Theorem elqaalem3 24076

Description: Lemma for elqaa 24077. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by AV, 3-Oct-2020.)

Hypotheses

Ref	Expression
elqaa.1	|- ( ph -> A e. CC )
elqaa.2	|- ( ph -> F e. ( (Poly ` QQ ) \ { 0p } ) )
elqaa.3	|- ( ph -> ( F ` A ) = 0 )
elqaa.4	|- B = (coeff ` F )
elqaa.5	|- N = ( k e. NN0 |-> inf ( { n e. NN | ( ( B ` k ) x. n ) e. ZZ } , RR , < ) )
elqaa.6	|- R = ( seq 0 ( x. , N ) ` (deg ` F ) )

Assertion

Ref	Expression
elqaalem3	|- ( ph -> A e. AA )

Distinct variable groups:   k, n, A   B, k, n   ph, k   k, N, n   R, k

Allowed substitution hints:   ph( n)   R( n)   F( k, n)

Proof of Theorem elqaalem3

Dummy variables f m x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elqaa.1	. 2 |- ( ph -> A e. CC )
2		cnex 10017	. . . . . . . 8 |- CC e. _V
3	2	a1i 11	. . . . . . 7 |- ( ph -> CC e. _V )
4		elqaa.6	. . . . . . . . 9 |- R = ( seq 0 ( x. , N ) ` (deg ` F ) )
5		fvex 6201	. . . . . . . . 9 |- ( seq 0 ( x. , N ) ` (deg ` F ) ) e. _V
6	4, 5	eqeltri 2697	. . . . . . . 8 |- R e. _V
7	6	a1i 11	. . . . . . 7 |- ( ( ph /\ z e. CC ) -> R e. _V )
8		fvexd 6203	. . . . . . 7 |- ( ( ph /\ z e. CC ) -> ( F ` z ) e. _V )
9		fconstmpt 5163	. . . . . . . 8 |- ( CC X. { R } ) = ( z e. CC |-> R )
10	9	a1i 11	. . . . . . 7 |- ( ph -> ( CC X. { R } ) = ( z e. CC |-> R ) )
11		elqaa.2	. . . . . . . . . 10 |- ( ph -> F e. ( (Poly ` QQ ) \ { 0p } ) )
12	11	eldifad 3586	. . . . . . . . 9 |- ( ph -> F e. (Poly ` QQ ) )
13		plyf 23954	. . . . . . . . 9 |- ( F e. (Poly ` QQ ) -> F : CC --> CC )
14	12, 13	syl 17	. . . . . . . 8 |- ( ph -> F : CC --> CC )
15	14	feqmptd 6249	. . . . . . 7 |- ( ph -> F = ( z e. CC |-> ( F ` z ) ) )
16	3, 7, 8, 10, 15	offval2 6914	. . . . . 6 |- ( ph -> ( ( CC X. { R } ) oF x. F ) = ( z e. CC |-> ( R x. ( F ` z ) ) ) )
17		fzfid 12772	. . . . . . . . 9 |- ( ( ph /\ z e. CC ) -> ( 0 ... (deg ` F ) ) e. Fin )
18		nn0uz 11722	. . . . . . . . . . . . . 14 |- NN0 = ( ZZ>= ` 0 )
19		0zd 11389	. . . . . . . . . . . . . 14 |- ( ph -> 0 e. ZZ )
20		ssrab2 3687	. . . . . . . . . . . . . . 15 |- { n e. NN | ( ( B ` m ) x. n ) e. ZZ } C_ NN
21		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . 22 |- ( k = m -> ( B ` k ) = ( B ` m ) )
22	21	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . 21 |- ( k = m -> ( ( B ` k ) x. n ) = ( ( B ` m ) x. n ) )
23	22	eleq1d 2686	. . . . . . . . . . . . . . . . . . . 20 |- ( k = m -> ( ( ( B ` k ) x. n ) e. ZZ <-> ( ( B ` m ) x. n ) e. ZZ ) )
