1 / A ) e. AA )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   C_ wss 3574   ifcif 4086   {csn 4177   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   <_ cle 10075   - cmin 10266   / cdiv 10684   NN0cn0 11292   ZZcz 11377   ...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: (None)