24	23	rabbidv 3189	. . . . . . . . . . . . . . . . . . 19 |- ( k = m -> { n e. NN | ( ( B ` k ) x. n ) e. ZZ } = { n e. NN | ( ( B ` m ) x. n ) e. ZZ } )
25	24	infeq1d 8383	. . . . . . . . . . . . . . . . . 18 |- ( k = m -> inf ( { n e. NN | ( ( B ` k ) x. n ) e. ZZ } , RR , < ) = inf ( { n e. NN | ( ( B ` m ) x. n ) e. ZZ } , RR , < ) )
26		elqaa.5	. . . . . . . . . . . . . . . . . 18 |- N = ( k e. NN0 |-> inf ( { n e. NN | ( ( B ` k ) x. n ) e. ZZ } , RR , < ) )
27		ltso 10118	. . . . . . . . . . . . . . . . . . 19 |- < Or RR
28	27	infex 8399	. . . . . . . . . . . . . . . . . 18 |- inf ( { n e. NN | ( ( B ` m ) x. n ) e. ZZ } , RR , < ) e. _V
29	25, 26, 28	fvmpt 6282	. . . . . . . . . . . . . . . . 17 |- ( m e. NN0 -> ( N ` m ) = inf ( { n e. NN | ( ( B ` m ) x. n ) e. ZZ } , RR , < ) )
30	29	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ m e. NN0 ) -> ( N ` m ) = inf ( { n e. NN | ( ( B ` m ) x. n ) e. ZZ } , RR , < ) )
31		nnuz 11723	. . . . . . . . . . . . . . . . . 18 |- NN = ( ZZ>= ` 1 )
32	20, 31	sseqtri 3637	. . . . . . . . . . . . . . . . 17 |- { n e. NN | ( ( B ` m ) x. n ) e. ZZ } C_ ( ZZ>= ` 1 )
33		0z 11388	. . . . . . . . . . . . . . . . . . . . . 22 |- 0 e. ZZ
34		zq 11794	. . . . . . . . . . . . . . . . . . . . . 22 |- ( 0 e. ZZ -> 0 e. QQ )
35	33, 34	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 |- 0 e. QQ
36		elqaa.4	. . . . . . . . . . . . . . . . . . . . . 22 |- B = (coeff ` F )
37	36	coef2 23987	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( F e. (Poly ` QQ ) /\ 0 e. QQ ) -> B : NN0 --> QQ )
38	12, 35, 37	sylancl 694	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> B : NN0 --> QQ )
39	38	ffvelrnda 6359	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ m e. NN0 ) -> ( B ` m ) e. QQ )
40		qmulz 11791	. . . . . . . . . . . . . . . . . . 19 |- ( ( B ` m ) e. QQ -> E. n e. NN ( ( B ` m ) x. n ) e. ZZ )
41	39, 40	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ m e. NN0 ) -> E. n e. NN ( ( B ` m ) x. n ) e. ZZ )
42		rabn0 3958	. . . . . . . . . . . . . . . . . 18 |- ( { n e. NN | ( ( B ` m ) x. n ) e. ZZ } =/= (/) <-> E. n e. NN ( ( B ` m ) x. n ) e. ZZ )
43	41, 42	sylibr 224	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ m e. NN0 ) -> { n e. NN | ( ( B ` m ) x. n ) e. ZZ } =/= (/) )
44		infssuzcl 11772	. . . . . . . . . . . . . . . . 17 |- ( ( { n e. NN | ( ( B ` m ) x. n ) e. ZZ } C_ ( ZZ>= ` 1 ) /\ { n e. NN | ( ( B ` m ) x. n ) e. ZZ } =/= (/) ) -> inf ( { n e. NN | ( ( B ` m ) x. n ) e. ZZ } , RR , < ) e. { n e. NN | ( ( B ` m ) x. n ) e. ZZ } )
45	32, 43, 44	sylancr 695	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ m e. NN0 ) -> inf ( { n e. NN | ( ( B ` m ) x. n ) e. ZZ } , RR , < ) e. { n e. NN | ( ( B ` m ) x. n ) e. ZZ } )
46	30, 45	eqeltrd 2701	. . . . . . . . . . . . . . 15 |- ( ( ph /\ m e. NN0 ) -> ( N ` m ) e. { n e. NN | ( ( B ` m ) x. n ) e. ZZ } )
47	20, 46	sseldi 3601	. . . . . . . . . . . . . 14 |- ( ( ph /\ m e. NN0 ) -> ( N ` m ) e. NN )
48		nnmulcl 11043	. . . . . . . . . . . . . . 15 |- ( ( m e. NN /\ k e. NN ) -> ( m x. k ) e. NN )
49	48	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( m e. NN /\ k e. NN ) ) -> ( m x. k ) e. NN )
50	18, 19, 47, 49	seqf 12822	. . . . . . . . . . . . 13 |- ( ph -> seq 0 ( x. , N ) : NN0 --> NN )
51		dgrcl 23989	. . . . . . . . . . . . . 14 |- ( F e. (Poly ` QQ ) -> (deg ` F ) e. NN0 )
52	12, 51	syl 17	. . . . . . . . . . . . 13 |- ( ph -> (deg ` F ) e. NN0 )
53	50, 52	ffvelrnd 6360	. . . . . . . . . . . 12 |- ( ph -> ( seq 0 ( x. , N ) ` (deg ` F ) ) e. NN )
54	4, 53	syl5eqel 2705	. . . . . . . . . . 11 |- ( ph -> R e. NN )
55	54	nncnd 11036	. . . . . . . . . 10 |- ( ph -> R e. CC )
56	55	adantr 481	. . . . . . . . 9 |- ( ( ph /\ z e. CC ) -> R e. CC )
57		elfznn0 12433	. . . . . . . . . 10 |- ( m e. ( 0 ... (deg ` F ) ) -> m e. NN0 )
58	36	coef3 23988	. . . . . . . . . . . . . 14 |- ( F e. (Poly ` QQ ) -> B : NN0 --> CC )
59	12, 58	syl 17	. . . . . . . . . . . . 13 |- ( ph -> B : NN0 --> CC )
60	59	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ z e. CC ) -> B : NN0 --> CC )
61	60	ffvelrnda 6359	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ m e. NN0 ) -> ( B ` m ) e. CC )
62		expcl 12878	. . . . . . . . . . . 12 |- ( ( z e. CC /\ m e. NN0 ) -> ( z ^ m ) e. CC )
63	62	adantll 750	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ m e. NN0 ) -> ( z ^ m ) e. CC )
64	61, 63	mulcld 10060	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ m e. NN0 ) -> ( ( B ` m ) x. ( z ^ m ) ) e. CC )
65	57, 64	sylan2 491	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ m e. ( 0 ... (deg ` F ) ) ) -> ( ( B ` m ) x. ( z ^ m ) ) e. CC )
66	17, 56, 65	fsummulc2 14516	. . . . . . . 8 |- ( ( ph /\ z e. CC ) -> ( R x. sum_ m e. ( 0 ... (deg ` F ) ) ( ( B ` m ) x. ( z ^ m ) ) ) = sum_ m e. ( 0 ... (deg ` F ) ) ( R x. ( ( B ` m ) x. ( z ^ m ) ) ) )
67		eqid 2622	. . . . . . . . . . 11 |- (deg ` F ) = (deg ` F )
68	36, 67	coeid2 23995	. . . . . . . . . 10 |- ( ( F e. (Poly ` QQ ) /\ z e. CC ) -> ( F ` z ) = sum_ m e. ( 0 ... (deg ` F ) ) ( ( B ` m ) x. ( z ^ m ) ) )
69	12, 68	sylan 488	. . . . . . . . 9 |- ( ( ph /\ z e. CC ) -> ( F ` z ) = sum_ m e. ( 0 ... (deg ` F ) ) ( ( B ` m ) x. ( z ^ m ) ) )
70	69	oveq2d 6666	. . . . . . . 8 |- ( ( ph /\ z e. CC ) -> ( R x. ( F ` z ) ) = ( R x. sum_ m e. ( 0 ... (deg ` F ) ) ( ( B ` m ) x. ( z ^ m ) ) ) )
71	56	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. CC ) /\ m e. NN0 ) -> R e. CC )
72	71, 61, 63	mulassd 10063	. . . . . . . . . 10 |- ( ( ( ph /\ z e. CC ) /\ m e. NN0 ) -> ( ( R x. ( B ` m ) ) x. ( z ^ m ) ) = ( R x. ( ( B ` m ) x. ( z ^ m ) ) ) )
73	57, 72	sylan2 491	. . . . . . . . 9 |- ( ( ( ph /\ z e. CC ) /\ m e. ( 0 ... (deg ` F ) ) ) -> ( ( R x. ( B ` m ) ) x. ( z ^ m ) ) = ( R x. ( ( B ` m ) x. ( z ^ m ) ) ) )
74	73	sumeq2dv 14433	. . . . . . . 8 |- ( ( ph /\ z e. CC ) -> sum_ m e. ( 0 ... (deg ` F ) ) ( ( R x. ( B ` m ) ) x. ( z ^ m ) ) = sum_ m e. ( 0 ... (deg ` F ) ) ( R x. ( ( B ` m ) x. ( z ^ m ) ) ) )
75	66, 70, 74	3eqtr4d 2666	. . . . . . 7 |- ( ( ph /\ z e. CC ) -> ( R x. ( F ` z ) ) = sum_ m e. ( 0 ... (deg ` F ) ) ( ( R x. ( B ` m ) ) x. ( z ^ m ) ) )
76	75	mpteq2dva 4744	. . . . . 6 |- ( ph -> ( z e. CC |-> ( R x. ( F ` z ) ) ) = ( z e. CC |-> sum_ m e. ( 0 ... (deg ` F ) ) ( ( R x. ( B ` m ) ) x. ( z ^ m ) ) ) )
77	16, 76	eqtrd 2656	. . . . 5 |- ( ph -> ( ( CC X. { R } ) oF x. F ) = ( z e. CC |-> sum_ m e. ( 0 ... (deg ` F ) ) ( ( R x. ( B ` m ) ) x. ( z ^ m ) ) ) )
78		zsscn 11385	. . . . . . 7 |- ZZ C_ CC
79	78	a1i 11	. . . . . 6 |- ( ph -> ZZ C_ CC )
80	55	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN0 ) -> R e. CC )
81	47	nncnd 11036	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN0 ) -> ( N ` m ) e. CC )
82	47	nnne0d 11065	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN0 ) -> ( N ` m ) =/= 0 )
83	80, 81, 82	divcan2d 10803	. . . . . . . . . 10 |- ( ( ph /\ m e. NN0 ) -> ( ( N ` m ) x. ( R / ( N ` m ) ) ) = R )
84	83	oveq2d 6666	. . . . . . . . 9 |- ( ( ph /\ m e. NN0 ) -> ( ( B ` m ) x. ( ( N ` m ) x. ( R / ( N ` m ) ) ) ) = ( ( B ` m ) x. R ) )
85	59	ffvelrnda 6359	. . . . . . . . . 10 |- ( ( ph /\ m e. NN0 ) -> ( B ` m ) e. CC )
86	80, 81, 82	divcld 10801	. . . . . . . . . 10 |- ( ( ph /\ m e. NN0 ) -> ( R / ( N ` m ) ) e. CC )
87	85, 81, 86	mulassd 10063	. . . . . . . . 9 |- ( ( ph /\ m e. NN0 ) -> ( ( ( B ` m ) x. ( N ` m ) ) x. ( R / ( N ` m ) ) ) = ( ( B ` m ) x. ( ( N ` m ) x. ( R / ( N ` m ) ) ) ) )
88	80, 85	mulcomd 10061	. . . . . . . . 9 |- ( ( ph /\ m e. NN0 ) -> ( R x. ( B ` m ) ) = ( ( B ` m ) x. R ) )
89	84, 87, 88	3eqtr4rd 2667	. . . . . . . 8 |- ( ( ph /\ m e. NN0 ) -> ( R x. ( B ` m ) ) = ( ( ( B ` m ) x. ( N ` m ) ) x. ( R / ( N ` m ) ) ) )
90	57, 89	sylan2 491	. . . . . . 7 |- ( ( ph /\ m e. ( 0 ... (deg ` F ) ) ) -> ( R x. ( B ` m ) ) = ( ( ( B ` m ) x. ( N ` m ) ) x. ( R / ( N ` m ) ) ) )
91		oveq2 6658	. . . . . . . . . . . . 13 |- ( n = ( N ` m ) -> ( ( B ` m ) x. n ) = ( ( B ` m ) x. ( N ` m ) ) )
92	91	eleq1d 2686	. . . . . . . . . . . 12 |- ( n = ( N ` m ) -> ( ( ( B ` m ) x. n ) e. ZZ <-> ( ( B ` m ) x. ( N ` m ) ) e. ZZ ) )
93	92	elrab 3363	. . . . . . . . . . 11 |- ( ( N ` m ) e. { n e. NN | ( ( B ` m ) x. n ) e. ZZ } <-> ( ( N ` m ) e. NN /\ ( ( B ` m ) x. ( N ` m ) ) e. ZZ ) )
94	93	simprbi 480	. . . . . . . . . 10 |- ( ( N ` m ) e. { n e. NN | ( ( B ` m ) x. n ) e. ZZ } -> ( ( B ` m ) x. ( N ` m ) ) e. ZZ )
95	46, 94	syl 17	. . . . . . . . 9 |- ( ( ph /\ m e. NN0 ) -> ( ( B ` m ) x. ( N ` m ) ) e. ZZ )
96	57, 95	sylan2 491	. . . . . . . 8 |- ( ( ph /\ m e. ( 0 ... (deg ` F ) ) ) -> ( ( B ` m ) x. ( N ` m ) ) e. ZZ )
97		elqaa.3	. . . . . . . . . 10 |- ( ph -> ( F ` A ) = 0 )
98		eqid 2622	. . . . . . . . . 10 |- ( x e. _V , y e. _V |-> ( ( x x. y ) mod ( N ` m ) ) ) = ( x e. _V , y e. _V |-> ( ( x x. y ) mod ( N ` m ) ) )
99	1, 11, 97, 36, 26, 4, 98	elqaalem2 24075	. . . . . . . . 9 |- ( ( ph /\ m e. ( 0 ... (deg ` F ) ) ) -> ( R mod ( N ` m ) ) = 0 )
100	54	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ m e. ( 0 ... (deg ` F ) ) ) -> R e. NN )
101	57, 47	sylan2 491	. . . . . . . . . 10 |- ( ( ph /\ m e. ( 0 ... (deg ` F ) ) ) -> ( N ` m ) e. NN )
102		nnre 11027	. . . . . . . . . . 11 |- ( R e. NN -> R e. RR )
103		nnrp 11842	. . . . . . . . . . 11 |- ( ( N ` m ) e. NN -> ( N ` m ) e. RR+ )
104		mod0 12675	. . . . . . . . . . 11 |- ( ( R e. RR /\ ( N ` m ) e. RR+ ) -> ( ( R mod ( N ` m ) ) = 0 <-> ( R / ( N ` m ) ) e. ZZ ) )
105	102, 103, 104	syl2an 494	. . . . . . . . . 10 |- ( ( R e. NN /\ ( N ` m ) e. NN ) -> ( ( R mod ( N ` m ) ) = 0 <-> ( R / ( N ` m ) ) e. ZZ ) )
106	100, 101, 105	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ m e. ( 0 ... (deg ` F ) ) ) -> ( ( R mod ( N ` m ) ) = 0 <-> ( R / ( N ` m ) ) e. ZZ ) )
107	99, 106	mpbid 222	. . . . . . . 8 |- ( ( ph /\ m e. ( 0 ... (deg ` F ) ) ) -> ( R / ( N ` m ) ) e. ZZ )
108	96, 107	zmulcld 11488	. . . . . . 7 |- ( ( ph /\ m e. ( 0 ... (deg ` F ) ) ) -> ( ( ( B ` m ) x. ( N ` m ) ) x. ( R / ( N ` m ) ) ) e. ZZ )
109	90, 108	eqeltrd 2701	. . . . . 6 |- ( ( ph /\ m e. ( 0 ... (deg ` F ) ) ) -> ( R x. ( B ` m ) ) e. ZZ )
110	79, 52, 109	elplyd 23958	. . . . 5 |- ( ph -> ( z e. CC |-> sum_ m e. ( 0 ... (deg ` F ) ) ( ( R x. ( B ` m ) ) x. ( z ^ m ) ) ) e. (Poly ` ZZ ) )
111	77, 110	eqeltrd 2701	. . . 4 |- ( ph -> ( ( CC X. { R } ) oF x. F ) e. (Poly ` ZZ ) )
112		eldifsn 4317	. . . . . . 7 |- ( F e. ( (Poly ` QQ ) \ { 0p } ) <-> ( F e. (Poly ` QQ ) /\ F =/= 0p ) )
113	11, 112	sylib 208	. . . . . 6 |- ( ph -> ( F e. (Poly ` QQ ) /\ F =/= 0p ) )
114	113	simprd 479	. . . . 5 |- ( ph -> F =/= 0p )
115		oveq1 6657	. . . . . . 7 |- ( ( ( CC X. { R } ) oF x. F ) = 0p -> ( ( ( CC X. { R } ) oF x. F ) oF / ( CC X. { R } ) ) = ( 0p oF / ( CC X. { R } ) ) )
116	14	ffvelrnda 6359	. . . . . . . . . . 11 |- ( ( ph /\ z e. CC ) -> ( F ` z ) e. CC )
117	54	nnne0d 11065	. . . . . . . . . . . 12 |- ( ph -> R =/= 0 )
118	117	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ z e. CC ) -> R =/= 0 )
119	116, 56, 118	divcan3d 10806	. . . . . . . . . 10 |- ( ( ph /\ z e. CC ) -> ( ( R x. ( F ` z ) ) / R ) = ( F ` z ) )
120	119	mpteq2dva 4744	. . . . . . . . 9 |- ( ph -> ( z e. CC |-> ( ( R x. ( F ` z ) ) / R ) ) = ( z e. CC |-> ( F ` z ) ) )
121		ovexd 6680	. . . . . . . . . 10 |- ( ( ph /\ z e. CC ) -> ( R x. ( F ` z ) ) e. _V )
122	3, 121, 7, 16, 10	offval2 6914	. . . . . . . . 9 |- ( ph -> ( ( ( CC X. { R } ) oF x. F ) oF / ( CC X. { R } ) ) = ( z e. CC |-> ( ( R x. ( F ` z ) ) / R ) ) )
123	120, 122, 15	3eqtr4d 2666	. . . . . . . 8 |- ( ph -> ( ( ( CC X. { R } ) oF x. F ) oF / ( CC X. { R } ) ) = F )
124	55, 117	div0d 10800	. . . . . . . . . 10 |- ( ph -> ( 0 / R ) = 0 )
125	124	mpteq2dv 4745	. . . . . . . . 9 |- ( ph -> ( z e. CC |-> ( 0 / R ) ) = ( z e. CC |-> 0 ) )
126		0cnd 10033	. . . . . . . . . 10 |- ( ( ph /\ z e. CC ) -> 0 e. CC )
127		df-0p 23437	. . . . . . . . . . . 12 |- 0p = ( CC X. { 0 } )
128		fconstmpt 5163	. . . . . . . . . . . 12 |- ( CC X. { 0 } ) = ( z e. CC |-> 0 )
129	127, 128	eqtri 2644	. . . . . . . . . . 11 |- 0p = ( z e. CC |-> 0 )
130	129	a1i 11	. . . . . . . . . 10 |- ( ph -> 0p = ( z e. CC |-> 0 ) )
131	3, 126, 7, 130, 10	offval2 6914	. . . . . . . . 9 |- ( ph -> ( 0p oF / ( CC X. { R } ) ) = ( z e. CC |-> ( 0 / R ) ) )
132	125, 131, 130	3eqtr4d 2666	. . . . . . . 8 |- ( ph -> ( 0p oF / ( CC X. { R } ) ) = 0p )
133	123, 132	eqeq12d 2637	. . . . . . 7 |- ( ph -> ( ( ( ( CC X. { R } ) oF x. F ) oF / ( CC X. { R } ) ) = ( 0p oF / ( CC X. { R } ) ) <-> F = 0p ) )
134	115, 133	syl5ib 234	. . . . . 6 |- ( ph -> ( ( ( CC X. { R } ) oF x. F ) = 0p -> F = 0p ) )
135	134	necon3d 2815	. . . . 5 |- ( ph -> ( F =/= 0p -> ( ( CC X. { R } ) oF x. F ) =/= 0p ) )
136	114, 135	mpd 15	. . . 4 |- ( ph -> ( ( CC X. { R } ) oF x. F ) =/= 0p )
137		eldifsn 4317	. . . 4 |- ( ( ( CC X. { R } ) oF x. F ) e. ( (Poly ` ZZ ) \ { 0p } ) <-> ( ( ( CC X. { R } ) oF x. F ) e. (Poly ` ZZ ) /\ ( ( CC X. { R } ) oF x. F ) =/= 0p ) )
138	111, 136, 137	sylanbrc 698	. . 3 |- ( ph -> ( ( CC X. { R } ) oF x. F ) e. ( (Poly ` ZZ ) \ { 0p } ) )
139	6	fconst 6091	. . . . . . 7 |- ( CC X. { R } ) : CC --> { R }
140		ffn 6045	. . . . . . 7 |- ( ( CC X. { R } ) : CC --> { R } -> ( CC X. { R } ) Fn CC )
141	139, 140	mp1i 13	. . . . . 6 |- ( ph -> ( CC X. { R } ) Fn CC )
142		ffn 6045	. . . . . . 7 |- ( F : CC --> CC -> F Fn CC )
143	14, 142	syl 17	. . . . . 6 |- ( ph -> F Fn CC )
144		inidm 3822	. . . . . 6 |- ( CC i^i CC ) = CC
145	6	fvconst2 6469	. . . . . . 7 |- ( A e. CC -> ( ( CC X. { R } ) ` A ) = R )
146	145	adantl 482	. . . . . 6 |- ( ( ph /\ A e. CC ) -> ( ( CC X. { R } ) ` A ) = R )
147	97	adantr 481	. . . . . 6 |- ( ( ph /\ A e. CC ) -> ( F ` A ) = 0 )
148	141, 143, 3, 3, 144, 146, 147	ofval 6906	. . . . 5 |- ( ( ph /\ A e. CC ) -> ( ( ( CC X. { R } ) oF x. F ) ` A ) = ( R x. 0 ) )
149	1, 148	mpdan 702	. . . 4 |- ( ph -> ( ( ( CC X. { R } ) oF x. F ) ` A ) = ( R x. 0 ) )
150	55	mul01d 10235	. . . 4 |- ( ph -> ( R x. 0 ) = 0 )
151	149, 150	eqtrd 2656	. . 3 |- ( ph -> ( ( ( CC X. { R } ) oF x. F ) ` A ) = 0 )
152		fveq1 6190	. . . . 5 |- ( f = ( ( CC X. { R } ) oF x. F ) -> ( f ` A ) = ( ( ( CC X. { R } ) oF x. F ) ` A ) )
153	152	eqeq1d 2624	. . . 4 |- ( f = ( ( CC X. { R } ) oF x. F ) -> ( ( f ` A ) = 0 <-> ( ( ( CC X. { R } ) oF x. F ) ` A ) = 0 ) )
154	153	rspcev 3309	. . 3 |- ( ( ( ( CC X. { R } ) oF x. F ) e. ( (Poly ` ZZ ) \ { 0p } ) /\ ( ( ( CC X. { R } ) oF x. F ) ` A ) = 0 ) -> E. f e. ( (Poly ` ZZ ) \ { 0p } ) ( f ` A ) = 0 )
155	138, 151, 154	syl2anc 693	. 2 |- ( ph -> E. f e. ( (Poly ` ZZ ) \ { 0p } ) ( f ` A ) = 0 )
156		elaa 24071	. 2 |- ( A e. AA <-> ( A e. CC /\ E. f e. ( (Poly ` ZZ ) \ { 0p } ) ( f ` A ) = 0 ) )
157	1, 155, 156	sylanbrc 698	1 |- ( ph -> A e. AA )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   {csn 4177   |-> cmpt 4729   X. cxp 5112   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   oFcof 6895  infcinf 8347   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   < clt 10074   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   QQcq 11788   RR+crp 11832   ...cfz 12326   mod cmo 12668   seqcseq 12801   ^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-q 11789  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  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:  elqaa  24077